In response to chapters two and three, let me begin by saying that this semester my mathematics text is my most favorite one to read. I love the language used in this text! You do not have to be a math genius to figure it out. For me, that is a BIG statement, because normally I would try to avoid math if at all possible. I must say that from the way I was taught mathematics it is very difficult to not be provided the ‘right’ answers. It literally makes me stop for a second and I have to reemphasize the entire point of the text to myself- that learning is through exploration- and then I can proceed. It’s a difficult task to master but I am working on it. That is the most frustrating aspect for me in this process, that ‘correct’ answers are not provided. I do agree that the learning is more concrete if students make the connections and go through the process of learning to solve problems on their own. I know I would have learned better in this manner, but it leads me to have one nagging question in my head. Where do formulas like the Pythagorean theorem develop? Surely, at some point this information that these students are developing gets filtered into a concrete formula- but at what point does that happen?
Today was our first day of school. I work as a special education paraprofessional at the high school level. I noticed today among students getting their schedules how many made negative comments about math class. I wondered where this feeling came from. Is it from a bad experience back in 1st grade or just coming they picked up from peers. Anyways, it reminded me that I want to be the kind of teacher who instills excitement about math. I realize not all kids are going to enjoy the subject, but I would at least hope they could keep a neutral attitude and learn the skills that were discussed in Chapter 2-3 about solving problems. For instance, the text states, "The teacher's role is to create this spirit of inquiry, trust and experience." I think it is too easy to quit when something becomes hard. I don't want my children to have this attitude or my students. I agree with the text about building on prior knowledge. I believe that if we can relate a topic to something they know about it is easier to understand. I know this is true for myself. Another, topic I really enjoyed was encouraging multiple approaches. Having worked in special education for 7 years I have seen first hand how students learn differently, but yet come to the same conclusion. As teachers we need to encourage every learning style and celebrate diversity. As I continue to read through this text it is so easy to see how math is involved in everyday life and why it needs to be integrated across the curriculum.
To Adrianne: I wish I could answer the question about the pythagorean theorem. I tell myself that it is an important theorem for people who do research. I believe that math is a subject that is very important for students to learn. Math is a process and if one piece (such as multiplication) is missing it messes the whole process up. I worked summer school this year and tried to stress to elementary students how imperative it is for them to learn their basic addition, subtraction, multiplication and division facts. I like this textbook as well.
When reading Chapter 2 and 3 there was a lot of great information that I was able to gain from reading the chapters. I loved the part of Chapter 2 that talked about the four different features of a productive classroom culture in relationship to math. If as a future teacher I am able to implement these features into my classroom I will be able to have a productive math environment in my classroom. I also found the part of the chapter that talked about the five strands of mathematical proficiency to be interesting as well. It talked about how all of the five strands need to be used together in order for the classroom to be successful. I also learned about how the five strands are interrelated and interwoven together in order to show how the five strands are so closely related. When reading Chapter 3 about problem solving the item that popped out of the text to me was to teach through problem solving. Math and problem solving needs to be taught through real life contexts and situations as much as possible. The students will be able to remember the information better if they are able to relate math to the world around them rather than just filling out a math worksheet and forgetting the concepts learned. I also enjoyed reading about different types of problem solving strategies. I found the list to be very helpful for me as a future teacher and I can see myself using some of these strategies in my future classroom.
I have also always wondered where the dislike for math in students comes from as well. It could be a number of different reasons and as a future teacher I feel that it is my responsibility to figure out the reasons why they dislike math and then try to make the experience with math in my classroom a great experience. I also agree with you that it is so important that teachers build on their students prior knowledge in all subjects especially math. It is a lot easier for students to gain a new concept if they are able to relate back to their prior knowledge. Lastly I agree that it is important to have multiple approaches to teaching each math lesson because all students learn in different ways and you want to teach the students in the way that they will learn best.
In response to Brooke M, I also thought the four features of a productive classroom were very crucial to success in mathematics. I especially like the attitude of this text about math being a product of discovery. I too thought these chapters were filled with valuable ‘gems’ of information to future teachers. I also love the way they focus on making the mathematics applicable to the ‘real world’.
I really enjoyed the activities we did in class in chapter two. I know that the group I was in was looking for an answer, but there wasn’t one. Sometimes this is so hard to understand, because we often think that everything must have an answer. One of the girls in class said she understood the process of “one up, one down” but didn’t know how or when she would use it. For example, 7+7=14, but when are we going to face a situation is which we remember that, and then add one and subtract one from each number to obtain our answer? I thought this was a very valid point. I think that sometimes as teachers, we may forget that the students don’t know and that we often assume they do. Just because this technique may not help us, it may help a child better understand addition facts or number patterns. Problem solving is probably one of the difficult for children to learn. As a student, I was so used to number problems that it was almost terrifying to work a word problem. I was taught to find the question or the problem, pick out the important information that you must have to solve the problem, and then finally solve it. I remember by the time you advanced to a certain grade, you no longer were allowed to draw pictures to help you. You had to use the equations and do the problem solving “the teacher’s way”. As a future teacher, I want to better myself and allow my students to complete and solve problems in a way they understand; not in a way that they can’t even read or comprehend.
In reply to Brooke: As I was reading your post I remembered back to when I was in grade school. Several of the story problems were about things and ideas I had no idea what they were. I think having problems relating to the children and their background knowledge is so important. This is also an example of why relying on textbooks and worksheets is not always a good idea. A lot of times they contain content that is not appropriate for the culture, setting, or students you may be teaching.
What we did in class for Chapter 3 was what I really enjoyed this week. Problem solving, to me, has always been kind of entertaining and fun. It was fun working the bats and cats problem in class and then going over it today because there were several different ways to go about solving the problem. Three of the ways were draw a picture with the legs and then start at one end circling two legs and the other side circling four legs until you meet in the middle and run out of legs and see how many of each one you have. The second way was guess and check which is the method I went with. It’s not the fastest way to solve a problem but it works and sometimes you get lucky and solve it on the first try or two. The last way that was used was the formula. After I solved the problem using guess and check I tried to come up with a formula to solve it and just flat out couldn’t think of what it would be. After going over the formula today in class I feel like it should of come naturally to me that’s just how it works sometimes. Everybody has their own way of being able to solve a problem. Just because it’s not the way you were taught doesn’t mean that it’s the wrong way as long as you can support your answer.
Adrianne Hoefler I complete agree that the statement, “You don’t have to be a math genius to figure it out.” Is a huge statement. I am living proof of it. When it comes to solving problems I am definitely not the fastest at it. But I will eventually solve the problem. I remember when I was in Mathcounts in Jr. High, we would go to competitions and there would be kids who could solve the problem before I was even done reading it. It doesn’t necessarily mean that they were better at it than me, it just means that they have the ability to go through most of the information quicker than me.
I found the vocabulary words used in the beginning of Chapter 2 to be important to know. As teachers I believe we should all be on the same page when it comes to math vocabulary. Do our first grade students know that "take away" is the same thing as "less than" or "subtract?" Explaining real world problems in terms that our students understand is a way of connecting their worlds to math. For example, using M&M's when adding or subtracting. Grouping by the color of M&M's and then using the candy for predictions. When they can solve the problem then allow the student to have a sweet treat. I have enjoyed reading about math in these chapters. The authors have written this textbook in the same way they are teaching us how to teach math. In a real world setting. The answers to questions are some times not given rather questions are asked then examples are given to help understand the concept. I had a hard time with problem solving when I was an elementary school student. I found myself frozen in fear when a worksheet was presented with a page full of words. I was unable to decipher though the words to get to the problem. In order to teach effectively then asking questions about the problems before asking students to solve will help them work through what is being asked. Allowing the students to ask the questions and do the talking will help them work through the problem before they even see the words and numbers. Learning about the lesson plan phases was informative for me in chapter 3.
Deidre, your experience is shared by many. Terrifying is a great word to describe what I felt as well. As future teachers I think we will have the knowledge and skills to approach teaching differently. These chapters talked a lot about how to communicate with the students and how to relay the math message without using intimidating language or strategies. The Frequently Asked Questions at the end of chapter 3 were great to read through and get some real life answers to real life problems.
I am really liking this book so far, I hope it continues. The activities we did for Chapter 2 were fun. I told my boss and her daughter about the Start and Jump Numbers and the One Up, One Down. I thought they were so neat that I had to share them. I think what it means to do and know mathematics is to understand what you are doing and how to explain your reasoning. I don't think you have to be fast at solving problems in order to be good you just have to be able to understand and prove your answers. I feel incorporating real world situations in math for students will make math not seem so overwhelming. I think most students are intimidated by math because of the thinking that goes into it and the fact that there is one correct answer.
Chapter 3 to me required more thought. I am still a little unsure as how to explain what problem solving is. I would have to say its being able to solve different types of problems. It talked about letting the students do the talking. Allowing the students to discuss problems and ideas with other classmates learning will occur in different ways. The chapter also mentioned different problem solving strategies. There seven different ones; draw a picture, act it out, use a model; look for a pattern; guess and check; make a table or chart; try a simpler form of the problem; make and organized list; and write an equation. Who would of thought writing was important in math, it is. It allows students to be reflective, a rehearsal for discussion and its also a written record with the lesson is finished.
Jena, I didn’t talk about the vocabulary, but I agree with you in them being important. Students do need to know that there are multiple terms for subtraction, adding and so on. I like your examples of the use of M&Ms. Math was always more fun and exciting for me when we used fun objects or incorporated it with real life. I too had a hard time with problem solving and feared word problems. I would sometimes sit there and make it look like I was working on the problem because I didn’t want to do them. I agree to that children need to be able to do the talking as well and not just the teacher.
There was some very good information in chapters 2 and 3, but my a-ha moment came with the cats and bats problem. I have said it before and I will say it again that I am very intimidated by math. I always feel like the clock is ticking and I have to find a solution fast. I heard the problem and thought immediately that I couldn't do it. Then my moment came, slow down, take your time, find the best way for you to work it, and you can do it, and I did. This little problem might just have increased my confidence a little bit, I felt that moment of success, and this is what students need to feel. Chapter two provided implications for teaching mathematics. One of these is to build in opportunities for reflective thought. I think this is very important because it will help students understand where their answer came from, how they got there. By doing this the students will be more likely to remember what they have just learned. Chapter 3 was about teaching through problem solving and I found this chapter to have some very useful information and it will be a great resource in the future. This is so important because it helps students in finding their own strategies to solve problems, it doesn't just say this is how it must be done. I also liked that this chapter gave an activity evaluation and selection guide. This helps teachers find worthwhile activities, and ensure the students will see the importance of it. Overall, I got some very useful information from both chapters.
I am also finally discovering that just because you might not be really fast at solving problems doesn't mean that you aren't good at math. If I would have figured this out a long time ago, I might like math a little more. I also like that you are sharing the problems we are doing in class with others, I agree that the activities in chapter 2 were fun!
Elizabeth, I, too, like that discovering that just because I can't do math as fast as my husband, does not mean that I am not good at math. He is just quicker! Maybe that has to do with me being a more visual person and I have to "see" the problems, sometimes, to figure them out.
I had never considered myself "good" in math. However, I can do all types of calculations when needed in life. Maybe not as fast as other, but I can do them. I can do simple ones in my head; but harder ones, I need to write down and work on paper. It does help to have real life examples or experience. When teaching, I try to bring real life examples into my teaching. Things that the students can relate to. This also helps to activate prior knowledge. I tend to be a "hands-on" and "visual" learner. It is for this reason, I believe, that I lean toward using manipulatives and pictures when I am teaching, as well, as putting lessons in terms/scenarios that students can relate to. This seems to fall into the multiple representations of math ideas.
Lacey Keller While reading chapters 2 and 3, I could relate, not from my own personal math story, but from another student. You see, I am a special education paraprofessional here in our local community. Last year, a student came into the program from way on the eastern coast (I live in western KS). This young gal was frustrated with math at her new school. However, when I sat down with her, and she opened up, stating she just wanted to do math her way, not the teacher's way! So, I let her. Oh, boy, this was the best thing I ever did for that student. Apparently in her old school, she could use different strategies(suggested in chapter 2)to solve problems. However, she felt threatened when her new teacher would not let her do that. I loved doing math with her because it was a challenge for me to see just how she was going to solve the problem! I may look at a problem and say, "Just add." However, she would would look at the problem, subtract, add, and multiply and still come up with the correct answer! I see now from reading this chapter that this gal had an teacher who valued the invented approach.
I think the most valuable reading from these chapters came when the author states the teacher must create meaningful contexts. You have to find links to other subjects to get students excited! I remember always asking my high school geometry teacher, "Am I going to use this stuff in the real world?"
Kim, Your comment reminds me of the science methods class where science is "hands on, mind on." This means we must provide manipulatives, experiments, and real-life experiences to get our students learning.
In response to everyone who has asked about the Pythagorean theorem, I once sat in on a math class where the students were making posters while learning the history of famous mathematicians. The students learned about people, theorists, and had fun while doing math! By the way, Pythagoras was the man who came up with the equation.
First, I must say that I am so grateful that we actually use the textbook during class. Working the problems has been so helpful. Typically, I skim or skip past applications like these. Allowing us to be math students has been a tremendous help. Seeing the discovery process in action has been an eye opener. I love hearing discussion about how each student or group worked to find the answer. I don't remember getting to work much with groups or partners during math lessons. It was mostly individual work. Creating this type of environment will encourage students to be engaged.
I never really put much thought into the learning process of mathematics. Such as learning about shapes in preschool that eventually go into higher geometry learning. Reading about the number 7 on page 25 helped reinforce that knowledge. 7 is more than 1 and less that 10, 2+5=7, it is odd, compared to 1/10 it is bigger, it is "lucky", it is prime, ect. In theory it is a progression of "connecting the dots."
Integration is something that I look forward to doing in my classroom. I loved the idea of using "Harry Potter and the Sorcerer's Stone" as a measurement lesson. How tall is Hagrid compared to me? Using adding machine paper the students measure their height. Then they compare it to Hagrid who is twice as tall and five times as wide. Great ideas!
@ Elizabeth As a child it was always great to be the first person done. Everyone would think that person was a genius. Immediately, frustration would come it to play, realizing that you were not even close to being done. In the future I want to create an environment where speed is not the case.
I think that the best thing I read from Chapters 2 and 3 was this; "Mathematics is the science of pattern and order...Science is a process of figuring things out or making since of things." Wow. Growing, up I was always taught formulas and the "how" to solving problems. It rarely made since to me. "Science is a process of figuring things out or making since of things." I wish that some of my science teachers would have considered this while they taught me. I like how Dr. Stramel told us Friday that it is ok for each students to figure the problem out differently. And that we could even have them share with us, as teachers, how they figured the problem out. I believe it is good for the process that they choose to be correct, of course, but I can definitely see how the students figuring it out themselves would help them to "make since out of things". I remember sitting in many different math classes and there was usually one student who could figure the math problems out very quickly and in a way that was different than the teacher. I remember them always getting marked off for not doing the problem the way that the teacher showed the class. I don't ever remember a student, myself or a friend, doing a problem differently than that way that the teacher taught and not getting marked off for it. It was nice for the chapters to reiterate that it is ok for students to figure things out for themselves, and that the process of figuring it out can bring that full understanding and help the process to make since to each student.
Brandi S. You said, "I don't remember getting to work much with groups or partners during math lessons. It was mostly individual work. Creating this type of environment will encourage students to be engaged." I completely agree with you! I am excited about trying this in my future classroom! I have always hated math because it was boring and frustrating. If I had the opportunity to discuss it with my peers and to work it out with them, I may truly feel that I would have had a different experience. I cannot wait to see my future students enjoying math as they work through the process in groups with different tools that will allow the learning process to be fun rather than stressful!
For the reading this week chapter three really caught my attention simply because it was all over problem solving and me and problem solving don’t go together. Me and math don’t go together but me and problem solving really don’t go together. Even after reading all the information I am not sure I feel very confident in myself to teach this standard or math itself. I enjoy the older grade school students so that is where I am hoping to one day teach and of course that means more in depth studies. I did enjoy reading about letting the students do the talking and you monitoring. In my technology class we have been talking about partnering teaching, which is allowing more student to student teaching through activities and groups. I like the idea of all of this. I think students need to be more challenged in that aspect. I think it helps them in the real world because they create the habit to analyze the situation and try several approaches without someone telling them how to do it. Of course the teacher needs to be present and active in the learning so they don’t head off course from the correct answer and I don’t think it makes teaching any easier but it does help your students.
I too like the concept of relating teaching with life examples. Student’s, even if they don’t really care at this point of their life, need to be aware and have heard situations they may one day encounter. As I look back on my education I wonder how many life stories I heard but didn’t really listen. I wonder how many situations would have turned out better had I only listened. School subjects can be related to almost everything and I too hope to be able to bring a variety of experiences and knowledge to my room. Using manipulatives and pictures and stories is a great way for a student to remember and grasp the concept being presented.
I really enjoyed reading chapters 2 and 3. I also enjoyed watching the face-to-face class. I am not all the comfortable with the idea of teaching math yet but I am constantly writing things down in order to refer back to them later to help in the classroom. I love all the information on problem solving. I think that students need to see the problem as well as read it on paper. For the younger students I think that having students pick out keywords and draw them and cut them out would be beneficial to solving the problem. I also like the information on the importance of pattern and order. Students need to strive to see these things when faced with a math problem. I think that it is important for students to know when, where, and why they will use these math skills outside of the classroom as well.
Katie- I love that you stated that you and problem solving don't go together. I feel the same way. I have never been good at breaking it down and seeing all of the little pieces of the problem. I read it and instantly feel overwhelmed with the amount of information in it. I think that it is great that we are learning about ways to avoid letting our students feel like that and how to help them "go together" with problem solving!
I thought it was important in Chapter 2 when they talked about how the teacher should have a good feel for mathematics. I also liked how it said that you should get together with a peer and work together so you can get some experience with sharing and exchanging ideas. I think that just goes along with being prepared. It's just like Dr. Stramel said, you should work the problems along with the students so you're prepared for anything. One of the things I got from chapter three was confidence in students. The more a student enjoys working on a problem, the more likely they will be in succeeding at it. Attitudes go a long way in math.
@ Lane A. I agree, the activity we did in class was a great exercise. I liked how it made people of our age level really think about the problem. It all goes back to attitude, in my opinion. We all had a good time working together and figuring out the problem. That makes things go so much smoother. That was one of the main things I picked up from Chapter 3. A good attitude can take you a long way!
While Reading through Chapters 2 and 3 I am unfortunately reminded of how much emphasis and time I spent trying to constantly find the “right” answer, not only in mathematics but throughout many subjects and situations in life. However, I have been working on this part of myself and finding that things become a lot easier when not always focusing on just getting the answer. I was thrilled to see that students are still using physical geo-boards and manipulatives within the classroom. I have seen some schools that have advanced very much with technology and only use those activities on the computers. I did enjoy the activities in chapter 2 that were done in class; I tried to do them on my own and did well but didn’t find as much information as the students in class since they had groups to work in. I also loved how the math verbs were listed for everyone to see in the beginning of chapter 2, made it much easier to picture and gave more options than I had originally thought of. Chapter 3 about problem solving was honestly something I was dreading reading when I saw the title because I remember hearing and learning about problem solving way too often when I was in elementary school. However, I enjoyed it because I learned the best ways to teach problem solving methods and help the students. The FAT( For, About and Through) is something that I will be able to easily remember. I definitely believe that all three of these problem solving situations are important and should be used but teaching the students to teach Through Problem Solving is the most important. I am enjoying reading this text much more than most textbooks I have ever read. It is not boring and it has fun activities to keep from getting bored.
In response to Kristie C... I am also a little weary about teaching math. I feel like if I screw up than these students will be screwed up for the rest of their mathematics career. I think more than anything else it is me putting stress on myself and I realize this class is really helping to prepare me, and we have only been in the class for a short while.
After reading chapters 2 and 3 I found there to be a lot of useful information. I think that everyone has a different opinion on what “to do math” and “to know math” means. I don’t really feel like the book is limiting us to what we need to know and where we need to be at in regards to math when we are educators. Instead it is giving us guidelines to follow, but how we teach the lesson is up to us. Which brings me to my next point.
I love that this textbook is more than just text, but throughout the it there are sample problems, ideas, and tons of examples. I think my favorite part of chapter 2 was the explanation on ineffective uses of models and manipulatives. I can see how this could be a problem.
Today was the first day I was in the math classroom since reading chapter 3. I think it will be interesting to progress through the semester. I did not see a whole lot of variety today due to a test they were studying for that they have tomorrow over unit one. However, there was a lot of problem solving. I was helping one student, and it was so hard for me to not direct him to the answer. In chapter 3 there was a section about how much to tell and not to tell a student. It was challenging for me to not just tell him the answer, but I knew I couldn’t because I know this math, he on the other is still learning. I think it’s kind of funny though because he saw I was willing to “help” him, and I think with that he thought I would just give him the answers. It will be interesting to see how I change throughout this course / the internship!
Within Chapter 2, I like the overall idea of always relating everything back to the real world. When applying this to mathematics, students will receive and comprehend more when knowing they can apply what they are learning to their everyday lives (with all subjects, not just mathematics). For example, you start off with 5 baseball cards and your friend asks to trade you for 2 of yours for 1 of his...how many do you end up with if you agree on the trade? (3 baseball cards) I want to keep this in mind while in my future classroom. I also enjoyed and related to the four features of a productive classroom. I especially appreciated "The classroom culture exhibits an appreciation for mistakes as opportunities to learn." I think it's crucial as a classroom teacher to provide room for growth with all students. It's only natural for us as humans to make mistakes, and I want students to know that they need to try, but it's okay to fail. That's how we improve! Especially with potential hard homes lives, I don't want students to look at failure as always a negative thing-it's important to encourage their efforts. Also, with the idea of patterns, I agree that it's important to change up patterns and see how it affects the outcome. This make mathematics fun for students and keeps them interested!
Within Chapter 3, I find out that there is teaching for, about, and through within problem solving. When teaching through, I feel it allows the students and teacher to connect on a different level. It's important to keep productivity flowing for the classroom environment... I have had teachers in the past that didn't elaborate on the mathematics lesson being taught. It's important to keep the domino affect rolling. I feel this makes learning not only more fun, but more beneficial for the future of the students. Especially going from elementary school mathematics to middle school mathematics.
I can completely understand how much focus is put on finding the correct answer... I feel like I've always looked for the correct answer because up until this course finding the right answer was important, versus understanding the concept that was being learned. I'm also not a fan of all of the technology that schools have switched to. I loved manipulatives when I was in elementary school, and although technology is great, sadly it's what is dominating our society so it only makes sense to incorporate it in several ways throughout the classroom!
These two chapters really open your eyes to the idea of teaching math. The concepts showed me that just because I know how to do it one way doesn’t mean a child will think the same way. Some children think differently and out of the box. This is not a bad thing at all. The concepts that children come up with are amazing. The concept of doing math is not giving children the problem and then telling them this is the answer. That is totally not helping the child understand. By giving them the problem and helping them problem solve it the way that is best understandable for them the children can learn better. I love the statement of “In the real world of problem solving outside the classroom, there are not teachers with answers and no answer books.” This is a great thought. Knowing the “answer” does not mean you know how to do the mathematics. Teaching problem solving is teaching children how to think out of the box. By letting the children think on their own they can solve the problems using their past knowledge. As I become a teacher I think it is important to know the concepts and the way my children do learn. I think this will help me in my future, just knowing how children learn differently.
I love your idea of letting the children do their own thing and having confidence in them. Just because you give them a little bit of freedom does not mean you cannot guide them a little. You can take their idea of solving the problem and expand on it.
I agree with you, it is important to know that every student who gets the correct answer will not find that answer the same way. This is hard for those of us who did learn it from teachers who did not take a problem solving approach to realize. What worked for us may not work for our students and what works for one student may not work for a different student.
First of all I like this book, it is actually doing its job, teaching me how to teach, or at least is trying to teach me how to teach. Some of my textbooks have not done a good job of doing that or even making it look like the book was written for that purpose.
These chapters have a lot to teach me, as someone who has always been good at math. I tend to see the whole picture without much effort and need to remind myself that others can not do that as well as I can. I like the fact that the book says to call on the students who are less likely to talk first. As a substitute teacher I often call on students who do not have their hand up, I have even asked students not to hold up their hand as I intended to simply pick one of them.
The problem solving approach is still work for the teacher, and lots of it. I do not really believe that one approach is easier for the teacher then the other. They require different things from the teacher, but they both require work.
Chapter 2 does a great job of breaking the process of completing a math problem down; which is important for a person like me who moves through the process quickly to remember and be reminded of.
Chapter 3 describes the basics of the problem solving based classroom, it is different but has potential for success in the classroom. At the same time I find it hard to believe that the traditional methods have as little merit as the chapter suggests. While the books methods may work better the chapter come close to suggesting that very few if any students will learn from the other method. This has a simple effect, the book does not have even the appearance of an objective comparison of the two methods. It simply suggests that they method that worked for me can not work.
In response to Elizabeth I couldn’t agree with you more in class when the cat and the bat problem come up, I instantly froze and was like I have to get the answer fast. I loved how we had time to work it out and could do the problem the way we wanted. There were no rules, which is great because as we are learning there is not a “right” answer. I also think that reflecting from what you just learned is great because like you said they get to see where and how they got the answer. As I was a slower math learner I remember I had one teacher that would always ask “how did you get that answer?” and everyone in the class would have to explain how they got their answer to the teacher.
First off I would like to start out by saying how much I love this reading! One thing that stood out to me was the statement that math is the science of pattern and order. Every part of math has orders and patterns and I think that is what I appreciate most about math! One other thing I really liked was about treating error as opportunities for learning. I think this is a super important concept. We CAN learn from our mistakes. Also teaching children different ways of figuring out a problem is great. The school I am working in has everyday math curriculum and they teach so many different ways for students to learn! I really enjoyed reading about the different phases in chapter 3! @Adrienne, I also love this text, it truly says it in everyday terms. I think so far it has given us some fantastic teaching strategies! AWESOME!
First off I really am enjoying reading this book. It had great ideas that I can use later on as a future teacher. In chapters 2 it just is reminding future teachers that all students don’t have to learn the one way to figure out the math problem. Each child is different and don’t all think or learn the same. I love that the book says “that mathematics is generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense.” In Chapter 3 the For, About, and Through method of problem solving is going to be something I refer to as a future teacher. I feel as thou the most important is the Through problem solving because students get to connect to the teacher on a different level.
Throughout chapter 2 & 3, I found a great deal of useful information that will definitely be helpful in the future. Math never was one of my favorite subject, so I admit, I am a little nervous to teach it to my students. I want to find new and exciting ways to help my students learn math besides just strictly math worksheets and working out of the textbook. I believe this textbook is going to be very helpful in helping me achieve this goal. I loved the section on having a productive classroom when it came to math. I will take this information with me. I can't wait to read more into this textbook and see everything it has to offer to have a successful mathematical classroom. Hopefully, I will soon be more comfortable with teaching math in my classroom!
In response to Megan B--I am glad you talked about chapter 3, the "For, About and Through Method of Problem Solving". I forgot to mention this in my post but I also found this section very helpful. You are completely correct, this is something that will definitely be helpful and a great piece to look back on. Problem solving with be used in all aspects of math. Great point Megan!
I enjoyed reading Chapters 2 & 3 and am continuing to learn valuable information every time I pick up this textbook. I really liked the section where it talked about how mathematics is the science of pattern and order. The quote, “even the youngest schoolchildren can and should be involved in the science of pattern and order,” came to mind when I watched the students in my mathematics internship class make an AB pattern out of paperclips. It seems like such a simple concept, but I honestly do not remember doing that in kindergarten. The school where I am interning has started to implement the Common Core standards in their classroom and it is obvious that there is a higher level of learning going on.
I really like how the text provides a list of verbs that can be used when talking about mathematics and it reiterates the fact students should be actively learning while doing mathematics. The fact that the answers are not given anywhere in this textbook for any of them problems made me feel a bit uneasy. If the answers are not in the book, how will I know if I solved the problem correctly? During high school, our textbooks had some of the answers in the back and I realize that I became very reliant on those answers. It’s important for students to know that the answers to real life problems will not come in the back of a textbook.
Reading Chapter 3 and learning about teaching mathematics through problem solving was very interesting. I had a lot of math teachers who would show us how to do a problem on the board and that’s how we were expected to do those types of problems from then on. I never had to think of an approach to solving the problem because the approach was already laid out for us. I wish I would’ve been given more opportunities to think about different ways to solve the problems myself.
I definitely got that opportunity during class while working on the bats and cats problem. I was expecting to get a formula or something to show me how to solve it but we had to figure it out for ourselves. I ended up using the guess and check method to solve the problem and looking back, it wasn’t really as difficult to solve as I had originally thought. It was nice to see the list of problem solving strategies listed in the text and I agree that students should be able to solve the problem however they want, even if it is not the exact way that the teacher showed them. It's great to be learning about how to teach math while being given an opportunity to solve problems during class.
I was also always terrified to see a page of word problems in front of me and I have to say that I still am a bit scared. Although "naked number" problems weren't really that much fun, I almost preferred them to word problems because at least I already had the exact problem that I had to solve in front of me. Hopefully through this class, we will both have a different attitude about them.
There were a lot of things about Chapters 2 and 3 that I liked, but mostly I am so happy for the changes that have been made to how math is taught. I loved the explanations in Chapter 2 about what it really means to do, understand, and learn mathematics. When I was taught math it seemed that too often I was expected to just accept what was taught without understanding it fully. I think this is where a lot of my frustration stemmed from. However, if a child is truly doing, understanding, and learning mathematics then tasks are worthwhile, promote discussion, and make meaningful connections to previous concepts. I feel it is important to create an environment in a classroom where students have the opportunities to do all these things. Problem-based learning as discussed in Chapter 3 is important because it makes math skills relevant. So many times in school I heard “I’m never going to use this.” Not only are problem based lessons and tasks engaging, but they are also useful. Today was the first day of my internship and in math the students learned a new skill (GCF) and then worked their way up to a problem solving exercise. My mentor teacher is exceptionally good at how she questions her students. She helps lead them to the answer, but never explicitly says yes or no when they ask if it is correct. I know I am going to learn so much from her this semester, especially when it comes to “how much to tell and not tell.” She makes this look so easy, but I find myself wanting to help too much or give too much information. I am anxious to work on this skill throughout the semester and going forward. I feel as though every chapter I read gives me something new I can use and it is exciting to be able to learn something and put it into practice almost immediately.
When I learned math I never had the opportunity to think about how to solve a problem either because it was also already laid out for us as well. I think it's great that students are taught to truly understand what they are learning in mathematics. I can't wait to see the future implications of this. I started my internship today as well and have an amazing mentor teacher (I feel so inadequate :)). I know I am going to learn so much from her this semester, including how teach through problem solving.
In Chapter 2, we really get challenged to think about what math is to us. What does it mean to learn, what does it mean to do, and what is the difference? I enjoyed that concept and it certainly got me thinking about math and my own personal experiences. To me learning math was all about the “light bulb” moment or the “ah ha” moment, then moment while in class that the switch goes from dazed and confused to place where it just makes sense. Doing math was more about with worksheets and problems to work out of the book. Everyone would try to get done with their math assignment first and it was almost like a race. Doing math was getting the answers right and I didn’t really think about how I was getting it. Learning and doing math really get broken down in this chapter to how we learn and why it is important to do that math. There was a bit in this chapter about making mistakes or errors and how it is okay. I really appreciated this part of the chapter because mistakes should not be embarrassing and we should not try to cover them up, instead we should be learning from them and making ourselves more aware of where and how the mistake occurred so we know what we need work on. I think this text is great and is already offering fantastic information to help us prepare for being out in the field. In Chapter 3 I felt that the text really dug into the details of the importance of problem solving in our lessons. I specifically liked the part about problem solving helps the students build meaning for the concept they are learning. That is such a key to learning. Students can gain success by easily seeing the relationship of the concept to their everyday life. I really enjoyed learning how the problems are designed and the thought process there. Creating or designing well written problems can be difficult if it is something you have not previously done before. So I enjoyed learning about what is important to look for selecting problems for the class. I also learned that allowing some conversation where the students do the talking can be beneficial to the learning of the classroom.
@ Kristi P I completely agree that getting out into the classroom to teach math makes me very nervous as well. I just finished my first day of internship and I was overwhelmed by the different levels of the students’ abilities. Some of the kids really struggled with the concept of basic addition and patterns. It really opened my eyes that I need to be very prepared before getting into the classroom on my own. I think this book will offer great benefits for us to continue to gain knowledge so that we will be much more prepared.
I will admit that I was a little stumped at figuring out the difference between knowing and doing math, they seem like one in the same. I now see that there is a big difference between knowing and doing math. I also liked having the idea confirmed that students can do math in their own unique way and that is okay as long as they are achieving the correct outcome. I was one of the students who took a little longer with math and I would have to draw pictures and write every single thing out in order to complete a problem. Some of my teachers were okay with this but others pushed for me to do it in my head or to use other problem solving practices. It was hard enough for me to understand math as it was so I didn’t want to have to try to do it in an entirely foreign way that I wouldn't understand. I ‘m glad that the fact students can do math however is best understood to them is supported. It is something I will relay to them when I begin teaching myself.
Reading about problems solving was a little less reassuring to me than chapter 2. Like the text says, teaching through problem solving requires a philosophy change in how teachers believe students learn. I’d like to think I could be one of those teachers easily but it seems very demanding as well. Teaching through problem solving allows the students to work though things more independently or in groups with one another but if they are to do this the teacher has to be very careful on the content they choose for the students to work through. I guess at this point I just don’t have the best understanding of teaching through problem solving but I’m hoping to gain more knowledge on in throughout this course. I’m already learning so much and becoming more confident in my math teaching skills. I’ve been in and out of the classroom subbing and in today for interning and I felt more relaxed than I thought I would while helping the students with their math. I concentrated a lot on not giving answers but more guiding them to work through it all themselves. I’m excited to continue learning through this text.
I really enjoyed thinking more in-depth about what math is in chapter 2 as well. When I think back on my math experiences it's not something I enjoy to much. I was always the one who finished in the back of the pack in the math race. I was never very fast at it and it took me a while to understand it. I couldn't agree with you more on the mistakes part, I think as professional educators it will be important for us to really stress that it is okay to make mistakes and that they are what help us learn! So many students are scared of mistakes and are so disappointed when they make them that they can't learn from them. Mistakes can be a great learning tool.
My favorite part of these two chapters was the before, during, and after. I think it sets a baseline or standard to follow. It also puts the learning into a step by step instructional sequence. Patterns and order is something that seems to continually be repeated in what we are working on in Everyday Mathematics. It could be; circle, circle, square, circle, circle, square, circle, ?, ? or it could be the +3 where you take a number and add three to fill in the box. It could be any numbers or any + or -. In fifth grade were doing a number line, red dots on all the multiples of two, yellow dot on all the multiples of 5 and so on. It seems every grade level is looking for patterns of some kind. I love how this chapter goes over language of mathematics because kids really don’t know much of the language and when you are looking at word problems reading it seems to be one of the most important first steps, so they have to have that knowledge of the words and what it means when you say one of them. If I say sum they have to know that that meant to add. I really enjoyed the games, and games keep children engaged. I also like how it breaks it down to the idea that it is not about the answer, so many times children will do whatever it takes to get an answer, guess, cheat, keep asking you, and just give up instead of really trying to figure it out. Math really isn’t just about math. I once asked by college algebra teacher, “Why are we learning college algebra? My students can’t add and subtract, when could I possibly teach them algebra?” He said it wasn’t to teach us college algebra, it was to teach us “TO THINK”. I want my children to be able to think and try to figure things out for themselves. There is nothing sadder than an adult who runs around always asking someone for help and winning because they can’t do it, before they even try themselves. These are really good chapters to read and this is definitely one of the keeper books to go in your classroom for reference.
Shannon H. I was one of those kids that would just stare and not have a clue. I was not a good reader. I really wish I had a teacher that would have worked me through the problems so that I could understand and give me other ways of learning. I also always seemed so scared of my teachers. I think teachers now do so much more at supporting and encouraging students that it would make it easier to learn. Great post.
Chapters 2 and 3 of this week’s reading changed my perspective on math as a whole. When I wrote what math means to me I put something along the lines of using numbers and formulas to solve a problem, and that it is important for everybody in society to know how to do at least the basic functions. I now look at it in a little different light, I still think that it is important for society to be able to do basic math functions. We’ve all have had that casher that clicked the wrong button and was unable to count back our change and we had to assister him/her so that she/he didn’t short change herself or us. It’s sad that so many people don’t know how to do this simple process. The differences in my ideas are that math is using critical thinking and problem solving and not always are you give a formula. I never could do the guess and check method in school; I was too much of a concrete thinker. I know students hate doing naked number problems but in my experience they hate word problems just as much. I think I have solved the mystery to this, the word problems in most math books are worded funny and have nothing to do with the students. My students don’t care how many people on an average day enter an imaginary museum. When giving word problems they need to relate to the students and be written to the students. I’ve watched students, they do 25 naked problems and then they do 5 word problems. They find the naked problems easier but boring and they find the word problems hard. They have trouble shifting from concrete thinking to critical thinking because the lesson wasn’t taught in a critical manor. I think this may be the reason so many students dislike math; its either too boring or too hard.
In response to Tammy, I also think it is very important for teachers to have a before during and after. It’s also very important for students to have this same idea. Students have a short attention span, and unless you give them an overview of that they will be doing and a time for when the lesson will stop they see it as an endless lesson and will not get motivated about it. My husband is/was ADHD in school and unless you tell him specifically what will happen during the lesson and tell him a time for when the lesson will be over (ex. you will do this worksheet and then we go out to recess) he wouldn’t be able to stay focused and do his work. In simple they have to know in the beginning when the end is. Patterns are very important in every grade, I know my seventh graders still do algebraic patterns. They don’t like them because they take problem solving and critical thinking skills and there not use to that kind of thinking in math.
I also agree with you on that after reading chapter 2 and 3 I have a whole new perspective on math. I myself have struggled with math, so this is new to me. I also agree that students to believe that math is too boring or hard so they do not apply themselves and try and learn math.
Honestly I have been dreading this class, because I have always struggled with math. But reading chapters 2 and 3 has really helped me realize what I need to do to over come that and help my students. Looking back I have come to realize I really didn't know what math was. I just learned the basics and prayed that I got the assignments right. These chapters proved to me that math can be done and you do not have to do the problem by the text book. I was always forced to do it just right or it would be counted wrong. Knowing this will help me tremendously in the classroom, because times have changed and students just need to know math and understand it. If they understand it, then they will know math. Now I am really looking forward to this class so I can better myself with math and use my experience to help the students who are like me!
In class, we did some problem solving that made me think about the practice of solving mathematics problems. We were broken up into groups, and were asked to solve a problem about cats and bats. We all had the same problem, but that was where our similarities ended. The groups that completed the task arrived at the answer different ways, which was alright. In Chapter 3, the text states the value of teaching through problem solving. One benefit is that it focuses student’s attention on ideas and sense making. In the class assignment, I kept asking myself if the answer I got made sense. It made me think of more than numbers. Another benefit of problem solving is that it allows for extension and elaborations. It can be easy for the teacher to go off of the information, and ask the “what if” questions. I didn’t realize how easy it is to apply children’s literature into the math curriculum. Later in Chapter 3, the text gives examples of books that can be in the math lesson at all levels. Also, I thought that the story on the Lottery ticket was interesting. It’s amazing how mathematically illiterate we can be.
In response to Cassandra S.- I was also dreading this class. There have been times in my educational career where I was frustrated with mathematics. As you said, you don’t have to do the problem by the text book. Providing real-world examples and being creative with problems can help students understand the process, even if they do not like math. As in our class problem solving example, one can arrive at an answer differently than there neighbor; and, they both can be right. It will be important to ask the students how they arrived at the answer they got, and have them explain why it worked or didn’t work.
I really enjoy reading this textbook. I have had a better mind towards math. I have even felt a lot better at helping my 8th grade son with his homework. I am really learning some interesting things in this book. I feel it will really help me when I am out in the field and as well as in my own classroom as the math teacher. I really enjoyed writing the personal mathematics story. It really made me think of my mathematics history, and after i thought about it, it has not been that bad. I just have to have a better open mind about the subject. I have also enjoyed the problems we have done in class. They really get me to think outside the box. I like how we have a power point to go with the chapters we read. This textbook has a great way of putting the information in there to make it interesting. I will admit I was not very happy that I was going to have to read a book about MATH. Now I am happier because it is teaching me tons of things, and we are just in the first 3 chapters. This will be a great asset to me. I can’t wait to learn to use manipulatives with these problems. I have also enjoyed the books the textbook mentions that we can use to integrate reading and language arts into math. I will feel so much better learning how to integrate all subjects into math. It was easier in my opinion to integrate math into the other subjects but I was not figuring out how to implement the other subject areas into math. I think this textbook will do just that for me.
In response to Cassandra S--- I agree with you completely that I really did not know what math really was. I just tried to get through the assignment and hoped I got it right or that I at least passed the assignment to get a good grade. I struggled with math all throughout my life. This class is also teaching me things to help my 8th grade son with is math. The teacher is hated by all students but she is doing it totally wrong compared to what Dr. Stramel has been teaching, she teaches the students that it is her way or no way and they get the problem marked wrong, so I have been having chats with the teacher. I have stressed to her that there may be more than one way to do a problem and she needs to let the students do it the way that makes more sense to them. I can’t wait to go out in the field and work with the second grade class I am assigned to. It will be fun.
Chapter two and three of the textbook “Elementary and Middle School Mathematics: Teaching Developmentally (2010) by J.A. Van De Walle, K.S. Karp and J.M. Bay-Williams is a book that I really enjoy reading. Chapter two talked about “knowing” and “doing” mathematics and its similarity and differences and chapter three talked about “problem solving”. I found many interesting and important facts, knowledge that made me this differently, points that made me think about my own experience and a few questions. An interesting fact in chapter two of the textbook by Van De Walle et al. was on page 13 of the textbook by Van De Walle et al. “it is important for you to have a chance to ‘do mathematics’ in a way that nurtures understanding and builds connections” (Van De Walle et al.,13). There are so many times that I have seen students just sit there and do worksheets. I have learned that even if worksheets do help students practice it is not the best way to teach students. Teaching must do activities with the students that keep them involved and active in the learning process. Even though I think that all the information in the textbook is important a few pieces of information that stuck out to me in chapter two was the list of verbs on page 14 of the Van De Walle et al. textbook. According to the textbook “These verbs require higher-level thinking and encompass ‘making sense’ and ‘figuring out’” (Van De Walle et al.,14). This is important to remember because teachers do want students to have a high level of learning and sometimes that is not achieved in the classroom. If students are “doing” these words then they become more active than they otherwise would be.
Something that I learned in chapter two of the Van De Walle et al. text book that made me think differently was on page 13 when it states that “Doing mathematics in classrooms should closely model the act of doing mathematics in the real world” (Van De Walle et al.,13). I did not really think about how important it is that that there is a purpose to the math that students are doing. This does make sense because I feel the same way. Students will be more engaged is they are doing the math problems to find a solution to a real problem they may have because they can relate to it. Something that I learned in chapter three of the text book that made me think differently was when the textbook talked about how some teacher may think it is best to show the students answers to problems. According to the textbook by Van De Walle et al. “However, students are not learning content with deep understanding, often forgetting what they have learned; they need a more effective approach to learning mathematics” (Van De Walle et al.,33). I then started to think about all the times I have don’t this in the classroom and I think it will be a hard habit to break but it will benefit the students. A question that I had for chapter two of the text book was how do people feel about talking in the classroom? I was just reading the statement on page 21 of the Van De Walle et al. text book stating “Classroom discussion based on students’ own ideas and solutions to problems is absolutely ‘foundational to children’s learning’” (Van De Walle et al.,21). Some teachers that I have worked with in the past will tell their students that their work was independent and if they needed help to ask the teacher not another student, but I think that it is good when students get a peer feedback. Not only does the student learn from explaining it but the student listening might be able to understand the material from a student’s point of view. References: Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7 th ed.). Boston: Allyn & Bacon.
Andrew, I think that problem solving is a great tool to use in the classroom with every subject. Just like you did with the problem of bats and cats students will have to think on their own if that works or doesn't work. Even if the students get the wrong answer that is OK because at least they are thinking about it and they are more likely to remember it later.
Math has always been my favorite subject and for me learning ways to teach math is so reassuring. I have always been one of those people who just "get it" & when I am trying to teach someone how to do a problem I find myself just repeating what I already said until they say they "get it" too. I instantly thought of this when I read the opening quote on page 13 "No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students." That is what I find myself doing when I am helping my child with math, I just wish I could understand for them so they will get it and move on. I like the idea of teaching multiple approaches like when the book said 7 x 8 can be solved in so many ways, I would have never thought to teach some of the ways like to say if you know 7 x 4 which is have of the sevens, then you can double that and come up with 7 x 8 is 56. I appreciate thinking differently for my own sake because I have always just done math and not thought through math. I think that will make a huge difference in the way I teach math.
Problem solving will also be easier to teach when I understand how to think through math. I have always had a harder time with problem solving because it can't just be done without thinking about it. I like the statement that "effective lessons begin where the students are, not where the teachers are." That is such a basic idea but it is one that I think is lost on teachers a lot. It is hard to remember that they already have the knowledge that's why they know the answer, and maybe they don't even have the knowledge they just have the answer. I like how the book used literature in problem solving. This is a great idea of cross curriculum. I like the idea of reading the book "The Doorbell Rang" to the children to help them understand about sharing and redistribution. This would be a great activity to "act out" in class as well.
I so agree with you on the word problems. I used to dread word problems so I know exactly what students are feeling when they have to do one. I am excited to learn different methods of actually talking through problems and trying to do problems in a way that makes sense to everyone. I agree that most word problems seem to not apply to most people so it is hard to comprehend how to do them.
Chapters two and three are all about getting students actively doing math instead of just memorizing facts, or watching the teacher solve a problem approach to teaching mathematics. I love the ideas of using problem solving to teach concepts and allowing students to use varied strategies to solve those problems. When the main focus of math is merely getting a right answer the students lose the opportunity to gain a deeper understanding of the concepts, and find new ways to attain a solution. When I was in school we were supposed to do the math just as the teacher said. No one ever asked, “does anyone have a different idea on how to solve this problem”. How much more could students in classes like mine have learned if given the chance to explore and discuss different ways to frame problems and different strategies for solving them. For this reason I like the idea of not providing the answers to the problems as discussed in the text. We need to focus on doing the math, not just getting the answer.
One thing that stood out to me in Chapter 2 was that we should value different processes that lead to correct results. I remember in algebra I that I did problems differently than my classmates, but I got to the right answer. I was told that it was still wrong because it wasn’t the way the teacher had taught it. I remember how much that undermined my confidence to be able to solve equations. But in our text, two children used unique problem solving techniques and it is applauded. I think the teaching of mathematics has come a long way! Figure 2.11 on p. 24 really helped me to understand the importance of connecting as many ways of understanding as possible to the learning of new concepts. I also have a great appreciation for the concept of scaffolding. I can see where scaffolding has helped me learn many times, I also see where means to scaffolding can become a crutch and it is important to guard against teaching mindless us of manipulatives. I see where manipulative use can help learners, especially those who are struggling with a concept and yet that it is important not to teach using manipulatives in a way that as ineffective as memorization or learning by rote. A paragraph on p. 22 stood out to me when it addressed learning by rote (memorization). It says “Rote learning can be thought of as a “weak construction.” This is because the learning is not connected to other knowledge. Maybe this stood out to me because I remember all the flash card drills we had in elementary school memorizing multiplication facts! The shift in teaching to Teaching through problem solving is both exciting and overwhelming to me. One of the frequently asked questions at the back of the chapter is: How can I teach all the basic skills I have to teach? This sums up how I feel. The chapter stressed the importance of the “after” phase of the lesson format and how this can easily take 20 minutes. I think this tends to get squeezed out in classrooms because of time issues. I can see how this will take time to implement and to set up a teaching style that allows time for this. I particularly liked that the chapter laid out some problem-solving strategies on p. 43. I think the method I use most is guess and check. That is the method I have used every time so far when we’ve done problems together “in class.” I learned a lot from these two chapters. I feel a little overwhelmed and exhausted, but I am starting to feel like all the teaching knowledge I am taking in is coming together, I am starting to feel more and more equipped to teach. I love that feeling of empowerment. I hope I can give that many students!
April B.~ I could really relate to your post. I was not very excited to realize I would be learning about math. Teaching math seemed really daunting too, but the text is a great tool. I see a lot of correlation between the information in the text and the way that my mentor teacher teaches, so I already feel like what I am reading is being reinforced in the classroom. I am like you, I can't wait to play with my manipulatives! Once I opened the bag and looked inside, it was like Christmas in August!
Adrianne, in response to your question about the Pythagorean theorem, I feel that we need to allow children to “learn through exploration” as you put it, but we can give them tools with which they can better navigate through their explorations. Formulas such as these allow students to use skills they have already learned to delve even deeper into mathematical concepts. For instance, students must already have learned how to use square roots before using the Pythagorean theorem, but in using it they can discover new information. The students can use tools such as this formula and others to help them explore in a different way. Using “concrete formulas” doesn’t mean the learning experience has to be set in stone as well.
Chapter 2 was all about knowing and doing mathematics. I have now come to learn that these are two completely different things. I have always been pretty good at math, but after reading this chapter I would definitely say I am better at doing math than knowing or understanding it. I completely understand what the text is saying about how in the past we have specifically learned operations. I can’t remember really ever exploring more into math equations or finding out why they work the way they do. I think that this is definitely something that needs to change, because as I got older and into more advanced math concepts I began to struggle. Now I know why. Who knew there was more than one way to do a math problem? I found it to be very interesting that almost every group in class solved the cats and bats problem in a different way. I have some serious view changing to do about math.
Chapter 3 focuses on using problem solving to teach mathematics. After completing an action research project over effective math strategies last year, I 100% agree with using problem solving in mathematics. Students learn better when math is given real world applications. How many times have you asked or heard a fellow student ask, why do we have to learn this stuff I’m never going to use it? Putting math into problem solving lets students see where this skill will help them later in life and makes them more motivated to learn it.
I like the point you made about the problem solving activity we did in class. You’re right; doing that problem got us thinking about more than just coming up with an answer. We had to really think about the situation and if I answer made sense. I hadn’t thought about the extension idea from problem solving but that is true as well. Problem solving in math can easily lend itself to integration into other subjects.
I really enjoyed chapter 2 and the message it sends to us as future teachers about how not all of our students are going to learn the same way. Chapter 2 provided examples describing how students may interpret and work out a math problem differently, but still come up with the same answer. As future teachers we need to be mindful and aware that not all of our students are going to understand every lesson the same way as their classmates. When prepare class lesson we should have an alternative way to explain the lesson to those who may not understand our initial explanation. As teachers we should also be open to new ways of solving math problems; encouraging our students to work out math problems the way they are most comfortable may result in us finding a new way of teaching mathematics. In chapter 3 different ways of teaching about problem solving are discussed. As mentioned before, not all children learn and understand the same way as their classmates and this chapter will be wonderful to refer back to when looking for alternative ways to teach my students about problem solving.
Jen W. I can relate to your “I get it” story, just in a different way. I have always been one of those “I get it in my own way” kinds of people when it comes to math. I was working with a student a few years ago who just didn’t understand her math assignment and so I showed her an alternative way she could work out the problems; the way I understood how to do them. She “got it” and finished her assignment with very little assistance. It is because of that student that I try to think about different methods and examples I can use when teaching.
First let me say how much I enjoy the Pause and Reflect sections of the text. I really think it puts me in a place to think like a child and go through problems as they would approach them. I really enjoyed the discussion in class about problem solving. I thought it was very interesting the different ways people problem solve. I think the section of chapter three that discusses teaching through problem solving was particularly interesting. I remember going through math and just focusing on one problem at a time. And not really paying attention to the concept. I think its a great idea to teach students to focus on the concept and be able to solve many problems, instead of just the one at hand.
Allison G. I am the same way when it comes to knowing and doing Math. It has always come easy to me, but I would always do the math and not necessarily know what I was doing. It was easier to memorize the steps than the whole concept.
I am one of the 'geeks' that really enjoys math. When I work with a student, quite often I will work out the problem the same time as the student. I like to try and find the answer in my head because I want the students to find the answer without looking at my paper and it keeps me fresh and practiced. I have always told the students who struggle that it is a matter of understanding the the why. Why does 2 x 2 = 4, when 2 + 2 also = 4. Because 3 x 3 = 9, but 3 + 3 only = 6! Our text says "One way that we can think about understanding is that it exists along a continuum from a relational understanding - knowing what to do and why" (text pg 23)In other words: 2 + 2 = 4 because when you have 2 pencils and I give you 2 more pencils, you will have 4 pencils. So why does 2 x 2 also = 4? Because if I have 2 Mounds candy bars I have 4 candy bars because there are 2 bars in each packaged Mounds candy bar. (2 sets of 2 = 4)
Why is there two pieces in a Mounds candy bar? "So I can shaaaaaare" (I think I am dating myself)
In response to Tammi Whi, Just to reiterate; to focus on one problem at a time and not pay any attention to the concept is extremely time consuming. When the student concentrates on the concept it can help them to understand the Why, thereby understand the concept and the answer to the problem can come much easier. Good post, I enjoyed reading it.
I attended my first math internship this morning and constantly reflected back on what I had learned in chapters 2&3. In our book we are taught to teach through problem solving, but what I saw today was the old math. I enjoyed learning about ways to engage children to problem solve in math. If I had learned math this way, I would be a much better math student today. I also enjoyed how they integrated Harry Potter with doing measurements. I already feel more confident about teaching math after reading the Three-Phase Lesson Format. I can't wait to use some of the ideas from the textbooks on my students.
Very good post. It is so true that children may learn math different ways. Being a SPED minor, I agree that teachers should have alternative ways to teach the math lesson. Teachers should also give students the chance to show different ways a problem can be solved, if applicable, to help address various types of learners.
I really enjoyed the activites we did in class talking about problem solving. The question about the cats and the bats really got my group thinking and we actually thought harder than we needed to. Also, the activity where we used the one up one down method to solve an addition problem was great. It was nice to see a different method that could be used to teach math. Some students might not use this way, but as a teacher I need to know that this might be a way that a student who is struggling can use and it might make the most sense to them! I had a lot of trouble with problem solving starting out. I did not like word problems at all. I did not like having to figure out the equation and then solving the problem. I always felt like i wasn't doing it right. But, the more we did them the better I got and the more I realized that the word problems can be like real life situations. I enjoyed reading this part of the text!
I also think it is great when a student can figure out how to solve a problem in a different way that was used by the teacher. This way may be easier for that student and it may even be easier for other students as well. Students who think outside the box like this need to be applauded and praised!
Chapter 2 and 3 does a very good job of explaining mathematics and how to teach it. Chapter 2 describes how to create an environment or classroom setting and how it is the teacher’s responsibility to make it fun, inviting and excitable. This is very important as math is not a likeable subject. So if a teacher can make it fun then students will enjoy learning about math and doing the problems. Chapter 2 also goes over a few problems and lets you see what it is like to be a student trying to figure out the problems. You can see the struggles the students may face or any situations that may come into play when trying to figure out these problems. Now, in chapter 3, it talks about problem solving and how to teach it. A main point stated in the book is that the problem must begin where the students are. If it does not then the students will either not understand or it will be too easy for them to figure out. I like how it covers the topic about letting the students talk about the problems to each other instead of having the teaching doing all the talking. Students learn best from one another. I will definitely have to revisit this chapter so I can remember the before, during and after phases.
The bats and cats problem also got me thinking and made me realize what it is like for the students to figure this out. As a teacher we need to make sure that they are solving problems that is neither to hard for them to understand nor too easy that they get bored. I too enjoyed reading about this in the text and liked how the more problems you do the easier they get to figure out the answer.
I really learned a lot from the combination of reading and how it was introduced in class. I like what was added from Dewey about what the teacher should be like I want to be like that for my students. I liked the teacher saying it is okay to make a mistake. I have seen and experienced moments of feeling terrible over a failed assignment. The four features for having a productive classroom fit right in. The first is Ideas. The second is student respect for different methods to solve problems. The one I mentioned earlier be able to make a mistake to learn from it. The fourth being the logic and structure of the subject.
Adrianne, I totally agree it is so hard to work on a problem without a certain answer. Just one of those human error, logic, made you think kind of problems. I do like the way this class is ran.
While reading in Chapter 2 I thought about how setting up the mood/atmosphere is a very important role of teaching math in any classroom. Starting out saying that "you don't like math either" or that "math is hard" makes Math sound like the bad guy. Setting up math in a fun and enjoyable way is a perfect way to encourage students. I never realized how important problem solving was until I read through ch. 3. Problem solving encourages students in so many ways that aren't usually seen. Problem solving allow children to be more confident, to see the connection between math and life, and so many more things. When students are able to connect school work to the "Real world," they are able to grow exponentially in great ways.
response to "tracyp": I really can relate to what you said about making mistakes. The U.S. is very guilty of making people so perfect so then everyone thinks they have to be right the first time or they will not make it. It is totally a lie. Mistakes are just anther way we learn!
In chapter 2, I thought it was interesting that the description of mathematics used is similar to what I’ve come to try to describe to my own children. I always tell them that it is about looking for patterns or order. I don’t know that I’d have had this description years ago, but in trying to help my own children, I’ve found that this seems to “click” with them. However, I had never been introduced to the patterns shown in Chapter 2. “Start and Jump Numbers” and “One Up, One Down” were definitely new concepts for me. It was a great reminder of how it feels to be introduced to a new math concept, especially when we participated in the activity in class. One item that really jumped out for me was the Ineffective Use of Models and Manipulatives. I am an ELL Para, and we have been using Base-ten blocks in our second-grade class. The teacher does demonstrate quite a bit by saying, “So if we have 46, how many ten blocks should we put down?” Usually, some of the more advanced students will immediately say or model the answer, and I can easily see that other students are just “doing as they see” instead of doing what they understand. I will try to visit with the general classroom teacher to explore other methods we might try. The Implications for Teaching Mathematics sections in this chapter are wonderful. I like how they are divided out and titled for quick reference. There is so much information to learn, and this book is outlined well and will be a terrific resource!
I also believe that the key to teaching students is to get them excited about the content, to make a connection with their prior knowledge, and to use multiple teaching approaches. I am also a Paraeducator, and I am finding that it can be a challenge to break away from the way I was raised and taught in the classroom – which was to teach it one way...and one way, ONLY! I am an ELL Para, and it is interesting that I find myself just as challenged with many non-ELL students in finding ways to “reach” them. Students do bring so many different backgrounds to the classroom, and it definitely requires changing up your teaching methods several times for one lesson. We are currently working with tens and ones digits, and I am amazed at how many approaches the general classroom teacher and I have taken and still had difficulty reaching some of the students. I am excited about this text and plan to use what I am reading to help my students in every way I can.
I feel that Dr. Stramel’s “Bats and Cats” example used in class was a good representation of what Chapter 3 is all about. It is important for students to do the thinking when problem-solving, instead of just modeling what they see. If the students don’t actually think through what they’ve done and why they’ve done it, they won’t fully understand the process behind it for future use. It also goes hand in hand with the idea of multiple entry points. When she asked each of us to provide ideas on how we might solve it, there were many ideas that were suggested. Each idea came from a different background of thought, and, ultimately, none of us could come up with a solution in class. When we returned to class, the majority of us had solved the problem through the use of a starting point and then a process of elimination by figuring out which combination of numbers worked. We all knew that there must be a formula to solve the problem, but most of us could not come up with it. ONE student in the whole class knew how to solve the problem with a formula of variables, solving for one variable and then substituting that value to find the overall answer. This was a college class, and only one student knew the most advanced way to solve it. Just imagine what this means for an elementary level class! Dr. Stramel was careful to listen to all suggestions and have us “guide” her through our thought process, instead of just giving us the answer. She was also sure to allow us the opportunity to talk to each other, as the chapter encourages. I loved the “Phases” section of this chapter, too. This is such a great resource!
I completely agree that students need to be able to relate the problem-solving to real-life situations. I have heard (and said myself) many times, “What is the point of doing it if we are never going to use this stuff? Students need to understand that there is a very real reason for learning math, and when they can relate math to their everyday lives, I truly believe that they become so much more engaged and involved, and they learn and understand so much better.
Chapter 2 hit the nail on the head in the title. Exploring What It Means to Know and Do Mathematics, wow what a concept. Early in the chapter mathematics is defined as a science. The text stated that much of the time, mathematics is limited and the students don’t have the opportunity to deeply learn about different topics. The chapter went on to discuss the wordage associated with mathematics. I appreciate this book for the fact that it lets us get into the experience. I like the invitation to do mathematics; it is so user friendly and keeps my interest going! The theories discussed making it so easy to understand the ideas behind the teaching! Chapter 3 on problem solving was interesting. I have always struggled with problem solving and teaching it will be so much easier with the help of this text. I value the paragraph that discussed that teacher will be changing the philosophy of how she thinks and how her students will learn best. Involving children’s literature in math is a valid concept, but something that new teachers might not think about. After reading this section, I have so many great ideas!! The four step problem solving method is used in many different ways; however the concepts are the same. I information on teaching in a problem based classroom will be helpful, it is important that students like to talk, so let them!! It is nice to know how to teach in phases. I feel that it is so beneficial to our students to be able to break up the information that they have to eventually put back together!!
In response to Jeanette, I have also worked as a high school paraprofessional and I will completely agree with you that most of them have negative feelings when it comes to Math. I was one of those students too. I can honestly say I had some teachers for Math that I thought were bad teachers. Math tends to be somewhat boring anyway and to have a teacher go in and read a lesson then give you assignments leaves no room for anything positive. I want to be one of those teachers too, that makes it enjoyable for students of all ages. There are even activities that can be done at the high school level and I think it would make the student and teacher attitude much more pleasant!
I really enjoyed this chapter. It discusses that we must build on prior knowledge and this is so true. We have to have some knowledge in the area we are focusing on in order to build and expand on it. Math does not come easy to many students and this is no secret! We as educators must take the basic knowledge that the students do know and build from there. It is too difficult to try and teach something when the basic background knowledge isn't even there! I also like in Chapter 3 the information on Problem Solving. I was horrible at the problems and to this day still tend to struggle at times. I think it is imperative that there are different examples given to let students know that there is more than one way of doing something. What works for one many times does not work for others. This is okay and I don't like when teachers in my past would explain something only one way. I still remained clueless as on what to do. I feel I gained great information from this chapter and took notes for my folder to refer back to in the future!
You are all AWESOME!!! I have really enjoyed reading your posts, and the things that you gleaned from the chapters and our class discussions. Keep up the good work!!!
This chapter was really fun to read. I can remember in my days of elementary school using the tools and how much of a benefit they were to use these manipulatives. Once I could visualize them, I could usually do a problem in my head again visualizing that tool that was used. I think probably one of the most important parts of math is problem solving. Problem solving is an every day skill that everyone needs to learn. Also, as a teacher, the most important part of teaching is reflecting. As we have learned in many other courses we must reflect and think of what we did well, what we can do better, and what will we do to make it better. Technology is a wonderful tool that God has given us to use. I am excited about these tools to use in the classroom someday.
In response to chapters two and three, let me begin by saying that this semester my mathematics text is my most favorite one to read. I love the language used in this text! You do not have to be a math genius to figure it out. For me, that is a BIG statement, because normally I would try to avoid math if at all possible. I must say that from the way I was taught mathematics it is very difficult to not be provided the ‘right’ answers. It literally makes me stop for a second and I have to reemphasize the entire point of the text to myself- that learning is through exploration- and then I can proceed. It’s a difficult task to master but I am working on it. That is the most frustrating aspect for me in this process, that ‘correct’ answers are not provided. I do agree that the learning is more concrete if students make the connections and go through the process of learning to solve problems on their own. I know I would have learned better in this manner, but it leads me to have one nagging question in my head. Where do formulas like the Pythagorean theorem develop? Surely, at some point this information that these students are developing gets filtered into a concrete formula- but at what point does that happen?
ReplyDeleteToday was our first day of school. I work as a special education paraprofessional at the high school level. I noticed today among students getting their schedules how many made negative comments about math class. I wondered where this feeling came from. Is it from a bad experience back in 1st grade or just coming they picked up from peers. Anyways, it reminded me that I want to be the kind of teacher who instills excitement about math. I realize not all kids are going to enjoy the subject, but I would at least hope they could keep a neutral attitude and learn the skills that were discussed in Chapter 2-3 about solving problems. For instance, the text states, "The teacher's role is to create this spirit of inquiry, trust and experience." I think it is too easy to quit when something becomes hard. I don't want my children to have this attitude or my students. I agree with the text about building on prior knowledge. I believe that if we can relate a topic to something they know about it is easier to understand. I know this is true for myself. Another, topic I really enjoyed was encouraging multiple approaches. Having worked in special education for 7 years I have seen first hand how students learn differently, but yet come to the same conclusion. As teachers we need to encourage every learning style and celebrate diversity. As I continue to read through this text it is so easy to see how math is involved in everyday life and why it needs to be integrated across the curriculum.
ReplyDeleteTo Adrianne: I wish I could answer the question about the pythagorean theorem. I tell myself that it is an important theorem for people who do research. I believe that math is a subject that is very important for students to learn. Math is a process and if one piece (such as multiplication) is missing it messes the whole process up. I worked summer school this year and tried to stress to elementary students how imperative it is for them to learn their basic addition, subtraction, multiplication and division facts. I like this textbook as well.
ReplyDeleteWhen reading Chapter 2 and 3 there was a lot of great information that I was able to gain from reading the chapters. I loved the part of Chapter 2 that talked about the four different features of a productive classroom culture in relationship to math. If as a future teacher I am able to implement these features into my classroom I will be able to have a productive math environment in my classroom. I also found the part of the chapter that talked about the five strands of mathematical proficiency to be interesting as well. It talked about how all of the five strands need to be used together in order for the classroom to be successful. I also learned about how the five strands are interrelated and interwoven together in order to show how the five strands are so closely related.
ReplyDeleteWhen reading Chapter 3 about problem solving the item that popped out of the text to me was to teach through problem solving. Math and problem solving needs to be taught through real life contexts and situations as much as possible. The students will be able to remember the information better if they are able to relate math to the world around them rather than just filling out a math worksheet and forgetting the concepts learned. I also enjoyed reading about different types of problem solving strategies. I found the list to be very helpful for me as a future teacher and I can see myself using some of these strategies in my future classroom.
In response to Jeanette
ReplyDeleteI have also always wondered where the dislike for math in students comes from as well. It could be a number of different reasons and as a future teacher I feel that it is my responsibility to figure out the reasons why they dislike math and then try to make the experience with math in my classroom a great experience. I also agree with you that it is so important that teachers build on their students prior knowledge in all subjects especially math. It is a lot easier for students to gain a new concept if they are able to relate back to their prior knowledge. Lastly I agree that it is important to have multiple approaches to teaching each math lesson because all students learn in different ways and you want to teach the students in the way that they will learn best.
In response to Brooke M,
ReplyDeleteI also thought the four features of a productive classroom were very crucial to success in mathematics. I especially like the attitude of this text about math being a product of discovery. I too thought these chapters were filled with valuable ‘gems’ of information to future teachers. I also love the way they focus on making the mathematics applicable to the ‘real world’.
I really enjoyed the activities we did in class in chapter two. I know that the group I was in was looking for an answer, but there wasn’t one. Sometimes this is so hard to understand, because we often think that everything must have an answer. One of the girls in class said she understood the process of “one up, one down” but didn’t know how or when she would use it. For example, 7+7=14, but when are we going to face a situation is which we remember that, and then add one and subtract one from each number to obtain our answer? I thought this was a very valid point. I think that sometimes as teachers, we may forget that the students don’t know and that we often assume they do. Just because this technique may not help us, it may help a child better understand addition facts or number patterns.
ReplyDeleteProblem solving is probably one of the difficult for children to learn. As a student, I was so used to number problems that it was almost terrifying to work a word problem. I was taught to find the question or the problem, pick out the important information that you must have to solve the problem, and then finally solve it. I remember by the time you advanced to a certain grade, you no longer were allowed to draw pictures to help you. You had to use the equations and do the problem solving “the teacher’s way”. As a future teacher, I want to better myself and allow my students to complete and solve problems in a way they understand; not in a way that they can’t even read or comprehend.
In reply to Brooke: As I was reading your post I remembered back to when I was in grade school. Several of the story problems were about things and ideas I had no idea what they were. I think having problems relating to the children and their background knowledge is so important. This is also an example of why relying on textbooks and worksheets is not always a good idea. A lot of times they contain content that is not appropriate for the culture, setting, or students you may be teaching.
ReplyDeleteWhat we did in class for Chapter 3 was what I really enjoyed this week. Problem solving, to me, has always been kind of entertaining and fun. It was fun working the bats and cats problem in class and then going over it today because there were several different ways to go about solving the problem. Three of the ways were draw a picture with the legs and then start at one end circling two legs and the other side circling four legs until you meet in the middle and run out of legs and see how many of each one you have. The second way was guess and check which is the method I went with. It’s not the fastest way to solve a problem but it works and sometimes you get lucky and solve it on the first try or two. The last way that was used was the formula. After I solved the problem using guess and check I tried to come up with a formula to solve it and just flat out couldn’t think of what it would be. After going over the formula today in class I feel like it should of come naturally to me that’s just how it works sometimes. Everybody has their own way of being able to solve a problem. Just because it’s not the way you were taught doesn’t mean that it’s the wrong way as long as you can support your answer.
ReplyDeleteAdrianne Hoefler
ReplyDeleteI complete agree that the statement, “You don’t have to be a math genius to figure it out.” Is a huge statement. I am living proof of it. When it comes to solving problems I am definitely not the fastest at it. But I will eventually solve the problem. I remember when I was in Mathcounts in Jr. High, we would go to competitions and there would be kids who could solve the problem before I was even done reading it. It doesn’t necessarily mean that they were better at it than me, it just means that they have the ability to go through most of the information quicker than me.
I found the vocabulary words used in the beginning of Chapter 2 to be important to know. As teachers I believe we should all be on the same page when it comes to math vocabulary. Do our first grade students know that "take away" is the same thing as "less than" or "subtract?" Explaining real world problems in terms that our students understand is a way of connecting their worlds to math. For example, using M&M's when adding or subtracting. Grouping by the color of M&M's and then using the candy for predictions. When they can solve the problem then allow the student to have a sweet treat.
ReplyDeleteI have enjoyed reading about math in these chapters. The authors have written this textbook in the same way they are teaching us how to teach math. In a real world setting. The answers to questions are some times not given rather questions are asked then examples are given to help understand the concept.
I had a hard time with problem solving when I was an elementary school student. I found myself frozen in fear when a worksheet was presented with a page full of words. I was unable to decipher though the words to get to the problem. In order to teach effectively then asking questions about the problems before asking students to solve will help them work through what is being asked. Allowing the students to ask the questions and do the talking will help them work through the problem before they even see the words and numbers.
Learning about the lesson plan phases was informative for me in chapter 3.
Deidre, your experience is shared by many. Terrifying is a great word to describe what I felt as well. As future teachers I think we will have the knowledge and skills to approach teaching differently. These chapters talked a lot about how to communicate with the students and how to relay the math message without using intimidating language or strategies. The Frequently Asked Questions at the end of chapter 3 were great to read through and get some real life answers to real life problems.
ReplyDeleteI am really liking this book so far, I hope it continues. The activities we did for Chapter 2 were fun. I told my boss and her daughter about the Start and Jump Numbers and the One Up, One Down. I thought they were so neat that I had to share them. I think what it means to do and know mathematics is to understand what you are doing and how to explain your reasoning. I don't think you have to be fast at solving problems in order to be good you just have to be able to understand and prove your answers. I feel incorporating real world situations in math for students will make math not seem so overwhelming. I think most students are intimidated by math because of the thinking that goes into it and the fact that there is one correct answer.
ReplyDeleteChapter 3 to me required more thought. I am still a little unsure as how to explain what problem solving is. I would have to say its being able to solve different types of problems. It talked about letting the students do the talking. Allowing the students to discuss problems and ideas with other classmates learning will occur in different ways. The chapter also mentioned different problem solving strategies. There seven different ones; draw a picture, act it out, use a model; look for a pattern; guess and check; make a table or chart; try a simpler form of the problem; make and organized list; and write an equation. Who would of thought writing was important in math, it is. It allows students to be reflective, a rehearsal for discussion and its also a written record with the lesson is finished.
Both chapters had lots of goo information.
Jena,
ReplyDeleteI didn’t talk about the vocabulary, but I agree with you in them being important. Students do need to know that there are multiple terms for subtraction, adding and so on. I like your examples of the use of M&Ms. Math was always more fun and exciting for me when we used fun objects or incorporated it with real life. I too had a hard time with problem solving and feared word problems. I would sometimes sit there and make it look like I was working on the problem because I didn’t want to do them. I agree to that children need to be able to do the talking as well and not just the teacher.
There was some very good information in chapters 2 and 3, but my a-ha moment came with the cats and bats problem. I have said it before and I will say it again that I am very intimidated by math. I always feel like the clock is ticking and I have to find a solution fast. I heard the problem and thought immediately that I couldn't do it. Then my moment came, slow down, take your time, find the best way for you to work it, and you can do it, and I did. This little problem might just have increased my confidence a little bit, I felt that moment of success, and this is what students need to feel. Chapter two provided implications for teaching mathematics. One of these is to build in opportunities for reflective thought. I think this is very important because it will help students understand where their answer came from, how they got there. By doing this the students will be more likely to remember what they have just learned. Chapter 3 was about teaching through problem solving and I found this chapter to have some very useful information and it will be a great resource in the future. This is so important because it helps students in finding their own strategies to solve problems, it doesn't just say this is how it must be done. I also liked that this chapter gave an activity evaluation and selection guide. This helps teachers find worthwhile activities, and ensure the students will see the importance of it. Overall, I got some very useful information from both chapters.
ReplyDeleteIn response to Kayla R.
ReplyDeleteI am also finally discovering that just because you might not be really fast at solving problems doesn't mean that you aren't good at math. If I would have figured this out a long time ago, I might like math a little more. I also like that you are sharing the problems we are doing in class with others, I agree that the activities in chapter 2 were fun!
Elizabeth, I, too, like that discovering that just because I can't do math as fast as my husband, does not mean that I am not good at math. He is just quicker! Maybe that has to do with me being a more visual person and I have to "see" the problems, sometimes, to figure them out.
ReplyDeleteI had never considered myself "good" in math. However, I can do all types of calculations when needed in life. Maybe not as fast as other, but I can do them. I can do simple ones in my head; but harder ones, I need to write down and work on paper. It does help to have real life examples or experience. When teaching, I try to bring real life examples into my teaching. Things that the students can relate to. This also helps to activate prior knowledge. I tend to be a "hands-on" and "visual" learner. It is for this reason, I believe, that I lean toward using manipulatives and pictures when I am teaching, as well, as putting lessons in terms/scenarios that students can relate to. This seems to fall into the multiple representations of math ideas.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteLacey Keller
ReplyDeleteWhile reading chapters 2 and 3, I could relate, not from my own personal math story, but from another student. You see, I am a special education paraprofessional here in our local community. Last year, a student came into the program from way on the eastern coast (I live in western KS). This young gal was frustrated with math at her new school. However, when I sat down with her, and she opened up, stating she just wanted to do math her way, not the teacher's way! So, I let her. Oh, boy, this was the best thing I ever did for that student. Apparently in her old school, she could use different strategies(suggested in chapter 2)to solve problems. However, she felt threatened when her new teacher would not let her do that. I loved doing math with her because it was a challenge for me to see just how she was going to solve the problem! I may look at a problem and say, "Just add." However, she would would look at the problem, subtract, add, and multiply and still come up with the correct answer! I see now from reading this chapter that this gal had an teacher who valued the invented approach.
I think the most valuable reading from these chapters came when the author states the teacher must create meaningful contexts. You have to find links to other subjects to get students excited! I remember always asking my high school geometry teacher, "Am I going to use this stuff in the real world?"
Lacey Keller
ReplyDeleteKim,
Your comment reminds me of the science methods class where science is "hands on, mind on." This means we must provide manipulatives, experiments, and real-life experiences to get our students learning.
In response to everyone who has asked about the Pythagorean theorem, I once sat in on a math class where the students were making posters while learning the history of famous mathematicians. The students learned about people, theorists, and had fun while doing math! By the way, Pythagoras was the man who came up with the equation.
First, I must say that I am so grateful that we actually use the textbook during class. Working the problems has been so helpful. Typically, I skim or skip past applications like these. Allowing us to be math students has been a tremendous help. Seeing the discovery process in action has been an eye opener. I love hearing discussion about how each student or group worked to find the answer. I don't remember getting to work much with groups or partners during math lessons. It was mostly individual work. Creating this type of environment will encourage students to be engaged.
ReplyDeleteI never really put much thought into the learning process of mathematics. Such as learning about shapes in preschool that eventually go into higher geometry learning. Reading about the number 7 on page 25 helped reinforce that knowledge. 7 is more than 1 and less that 10, 2+5=7, it is odd, compared to 1/10 it is bigger, it is "lucky", it is prime, ect. In theory it is a progression of "connecting the dots."
Integration is something that I look forward to doing in my classroom. I loved the idea of using "Harry Potter and the Sorcerer's Stone" as a measurement lesson. How tall is Hagrid compared to me? Using adding machine paper the students measure their height. Then they compare it to Hagrid who is twice as tall and five times as wide. Great ideas!
@ Elizabeth
ReplyDeleteAs a child it was always great to be the first person done. Everyone would think that person was a genius. Immediately, frustration would come it to play, realizing that you were not even close to being done. In the future I want to create an environment where speed is not the case.
I think that the best thing I read from Chapters 2 and 3 was this; "Mathematics is the science of pattern and order...Science is a process of figuring things out or making since of things." Wow. Growing, up I was always taught formulas and the "how" to solving problems. It rarely made since to me. "Science is a process of figuring things out or making since of things." I wish that some of my science teachers would have considered this while they taught me. I like how Dr. Stramel told us Friday that it is ok for each students to figure the problem out differently. And that we could even have them share with us, as teachers, how they figured the problem out. I believe it is good for the process that they choose to be correct, of course, but I can definitely see how the students figuring it out themselves would help them to "make since out of things". I remember sitting in many different math classes and there was usually one student who could figure the math problems out very quickly and in a way that was different than the teacher. I remember them always getting marked off for not doing the problem the way that the teacher showed the class. I don't ever remember a student, myself or a friend, doing a problem differently than that way that the teacher taught and not getting marked off for it. It was nice for the chapters to reiterate that it is ok for students to figure things out for themselves, and that the process of figuring it out can bring that full understanding and help the process to make since to each student.
ReplyDeleteBrandi S.
ReplyDeleteYou said, "I don't remember getting to work much with groups or partners during math lessons. It was mostly individual work. Creating this type of environment will encourage students to be engaged." I completely agree with you! I am excited about trying this in my future classroom! I have always hated math because it was boring and frustrating. If I had the opportunity to discuss it with my peers and to work it out with them, I may truly feel that I would have had a different experience. I cannot wait to see my future students enjoying math as they work through the process in groups with different tools that will allow the learning process to be fun rather than stressful!
Katie Coulter
ReplyDeleteChapters 2 & 3
Blog 2
For the reading this week chapter three really caught my attention simply because it was all over problem solving and me and problem solving don’t go together. Me and math don’t go together but me and problem solving really don’t go together. Even after reading all the information I am not sure I feel very confident in myself to teach this standard or math itself. I enjoy the older grade school students so that is where I am hoping to one day teach and of course that means more in depth studies. I did enjoy reading about letting the students do the talking and you monitoring. In my technology class we have been talking about partnering teaching, which is allowing more student to student teaching through activities and groups. I like the idea of all of this. I think students need to be more challenged in that aspect. I think it helps them in the real world because they create the habit to analyze the situation and try several approaches without someone telling them how to do it. Of course the teacher needs to be present and active in the learning so they don’t head off course from the correct answer and I don’t think it makes teaching any easier but it does help your students.
In response to Kim Markham,
ReplyDeleteI too like the concept of relating teaching with life examples. Student’s, even if they don’t really care at this point of their life, need to be aware and have heard situations they may one day encounter. As I look back on my education I wonder how many life stories I heard but didn’t really listen. I wonder how many situations would have turned out better had I only listened. School subjects can be related to almost everything and I too hope to be able to bring a variety of experiences and knowledge to my room. Using manipulatives and pictures and stories is a great way for a student to remember and grasp the concept being presented.
I really enjoyed reading chapters 2 and 3. I also enjoyed watching the face-to-face class. I am not all the comfortable with the idea of teaching math yet but I am constantly writing things down in order to refer back to them later to help in the classroom. I love all the information on problem solving. I think that students need to see the problem as well as read it on paper. For the younger students I think that having students pick out keywords and draw them and cut them out would be beneficial to solving the problem. I also like the information on the importance of pattern and order. Students need to strive to see these things when faced with a math problem. I think that it is important for students to know when, where, and why they will use these math skills outside of the classroom as well.
ReplyDeleteKatie-
ReplyDeleteI love that you stated that you and problem solving don't go together. I feel the same way. I have never been good at breaking it down and seeing all of the little pieces of the problem. I read it and instantly feel overwhelmed with the amount of information in it. I think that it is great that we are learning about ways to avoid letting our students feel like that and how to help them "go together" with problem solving!
I thought it was important in Chapter 2 when they talked about how the teacher should have a good feel for mathematics. I also liked how it said that you should get together with a peer and work together so you can get some experience with sharing and exchanging ideas. I think that just goes along with being prepared. It's just like Dr. Stramel said, you should work the problems along with the students so you're prepared for anything. One of the things I got from chapter three was confidence in students. The more a student enjoys working on a problem, the more likely they will be in succeeding at it. Attitudes go a long way in math.
ReplyDeleteThis comment has been removed by the author.
ReplyDelete@ Lane A.
ReplyDeleteI agree, the activity we did in class was a great exercise. I liked how it made people of our age level really think about the problem. It all goes back to attitude, in my opinion. We all had a good time working together and figuring out the problem. That makes things go so much smoother. That was one of the main things I picked up from Chapter 3. A good attitude can take you a long way!
Chapters 2 and 3
ReplyDeleteWhile Reading through Chapters 2 and 3 I am unfortunately reminded of how much emphasis and time I spent trying to constantly find the “right” answer, not only in mathematics but throughout many subjects and situations in life. However, I have been working on this part of myself and finding that things become a lot easier when not always focusing on just getting the answer. I was thrilled to see that students are still using physical geo-boards and manipulatives within the classroom. I have seen some schools that have advanced very much with technology and only use those activities on the computers. I did enjoy the activities in chapter 2 that were done in class; I tried to do them on my own and did well but didn’t find as much information as the students in class since they had groups to work in. I also loved how the math verbs were listed for everyone to see in the beginning of chapter 2, made it much easier to picture and gave more options than I had originally thought of. Chapter 3 about problem solving was honestly something I was dreading reading when I saw the title because I remember hearing and learning about problem solving way too often when I was in elementary school. However, I enjoyed it because I learned the best ways to teach problem solving methods and help the students. The FAT( For, About and Through) is something that I will be able to easily remember. I definitely believe that all three of these problem solving situations are important and should be used but teaching the students to teach Through Problem Solving is the most important. I am enjoying reading this text much more than most textbooks I have ever read. It is not boring and it has fun activities to keep from getting bored.
In response to Kristie C...
ReplyDeleteI am also a little weary about teaching math. I feel like if I screw up than these students will be screwed up for the rest of their mathematics career. I think more than anything else it is me putting stress on myself and I realize this class is really helping to prepare me, and we have only been in the class for a short while.
After reading chapters 2 and 3 I found there to be a lot of useful information. I think that everyone has a different opinion on what “to do math” and “to know math” means. I don’t really feel like the book is limiting us to what we need to know and where we need to be at in regards to math when we are educators. Instead it is giving us guidelines to follow, but how we teach the lesson is up to us. Which brings me to my next point.
ReplyDeleteI love that this textbook is more than just text, but throughout the it there are sample problems, ideas, and tons of examples. I think my favorite part of chapter 2 was the explanation on ineffective uses of models and manipulatives. I can see how this could be a problem.
Today was the first day I was in the math classroom since reading chapter 3. I think it will be interesting to progress through the semester. I did not see a whole lot of variety today due to a test they were studying for that they have tomorrow over unit one. However, there was a lot of problem solving. I was helping one student, and it was so hard for me to not direct him to the answer. In chapter 3 there was a section about how much to tell and not to tell a student. It was challenging for me to not just tell him the answer, but I knew I couldn’t because I know this math, he on the other is still learning. I think it’s kind of funny though because he saw I was willing to “help” him, and I think with that he thought I would just give him the answers. It will be interesting to see how I change throughout this course / the internship!
Within Chapter 2, I like the overall idea of always relating everything back to the real world. When applying this to mathematics, students will receive and comprehend more when knowing they can apply what they are learning to their everyday lives (with all subjects, not just mathematics). For example, you start off with 5 baseball cards and your friend asks to trade you for 2 of yours for 1 of his...how many do you end up with if you agree on the trade? (3 baseball cards) I want to keep this in mind while in my future classroom. I also enjoyed and related to the four features of a productive classroom. I especially appreciated "The classroom culture exhibits an appreciation for mistakes as opportunities to learn." I think it's crucial as a classroom teacher to provide room for growth with all students. It's only natural for us as humans to make mistakes, and I want students to know that they need to try, but it's okay to fail. That's how we improve! Especially with potential hard homes lives, I don't want students to look at failure as always a negative thing-it's important to encourage their efforts. Also, with the idea of patterns, I agree that it's important to change up patterns and see how it affects the outcome. This make mathematics fun for students and keeps them interested!
ReplyDeleteWithin Chapter 3, I find out that there is teaching for, about, and through within problem solving. When teaching through, I feel it allows the students and teacher to connect on a different level. It's important to keep productivity flowing for the classroom environment... I have had teachers in the past that didn't elaborate on the mathematics lesson being taught. It's important to keep the domino affect rolling. I feel this makes learning not only more fun, but more beneficial for the future of the students. Especially going from elementary school mathematics to middle school mathematics.
@ Karissa C:
ReplyDeleteI can completely understand how much focus is put on finding the correct answer... I feel like I've always looked for the correct answer because up until this course finding the right answer was important, versus understanding the concept that was being learned. I'm also not a fan of all of the technology that schools have switched to. I loved manipulatives when I was in elementary school, and although technology is great, sadly it's what is dominating our society so it only makes sense to incorporate it in several ways throughout the classroom!
These two chapters really open your eyes to the idea of teaching math. The concepts showed me that just because I know how to do it one way doesn’t mean a child will think the same way. Some children think differently and out of the box. This is not a bad thing at all. The concepts that children come up with are amazing. The concept of doing math is not giving children the problem and then telling them this is the answer. That is totally not helping the child understand. By giving them the problem and helping them problem solve it the way that is best understandable for them the children can learn better. I love the statement of “In the real world of problem solving outside the classroom, there are not teachers with answers and no answer books.” This is a great thought. Knowing the “answer” does not mean you know how to do the mathematics. Teaching problem solving is teaching children how to think out of the box. By letting the children think on their own they can solve the problems using their past knowledge. As I become a teacher I think it is important to know the concepts and the way my children do learn. I think this will help me in my future, just knowing how children learn differently.
ReplyDeleteMatt B.
ReplyDeleteI love your idea of letting the children do their own thing and having confidence in them. Just because you give them a little bit of freedom does not mean you cannot guide them a little. You can take their idea of solving the problem and expand on it.
Amanda Lewallen
ReplyDeleteI agree with you, it is important to know that every student who gets the correct answer will not find that answer the same way. This is hard for those of us who did learn it from teachers who did not take a problem solving approach to realize. What worked for us may not work for our students and what works for one student may not work for a different student.
Jeremiah Gramkow
First of all I like this book, it is actually doing its job, teaching me how to teach, or at least is trying to teach me how to teach. Some of my textbooks have not done a good job of doing that or even making it look like the book was written for that purpose.
ReplyDeleteThese chapters have a lot to teach me, as someone who has always been good at math. I tend to see the whole picture without much effort and need to remind myself that others can not do that as well as I can. I like the fact that the book says to call on the students who are less likely to talk first. As a substitute teacher I often call on students who do not have their hand up, I have even asked students not to hold up their hand as I intended to simply pick one of them.
The problem solving approach is still work for the teacher, and lots of it. I do not really believe that one approach is easier for the teacher then the other. They require different things from the teacher, but they both require work.
Chapter 2 does a great job of breaking the process of completing a math problem down; which is important for a person like me who moves through the process quickly to remember and be reminded of.
Chapter 3 describes the basics of the problem solving based classroom, it is different but has potential for success in the classroom. At the same time I find it hard to believe that the traditional methods have as little merit as the chapter suggests. While the books methods may work better the chapter come close to suggesting that very few if any students will learn from the other method. This has a simple effect, the book does not have even the appearance of an objective comparison of the two methods. It simply suggests that they method that worked for me can not work.
Jeremiah Gramkow
In response to Elizabeth
ReplyDeleteI couldn’t agree with you more in class when the cat and the bat problem come up, I instantly froze and was like I have to get the answer fast. I loved how we had time to work it out and could do the problem the way we wanted. There were no rules, which is great because as we are learning there is not a “right” answer. I also think that reflecting from what you just learned is great because like you said they get to see where and how they got the answer. As I was a slower math learner I remember I had one teacher that would always ask “how did you get that answer?” and everyone in the class would have to explain how they got their answer to the teacher.
First off I would like to start out by saying how much I love this reading! One thing that stood out to me was the statement that math is the science of pattern and order. Every part of math has orders and patterns and I think that is what I appreciate most about math! One other thing I really liked was about treating error as opportunities for learning. I think this is a super important concept. We CAN learn from our mistakes. Also teaching children different ways of figuring out a problem is great. The school I am working in has everyday math curriculum and they teach so many different ways for students to learn! I really enjoyed reading about the different phases in chapter 3!
ReplyDelete@Adrienne, I also love this text, it truly says it in everyday terms. I think so far it has given us some fantastic teaching strategies! AWESOME!
First off I really am enjoying reading this book. It had great ideas that I can use later on as a future teacher. In chapters 2 it just is reminding future teachers that all students don’t have to learn the one way to figure out the math problem. Each child is different and don’t all think or learn the same. I love that the book says “that mathematics is generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense.”
ReplyDeleteIn Chapter 3 the For, About, and Through method of problem solving is going to be something I refer to as a future teacher. I feel as thou the most important is the Through problem solving because students get to connect to the teacher on a different level.
Throughout chapter 2 & 3, I found a great deal of useful information that will definitely be helpful in the future. Math never was one of my favorite subject, so I admit, I am a little nervous to teach it to my students. I want to find new and exciting ways to help my students learn math besides just strictly math worksheets and working out of the textbook. I believe this textbook is going to be very helpful in helping me achieve this goal. I loved the section on having a productive classroom when it came to math. I will take this information with me. I can't wait to read more into this textbook and see everything it has to offer to have a successful mathematical classroom. Hopefully, I will soon be more comfortable with teaching math in my classroom!
ReplyDeleteIn response to Megan B--I am glad you talked about chapter 3, the "For, About and Through Method of Problem Solving". I forgot to mention this in my post but I also found this section very helpful. You are completely correct, this is something that will definitely be helpful and a great piece to look back on. Problem solving with be used in all aspects of math. Great point Megan!
ReplyDeleteI enjoyed reading Chapters 2 & 3 and am continuing to learn valuable information every time I pick up this textbook. I really liked the section where it talked about how mathematics is the science of pattern and order. The quote, “even the youngest schoolchildren can and should be involved in the science of pattern and order,” came to mind when I watched the students in my mathematics internship class make an AB pattern out of paperclips. It seems like such a simple concept, but I honestly do not remember doing that in kindergarten. The school where I am interning has started to implement the Common Core standards in their classroom and it is obvious that there is a higher level of learning going on.
ReplyDeleteI really like how the text provides a list of verbs that can be used when talking about mathematics and it reiterates the fact students should be actively learning while doing mathematics. The fact that the answers are not given anywhere in this textbook for any of them problems made me feel a bit uneasy. If the answers are not in the book, how will I know if I solved the problem correctly? During high school, our textbooks had some of the answers in the back and I realize that I became very reliant on those answers. It’s important for students to know that the answers to real life problems will not come in the back of a textbook.
Reading Chapter 3 and learning about teaching mathematics through problem solving was very interesting. I had a lot of math teachers who would show us how to do a problem on the board and that’s how we were expected to do those types of problems from then on. I never had to think of an approach to solving the problem because the approach was already laid out for us. I wish I would’ve been given more opportunities to think about different ways to solve the problems myself.
I definitely got that opportunity during class while working on the bats and cats problem. I was expecting to get a formula or something to show me how to solve it but we had to figure it out for ourselves. I ended up using the guess and check method to solve the problem and looking back, it wasn’t really as difficult to solve as I had originally thought. It was nice to see the list of problem solving strategies listed in the text and I agree that students should be able to solve the problem however they want, even if it is not the exact way that the teacher showed them. It's great to be learning about how to teach math while being given an opportunity to solve problems during class.
Jena,
ReplyDeleteI was also always terrified to see a page of word problems in front of me and I have to say that I still am a bit scared. Although "naked number" problems weren't really that much fun, I almost preferred them to word problems because at least I already had the exact problem that I had to solve in front of me. Hopefully through this class, we will both have a different attitude about them.
There were a lot of things about Chapters 2 and 3 that I liked, but mostly I am so happy for the changes that have been made to how math is taught. I loved the explanations in Chapter 2 about what it really means to do, understand, and learn mathematics. When I was taught math it seemed that too often I was expected to just accept what was taught without understanding it fully. I think this is where a lot of my frustration stemmed from. However, if a child is truly doing, understanding, and learning mathematics then tasks are worthwhile, promote discussion, and make meaningful connections to previous concepts. I feel it is important to create an environment in a classroom where students have the opportunities to do all these things. Problem-based learning as discussed in Chapter 3 is important because it makes math skills relevant. So many times in school I heard “I’m never going to use this.” Not only are problem based lessons and tasks engaging, but they are also useful. Today was the first day of my internship and in math the students learned a new skill (GCF) and then worked their way up to a problem solving exercise. My mentor teacher is exceptionally good at how she questions her students. She helps lead them to the answer, but never explicitly says yes or no when they ask if it is correct. I know I am going to learn so much from her this semester, especially when it comes to “how much to tell and not tell.” She makes this look so easy, but I find myself wanting to help too much or give too much information. I am anxious to work on this skill throughout the semester and going forward. I feel as though every chapter I read gives me something new I can use and it is exciting to be able to learn something and put it into practice almost immediately.
ReplyDelete@SarahRob
ReplyDeleteWhen I learned math I never had the opportunity to think about how to solve a problem either because it was also already laid out for us as well. I think it's great that students are taught to truly understand what they are learning in mathematics. I can't wait to see the future implications of this. I started my internship today as well and have an amazing mentor teacher (I feel so inadequate :)). I know I am going to learn so much from her this semester, including how teach through problem solving.
In Chapter 2, we really get challenged to think about what math is to us. What does it mean to learn, what does it mean to do, and what is the difference? I enjoyed that concept and it certainly got me thinking about math and my own personal experiences. To me learning math was all about the “light bulb” moment or the “ah ha” moment, then moment while in class that the switch goes from dazed and confused to place where it just makes sense. Doing math was more about with worksheets and problems to work out of the book. Everyone would try to get done with their math assignment first and it was almost like a race. Doing math was getting the answers right and I didn’t really think about how I was getting it. Learning and doing math really get broken down in this chapter to how we learn and why it is important to do that math. There was a bit in this chapter about making mistakes or errors and how it is okay. I really appreciated this part of the chapter because mistakes should not be embarrassing and we should not try to cover them up, instead we should be learning from them and making ourselves more aware of where and how the mistake occurred so we know what we need work on. I think this text is great and is already offering fantastic information to help us prepare for being out in the field.
ReplyDeleteIn Chapter 3 I felt that the text really dug into the details of the importance of problem solving in our lessons. I specifically liked the part about problem solving helps the students build meaning for the concept they are learning. That is such a key to learning. Students can gain success by easily seeing the relationship of the concept to their everyday life. I really enjoyed learning how the problems are designed and the thought process there. Creating or designing well written problems can be difficult if it is something you have not previously done before. So I enjoyed learning about what is important to look for selecting problems for the class. I also learned that allowing some conversation where the students do the talking can be beneficial to the learning of the classroom.
@ Kristi P
ReplyDeleteI completely agree that getting out into the classroom to teach math makes me very nervous as well. I just finished my first day of internship and I was overwhelmed by the different levels of the students’ abilities. Some of the kids really struggled with the concept of basic addition and patterns. It really opened my eyes that I need to be very prepared before getting into the classroom on my own. I think this book will offer great benefits for us to continue to gain knowledge so that we will be much more prepared.
I will admit that I was a little stumped at figuring out the difference between knowing and doing math, they seem like one in the same. I now see that there is a big difference between knowing and doing math. I also liked having the idea confirmed that students can do math in their own unique way and that is okay as long as they are achieving the correct outcome. I was one of the students who took a little longer with math and I would have to draw pictures and write every single thing out in order to complete a problem. Some of my teachers were okay with this but others pushed for me to do it in my head or to use other problem solving practices. It was hard enough for me to understand math as it was so I didn’t want to have to try to do it in an entirely foreign way that I wouldn't understand. I ‘m glad that the fact students can do math however is best understood to them is supported. It is something I will relay to them when I begin teaching myself.
ReplyDeleteReading about problems solving was a little less reassuring to me than chapter 2. Like the text says, teaching through problem solving requires a philosophy change in how teachers believe students learn. I’d like to think I could be one of those teachers easily but it seems very demanding as well. Teaching through problem solving allows the students to work though things more independently or in groups with one another but if they are to do this the teacher has to be very careful on the content they choose for the students to work through. I guess at this point I just don’t have the best understanding of teaching through problem solving but I’m hoping to gain more knowledge on in throughout this course. I’m already learning so much and becoming more confident in my math teaching skills. I’ve been in and out of the classroom subbing and in today for interning and I felt more relaxed than I thought I would while helping the students with their math. I concentrated a lot on not giving answers but more guiding them to work through it all themselves. I’m excited to continue learning through this text.
Shawna,
ReplyDeleteI really enjoyed thinking more in-depth about what math is in chapter 2 as well. When I think back on my math experiences it's not something I enjoy to much. I was always the one who finished in the back of the pack in the math race. I was never very fast at it and it took me a while to understand it. I couldn't agree with you more on the mistakes part, I think as professional educators it will be important for us to really stress that it is okay to make mistakes and that they are what help us learn! So many students are scared of mistakes and are so disappointed when they make them that they can't learn from them. Mistakes can be a great learning tool.
My favorite part of these two chapters was the before, during, and after. I think it sets a baseline or standard to follow. It also puts the learning into a step by step instructional sequence. Patterns and order is something that seems to continually be repeated in what we are working on in Everyday Mathematics. It could be; circle, circle, square, circle, circle, square, circle, ?, ? or it could be the +3 where you take a number and add three to fill in the box. It could be any numbers or any + or -. In fifth grade were doing a number line, red dots on all the multiples of two, yellow dot on all the multiples of 5 and so on. It seems every grade level is looking for patterns of some kind. I love how this chapter goes over language of mathematics because kids really don’t know much of the language and when you are looking at word problems reading it seems to be one of the most important first steps, so they have to have that knowledge of the words and what it means when you say one of them. If I say sum they have to know that that meant to add. I really enjoyed the games, and games keep children engaged. I also like how it breaks it down to the idea that it is not about the answer, so many times children will do whatever it takes to get an answer, guess, cheat, keep asking you, and just give up instead of really trying to figure it out. Math really isn’t just about math. I once asked by college algebra teacher, “Why are we learning college algebra? My students can’t add and subtract, when could I possibly teach them algebra?” He said it wasn’t to teach us college algebra, it was to teach us “TO THINK”. I want my children to be able to think and try to figure things out for themselves. There is nothing sadder than an adult who runs around always asking someone for help and winning because they can’t do it, before they even try themselves. These are really good chapters to read and this is definitely one of the keeper books to go in your classroom for reference.
ReplyDeleteShannon H.
ReplyDeleteI was one of those kids that would just stare and not have a clue. I was not a good reader. I really wish I had a teacher that would have worked me through the problems so that I could understand and give me other ways of learning. I also always seemed so scared of my teachers. I think teachers now do so much more at supporting and encouraging students that it would make it easier to learn. Great post.
Chapters 2 and 3 of this week’s reading changed my perspective on math as a whole. When I wrote what math means to me I put something along the lines of using numbers and formulas to solve a problem, and that it is important for everybody in society to know how to do at least the basic functions. I now look at it in a little different light, I still think that it is important for society to be able to do basic math functions. We’ve all have had that casher that clicked the wrong button and was unable to count back our change and we had to assister him/her so that she/he didn’t short change herself or us. It’s sad that so many people don’t know how to do this simple process. The differences in my ideas are that math is using critical thinking and problem solving and not always are you give a formula. I never could do the guess and check method in school; I was too much of a concrete thinker. I know students hate doing naked number problems but in my experience they hate word problems just as much. I think I have solved the mystery to this, the word problems in most math books are worded funny and have nothing to do with the students. My students don’t care how many people on an average day enter an imaginary museum. When giving word problems they need to relate to the students and be written to the students. I’ve watched students, they do 25 naked problems and then they do 5 word problems. They find the naked problems easier but boring and they find the word problems hard. They have trouble shifting from concrete thinking to critical thinking because the lesson wasn’t taught in a critical manor. I think this may be the reason so many students dislike math; its either too boring or too hard.
ReplyDeleteIn response to Tammy,
ReplyDeleteI also think it is very important for teachers to have a before during and after. It’s also very important for students to have this same idea. Students have a short attention span, and unless you give them an overview of that they will be doing and a time for when the lesson will stop they see it as an endless lesson and will not get motivated about it. My husband is/was ADHD in school and unless you tell him specifically what will happen during the lesson and tell him a time for when the lesson will be over (ex. you will do this worksheet and then we go out to recess) he wouldn’t be able to stay focused and do his work. In simple they have to know in the beginning when the end is. Patterns are very important in every grade, I know my seventh graders still do algebraic patterns. They don’t like them because they take problem solving and critical thinking skills and there not use to that kind of thinking in math.
Rebecca B.
ReplyDeleteI also agree with you on that after reading chapter 2 and 3 I have a whole new perspective on math. I myself have struggled with math, so this is new to me. I also agree that students to believe that math is too boring or hard so they do not apply themselves and try and learn math.
Honestly I have been dreading this class, because I have always struggled with math. But reading chapters 2 and 3 has really helped me realize what I need to do to over come that and help my students. Looking back I have come to realize I really didn't know what math was. I just learned the basics and prayed that I got the assignments right. These chapters proved to me that math can be done and you do not have to do the problem by the text book. I was always forced to do it just right or it would be counted wrong. Knowing this will help me tremendously in the classroom, because times have changed and students just need to know math and understand it. If they understand it, then they will know math. Now I am really looking forward to this class so I can better myself with math and use my experience to help the students who are like me!
ReplyDeleteIn class, we did some problem solving that made me think about the practice of solving mathematics problems. We were broken up into groups, and were asked to solve a problem about cats and bats. We all had the same problem, but that was where our similarities ended. The groups that completed the task arrived at the answer different ways, which was alright.
ReplyDeleteIn Chapter 3, the text states the value of teaching through problem solving. One benefit is that it focuses student’s attention on ideas and sense making. In the class assignment, I kept asking myself if the answer I got made sense. It made me think of more than numbers. Another benefit of problem solving is that it allows for extension and elaborations. It can be easy for the teacher to go off of the information, and ask the “what if” questions.
I didn’t realize how easy it is to apply children’s literature into the math curriculum. Later in Chapter 3, the text gives examples of books that can be in the math lesson at all levels. Also, I thought that the story on the Lottery ticket was interesting. It’s amazing how mathematically illiterate we can be.
In response to Cassandra S.-
ReplyDeleteI was also dreading this class. There have been times in my educational career where I was frustrated with mathematics. As you said, you don’t have to do the problem by the text book. Providing real-world examples and being creative with problems can help students understand the process, even if they do not like math. As in our class problem solving example, one can arrive at an answer differently than there neighbor; and, they both can be right. It will be important to ask the students how they arrived at the answer they got, and have them explain why it worked or didn’t work.
I really enjoy reading this textbook. I have had a better mind towards math. I have even felt a lot better at helping my 8th grade son with his homework. I am really learning some interesting things in this book. I feel it will really help me when I am out in the field and as well as in my own classroom as the math teacher. I really enjoyed writing the personal mathematics story. It really made me think of my mathematics history, and after i thought about it, it has not been that bad. I just have to have a better open mind about the subject. I have also enjoyed the problems we have done in class. They really get me to think outside the box. I like how we have a power point to go with the chapters we read. This textbook has a great way of putting the information in there to make it interesting. I will admit I was not very happy that I was going to have to read a book about MATH. Now I am happier because it is teaching me tons of things, and we are just in the first 3 chapters. This will be a great asset to me. I can’t wait to learn to use manipulatives with these problems. I have also enjoyed the books the textbook mentions that we can use to integrate reading and language arts into math. I will feel so much better learning how to integrate all subjects into math. It was easier in my opinion to integrate math into the other subjects but I was not figuring out how to implement the other subject areas into math. I think this textbook will do just that for me.
ReplyDeleteIn response to Cassandra S---
ReplyDeleteI agree with you completely that I really did not know what math really was. I just tried to get through the assignment and hoped I got it right or that I at least passed the assignment to get a good grade. I struggled with math all throughout my life. This class is also teaching me things to help my 8th grade son with is math. The teacher is hated by all students but she is doing it totally wrong compared to what Dr. Stramel has been teaching, she teaches the students that it is her way or no way and they get the problem marked wrong, so I have been having chats with the teacher. I have stressed to her that there may be more than one way to do a problem and she needs to let the students do it the way that makes more sense to them. I can’t wait to go out in the field and work with the second grade class I am assigned to. It will be fun.
Chapter two and three of the textbook “Elementary and Middle School Mathematics: Teaching Developmentally (2010) by J.A. Van De Walle, K.S. Karp and J.M. Bay-Williams is a book that I really enjoy reading. Chapter two talked about “knowing” and “doing” mathematics and its similarity and differences and chapter three talked about “problem solving”. I found many interesting and important facts, knowledge that made me this differently, points that made me think about my own experience and a few questions.
ReplyDeleteAn interesting fact in chapter two of the textbook by Van De Walle et al. was on page 13 of the textbook by Van De Walle et al. “it is important for you to have a chance to ‘do mathematics’ in a way that nurtures understanding and builds connections” (Van De Walle et al.,13). There are so many times that I have seen students just sit there and do worksheets. I have learned that even if worksheets do help students practice it is not the best way to teach students. Teaching must do activities with the students that keep them involved and active in the learning process.
Even though I think that all the information in the textbook is important a few pieces of information that stuck out to me in chapter two was the list of verbs on page 14 of the Van De Walle et al. textbook. According to the textbook “These verbs require higher-level thinking and encompass ‘making sense’ and ‘figuring out’” (Van De Walle et al.,14). This is important to remember because teachers do want students to have a high level of learning and sometimes that is not achieved in the classroom. If students are “doing” these words then they become more active than they otherwise would be.
This comment has been removed by the author.
ReplyDeleteContinued...
ReplyDeleteSomething that I learned in chapter two of the Van De Walle et al. text book that made me think differently was on page 13 when it states that “Doing mathematics in classrooms should closely model the act of doing mathematics in the real world” (Van De Walle et al.,13). I did not really think about how important it is that that there is a purpose to the math that students are doing. This does make sense because I feel the same way. Students will be more engaged is they are doing the math problems to find a solution to a real problem they may have because they can relate to it.
Something that I learned in chapter three of the text book that made me think differently was when the textbook talked about how some teacher may think it is best to show the students answers to problems. According to the textbook by Van De Walle et al. “However, students are not learning content with deep understanding, often forgetting what they have learned; they need a more effective approach to learning mathematics” (Van De Walle et al.,33). I then started to think about all the times I have don’t this in the classroom and I think it will be a hard habit to break but it will benefit the students.
A question that I had for chapter two of the text book was how do people feel about talking in the classroom? I was just reading the statement on page 21 of the Van De Walle et al. text book stating “Classroom discussion based on students’ own ideas and solutions to problems is absolutely ‘foundational to children’s learning’” (Van De Walle et al.,21). Some teachers that I have worked with in the past will tell their students that their work was independent and if they needed help to ask the teacher not another student, but I think that it is good when students get a peer feedback. Not only does the student learn from explaining it but the student listening might be able to understand the material from a student’s point of view.
References:
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7 th ed.). Boston: Allyn & Bacon.
Andrew,
ReplyDeleteI think that problem solving is a great tool to use in the classroom with every subject. Just like you did with the problem of bats and cats students will have to think on their own if that works or doesn't work. Even if the students get the wrong answer that is OK because at least they are thinking about it and they are more likely to remember it later.
Math has always been my favorite subject and for me learning ways to teach math is so reassuring. I have always been one of those people who just "get it" & when I am trying to teach someone how to do a problem I find myself just repeating what I already said until they say they "get it" too. I instantly thought of this when I read the opening quote on page 13 "No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students." That is what I find myself doing when I am helping my child with math, I just wish I could understand for them so they will get it and move on. I like the idea of teaching multiple approaches like when the book said 7 x 8 can be solved in so many ways, I would have never thought to teach some of the ways like to say if you know 7 x 4 which is have of the sevens, then you can double that and come up with 7 x 8 is 56. I appreciate thinking differently for my own sake because I have always just done math and not thought through math. I think that will make a huge difference in the way I teach math.
ReplyDeleteProblem solving will also be easier to teach when I understand how to think through math. I have always had a harder time with problem solving because it can't just be done without thinking about it. I like the statement that "effective lessons begin where the students are, not where the teachers are." That is such a basic idea but it is one that I think is lost on teachers a lot. It is hard to remember that they already have the knowledge that's why they know the answer, and maybe they don't even have the knowledge they just have the answer. I like how the book used literature in problem solving. This is a great idea of cross curriculum. I like the idea of reading the book "The Doorbell Rang" to the children to help them understand about sharing and redistribution. This would be a great activity to "act out" in class as well.
Rebecca B,
ReplyDeleteI so agree with you on the word problems. I used to dread word problems so I know exactly what students are feeling when they have to do one. I am excited to learn different methods of actually talking through problems and trying to do problems in a way that makes sense to everyone. I agree that most word problems seem to not apply to most people so it is hard to comprehend how to do them.
Chapters two and three are all about getting students actively doing math instead of just memorizing facts, or watching the teacher solve a problem approach to teaching mathematics. I love the ideas of using problem solving to teach concepts and allowing students to use varied strategies to solve those problems. When the main focus of math is merely getting a right answer the students lose the opportunity to gain a deeper understanding of the concepts, and find new ways to attain a solution. When I was in school we were supposed to do the math just as the teacher said. No one ever asked, “does anyone have a different idea on how to solve this problem”. How much more could students in classes like mine have learned if given the chance to explore and discuss different ways to frame problems and different strategies for solving them. For this reason I like the idea of not providing the answers to the problems as discussed in the text. We need to focus on doing the math, not just getting the answer.
ReplyDeleteOne thing that stood out to me in Chapter 2 was that we should value different processes that lead to correct results. I remember in algebra I that I did problems differently than my classmates, but I got to the right answer. I was told that it was still wrong because it wasn’t the way the teacher had taught it. I remember how much that undermined my confidence to be able to solve equations. But in our text, two children used unique problem solving techniques and it is applauded. I think the teaching of mathematics has come a long way! Figure 2.11 on p. 24 really helped me to understand the importance of connecting as many ways of understanding as possible to the learning of new concepts. I also have a great appreciation for the concept of scaffolding. I can see where scaffolding has helped me learn many times, I also see where means to scaffolding can become a crutch and it is important to guard against teaching mindless us of manipulatives. I see where manipulative use can help learners, especially those who are struggling with a concept and yet that it is important not to teach using manipulatives in a way that as ineffective as memorization or learning by rote. A paragraph on p. 22 stood out to me when it addressed learning by rote (memorization). It says “Rote learning can be thought of as a “weak construction.” This is because the learning is not connected to other knowledge. Maybe this stood out to me because I remember all the flash card drills we had in elementary school memorizing multiplication facts!
ReplyDeleteThe shift in teaching to Teaching through problem solving is both exciting and overwhelming to me. One of the frequently asked questions at the back of the chapter is: How can I teach all the basic skills I have to teach? This sums up how I feel. The chapter stressed the importance of the “after” phase of the lesson format and how this can easily take 20 minutes. I think this tends to get squeezed out in classrooms because of time issues. I can see how this will take time to implement and to set up a teaching style that allows time for this. I particularly liked that the chapter laid out some problem-solving strategies on p. 43. I think the method I use most is guess and check. That is the method I have used every time so far when we’ve done problems together “in class.” I learned a lot from these two chapters. I feel a little overwhelmed and exhausted, but I am starting to feel like all the teaching knowledge I am taking in is coming together, I am starting to feel more and more equipped to teach. I love that feeling of empowerment. I hope I can give that many students!
April B.~
ReplyDeleteI could really relate to your post. I was not very excited to realize I would be learning about math. Teaching math seemed really daunting too, but the text is a great tool. I see a lot of correlation between the information in the text and the way that my mentor teacher teaches, so I already feel like what I am reading is being reinforced in the classroom. I am like you, I can't wait to play with my manipulatives! Once I opened the bag and looked inside, it was like Christmas in August!
Adrianne, in response to your question about the Pythagorean theorem, I feel that we need to allow children to “learn through exploration” as you put it, but we can give them tools with which they can better navigate through their explorations. Formulas such as these allow students to use skills they have already learned to delve even deeper into mathematical concepts. For instance, students must already have learned how to use square roots before using the Pythagorean theorem, but in using it they can discover new information. The students can use tools such as this formula and others to help them explore in a different way. Using “concrete formulas” doesn’t mean the learning experience has to be set in stone as well.
ReplyDeleteChapter 2 was all about knowing and doing mathematics. I have now come to learn that these are two completely different things. I have always been pretty good at math, but after reading this chapter I would definitely say I am better at doing math than knowing or understanding it. I completely understand what the text is saying about how in the past we have specifically learned operations. I can’t remember really ever exploring more into math equations or finding out why they work the way they do. I think that this is definitely something that needs to change, because as I got older and into more advanced math concepts I began to struggle. Now I know why. Who knew there was more than one way to do a math problem? I found it to be very interesting that almost every group in class solved the cats and bats problem in a different way. I have some serious view changing to do about math.
ReplyDeleteChapter 3 focuses on using problem solving to teach mathematics. After completing an action research project over effective math strategies last year, I 100% agree with using problem solving in mathematics. Students learn better when math is given real world applications. How many times have you asked or heard a fellow student ask, why do we have to learn this stuff I’m never going to use it? Putting math into problem solving lets students see where this skill will help them later in life and makes them more motivated to learn it.
Andrew D –
ReplyDeleteI like the point you made about the problem solving activity we did in class. You’re right; doing that problem got us thinking about more than just coming up with an answer. We had to really think about the situation and if I answer made sense. I hadn’t thought about the extension idea from problem solving but that is true as well. Problem solving in math can easily lend itself to integration into other subjects.
I really enjoyed chapter 2 and the message it sends to us as future teachers about how not all of our students are going to learn the same way. Chapter 2 provided examples describing how students may interpret and work out a math problem differently, but still come up with the same answer. As future teachers we need to be mindful and aware that not all of our students are going to understand every lesson the same way as their classmates. When prepare class lesson we should have an alternative way to explain the lesson to those who may not understand our initial explanation. As teachers we should also be open to new ways of solving math problems; encouraging our students to work out math problems the way they are most comfortable may result in us finding a new way of teaching mathematics.
ReplyDeleteIn chapter 3 different ways of teaching about problem solving are discussed. As mentioned before, not all children learn and understand the same way as their classmates and this chapter will be wonderful to refer back to when looking for alternative ways to teach my students about problem solving.
Jen W.
ReplyDeleteI can relate to your “I get it” story, just in a different way. I have always been one of those “I get it in my own way” kinds of people when it comes to math. I was working with a student a few years ago who just didn’t understand her math assignment and so I showed her an alternative way she could work out the problems; the way I understood how to do them. She “got it” and finished her assignment with very little assistance. It is because of that student that I try to think about different methods and examples I can use when teaching.
First let me say how much I enjoy the Pause and Reflect sections of the text. I really think it puts me in a place to think like a child and go through problems as they would approach them. I really enjoyed the discussion in class about problem solving. I thought it was very interesting the different ways people problem solve. I think the section of chapter three that discusses teaching through problem solving was particularly interesting. I remember going through math and just focusing on one problem at a time. And not really paying attention to the concept. I think its a great idea to teach students to focus on the concept and be able to solve many problems, instead of just the one at hand.
ReplyDeleteAllison G.
I am the same way when it comes to knowing and doing Math. It has always come easy to me, but I would always do the math and not necessarily know what I was doing. It was easier to memorize the steps than the whole concept.
I am one of the 'geeks' that really enjoys math. When I work with a student, quite often I will work out the problem the same time as the student. I like to try and find the answer in my head because I want the students to find the answer without looking at my paper and it keeps me fresh and practiced.
ReplyDeleteI have always told the students who struggle that it is a matter of understanding the the why. Why does 2 x 2 = 4, when 2 + 2 also = 4. Because 3 x 3 = 9, but 3 + 3 only = 6!
Our text says "One way that we can think about understanding is that it exists along a continuum from a relational understanding - knowing what to do and why" (text pg 23)In other words:
2 + 2 = 4 because when you have 2 pencils and I give you 2 more pencils, you will have 4 pencils. So why does 2 x 2 also = 4? Because if I have 2 Mounds candy bars I have 4 candy bars because there are 2 bars in each packaged Mounds candy bar. (2 sets of 2 = 4)
Why is there two pieces in a Mounds candy bar? "So I can shaaaaaare" (I think I am dating myself)
In response to Tammi Whi,
ReplyDeleteJust to reiterate; to focus on one problem at a time and not pay any attention to the concept is extremely time consuming. When the student concentrates on the concept it can help them to understand the Why, thereby understand the concept and the answer to the problem can come much easier.
Good post, I enjoyed reading it.
I attended my first math internship this morning and constantly reflected back on what I had learned in chapters 2&3. In our book we are taught to teach through problem solving, but what I saw today was the old math. I enjoyed learning about ways to engage children to problem solve in math. If I had learned math this way, I would be a much better math student today. I also enjoyed how they integrated Harry Potter with doing measurements. I already feel more confident about teaching math after reading the Three-Phase Lesson Format. I can't wait to use some of the ideas from the textbooks on my students.
ReplyDeleteIn response to Stephanie Potter,
ReplyDeleteVery good post. It is so true that children may learn math different ways. Being a SPED minor, I agree that teachers should have alternative ways to teach the math lesson. Teachers should also give students the chance to show different ways a problem can be solved, if applicable, to help address various types of learners.
I really enjoyed the activites we did in class talking about problem solving. The question about the cats and the bats really got my group thinking and we actually thought harder than we needed to. Also, the activity where we used the one up one down method to solve an addition problem was great. It was nice to see a different method that could be used to teach math. Some students might not use this way, but as a teacher I need to know that this might be a way that a student who is struggling can use and it might make the most sense to them! I had a lot of trouble with problem solving starting out. I did not like word problems at all. I did not like having to figure out the equation and then solving the problem. I always felt like i wasn't doing it right. But, the more we did them the better I got and the more I realized that the word problems can be like real life situations. I enjoyed reading this part of the text!
ReplyDeleteCarrie H,
ReplyDeleteI also think it is great when a student can figure out how to solve a problem in a different way that was used by the teacher. This way may be easier for that student and it may even be easier for other students as well. Students who think outside the box like this need to be applauded and praised!
Chapter 2 and 3 does a very good job of explaining mathematics and how to teach it. Chapter 2 describes how to create an environment or classroom setting and how it is the teacher’s responsibility to make it fun, inviting and excitable. This is very important as math is not a likeable subject. So if a teacher can make it fun then students will enjoy learning about math and doing the problems. Chapter 2 also goes over a few problems and lets you see what it is like to be a student trying to figure out the problems. You can see the struggles the students may face or any situations that may come into play when trying to figure out these problems. Now, in chapter 3, it talks about problem solving and how to teach it. A main point stated in the book is that the problem must begin where the students are. If it does not then the students will either not understand or it will be too easy for them to figure out. I like how it covers the topic about letting the students talk about the problems to each other instead of having the teaching doing all the talking. Students learn best from one another. I will definitely have to revisit this chapter so I can remember the before, during and after phases.
ReplyDeleteIn response to Jordan O. -
ReplyDeleteThe bats and cats problem also got me thinking and made me realize what it is like for the students to figure this out. As a teacher we need to make sure that they are solving problems that is neither to hard for them to understand nor too easy that they get bored. I too enjoyed reading about this in the text and liked how the more problems you do the easier they get to figure out the answer.
I really learned a lot from the combination of reading and how it was introduced in class. I like what was added from Dewey about what the teacher should be like I want to be like that for my students. I liked the teacher saying it is okay to make a mistake. I have seen and experienced moments of feeling terrible over a failed assignment. The four features for having a productive classroom fit right in. The first is Ideas. The second is student respect for different methods to solve problems. The one I mentioned earlier be able to make a mistake to learn from it. The fourth being the logic and structure of the subject.
ReplyDeleteAdrianne, I totally agree it is so hard to work on a problem without a certain answer. Just one of those human error, logic, made you think kind of problems. I do like the way this class is ran.
ReplyDeleteWhile reading in Chapter 2 I thought about how setting up the mood/atmosphere is a very important role of teaching math in any classroom. Starting out saying that "you don't like math either" or that "math is hard" makes Math sound like the bad guy. Setting up math in a fun and enjoyable way is a perfect way to encourage students. I never realized how important problem solving was until I read through ch. 3. Problem solving encourages students in so many ways that aren't usually seen. Problem solving allow children to be more confident, to see the connection between math and life, and so many more things. When students are able to connect school work to the "Real world," they are able to grow exponentially in great ways.
ReplyDeleteresponse to "tracyp":
ReplyDeleteI really can relate to what you said about making mistakes. The U.S. is very guilty of making people so perfect so then everyone thinks they have to be right the first time or they will not make it. It is totally a lie. Mistakes are just anther way we learn!
In chapter 2, I thought it was interesting that the description of mathematics used is similar to what I’ve come to try to describe to my own children. I always tell them that it is about looking for patterns or order. I don’t know that I’d have had this description years ago, but in trying to help my own children, I’ve found that this seems to “click” with them. However, I had never been introduced to the patterns shown in Chapter 2. “Start and Jump Numbers” and “One Up, One Down” were definitely new concepts for me. It was a great reminder of how it feels to be introduced to a new math concept, especially when we participated in the activity in class. One item that really jumped out for me was the Ineffective Use of Models and Manipulatives. I am an ELL Para, and we have been using Base-ten blocks in our second-grade class. The teacher does demonstrate quite a bit by saying, “So if we have 46, how many ten blocks should we put down?” Usually, some of the more advanced students will immediately say or model the answer, and I can easily see that other students are just “doing as they see” instead of doing what they understand. I will try to visit with the general classroom teacher to explore other methods we might try. The Implications for Teaching Mathematics sections in this chapter are wonderful. I like how they are divided out and titled for quick reference. There is so much information to learn, and this book is outlined well and will be a terrific resource!
ReplyDeleteIn response to Jeanette (for Chapter 2):
ReplyDeleteI also believe that the key to teaching students is to get them excited about the content, to make a connection with their prior knowledge, and to use multiple teaching approaches. I am also a Paraeducator, and I am finding that it can be a challenge to break away from the way I was raised and taught in the classroom – which was to teach it one way...and one way, ONLY! I am an ELL Para, and it is interesting that I find myself just as challenged with many non-ELL students in finding ways to “reach” them. Students do bring so many different backgrounds to the classroom, and it definitely requires changing up your teaching methods several times for one lesson. We are currently working with tens and ones digits, and I am amazed at how many approaches the general classroom teacher and I have taken and still had difficulty reaching some of the students. I am excited about this text and plan to use what I am reading to help my students in every way I can.
I feel that Dr. Stramel’s “Bats and Cats” example used in class was a good representation of what Chapter 3 is all about. It is important for students to do the thinking when problem-solving, instead of just modeling what they see. If the students don’t actually think through what they’ve done and why they’ve done it, they won’t fully understand the process behind it for future use. It also goes hand in hand with the idea of multiple entry points. When she asked each of us to provide ideas on how we might solve it, there were many ideas that were suggested. Each idea came from a different background of thought, and, ultimately, none of us could come up with a solution in class. When we returned to class, the majority of us had solved the problem through the use of a starting point and then a process of elimination by figuring out which combination of numbers worked. We all knew that there must be a formula to solve the problem, but most of us could not come up with it. ONE student in the whole class knew how to solve the problem with a formula of variables, solving for one variable and then substituting that value to find the overall answer. This was a college class, and only one student knew the most advanced way to solve it. Just imagine what this means for an elementary level class! Dr. Stramel was careful to listen to all suggestions and have us “guide” her through our thought process, instead of just giving us the answer. She was also sure to allow us the opportunity to talk to each other, as the chapter encourages. I loved the “Phases” section of this chapter, too. This is such a great resource!
ReplyDeleteIn response to Allison G (for Chapter3):
ReplyDeleteI completely agree that students need to be able to relate the problem-solving to real-life situations. I have heard (and said myself) many times, “What is the point of doing it if we are never going to use this stuff? Students need to understand that there is a very real reason for learning math, and when they can relate math to their everyday lives, I truly believe that they become so much more engaged and involved, and they learn and understand so much better.
Chapter 2 hit the nail on the head in the title. Exploring What It Means to Know and Do Mathematics, wow what a concept. Early in the chapter mathematics is defined as a science. The text stated that much of the time, mathematics is limited and the students don’t have the opportunity to deeply learn about different topics. The chapter went on to discuss the wordage associated with mathematics. I appreciate this book for the fact that it lets us get into the experience. I like the invitation to do mathematics; it is so user friendly and keeps my interest going! The theories discussed making it so easy to understand the ideas behind the teaching!
ReplyDeleteChapter 3 on problem solving was interesting. I have always struggled with problem solving and teaching it will be so much easier with the help of this text. I value the paragraph that discussed that teacher will be changing the philosophy of how she thinks and how her students will learn best. Involving children’s literature in math is a valid concept, but something that new teachers might not think about. After reading this section, I have so many great ideas!! The four step problem solving method is used in many different ways; however the concepts are the same. I information on teaching in a problem based classroom will be helpful, it is important that students like to talk, so let them!! It is nice to know how to teach in phases. I feel that it is so beneficial to our students to be able to break up the information that they have to eventually put back together!!
@ Tracy P.
ReplyDeleteI agree, the combination is perfect! I plan on investing in the Dewey book I am very interested in him now.
In response to Jeanette,
ReplyDeleteI have also worked as a high school paraprofessional and I will completely agree with you that most of them have negative feelings when it comes to Math. I was one of those students too. I can honestly say I had some teachers for Math that I thought were bad teachers. Math tends to be somewhat boring anyway and to have a teacher go in and read a lesson then give you assignments leaves no room for anything positive. I want to be one of those teachers too, that makes it enjoyable for students of all ages. There are even activities that can be done at the high school level and I think it would make the student and teacher attitude much more pleasant!
I really enjoyed this chapter. It discusses that we must build on prior knowledge and this is so true. We have to have some knowledge in the area we are focusing on in order to build and expand on it. Math does not come easy to many students and this is no secret! We as educators must take the basic knowledge that the students do know and build from there. It is too difficult to try and teach something when the basic background knowledge isn't even there! I also like in Chapter 3 the information on Problem Solving. I was horrible at the problems and to this day still tend to struggle at times. I think it is imperative that there are different examples given to let students know that there is more than one way of doing something. What works for one many times does not work for others. This is okay and I don't like when teachers in my past would explain something only one way. I still remained clueless as on what to do. I feel I gained great information from this chapter and took notes for my folder to refer back to in the future!
ReplyDeleteYou are all AWESOME!!! I have really enjoyed reading your posts, and the things that you gleaned from the chapters and our class discussions. Keep up the good work!!!
ReplyDeleteThis chapter was really fun to read. I can remember in my days of elementary school using the tools and how much of a benefit they were to use these manipulatives. Once I could visualize them, I could usually do a problem in my head again visualizing that tool that was used.
ReplyDeleteI think probably one of the most important parts of math is problem solving. Problem solving is an every day skill that everyone needs to learn. Also, as a teacher, the most important part of teaching is reflecting. As we have learned in many other courses we must reflect and think of what we did well, what we can do better, and what will we do to make it better.
Technology is a wonderful tool that God has given us to use. I am excited about these tools to use in the classroom someday.
Dina,
ReplyDeleteBackground is the most important part of teaching math. It is so true that without a background there is nothing to move forward with.