I was familiar with each of the four problem structures described at the beginning of Chapter 9; however, I wasn’t surprised that I didn’t know the technical terms for them. Although I wasn’t surprised, I still wonder why I never learned these terms (even in a previous Math for Elementary Students course). The textbook emphasized the importance of teachers knowing this, and I was confused as to why any educator would not be able to understand or successfully teach these concepts. Maybe I am close-minded, but it seems pretty elementary and I can’t imagine a teaching not being able to fully understand.
I also never knew not to use the words “take away” when talking about subtraction to a group of students. Minus or subtract are acceptable, but never take away. I think I hear my mentor teacher use “take away” on a daily basis when I’m there for my math internship. I also liked the way described to selecting numbers to use: numbers that are in the student’s grasp. Students know certain numbers at different ages, so as an educator you need to plan to use what is within that grasp versus below or above, which could make a problem too hard or too easy.
One of the very first things mentioned in Chapter 10 was that, “teaching basic facts well requires that essential understanding that students progress through stages that eventually results in just knowing.” I think this is important to keep in mind not only with math materials, but also with every subject matter. Students do not just know something, but rather they acquire it over time. This is important for teachers to keep in mind.
Knowing basic facts is an important part of mathematics. I think the most beneficial aspect of this chapter was all of the great ideas to use to teach basic facts (and also the ideas Dr. Stramel showed us in class a week or two ago)!
Developing Meanings for the Operations Chapter 9 was a whole new thing to me. I had never seen or heard of the four problem structures before. I just thought there was a variety of ways you could write problems, pick one. I am glad I became aware of this now as these structures do have an ordinance. As I continued on into the contextual problems I couldn’t help but laugh when I came across the fact about how Japan attacks this. As competitive as the U.S. is, I would have thought for sure by now we would be adopting more of their habits to increase our scores to compete with them. The chapter does a really good job walking you through each type of problem you can face with the four structures. Examples are always nice!
Helping Children Master the Basic Facts
Chapter 10 gives some great foundation material. When you really think about how you learned math and reflect upon personal success to me it was plain to see the 3 phases through my math knowledge. Back them it was all about memorizing and competition. Which we have been told by this section is not the way to successful math skills or an appropriate route as a teacher to put your students through that. To some extent I believe that but on other basis children will deal with competition and comparison the rest of their life. The chapter once again does a great job giving in-depth focus on each particular phase especially reasoning. They give some great insight to how reasoning works for addition, subtraction and multiplication. I love the Nifty Nine activity. I have shown that to numerous students who have ever asked me a multiple of nine. Towards the end of the section it gives a list of some basic do’s and don’ts of math. One in particular that surprised me was how they discouraged you focused on the amount of time you made them practice and drill facts. I knew after a while students would obviously become bored, especially those who are advanced but I never thought it would be a bad thing for a student who really needed more time and focus. The book makes a good point stating they become frustrating and fatigued by repeatedly viewing the facts.
In response to Emily M,
Isn’t it crazy, I would have never thought twice about saying “take away”? It amazes me how we use a term or process for years and years only to find out that it is “not proper”. It’s not just in education but in many topics in life. I can understand how these words might confuse someone but at the same time I think to myself well I learned it like that can’t they? My stubbornness of following how I was taught is going to be my enemy teaching in today’s society with all these changes.
When I was reading Chapter 9 about developing mathematics for the operations there was lots of different areas that I was able to become more informed of. The title of the chapter says a lot about what the chapter hoped to accomplish which is making math meaningful for the students and helping to apply the math to real world settings. I liked how the chapter hit on the area that it is commonly thought of in the United States to have the children do lots of math problems in a single setting when in reality the teacher should have the student only focus a few math problems each class period. I liked how the chapter focused on the properties of addition and subtraction as well as the properties of multiplication and division. The last area I found interesting when reading chapter 9 was the area of not always relying on the key word strategy in order to solve a problem. The book went into detail about the three reasons why to not rely on the key word strategy which are key words are often misleading, many problems have no key words and lastly it sends a wrong message about doing math. Overall I thought this chapter was very beneficial to me as a future teacher. When I was reading Chapter 10 which focused on helping children master the basic facts I was able to learn some tips to use in my future classroom as well as different strategies. I liked how the book gave the definition of mastery of a basic fact which means that the student can give the correct response within 3 seconds. I was able to learn that there are three phases that students go through in order to become a master of a fact which are counting strategies, reasoning strategies and lastly mastery. I liked how the chapter gave different reasoning strategies that a classroom teacher could use in all fours areas including addition, subtraction, multiplication and division. The section in the chapter that also talked about the do’s and don’ts when teaching basic facts in the classroom was helpful for me as a future teacher. Lastly I enjoyed reading about the fact remediation section at the end of chapter 10 which will be helpful for me in the near future when I begin teaching.
In response to Katie Coulter I agree with you when I was reading chapter 9 I was not aware of the four problem structures either. I am glad that the book explained the four problem structures so that I was able to become better informed of them and learn the structures. I also found it interesting when the book was comparing the United States to Japan and how the United States focuses on lots of math problems in a single setting and Japan only focuses on a few problems in each setting. I also agree that the book does a great job of giving examples along the way. I agree with you that when I look back on my schooling a lot of the math instruction was focused on memorization of facts which was talked about in chapter 10. I am glad that the book is pushing towards teachers using more reasoning strategies to get students to move from counting into reasoning which leads to mastery. The Nifty Nines strategy was by far my favorite part of chapter 10 because I think that it is a great strategy and I have never seen it done before using your fingers.
I am on the same page as you with the four problem structure. I am familiar with each of the processes or types of problems, but I never knew that they were a part of something called the four problem structure.
I also agree on looking back on your own math successes. Everything seems to make more sense and give me an idea about how learning math happens, etc. I wish I remembered more about specific techniques and other ways I was taught math. I remember major things, but some times I find myself asking “Did I learn A, B, and C this way?!”
Chapter 9- Developing Meanings for the Operations- While i found the information in chapter 9 to be informative also found some of the teaching strategies to be a bit complicated. It could be because I am interning in kindergarten. I think that the very best way out there to teach children addition and subtraction is HANDS ON!!!! They need for all of their senses to be activated in order to really get it. I think that by giving them something to actually count and add to and subtract from you are really keeping every sense activated.
Chapter 10- Helping Children Master the Basic Facts-I found the information in chapter 10 to be informative and useful. Along with giving students the opportunity of learning hands-on you need to teach them short cuts to help them "just know" the basics. I subbed in second grade last week and got the opportunity to teach the kids how to do the "9 shortcut" in which you add 10 to a number and take away 1. They apparently had been struggling with learning to add 9 and after teaching them the shortcut they were easily able to add the number in their head.
Emily- I also found it extremely interesting that we are no longer supposed to use the words "take away" as I hear it in the kindergarten classroom I intern in as well as the classrooms I sub in. Even when I am doing math in my head (and I am 27 years old) I still sometimes find myself thinking the words "take away". Looks like I am going to have to work on that one! =)
With regard to chapters 9 and 10, I truly enjoyed these chapters. I found them exceptionally informative and thought provoking. I had no idea about the four basic problem structures, so this was all new to me- of course once it was explained I realized I have know it for a long time, but it is good to know how to actually teach using these structures. I also found it interesting when they explained the way they work on mathematics in Japanese classrooms- I can't be the only one who thought about where most of our new technology originates and I wonder if that is because children learn at a young age to be true problem solvers and think outside the 'lines'. I also did not know what an array was, I have seen them before- but never really understood the concept. I think it is ingenious the way they discuss teaching the associative property using this type of manipulative. The example problems in this chapter were wonderful and I had several ah ha moments while reading them and can see myself utilizing these problems ( or one like them) in my eventual classroom. In chapter ten, I learned about ten frames. I'll shamefully admit that when Dr. Stramel first discussed them I had never seen them before and had no concept of what they were. This chapter enlightened me and allowed me to see them as a very useful tool. I also like that in this chapter they specifically mentioned things to avoid, such as flashcard memorization of facts, after reading this book I am sure we are all convinced that a problem based learning environment is the best- but the little affirmations throughout the chapters certainly help to reconfirm our decision.
Kristie C, I like you am in an early elementary classroom and found some of the concepts harder for younger children to grasp, I had to intentionally remind myself while reading that I might not always be teaching very young students and to 'pay attention' to ALL the suggestions. I like though, that our text offers so many examples and ways to teach concepts. I, like you, think that HANDS ON is the best way for young ones to learn. I never learned the nine 'trick' that your taught to that second grade class, this book and class are teaching me so many ways of doing things. It is sad to say that I must have had an extremely sub par mathematics education. But, each day I learn something new and am looking forward to giving students what I never had- which is mathematical confidence!
The part of this chapter that I really liked was the section that talked about what to do when teaching basic facts. There is a list of six things to have them do: ask students to self-monitor, focus on self-improvement, drill in short time segments, work on facts over time, involve families, and make drill enjoyable. Three of these really stand out to me. The first one is to focus on self-improvement. One of the big problems no days is that everybody just focuses on where a student is at compared to the rest of the class. What they should be focusing on is whether the student is getting better or not. As a student that was always one of my major goals was to not worry about where everybody else was at and focus on whether or not I was improving at the subject. The second is to drill in short time segments. This, in my opinion, helps students to not get so worn out on a subject and doesn’t overwhelm them. Lastly is getting the family involved. A student’s biggest supporting cast should be his/her family. If a student gets support and help from their family then it will boost their confidence tremendously and the sky will be the limit.
Kristle C. Teaching students with hands on activities and teaching them shortcuts is extremely important. Students at the elementary level seem to respond best to learning with hands on materials but at the same time you want them to learn shortcuts so that after a while some things just seem so basic to them that all they have to do is look at a problem and know the answer without having to do any counting or anything like that.
@ Brooke M Brooke, I really liked your insights in to the chapters. I agree with the book whole heartedly. It is so important to make math meaningful for our students and relate it to real world experiences. I also liked the parts about the do’s and don’ts about teaching basic math facts.
Chapter 9 & 10 First of all I really liked how this chapter started. “This chapter is about helping children connect different meanings, interpretations, and relationships to the four operations of addition, subtraction, multiplication, and division.” I also liked the part about teaching division. I find remainders to be a nuisance to some students. I think it would be easier to teach younger students about “left over” instead of remainders. I loved all the different instruction methods for the 4 basic math concepts in chapter 10. I also really like the take from 10 strategies. Mostly because this is the strategy that I use all the time when I do mental math!
Chapter 9 is about developing meaning for operations. It's about helping students learn to connect addition, subtraction, multiplication, and division to help them use them in real-world situations. They go into how all the operations are connected. I see my mentor teacher doing this all the time. Having students check their division work by multiplying. I thought the information the book gave about introducing the = sign was interesting. I guess when I see that sign, I always assume the answer will follow, but not necessarily. The book suggests using the phrase "is the same as." I got the most out of the section about multiplication and division problem structures. I did not know there are problem structures that help teachers forming and assigning multiplication and division tasks. Another thing that I am taking away from this chapter is that when doing story problems, it is important to provide both multiplication and division problems in the same problem. This will prevent students from just using the day's operation, and help them to figure out which operation to use. Chapter 10 is about helping children master the basic facts. This means helping students give a quick response, in 3 seconds, without resorting to non-efficient means, like counting. Clearly, students need the knowledge of basic facts so they can continue to build on them. This chapter touched on memorizing facts. I always find it interesting to hear opinions on this. Some teachers don't teach strategies, they just teach memorization of facts. Research shows that this does not work, that students are not mastering the facts. It has also shown that by doing this, students are inefficient, they misapply the facts and don't check their work, and they don't learn flexible strategies. The text goes into guiding strategy development, and says that you need to give students as many strategies as possible. This is something that I feel is so important, and I see my mentor teacher doing this all the time. She gives them different strategies to solve problems, teaches them how to use them, but then lets them decide which strategy works best for them. I also will take a lot out of the sections about what to do and what not to do when teaching basic facts. One of the what not to do is using public comparison of mastery. I never liked it when teachers did this when I was in school. It was humiliating when you were at the bottom, and I think it puts too much pressure on kids. Overall, both chapters were very informative and full of great activities and examples that I will use in the years to come.
In response to Lane A, I also found the sections over what to do and what not to do when teaching basic facts informative. Like you, I had teachers that displayed students standings, and I think it had a bad effect on the students who were not doing as well as the rest of the class. I think that self-monitoring is very important. Teachers need to know where students feel they stand and if the feel like they are getting a concept. I also like the idea of making the drill enjoyable. I think this will help students succeed and help them retain the information.
Chapter 9 developing meanings for the operations- I found this chapter helpful but I thought that some of it was a bit complicating. I know that I learn better with hands-on materials and so do the kindergarten student that I work with in my internship. They can visually see what they have left in their addition and subtraction problems. Chapter 10 was the basic facts- I found this chapter was the most informational and helpful chapter. As well as students needed to know how to work out problems by hand, they also need to know how to use short cuts as well. I was so glad that when I was in third grade and someone showed me the 9 finger trick on the 9s multiplication.
To Kristle C. I couldn't agree more about students learning their basic facts, but also having those helpful short cuts. I also really enjoy that you bought up the fact of using hands on activities to learn math. Currently I'm working on a action research project of how using math manipulatives increases students grades in the curriculum. I was a visual and hands-on learner so these are very important to me, however I'm always reminding myself that not all children learn like I do.
I have to say I greatly enjoyed reading this chapter. I am currently working with an 8th grade student who scored on a 3rd grade level on his Star Math assessment. This student is way below grade level and has received academic warning every year on his state assessments. He is a new student to our school this year therefore he is extremely behind his fellow classmates in all of his academic areas. I have to do tier 3 interventions with him for Math, doing 3rd and 4th grade work, and these chapters provided me with wonderful material and strategies to use when teaching my student. My student does not even now his basic Math facts and I love how these chapters have given multiple strategies to teach him that hopefully he will become comfortable with one or more of them and can excel in his interventions. There were numerous strategies presented in chapter 10 for basic facts that I had never even heard of. Actually some of the strategies were quite confusing to me and I believe I would confuse a student more by trying to explain some of the strategies to them! Many of the strategies remind me of the intervention program I am using with my student. The program is called Camelot Learning Math Intervention Curriculum. With this intervention program each student receives their own bag of manipulatives and a workbook to complete the intervention assignments. I am using a 4th grade level book with my student and I also have my own teacher workbook that has each lesson planned out with the script to use to teach the lesson also. We are doing mental math strategies at the moment and many of the strategies in the Camelot book are just like the strategies presented in these chapters. I am anxious to take our text to school and do the activities provided with my student!
In response to Elizabeth- I agree that it is very important for teachers to teach students multiple strategies to use when solving math problems. I am a student of memorizing the facts and I have to say that I am not that good at even the basics of math. I love how these chapters provided numerous different strategies to use in relation to all of the math operations and basic facts. My mentor also gives his students multiple strategies to use for each of the concepts he is teaching. It is amazing when students cannot understand the lesson being done one way but then all of a sudden it is like a light bulb comes on and they can completely understand the lesson by using another strategy!
In Chapter 9, I do want to comment on the language used when teaching addition and subtraction problems, subtraction in particular. I was raised hearing the term “take away” when describing how to subtract, and I must admit that I did not understand what the problem was with using this terminology when describing subtraction. After reading page 152, I completely understand why it is so helpful to not use “take away” and instead use “Think-Addition” when subtracting. It is more important for students to think of the “missing piece” which will help them immensely with addition facts, as well. If you think in terms of “What is 9 – 6 equal to?” and then look at it as “What can we add to 6 in order to make 9?” then it will help with addition - instead of going to the extra work of “taking away 6 parts from 9,” only to count “what is left.” I also thought it was interesting to point out that = is what we use on a calculator to say “…and the answer is…” We need to be sure to teach our students that the “=” sign means “the same as” so that problems/statements like “5 + 4 = 6 +3” make sense. Chapter 10 is perfect timing for me! I am about to teach my formal teach this week, and my mentor teacher did not decide on my lesson until two days ago. I will be introducing a new unit which begins with a “Make a 10 to Add” strategy. Page 172 in our text says, “Perhaps the most important strategy for students to know is the Make 10 strategy, or the combinations that make 10.” In all truthfulness, the only strategy I was taught when growing up was the “Memorizing Facts” strategy. We learned math facts by memorizing. We did activities like racing at the chalkboard to see who could write the memorized answer faster. We practiced with flashcards, and we completed mad minute worksheets to show what we knew. The idea of “Making a Ten to Add” 9 plus 6 was completely foreign to me. As a matter of fact, my husband once tried to show one of our children this strategy at a young age, and I asked him to stop because I thought it would confuse her. I am truly embarrassed to even admit that now, but that was just how unfamiliar I was with the idea of using “guided invention” to teach math. When I grew up, a person memorized the fact, or there was “only one way” to do the math! I love the activities the chapter provides, and I love how all of this encourages reasoning strategies instead of just memorizing facts and strategies. Being able to reason is a skill that will be used throughout students’ lives in far more than just math problems. I also think that the “Fact Remediation” on page 184 is a very helpful guide to figure out where our students are and how to help them move forward with success. This book is excellent!!!
In response to Emily M…In chapter 9, I also did not know the technical terms for the four problem structures. I have never heard those terms used, either, but I do agree with you that I have seen all of those types and the names completely make sense to me. I just didn’t know that those terms were the ones used to describe these types of problems. Not using “Take away” to describe subtraction was also something new to me. I heard it my whole life, and I’m sure that it is still being used regularly today, but the book did a really good job of making sense as to why we should not use that term when relating to subtraction. In chapter 10, the activities shown are terrific! I can’t imagine getting rid of this book. The amount of useful information in it is incredible!!!
Chapter 9 Interestingly, my mentor teacher in mathematics has been teaching the properties of Addition to her 3rd graders. They have seem to catch on the concepts very quickly. I believe it is important to understand these concepts because they are used through out mathematics. I work as a high school para and I have seen students having to recall these same properties when doing math. This chapter also discusses multiplication and division. It says that division needs to be taught shortly after multiplication so that the students can see how they connect. Personally,I agree because the concept is new in their brains and I feel like it division helps reinforce multiplication. I liked how the text provided strategies for solving a problem and stating that students need to take their time before offering a solution. I think teachers sometimes put pressure on students to answer quickly because silence is awkward. I for me that is something I am working. I am trying to allow for more response time before I say something.
Chapter 10 I remember being in 2nd grade and learning my multiplication facts. We worked with a partner and flash cards. We memorized our facts and I don't remember any strategies that were used to help us. I was able to learn mine this way, but as I now work with special education students I have found that this method does not work for all students. Since taking this class, my attitude towards memorizing math facts, calculator use has changed. I have always advocated for students to concretely know their basic facts because it is necessity for upper level class such as algebra. However, as we are learning some students may not be able to memorize all these facts. I think the different strategies suggested in the chapter were great options to teach students. For some memorizing will work fine, while another child may be able to come to the same answer by Guided Invention. Chapter ten offers many great activities that I can implement into my own classroom to help students learn their basic facts.
To Angela R. I am so glad you are trying to help your student gain some ground in math. I too have seen older children (high school) still doing 3rd grade math. I want to help these students get to grade level because math is so important in everyday life. I hope you find a strategy that will work with this student. I would love to hear what progress you make by May.
Megan B: I would like to know the 9 finger trick for nines. I was never taught any tricks. It was memorization or nothing. I cringe when high school students do not know their multiplication facts because I believe math is a process that builds on each concept. It is so important to identify a struggling student early before they fall so far behind that it is too late to get them caught up to their peers.
Chapter 9: When reading this chapter I was able to connect with it in many different ways. It talked about the connections between addition and subtraction, as well as multiplication and division. I thought it was interesting that it said how all joining problems you don't have to be doing addition and for all take away problems you don't have to be doing subtraction. It was also interesting when I was reading about when teachers mention that division by 0 is not allowed. The point is that it just doesn't come out to anything. I also saw that they said- 'what if a teacher says to put 30 counters in to as many sets of zero as possible. Or put 12 blocks into 0 equal groups. It can make more sense! Chapter 10: Number relationships- With the idea of memorizing facts (multiplication/division). It can be good sometimes but sometimes students benefit more from using strategies to figure out problems and they learn how to be flexible and how to problem solve. I also read about story problems. I love story problems- I think that I learn best when problems are laid out in this format. They are great because they show a context in real life that each situation can be applied to. It teaches them how to problem solve as well as helps them to gain interest in the subject at hand. I really liked reading about strategies for subtractions (and everything else) it is important to know different strategies so that you can help each one of your students.
Response to Jeanette: Chapter 9: I liked how you mentioned how high school students still use these same strategies. I know that I still use strategies that I learned in grade school With out them I would have had a such harder time learning math. Chapter 10: I agree that my attitude towards memorizing math facts/calculator use has change also. I am very neutral on these ideas. I think at times they are great and at other times they may fail to help the students at all. I loved the strategies that were mentioned and will definitely try to use them.
The collection of strategies in both chapters is certainly interesting. I find the actual application of some of them to feel a little silly to me.
Everything in math is connected to one or more other things, only through an understanding of these connections can students really come to fully understand math.
Chapters nine and ten are full of meanings, structures, and strategies; they are all useful things that can help a student who struggles to learn math. At the same time, all students do not struggle to learn math. The book makes it clear that the study of these things is useful for students who do not struggle to learn/memorize their math facts. While this could be true a teacher might have a hard time explaining this to the student who is not having problems.
These chapters are certainly worth our time, but do not say enough about what to do for and about the student who does not need the strategies and has no interest in spending any time learning them.
These to chapters are full of so many useful ways to teach and complete math facts that they will continue to have something to teach me for a long time to come.
They make me think about people like my ex-wife, who continue to be bad at math all the way through the system. Maybe if she had spend more time with these ideas early in her school career she would not have had so many problems with her college math classes, never did get the C.
If a student does not h ave the basic concepts down in will be much harder for them to get down the more complicated things they will study later. while all students should be exposed to the more advanced and enjoyable activities those without a full knowledge of the basic facts will have a much harder time with the more advanced material.
In chapter 10, I really thought the the section, "What Not to Do When Teaching Basic Facts," was very interesting. The very first on the list was don't use lengthy timed tests. The reason behind this is that students will often ignore or abandon their strategies that they have learned as a result of the pressure of a timed test. I often still feel that way today. Whenever there is a timed test, even if it is more than long enough, I still have a little trouble with it. I also think that there is a time for students to be placed in an environment where there is a little pressure. Although, I think the level should be somewhat mastered before the student is placed under time restrictions.
I think you are right. If we, as teachers, can find the struggles in subjects that are at the heart of it all, we can make the difference of a student who is now successful in math instead of one that struggles.
Chapter 10 talks about children learning basic math facts and different ways for children to learn the facts. After reading the first part of the chapter I realized that my students in my internship are learning basic math facts. The math facts are not being memorized at this point because the students are still learning why 2+2=4. I think that memorization is important, but understanding what you are memorizing is also important. If a student knows that 2+2=4 but doesn’t know why it becomes just a fact not a problem. In every problem the chapter describes that story problems are the best for learning. Whether it is addition, subtraction, multiplication or division, the student still needs to learn the basis of the fact. There are so many activities in the chapter that students can use to learn the facts. I really enjoyed the last part of chapter 10. It talks about students who have not mastered the facts by the 5th or 6th grade. These students need help remembering their facts. Whether it is number touch points or visual aids that help them. There are a lot of tips to remember. I really liked the one that says stay positive and diagnose their strengths along with their weaknesses.
I feel the same way about time tests. Usually when I take times tests I don't really use strategies that I learned and I tend to forget a lot. Students I have seen taking time tests are normally counting on their fingers to find the answer instead of using memorization. If you give the students the time they need to complete the activity but still in a reasonable amount of time, I think the students will do better.
I found the Introducing Symbolism section in Chapter 9 to be great information. After observing a 1st grade teacher as a Para for three years now I understand how important this topic is to a teacher. Understanding the terms is the beginning of making sense of the math world. Model based problems are all over in the elementary school classrooms. I hear word problems being modeled as I walk down the school hallways. This seems to be the way more and more teachers are teaching math strategies. I found Chapter 10 to be very useful information. This is the type of information I want to save in a file folder named: Things To Use When Teaching... For example, up over 10 strategy uses 10 facts to figure out greater than 10 facts. I am a visual learner so the illustrations are helpful for me to understand the strategies.The nifty nines is a wonderful strategy for multiplying nines. I have used this strategy for years with students who are just learning their multiplication tables. I must say that the literature connection at the back of chapter 10 provides some good books to integrate within math lessons.
Just like many of the students that have already posted, chapter 9 contained a lot of material that was new to me. I had never heard of the Four Problem Structures, but after reading this section it made total sense. Properties of addition and subtraction was a good review. Of course my favorite part of each chapter is the activities. In St. Joseph, math is taught through investigations and activities and this book gives me a lot of tools for my toolbox. Chapter 10 begins with the developmental nature of basic fact mastery. Figure 10.1 has lots of good information for addition and subtraction basic fact mastery. And then the book discusses approaches to fact mastery and guiding strategy development. I liked the multiple reasoning strategies for addition, subtraction and multiplication. Many of these strategies I had never heard of. I had to learn math the old fashion way, just memorizing facts, which didn't work very well for me. I would have loved to have known other ways to find the same answer when I was younger. I agree with the book that children learn different ways. Towards the end of the chapter, it discusses mastering the basic facts and tells us as teachers what to do and not to do when teaching basic facts. Finally, the chapter has lots of good activities throughout for me to use in the future.
Comment for Joel Stucky: Yes, this section was very important and informative. Proceeding through facts from 0 to 9 is a sure fire way of teaching the importance of lower numbers verses the last in line being nines. I think that teaching memorization to fast is not good as well. Learning the meaning of why 6 X 6 = 36 is the goal. It all comes back to number sense...
I also agree that the student needs to understand what they are doing, not just memorize the facts. I am a very visual learner, so story problems always made more sense to me. Chapter 10 is filled with lots of activities that I believe will be helpful to me in the future.
I was glad to see that in chapter nine, the author discusses the use of a number line to add and subtract. I say this because in my first grade class, I will introduce the number line to the students as part of my formal observation!
Addition and subtraction go together like multiplication and division. The author provided many ways to teach these operations. Being a special education para presently, one thing that we always teach our (fourth grade and older) students is how to tell the which operation to perform. So many students see numbers and automatically beginning to compute. I think understanding and comprehending the problem is one of the big problems in my department.
Children need to master basic facts. Pure and simple, these skills will be used again and again throughout life. I remember that I worked with one gal in fourth grade who had dyscalculia. Recalling facts was extremely difficult. Her teacher utilized the use of timed tests; every student had one minute to complete forty multiplication facts. However, after visiting with the teacher, she allowed this student three minutes to complete the timed test so that she had an opportunity to complete the test. Without increasing the time, the student would probably have never mastered her facts.
Jena, I have been using the nifty nines strategy to my special education students for the past couple of years. It is a great lean-to strategy. Of course, you always want your students to be able to recall the facts instantaneously, but this trick is awesome. Not only do my students with special needs love the trick, but the regular education students just think it is the coolest trick in the books. Matter of fact, one of my classes composed of fourth graders have shown their PE, art, and music teacher the trick!
Chapter 10 talked about learning basic facts. I like how the book stated that the students need to know how to work out their basic facts by hand, but a short cut can really help when it comes to solving the problem. This is so true for me. Plus, it's like we say in class, mathematics is the pursuit of laziness. Shortcuts were always beneficial to me, I thought it was so much better than the long way of solving problems, and I'm sure everyone else does too. Using short cuts to learn the basic skill can go a long way and also help in learning down the road. The more you know the basics, the better off you will be.
I think that a lot of students our age were taught with memorization. I know I used flash cards for multiplication and division facts. I now know why these facts equal what they do, but when I was just learning them I feel like I was just memorizing and not really understanding the full meaning. As teachers we need to make sure that our students fully understand the meaning of a problem and how to solve it.
In Chapter 9 the section on teaching multiplication and division really stuck out to me because that is something some of the kids in my mentor classroom seem to really be struggling with. For me I am finding it difficult to help them understand some of the concepts because it has been so long since I have learned them. The problems seem so simple to me, so I am struggling with finding a way to explain what is going on. This chapter was helpful to me because it explains some of the reasons for the struggles as well and some of the verbiage I was taught with that needs to be abandoned. I really like the idea of having the students create word problems. I feel like this is something that may help some of the struggling students to understand word problems and how to handle them. This chapter was helpful to me because it really hit close to home. Chapter 10 – Again, this chapter made me stop to think, because I really do take for granted that “I just know it”. I have forgotten how I learned it. It was interesting to me to go back through some of the strategies for learning some basic addition facts. I enjoyed the activities that were provided. I actually use some of the strategies and didn’t even realize it. Funny. I like the idea of “think-addition” with the subtraction strategies. Once kids know their addition, it really makes sense to encourage them to use that knowledge when solving their subtraction questions. Really what this chapter comes down to is relating new facts to existing knowledge.
@Betsy Betsy, you bring us such a great point in the chapter 9 text. The terminology really has changed today for the ways that we were taught. It is amazing that we even learned to add and subtract. I think watching how we say things in our lessons and instruction is important so that we do not further confuse our students, but I also feel that if we slip up here or there, it probably won’t make or break their learning progress. Your Make 10 strategy is a great one I think. It is funny because I think I use this often when I am doing mental math and I don’t even realize it. I just am all the time trying to see where and how I can make groups of 10 and then add in any remainders I might have. I feel like this is a really important strategy for kids to learn. Good Luck with your formal lesson this week!!
Within Chapter 9, I was introduced to the four problem structures. I always feel that I teach, as well as learn, best with hands-on resources and tools. It's always ideal for me to also visually see the problems, or models/examples, of problems shown within textbooks, on the board, worksheets, etc. I feel that the more options and tools I have the better-this textbook seems to do an overall great job with that and I appreciated Chapter 9 for that as well. As stated within Chapter 9, "Cognitively Guided Instruction is not a curriculum program but a professional development program in which teachers learn to use students' thinking to guide instruction." I agree and feel this concept is beneficial to the classroom and student's learning experience. Along with this idea, I feel that scaffolding is similar to this concept. I have noticed within internships that if you show a student how to do a mathematics problem a few times, but then gradually allow the student to do the first step of the problem and then try to do the next step on the next problem. This helps the student not feel overwhelmed, but feel motivated that they can complete the mathematics problem and be successful at the same time. That is learning. I feel this chapter elaborates on this idea.
Chapter 10
Within Chapter 10, it discusses the overall basics and how crucial they are to a students' learning process. I feel especially with mathematics, if you do not understand and are unable to apply the first step to an equation or problem, then how can you go on to the very next step? It's definitely a domino effect of progress. I want to make sure all of my future students understand (whether one learns one way, and another learns differently) how to get the problem down.
I can remember memorizing and applying all of the fac family facts within mathematics. For example, for the multiples of "9", you can add the first digit to the second digit if you line up "0,1,2,3,4,5,6,7,8,9" up then back down. So that 0+9=9(9), 1+8=9 (18), 2+7=9 (27), etc. It was fun for myself and my classmates in elementary school to come up with creative ways to learn our fac families. I still use all of the fac families to this day. For example, buying a 24 pack of Dr.Pepper and knowing how to split it evenly with 3 roommates.
Overall, I feel both Chapters 9 and 10 were informative and helpful. I will be able to reference back to these chapters on how to set up equations...setting up those basics!
I agree, Chapter 10 was helpful with useful future ideas and equations set-ups. I too feel that there are more and more ways and techniques that are coming about for the mathematics world. This is great for students because it gives them more and more options as how to get the correct answers. Any way is okay, as long as the student understands how he or she got there. I don't know how many times I have heard my parents say while I was growing up,"Wow! This math problem is way advanced and you are only in junior high/high school! We learned these types of problems as a Senior in high school or in college." I think it will interesting to see how it continues to change and develop in the future...especially as teachers. Is it true that every 5 years, curriculum is updated and processed earlier and earlier...meaning to different ages/grade levels?
I liked how the chapter started off with a definition of basic facts. Basic facts are combinations where both addends and factors are less than ten. The book goes on further saying subtraction and division facts correspond with addition and subtraction facts, for example 15-7=8 is a basic fact because seven and eight are less than ten. I also liked the part where the book talks about memorizing facts. It said that if students were to memorize the facts they would have 100 separate addition facts and 100 separate multiplication facts and sometimes separate subtraction and division facts. However, most students do not know their addition facts in upper elementary and in middle school they do not have mastery of multiplication or division. I agree with the book. It is so hard for children to learn these days. Maybe folks thought that when I was in grade school and I just never heard it. But children of today struggle so much in school. My class consisted of 17 students from first to sixth grade. Everyone was an above average student. When we transferred to junior high school, the students over there were below us. Maybe that is the difference between private and public schooling.
In reply to Rachel: Your example of the nine fact family was confusing to me. This is a perfect example of Dr. Stramel's "It may be easy for me this way, but not easy for you." I think the hardest part about teaching mathematics is understanding the different ways of teaching it. I was taught my memorizing and doing naked number problems. But my class always looked at it as a competition so we pushed each other, which was very helpful. We did packs and packs of worksheets and that is how we learned.
Chapter 9 started off by showing examples of the four problem structures for addition and subtraction: join, separate, part-part-whole and compare.I enjoyed reading the difference in story problems and context problems. I would prefer to do context problems because they are connected as closely as possible to children's lives. When students are doing story problems the focus on getting the answer. The chapter also talked about the four problem structures for multiplication and division: equal-group, comparison,combination, and area and other product-of-measurement. Chapter 10 started off by talking about basic facts and that students eventually get to stage of "just knowing." I found myself just knowing how to do a problem and then when my teacher would ask well how do you know that I would say I just know. It also talked about the three different approaches to fact mastery: memorizing, explicit strategy instructions and guided invention. The book also talked about as a teacher you need to have good command of many strategies. That is something I am worried I won't have because math isn't my strongest subject. The chapter went on to talk about strategies for addition and subtraction and gave some activities. Then strategies for multiplication and division with activities to go along as well. the last part of the chapter talked about effective drill. Doing drills is effective when the student has a strategy they understand and can use.
Chapter 9 and 10 were very useful and informative I believe. Chapter 9 talked quite a bit about multiplication and division. I really enjoyed reading this because these are two very important strategies in mathematics. Students are starting to learn these at such young ages now because they are that important. Another thing I found to be very interesting in chapter 9 was the section it talked about different problem structures. Every student learns differently so I think this was great to read about because it really put things into perspective. Some are more visual than others, others like manipulatives while other just like the white board, etc. I know I am a very visual person so taking classes virtually can sometimes be very challenging to me because I do not have someone actually showing me how to write a lesson plan, etc. Chapter 10 was also interesting to me. It mainly talked about the overall basis of mathematics. I enjoyed how it talked about different methods to teach different lessons. This chapter, especially, will come in handy in the near future. I believe having different methods to teach different lessons will only make the students that more involved. Doing the same thing day after day while trying to learn math facts will not be as fun as trying out different ways to learn them, for instance: white board, online games, manipulatives, etc. Overall, I really enjoyed reading these two chapters and I believe they are both useful resources to use in the future.
In response to Rachel--Thanks for sharing your method of learning your multiplication facts of "9" that is a great method. I also remembering using your hands when multiplying "9's" for example if the problem was 9x3, you would hold up both hands out in front of you and put down your third finger from the left. Then you would count you have two fingers on the left of the finger you put down and 7 on the right. Put those numbers together and you got "27". I agree, there is so many fun different ways to teach things. Thanks for sharing!
In response to Deidre, I agree that it is so hard for children to learn these days. I feel like there isn't a sense of urgency. I feel it's also the attitudes of the students and I feel as though children these days are aloud to do what they want when they want. I feel since students are struggling to learn it is important to try and get them hooked on learning and continue to keep learning fun and interesting.
Chapter 10 was over how a teacher can help their students master basic math facts. I found this chapter to be the most beneficial for me thus far in this semester. In my internship class (4th grade), we have been working on how to add and subtract large numbers. It’s amazing how many students have difficulties subtracting large numbers. This is why I think this chapter was great. A student needs these basic math facts in order to achieve harder problems. At the end of the chapter, the author talks about what one can do to help students who have not mastered their math facts by 5th or 6th grade. One of the suggestions was to provide hope. I think that this in especially important in math because students can give up so quickly. It’s the teachers’ responsibility to encourage the students. It also says to provide new strategies for the student. This will help the student to not become overwhelmed. If something isn’t working, throw it out and try something else.
Response to Andrew Dempewolf: Like you, I also think that it's important to provide hope to students when they are struggling to master their math facts. As teachers, we should provide that hope in any subject a student in struggling with. If teacher's don't encourage their students to not give up then who else will? If the student is not understanding the concepts in one way it's the teachers responsibility to introduce the concept in a way that student will understand.
Chapter 10 was about helping students master their basic math facts. This is such an important skill in math, but is often taught in an ineffective manner. The book mentions that many teachers go straight to memorizing facts after they introduce the concepts. However, it is important to include a strategies stage. Not only does it help chunk math facts together for easier memorization. It helps students learn how to do higher order thinking, which they will need to complete higher level mathematics. The last part of the chapter really caught my eye and I tried to soak up that information. It covered what to do with students who reach the 5th and 6th grade and have not yet memorized their math facts. This was important to me, because I plan on teaching middle school math. I know I will have students that get to me and still need work on basic math facts. Now I have a resource to help me with these students.
I completely agree with you, hope is probably one of the most important things to give to a student that is struggling with mathematics. Math can be very frustrating and easy to give up on. As teachers, we have to keep our students believing that they can do it, so that they keep trying to learn and don't shut down.
I learned some new ideas in chapter 9. I have always said “take away” to describe subtraction. I think it will be a hard habit to break. I like the example for finding the difference in figure 9.5 that shows how to use the Think-Addition strategy. I like it because I can look at a visual example that makes sense to me. I am not the person that math concepts come to easily, so I need that visual, especially in order to make sense of a way of doing things that is different from how I have been doing them the past 35 years. The reference I am making is using a number line with the arrow hops to show the numbers in the problem and one big hop for the total number. In chapter 10 the section on Story Problems was most interesting to me because it shows that there is more than one way to solve a problem, which I love, but was never taught as a child. My daughter learned her “doubles” last year in first grade and she uses them for every kind of math that she does! So one day she had the math problem 6 + 8. She told me 14 and then said, you want to know how I got that? I know that double 6 is 12 and 8 is two more than 6 so 2 more is 14. This way of adding completely foreign to me so if I seem overly enthusiastic, maybe I am. To me, the concepts we read about in chapter after chapter drive home the Learning through Problem Solving idea, which I love. I love it because it means that even someone like me can do math! This class gets me excited about teaching math, because I will be teaching children possibilities, not just one way or no way like I learned.
Andrew, I liked your observation about how many children have a difficult time subtracting large numbers. The children in the 6th grade class I intern in know more ways to solve math problems than I ever dreamed of. One day a couple weeks ago, she put a sample problem up and four students demonstrated four different ways that they solved for the same solution. It was fascinating to me!
Chapter 10 provided many strategies for helping students master their basic facts. At the beginning of the chapter, three different approaches to fact mastery were mentioned. Those three approaches are memorizing facts, explicit strategy instruction, and guided invention. Like other have mentioned, I am most familiar with memorizing facts. In grade school we had math facts up on the wall. We were timed to see how fast we stated and answered the math fact. They were always in the same order so it was easy for people to memorize the math fact instead of actually learning and understanding how to get the answer. The next topic covered in this chapter was guiding strategy development and story problems and reasoning strategies were mentioned. Next, reasoning strategies for addiction, subtraction, multiplication, and division facts were covered. I found the sections on what to do/not do when teaching basic facts very helpful. This chapter was beneficial because knowing basics facts is very important. If students don't know their basic facts then it will be hard for them to learn other math concepts.
Chapter 10 focused on helping students master the basic facts so that they won’t have to result to using ineffective ways to solve problems. The addition and subtraction facts are extremely important and will continue to be used as the students increase their knowledge of mathematics in order to solve higher level problems. If students don’t have a good grasp of these basic facts, they are going to struggle as they get into more advanced mathematics classes. One of the approaches presented to help students with fact mastery was memorizing and this approach was used frequently in elementary mathematics classes. I also had a bit of trouble learning the multiplication/division facts and my parents used flash cards to help me. I also had trouble with story problems because I had to set up the equations myself and they weren’t already written out. The reasoning strategies presented in the text can be very helpful in assisting students write story problem equations and allow them to become more aware of the relationships between numbers.
Some of the strategies such as Using 5 as an Anchor and Up Over 10 seemed a bit difficult at first and I had to really think about them. I don’t remember being taught to do addition/subtraction this way so using these methods seems like it would take more time than simply memorizing the facts. This made me realize the importance of this chapter and the strategies & methods that were discussed. By simply memorizing facts, students don’t really have to think about why the answer to 6+4=10; that will always be the answer. By using these strategies, they will be thinking deeper about why that is the answer. I thought that using a clock to help students learn 5 facts was really cool. In regards to the strategies used to help students with multiplication facts, I did learn how to use Nifty Nines and I thought that it was awesome. The What To Do & What Not To Do sections at the end of this chapter were very helpful and overall I learned a lot of valuable methods and strategies to better teach students how to master the basic facts.
In response to Monse R.: I liked reading your post and I was also in a classroom where we were timed and were expected to answer the math questions as quickly as possible . I liked that the fact that you mentioned how the problems were always presented in the same order. This is the perfect example of how memorizing and drill is not very beneficial. The students are not even required to think about the problems in order to answer them. Once the students have learned the basic facts, they should be given more challenging and meaningful activities to help them practice these skills.
In response to Sarah R. I also think mastering the basic facts in so important. I work in the middle school and I can tell right away if a student knows their multiplication facts or not. They all seem to struggle more, and need more assistance. I have gotten where I just tell them to grab their agenda right away and use the multiplication chart rather than have them guess. My thought is if they keep looking at this chart slowly but surely they will start to pick up on them and it will make their lives a lot easier in high school. I get so upset when I hear that the elementary school teachers are not focusing on them. There are so many activities I have seen done to help students memorize them, when I was in school we had a banana split party at the end of the semester. Each week we focused on one fact family 1x2, 1x5, ect.. And we had a times test at the end of the week and you earned a part of the banana split. If you didn’t pass it you didn’t get that ingredient. I remember students that only had the banana and one scoop of ice cream. The whole goal was to make a complete banana split with a cherry on top…we loved it! I also learned another trick that is very similar to the nines trick but it has to do with 6,7, and 8. You use your fingers to help yourself solve the problems. It is to complicated to try to explain but I bet if you are really interested you could goggle it or YouTube it and find someone demonstrating it.
Chapter 9 and 10 was over helping students understand the operations addition, subtraction, multiplication, and division and to help them master the basic facts of these. Chapter 9 was over helping students have an understanding of the number sense and what it really means to add or subtract rather than just the process. Students first must know how to set the problem up, what symbols to use and how to solve for the answer. Then they must understand the why questions, in order for students to have a good foundation for number sense they have to understand the terminology, subtract means to take a way, the number will be getting smaller, you will have less than what you started with. These are just some points that a student must understand to have a full understanding of the operations. Chapter 10 was over teaching students their basic math facts. I am a strong believe that students need to learn their math facts (addition, multiplication) if they are ever going to be successful and enjoy math. I work in the middle school and I can point out the students that I know don’t know their multiplication facts, their assignments take longer, they get frustrated quicker, there are more mistakes in their answers and not over the material that was being covered. There are so many different ways to teach students their facts I think working on them for a few minutes of the day would help these students dramatically! When I was in school they taught us how to use our hands to find the multiplication facts of 6-9, and I still use them when my brain is working too slowly. When I was in school I had to use my hands a lot but with practice and a lot of time I ended up memorizing them from doing problems with them in it. This is a very easy way to help students learn them rather than just giving them the answer or telling them to look at the multiplication chart. Overall both of these chapters made me take a step back, I have a very high understanding of number sense, I have always had, I could just see why we would do it that way. But I liked how the chapter broke it down so that when I have a student or students that don’t have this sense I can help them develop it.
TracyPer Chapter 9 learning about the four basic operations, and the newer terms used to help in all to understand the concepts of basic addition, subtraction, multiplication, and division. I had not realized how important the terminology had such a effect on student learning until now. Chapter 10 really learning and memorizing like the multiplication facts has sure been useful for many years. Especially with the concept of mental math, like being able to count back change is a dieing art. We have become so accustomed to the computer cash register telling us what to do we no longer think for ourselves.
Chapter 10: Chapter ten of the text book Elementary and middle school mathematics: Teaching developmentally (7 th ed.) by Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. discussed how teachers assist students in doing math problems that require little thought process in basic math problems (Van de Walle et al., 2010, p. 167). This chapter was packed with new information that made me think of matters differently and information that made me think of my own experience. A piece of information that I learned from this text was “What not to do when teaching basic facts” (Van de Walle et al., 2010, p. 183). Van de Walle et al. states five different things not to do when teaching basic math, which are “Don’t use lenghthy timed tests”, “don’t use public comparisons of mastery”, “don’t proceed through facts in order from 0 to 9”, “don’t move to memorization too soon” and “don’t use facts as a barrier to good mathematics” (Van de Walle et al., 2010, p. 183-184). This is very good advice for any teacher especially new educators. A piece of information that made me look at my own experiences was on page 179 of the textbook that mentions a simple way to multiply 9s. Students can do this by putting down one finger on their to show the divide of tens and ones. This is a strategy that I use all the time. I teach students this strategy as much as I can because it will help some students. This strategy can be difficult for some students but it is worth showing the students in hopes it will befit them. If they do this enough times they can gradually start memorizing the answer and not rely on using their fingers.
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.
Tracy Per to Lane A I so agree with your idea of getting the family to help. I know without my families support I could have never gotten this far in my educational experiences. I think there are a lot of parents just waiting to be encouraged to get involved. I am a parent just like that as well.
I really enjoyed reading chapter 10. This chapter gives a lot of great examples and activities for teaching basic facts. In my internship classroom, when I am helping the third graders I sometimes have problems explaining to them the strategies that are second nature to me as an adult. I could definitely use some of the strategies like, near doubles, taking from the ten, and up/down over ten to explain things to the students. I am also realizing just how valuable the ten frames can be.
Jennifer Pen reply to Rebecca B. Great post. I agree with you. I had to use my fingers a lot when I was counting and figuring out math problems and I think that is OK because it became easier and easier the more I did it because I would start memorizing them instead. Some teachers that I have met do not like the students to use their fingers but I thin that they students should do what ever it takes to get the correct answer and assist them in problem solving.
Chapter nine of the text book Elementary and middle school mathematics: Teaching developmentally (7 th ed.) by Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. discussed how to teach students ways to add, subtract, divide and multiply. A piece of information that I learned from this text was there are "separated addition and subtraction problems into categories based on the kinds of relationships involved. These include join problems, separate problems, part-part-whole problems, and computer problems" (Van de Walle et al., 2010, p. 146). I was seen these different ways before in the classroom did not know that is what they were called. A piece of information that made me look at my own experiences was figure 9.6 on page 153. I remember learning this strategy when I was younger and find it very useful. Students combine numbers together (equal 10) that are easier to add and then add the rest. This strategy is easier for students because the can relate better to adding to 10. 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.
When subtracting, using take away is not really the best word. Minus or subtract are the best words to use for this. Children learn at different rates and at different ages. Educators need to remember this when teaching math facts. They will not all know the basics as soon as others may know them. Knowing the basic facts is very important when learning or teaching mathematics. Teachers have to help students acquire their knowledge over time, not just assume they know that facts. These chapters touched base on four problem structures. It was very interesting to me to read about these. The examples and walkthrough for the different type of problem structures was interesting to read about.
In response to Jennifer P--- I think the section of the separated addition and subtraction problems was very interesting. I am using the strategy on page 153 you talked about in my internship class. They want to just count all the time. The teacher is really trying to show them tricks for remembering certain addition and subtraction problems. The one I did not know about was the 9 trick. When adding, you just put a one and one less than the number. So for example: 9+6= 1 and one less than 6 which is 5 so the answer is 15. It was pretty neat to learn this. I wished I would have learned this when I was in school.
I liked the different concepts and strategies chapter 10 talked about. I liked how they showed different ways students might work out a problem. For example, the text had a problem that was 6+8. Some students may not know it off the top of their heads, so they might add 2 onto 8 to make 10, then they might subtract the 2 they added onto 8 from 2. Now it turned into a 10+4 problem. I can relate tot hat because I can remember doing that when I was younger. I found this chapter had many cool strategies that children can use if they're having trouble with their problems.
I sometimes have a similar problem like yours. I also struggle explaining problems to students at times. Like you said, it's like second nature to us and we don't even have to think about how we got the answer. We just know it. I feel like this chapter gave me some helpful ideas from this chapter.
I was a little surprised that they separated the four problems into four different structures, Where, we would have just called them word problems or story problems. I liked how the books suggested that story problems should be started as early as kindergarten and pre K. This is something that the children really struggle with so all the practice is very important. I think that the sooner children learn about symbols the better. The one thing I do see is sometimes when you say addition to a child they know it as plus and don’t recognize the difference in the name. The same thing happens with subtraction and take away, and multiplication and times. So I think it’s important to get them use to all the different ways we call the symbol. Remainders really drive me crazy when I’m walking into a classroom because it is hard to tell what they are doing with the remainder; leaving it as r. or making it round up or continue till we get through all the numbers by adding 0’s. Mastering math facts has 3 approaches; memorizing, explicit strategy instruction (teacher shows them a strategy and then they practice), guided invention (using number combinations and doing which ones make sense to them). This chapter has tones of activities I can use. I love that. I was trying to think of some good ways to store them for easy access. (on index cards ¬in one of those recipe boxes) or something like that. When I was looking at effective drill, I noticed the, what to do and what not to do when teaching basic facts. These are great and very important. I think making drill enjoyable and involve the family are most important. This is great car ride work. I can do multiplication facts with my children while I’m cooking dinner or doing the dishes. I can do it at the park or waiting in line at the grocery store. Children have to be reminded about this so they can remind the parents. There is not enough time in the day to teach everything so it’s great if they can learn something extra at home. Fact remediation, there are those kids that just don’t get facts no matter how much time they spend on them, they need something else to help them.
Rebecca B. I really agree with you children need to learn to love math if they will be really successful with it. They need to learn their facts. I'm glad the book gives us so many different ways to teach the same thing. These children really don't learn the same way. This book does get you so many opportunities to guide the children.
Chapter 10 covers the development nature of basic fact mastery. I was excited to read this chapter just by reading the chapter title. My thoughts before I read it were that it would be about memorizing mathematic facts and the different ways to teach that. One of the strategies the text lists is drill. While this strategy works for me, it does not work for everyone. I have seen in my internship class that some students don’t work well under pressure. I have worked with a student one on one and they get very nervous and I have to tell them to stay calm and just think. I definitely think they would benefit from a different strategy. In Reply to: Matthew B. I also thought this chapter had a lot of interesting strategies to use when working with students. I think that there were a lot of different ways to approach teaching facts when you have students that learn differently.
Chapter nine was very informative for me. I think a key to understanding mathematics is being able to connect different meanings and relationships to it. A group of numbers doesn’t make much sense on its own, it has to have some meaning or interpretation supporting it. I feel like I can do so much more in the classroom after reading this chapter. When I sub or get to help teach in my observation classroom I often struggle with math. I can present the facts and problem solving strategies fine but I have trouble when students need it more broken down. Information can’t just be presented to a student. A key to being an effective educator is presenting information in an understandable way. It’s being able to break down those facts and rules so all learners can grasp them. I didn’t realize that when I was showing them a simple addition word problem that I was using the join structure. These broken down structure examples really help me understand what I’m trying to say even better so I know they’ll help the students understand me better as well. I think when he students can see a concrete structure of something the numbers aren’t so hard to manipulate for the correct outcome.
Chapter ten was also very informative for me because it discussed what my mentor teacher stresses everyday. She works very hard to instill the basic math facts in her students. She uses tons of repetition when it comes to the basic facts. I’ve realized through this chapter and my observation that even the tiniest basic math fact plays a huge role in a student’s mathematical success. If they don’t understand even the smallest fact it could hinder their future progress. You’ve got to get the basics before progress can be made. I really enjoyed all of the activities listed throughout this chapter as well. I can use these even when working with a group of students during down time in my observations.
Matthew B, I also enjoyed learning about the concepts and strategies in chapter 10. Through my observation class I’ve seen students work out problems in so many different ways. It’s amazing how their minds put things together. One thing I noticed in the classroom is something you mentioned. The students were doing a math magician activity and they were just buzzing through it crazy fast. I asked how they did them in their heads so fast and many of them said they would round to ten and then subtract what they needed. I didn’t do that when I was younger but it makes perfect sense and it seems to work for so many. You’re right, there were lots of useful strategies for us to teach our future students in this chapter.
In Chapter 9 where it talks about addition being referred to as "put together" and subtraction as "take away" and using these definitions puts limitations on students and makes it difficult for them to do problems that do not use those exact words. Those words become their key and trigger words and if they are not there than they struggle. I would say I agree with this totally, I have always been good at math but horrible at story problems. I could not figure out what I was supposed to do it less it asked specific questions like all together or what's left over. I like that we are learning strategies to help children work through word problems because I think word problems scare a lot of students. I have also always thought of the equal sign as equal to, but I can see how it would be better to say "the same as" or to think of it as a balance. I think many of us have the traditional mind set of math and this book does a great job of getting us out of the traditional mind set and getting us to think of things differently so we can teach it more effectively.
Chapter 10 talks about how a teacher can use strategies to understand how to get to the answer. On page 169 it talks about guided strategy and I recognized a strategy that I used this week in my intern class to help a student. The student was trying to figure out 7 x 4, so I asked her what 7 x 3 is, she didn't know that so I asked her 7 x 2 is & then we built on that. She ended up getting the correct answer and it was rewarding to see her understand it. Another strategy that I used was in division, the worksheet looked like this 30/5= ? and one student asked me if their answer was right and I said "what # do you multiply by 5 to get 30" so just because you are trying to find the answer to a division problem doesn't mean you need to only use division, sometimes you can use multiplication to solve a division problem, you can use addition to solve a multiplication problem. I also love the Nifty Nines trick, and I think that will help some students struggling in my intern class. I want to use these strategies and help students learn.
I have a son who gets extremely nervous with math drills. He is a smart kid and gets A's all the time in math but give him a time test and he just falls apart. Last year in 2nd grade I think he only got to 5's in addition but he could add or subtract any other time no problem. I practiced one time with him at home and as soon as the timer went on he was shaking like a leaf and almost in tears from the very beginning. When he was done, he had only half the page left and I asked what was wrong and he just said he couldn't do it. He said at school it's frustrating because every slaps their pencils done real loud, and turns the papers over and he just can't think. I went and spoke to his teacher and she knew about the situation but she did not offer any alternative. I think there should be an alternative and for students like that all they are getting is how frustrated they are when doing math. My son says I hate math all the time and he is only in the 3rd grade. I think we need to re think the math time test drills.
There were a few things from Chapter 9 that I didn't quite understand. It says that you shouldn't use the word "take away". When I was in school, that is always what my teacher told us to do when subtracting. I just think it can be very confusing when you are taught one way and then later you result to another way. In my internship class I was somewhat confused at first too because the teacher would ask the class on some problems if they were to add up or subtract down. I understood it because I just knew in my head what to do, but to be honest I probably would have tried a different approach to teach them. Many seemed to be confused on it. Back to subtraction! I do agree that there are better choice of words than "take away."
I agree with you that basic math facts must be repetitive in the classroom. We don't realize that one tiny mistake makes a huge difference. My mentor teacher sounds much like yours. Every day she goes over the basics before beginning a lesson. We all know that you have to have the basics down before advancing up. Good comment!
Chapters 9 and 10 were very informative. I liked how they had teaching strategies for teaching the basics. Like most everyone else, the fact that using the words “take-away” isn’t considered appropriate anymore is strange to me. Take away, to me, simply turns it into more of a words problem and allows the students to immediately relate the term to taking away something. I definitely think it is important to use words such as minus and subtract are important because it is important to know the technical terms. I am confused about this because Dr. Stramel said using words like “8 and 8 is 16” rather than “8 plus 8 equals 16”. Using the words “and” and “is” seems very similar in nature to using the words, “take away”. Other than that, I thought the reading was very informative and helpful for the future.
The four problem structures were really confusing when I first read this section. However, if you really look at the various problems, it really makes sense. I think sometimes we try and make things easier to understand but what happens is it is made more difficult. The text labels the different 'parts' of the problems "Initial, Change, and Result" These are for the first two problem structures "Join" and "Separate". These two stuctures are simply add and subtract depending on how it is worded and what numbers are provided! The key word in ALL of these problems is MORE. This mean we add two things to find the result. If we have the result, we will need to subtract. "5 plus WHAT will equal 10?
To Jen Watson The strategy you mentioned in your post (how much is 7X3 and then 7X2) is, interestingly, exactly what I did today! Another strategy I use with a student in special ed is "How much is 3 groups of 7" That seems to work well with him. I then slip in the word "times" so he will eventually associate 'groups of' with 'times'. It is really cool when we find a way the kids understand.
I was familiar with each of the four problem structures described at the beginning of Chapter 9; however, I wasn’t surprised that I didn’t know the technical terms for them. Although I wasn’t surprised, I still wonder why I never learned these terms (even in a previous Math for Elementary Students course). The textbook emphasized the importance of teachers knowing this, and I was confused as to why any educator would not be able to understand or successfully teach these concepts. Maybe I am close-minded, but it seems pretty elementary and I can’t imagine a teaching not being able to fully understand.
ReplyDeleteI also never knew not to use the words “take away” when talking about subtraction to a group of students. Minus or subtract are acceptable, but never take away. I think I hear my mentor teacher use “take away” on a daily basis when I’m there for my math internship. I also liked the way described to selecting numbers to use: numbers that are in the student’s grasp. Students know certain numbers at different ages, so as an educator you need to plan to use what is within that grasp versus below or above, which could make a problem too hard or too easy.
One of the very first things mentioned in Chapter 10 was that, “teaching basic facts well requires that essential understanding that students progress through stages that eventually results in just knowing.” I think this is important to keep in mind not only with math materials, but also with every subject matter. Students do not just know something, but rather they acquire it over time. This is important for teachers to keep in mind.
Knowing basic facts is an important part of mathematics. I think the most beneficial aspect of this chapter was all of the great ideas to use to teach basic facts (and also the ideas Dr. Stramel showed us in class a week or two ago)!
Katie Coulter
ReplyDeleteChapter 9 & 10
Developing Meanings for the Operations
Chapter 9 was a whole new thing to me. I had never seen or heard of the four problem structures before. I just thought there was a variety of ways you could write problems, pick one. I am glad I became aware of this now as these structures do have an ordinance. As I continued on into the contextual problems I couldn’t help but laugh when I came across the fact about how Japan attacks this. As competitive as the U.S. is, I would have thought for sure by now we would be adopting more of their habits to increase our scores to compete with them. The chapter does a really good job walking you through each type of problem you can face with the four structures. Examples are always nice!
Helping Children Master the Basic Facts
Chapter 10 gives some great foundation material. When you really think about how you learned math and reflect upon personal success to me it was plain to see the 3 phases through my math knowledge. Back them it was all about memorizing and competition. Which we have been told by this section is not the way to successful math skills or an appropriate route as a teacher to put your students through that. To some extent I believe that but on other basis children will deal with competition and comparison the rest of their life. The chapter once again does a great job giving in-depth focus on each particular phase especially reasoning. They give some great insight to how reasoning works for addition, subtraction and multiplication. I love the Nifty Nine activity. I have shown that to numerous students who have ever asked me a multiple of nine. Towards the end of the section it gives a list of some basic do’s and don’ts of math. One in particular that surprised me was how they discouraged you focused on the amount of time you made them practice and drill facts. I knew after a while students would obviously become bored, especially those who are advanced but I never thought it would be a bad thing for a student who really needed more time and focus. The book makes a good point stating they become frustrating and fatigued by repeatedly viewing the facts.
In response to Emily M,
Isn’t it crazy, I would have never thought twice about saying “take away”? It amazes me how we use a term or process for years and years only to find out that it is “not proper”. It’s not just in education but in many topics in life. I can understand how these words might confuse someone but at the same time I think to myself well I learned it like that can’t they? My stubbornness of following how I was taught is going to be my enemy teaching in today’s society with all these changes.
When I was reading Chapter 9 about developing mathematics for the operations there was lots of different areas that I was able to become more informed of. The title of the chapter says a lot about what the chapter hoped to accomplish which is making math meaningful for the students and helping to apply the math to real world settings. I liked how the chapter hit on the area that it is commonly thought of in the United States to have the children do lots of math problems in a single setting when in reality the teacher should have the student only focus a few math problems each class period.
ReplyDeleteI liked how the chapter focused on the properties of addition and subtraction as well as the properties of multiplication and division. The last area I found interesting when reading chapter 9 was the area of not always relying on the key word strategy in order to solve a problem. The book went into detail about the three reasons why to not rely on the key word strategy which are key words are often misleading, many problems have no key words and lastly it sends a wrong message about doing math. Overall I thought this chapter was very beneficial to me as a future teacher.
When I was reading Chapter 10 which focused on helping children master the basic facts I was able to learn some tips to use in my future classroom as well as different strategies. I liked how the book gave the definition of mastery of a basic fact which means that the student can give the correct response within 3 seconds. I was able to learn that there are three phases that students go through in order to become a master of a fact which are counting strategies, reasoning strategies and lastly mastery.
I liked how the chapter gave different reasoning strategies that a classroom teacher could use in all fours areas including addition, subtraction, multiplication and division. The section in the chapter that also talked about the do’s and don’ts when teaching basic facts in the classroom was helpful for me as a future teacher. Lastly I enjoyed reading about the fact remediation section at the end of chapter 10 which will be helpful for me in the near future when I begin teaching.
In response to Katie Coulter
ReplyDeleteI agree with you when I was reading chapter 9 I was not aware of the four problem structures either. I am glad that the book explained the four problem structures so that I was able to become better informed of them and learn the structures. I also found it interesting when the book was comparing the United States to Japan and how the United States focuses on lots of math problems in a single setting and Japan only focuses on a few problems in each setting. I also agree that the book does a great job of giving examples along the way.
I agree with you that when I look back on my schooling a lot of the math instruction was focused on memorization of facts which was talked about in chapter 10. I am glad that the book is pushing towards teachers using more reasoning strategies to get students to move from counting into reasoning which leads to mastery. The Nifty Nines strategy was by far my favorite part of chapter 10 because I think that it is a great strategy and I have never seen it done before using your fingers.
@ Katie C
ReplyDeleteI am on the same page as you with the four problem structure. I am familiar with each of the processes or types of problems, but I never knew that they were a part of something called the four problem structure.
I also agree on looking back on your own math successes. Everything seems to make more sense and give me an idea about how learning math happens, etc. I wish I remembered more about specific techniques and other ways I was taught math. I remember major things, but some times I find myself asking “Did I learn A, B, and C this way?!”
Chapter 9-
ReplyDeleteDeveloping Meanings for the Operations- While i found the information in chapter 9 to be informative also found some of the teaching strategies to be a bit complicated. It could be because I am interning in kindergarten. I think that the very best way out there to teach children addition and subtraction is HANDS ON!!!! They need for all of their senses to be activated in order to really get it. I think that by giving them something to actually count and add to and subtract from you are really keeping every sense activated.
Chapter 10-
Helping Children Master the Basic Facts-I found the information in chapter 10 to be informative and useful. Along with giving students the opportunity of learning hands-on you need to teach them short cuts to help them "just know" the basics. I subbed in second grade last week and got the opportunity to teach the kids how to do the "9 shortcut" in which you add 10 to a number and take away 1. They apparently had been struggling with learning to add 9 and after teaching them the shortcut they were easily able to add the number in their head.
Emily- I also found it extremely interesting that we are no longer supposed to use the words "take away" as I hear it in the kindergarten classroom I intern in as well as the classrooms I sub in. Even when I am doing math in my head (and I am 27 years old) I still sometimes find myself thinking the words "take away". Looks like I am going to have to work on that one! =)
ReplyDeleteWith regard to chapters 9 and 10, I truly enjoyed these chapters. I found them exceptionally informative and thought provoking.
ReplyDeleteI had no idea about the four basic problem structures, so this was all new to me- of course once it was explained I realized I have know it for a long time, but it is good to know how to actually teach using these structures. I also found it interesting when they explained the way they work on mathematics in Japanese classrooms- I can't be the only one who thought about where most of our new technology originates and I wonder if that is because children learn at a young age to be true problem solvers and think outside the 'lines'. I also did not know what an array was, I have seen them before- but never really understood the concept. I think it is ingenious the way they discuss teaching the associative property using this type of manipulative. The example problems in this chapter were wonderful and I had several ah ha moments while reading them and can see myself utilizing these problems ( or one like them) in my eventual classroom. In chapter ten, I learned about ten frames. I'll shamefully admit that when Dr. Stramel first discussed them I had never seen them before and had no concept of what they were. This chapter enlightened me and allowed me to see them as a very useful tool. I also like that in this chapter they specifically mentioned things to avoid, such as flashcard memorization of facts, after reading this book I am sure we are all convinced that a problem based learning environment is the best- but the little affirmations throughout the chapters certainly help to reconfirm our decision.
Kristie C, I like you am in an early elementary classroom and found some of the concepts harder for younger children to grasp, I had to intentionally remind myself while reading that I might not always be teaching very young students and to 'pay attention' to ALL the suggestions. I like though, that our text offers so many examples and ways to teach concepts. I, like you, think that HANDS ON is the best way for young ones to learn.
ReplyDeleteI never learned the nine 'trick' that your taught to that second grade class, this book and class are teaching me so many ways of doing things. It is sad to say that I must have had an extremely sub par mathematics education. But, each day I learn something new and am looking forward to giving students what I never had- which is mathematical confidence!
The part of this chapter that I really liked was the section that talked about what to do when teaching basic facts. There is a list of six things to have them do: ask students to self-monitor, focus on self-improvement, drill in short time segments, work on facts over time, involve families, and make drill enjoyable. Three of these really stand out to me. The first one is to focus on self-improvement. One of the big problems no days is that everybody just focuses on where a student is at compared to the rest of the class. What they should be focusing on is whether the student is getting better or not. As a student that was always one of my major goals was to not worry about where everybody else was at and focus on whether or not I was improving at the subject. The second is to drill in short time segments. This, in my opinion, helps students to not get so worn out on a subject and doesn’t overwhelm them. Lastly is getting the family involved. A student’s biggest supporting cast should be his/her family. If a student gets support and help from their family then it will boost their confidence tremendously and the sky will be the limit.
ReplyDeleteKristle C.
ReplyDeleteTeaching students with hands on activities and teaching them shortcuts is extremely important. Students at the elementary level seem to respond best to learning with hands on materials but at the same time you want them to learn shortcuts so that after a while some things just seem so basic to them that all they have to do is look at a problem and know the answer without having to do any counting or anything like that.
@ Brooke M
ReplyDeleteBrooke, I really liked your insights in to the chapters. I agree with the book whole heartedly. It is so important to make math meaningful for our students and relate it to real world experiences. I also liked the parts about the do’s and don’ts about teaching basic math facts.
Chapter 9 & 10
First of all I really liked how this chapter started. “This chapter is about helping children connect different meanings, interpretations, and relationships to the four operations of addition, subtraction, multiplication, and division.” I also liked the part about teaching division. I find remainders to be a nuisance to some students. I think it would be easier to teach younger students about “left over” instead of remainders. I loved all the different instruction methods for the 4 basic math concepts in chapter 10. I also really like the take from 10 strategies. Mostly because this is the strategy that I use all the time when I do mental math!
Chapter 9 is about developing meaning for operations. It's about helping students learn to connect addition, subtraction, multiplication, and division to help them use them in real-world situations. They go into how all the operations are connected. I see my mentor teacher doing this all the time. Having students check their division work by multiplying. I thought the information the book gave about introducing the = sign was interesting. I guess when I see that sign, I always assume the answer will follow, but not necessarily. The book suggests using the phrase "is the same as." I got the most out of the section about multiplication and division problem structures. I did not know there are problem structures that help teachers forming and assigning multiplication and division tasks. Another thing that I am taking away from this chapter is that when doing story problems, it is important to provide both multiplication and division problems in the same problem. This will prevent students from just using the day's operation, and help them to figure out which operation to use.
ReplyDeleteChapter 10 is about helping children master the basic facts. This means helping students give a quick response, in 3 seconds, without resorting to non-efficient means, like counting. Clearly, students need the knowledge of basic facts so they can continue to build on them. This chapter touched on memorizing facts. I always find it interesting to hear opinions on this. Some teachers don't teach strategies, they just teach memorization of facts. Research shows that this does not work, that students are not mastering the facts. It has also shown that by doing this, students are inefficient, they misapply the facts and don't check their work, and they don't learn flexible strategies. The text goes into guiding strategy development, and says that you need to give students as many strategies as possible. This is something that I feel is so important, and I see my mentor teacher doing this all the time. She gives them different strategies to solve problems, teaches them how to use them, but then lets them decide which strategy works best for them. I also will take a lot out of the sections about what to do and what not to do when teaching basic facts. One of the what not to do is using public comparison of mastery. I never liked it when teachers did this when I was in school. It was humiliating when you were at the bottom, and I think it puts too much pressure on kids.
Overall, both chapters were very informative and full of great activities and examples that I will use in the years to come.
In response to Lane A,
ReplyDeleteI also found the sections over what to do and what not to do when teaching basic facts informative. Like you, I had teachers that displayed students standings, and I think it had a bad effect on the students who were not doing as well as the rest of the class. I think that self-monitoring is very important. Teachers need to know where students feel they stand and if the feel like they are getting a concept. I also like the idea of making the drill enjoyable. I think this will help students succeed and help them retain the information.
Chapter 9 developing meanings for the operations- I found this chapter helpful but I thought that some of it was a bit complicating. I know that I learn better with hands-on materials and so do the kindergarten student that I work with in my internship. They can visually see what they have left in their addition and subtraction problems.
ReplyDeleteChapter 10 was the basic facts- I found this chapter was the most informational and helpful chapter. As well as students needed to know how to work out problems by hand, they also need to know how to use short cuts as well. I was so glad that when I was in third grade and someone showed me the 9 finger trick on the 9s multiplication.
To Kristle C.
ReplyDeleteI couldn't agree more about students learning their basic facts, but also having those helpful short cuts. I also really enjoy that you bought up the fact of using hands on activities to learn math. Currently I'm working on a action research project of how using math manipulatives increases students grades in the curriculum. I was a visual and hands-on learner so these are very important to me, however I'm always reminding myself that not all children learn like I do.
I have to say I greatly enjoyed reading this chapter. I am currently working with an 8th grade student who scored on a 3rd grade level on his Star Math assessment. This student is way below grade level and has received academic warning every year on his state assessments. He is a new student to our school this year therefore he is extremely behind his fellow classmates in all of his academic areas. I have to do tier 3 interventions with him for Math, doing 3rd and 4th grade work, and these chapters provided me with wonderful material and strategies to use when teaching my student. My student does not even now his basic Math facts and I love how these chapters have given multiple strategies to teach him that hopefully he will become comfortable with one or more of them and can excel in his interventions. There were numerous strategies presented in chapter 10 for basic facts that I had never even heard of. Actually some of the strategies were quite confusing to me and I believe I would confuse a student more by trying to explain some of the strategies to them! Many of the strategies remind me of the intervention program I am using with my student. The program is called Camelot Learning Math Intervention Curriculum. With this intervention program each student receives their own bag of manipulatives and a workbook to complete the intervention assignments. I am using a 4th grade level book with my student and I also have my own teacher workbook that has each lesson planned out with the script to use to teach the lesson also. We are doing mental math strategies at the moment and many of the strategies in the Camelot book are just like the strategies presented in these chapters. I am anxious to take our text to school and do the activities provided with my student!
ReplyDeleteIn response to Elizabeth-
ReplyDeleteI agree that it is very important for teachers to teach students multiple strategies to use when solving math problems. I am a student of memorizing the facts and I have to say that I am not that good at even the basics of math. I love how these chapters provided numerous different strategies to use in relation to all of the math operations and basic facts. My mentor also gives his students multiple strategies to use for each of the concepts he is teaching. It is amazing when students cannot understand the lesson being done one way but then all of a sudden it is like a light bulb comes on and they can completely understand the lesson by using another strategy!
In Chapter 9, I do want to comment on the language used when teaching addition and subtraction problems, subtraction in particular. I was raised hearing the term “take away” when describing how to subtract, and I must admit that I did not understand what the problem was with using this terminology when describing subtraction. After reading page 152, I completely understand why it is so helpful to not use “take away” and instead use “Think-Addition” when subtracting. It is more important for students to think of the “missing piece” which will help them immensely with addition facts, as well. If you think in terms of “What is 9 – 6 equal to?” and then look at it as “What can we add to 6 in order to make 9?” then it will help with addition - instead of going to the extra work of “taking away 6 parts from 9,” only to count “what is left.” I also thought it was interesting to point out that = is what we use on a calculator to say “…and the answer is…” We need to be sure to teach our students that the “=” sign means “the same as” so that problems/statements like “5 + 4 = 6 +3” make sense.
ReplyDeleteChapter 10 is perfect timing for me! I am about to teach my formal teach this week, and my mentor teacher did not decide on my lesson until two days ago. I will be introducing a new unit which begins with a “Make a 10 to Add” strategy. Page 172 in our text says, “Perhaps the most important strategy for students to know is the Make 10 strategy, or the combinations that make 10.” In all truthfulness, the only strategy I was taught when growing up was the “Memorizing Facts” strategy. We learned math facts by memorizing. We did activities like racing at the chalkboard to see who could write the memorized answer faster. We practiced with flashcards, and we completed mad minute worksheets to show what we knew. The idea of “Making a Ten to Add” 9 plus 6 was completely foreign to me. As a matter of fact, my husband once tried to show one of our children this strategy at a young age, and I asked him to stop because I thought it would confuse her. I am truly embarrassed to even admit that now, but that was just how unfamiliar I was with the idea of using “guided invention” to teach math. When I grew up, a person memorized the fact, or there was “only one way” to do the math! I love the activities the chapter provides, and I love how all of this encourages reasoning strategies instead of just memorizing facts and strategies. Being able to reason is a skill that will be used throughout students’ lives in far more than just math problems. I also think that the “Fact Remediation” on page 184 is a very helpful guide to figure out where our students are and how to help them move forward with success. This book is excellent!!!
In response to Emily M…In chapter 9, I also did not know the technical terms for the four problem structures. I have never heard those terms used, either, but I do agree with you that I have seen all of those types and the names completely make sense to me. I just didn’t know that those terms were the ones used to describe these types of problems. Not using “Take away” to describe subtraction was also something new to me. I heard it my whole life, and I’m sure that it is still being used regularly today, but the book did a really good job of making sense as to why we should not use that term when relating to subtraction. In chapter 10, the activities shown are terrific! I can’t imagine getting rid of this book. The amount of useful information in it is incredible!!!
ReplyDeleteChapter 9
ReplyDeleteInterestingly, my mentor teacher in mathematics has been teaching the properties of Addition to her 3rd graders. They have seem to catch on the concepts very quickly. I believe it is important to understand these concepts because they are used through out mathematics. I work as a high school para and I have seen students having to recall these same properties when doing math. This chapter also discusses multiplication and division. It says that division needs to be taught shortly after multiplication so that the students can see how they connect. Personally,I agree because the concept is new in their brains and I feel like it division helps reinforce multiplication. I liked how the text provided strategies for solving a problem and stating that students need to take their time before offering a solution. I think teachers sometimes put pressure on students to answer quickly because silence is awkward. I for me that is something I am working. I am trying to allow for more response time before I say something.
Chapter 10
I remember being in 2nd grade and learning my multiplication facts. We worked with a partner and flash cards. We memorized our facts and I don't remember any strategies that were used to help us. I was able to learn mine this way, but as I now work with special education students I have found that this method does not work for all students. Since taking this class, my attitude towards memorizing math facts, calculator use has changed. I have always advocated for students to concretely know their basic facts because it is necessity for upper level class such as algebra. However, as we are learning some students may not be able to memorize all these facts. I think the different strategies suggested in the chapter were great options to teach students. For some memorizing will work fine, while another child may be able to come to the same answer by Guided Invention. Chapter ten offers many great activities that I can implement into my own classroom to help students learn their basic facts.
To Angela R. I am so glad you are trying to help your student gain some ground in math. I too have seen older children (high school) still doing 3rd grade math. I want to help these students get to grade level because math is so important in everyday life. I hope you find a strategy that will work with this student. I would love to hear what progress you make by May.
ReplyDeleteMegan B: I would like to know the 9 finger trick for nines. I was never taught any tricks. It was memorization or nothing. I cringe when high school students do not know their multiplication facts because I believe math is a process that builds on each concept. It is so important to identify a struggling student early before they fall so far behind that it is too late to get them caught up to their peers.
ReplyDeleteChapter 9:
ReplyDeleteWhen reading this chapter I was able to connect with it in many different ways. It talked about the connections between addition and subtraction, as well as multiplication and division. I thought it was interesting that it said how all joining problems you don't have to be doing addition and for all take away problems you don't have to be doing subtraction. It was also interesting when I was reading about when teachers mention that division by 0 is not allowed. The point is that it just doesn't come out to anything. I also saw that they said- 'what if a teacher says to put 30 counters in to as many sets of zero as possible. Or put 12 blocks into 0 equal groups. It can make more sense!
Chapter 10:
Number relationships- With the idea of memorizing facts (multiplication/division). It can be good sometimes but sometimes students benefit more from using strategies to figure out problems and they learn how to be flexible and how to problem solve. I also read about story problems. I love story problems- I think that I learn best when problems are laid out in this format. They are great because they show a context in real life that each situation can be applied to. It teaches them how to problem solve as well as helps them to gain interest in the subject at hand. I really liked reading about strategies for subtractions (and everything else) it is important to know different strategies so that you can help each one of your students.
Response to Jeanette:
Chapter 9: I liked how you mentioned how high school students still use these same strategies.
I know that I still use strategies that I learned in grade school With out them I would have had a such harder time learning math.
Chapter 10:
I agree that my attitude towards memorizing math facts/calculator use has change also. I am very neutral on these ideas. I think at times they are great and at other times they may fail to help the students at all. I loved the strategies that were mentioned and will definitely try to use them.
Response to Tessa W
ReplyDeleteThe collection of strategies in both chapters is certainly interesting. I find the actual application of some of them to feel a little silly to me.
Everything in math is connected to one or more other things, only through an understanding of these connections can students really come to fully understand math.
Chapters nine and ten are full of meanings, structures, and strategies; they are all useful things that can help a student who struggles to learn math. At the same time, all students do not struggle to learn math. The book makes it clear that the study of these things is useful for students who do not struggle to learn/memorize their math facts. While this could be true a teacher might have a hard time explaining this to the student who is not having problems.
ReplyDeleteThese chapters are certainly worth our time, but do not say enough about what to do for and about the student who does not need the strategies and has no interest in spending any time learning them.
These to chapters are full of so many useful ways to teach and complete math facts that they will continue to have something to teach me for a long time to come.
They make me think about people like my ex-wife, who continue to be bad at math all the way through the system. Maybe if she had spend more time with these ideas early in her school career she would not have had so many problems with her college math classes, never did get the C.
If a student does not h ave the basic concepts down in will be much harder for them to get down the more complicated things they will study later. while all students should be exposed to the more advanced and enjoyable activities those without a full knowledge of the basic facts will have a much harder time with the more advanced material.
In chapter 10, I really thought the the section, "What Not to Do When Teaching Basic Facts," was very interesting. The very first on the list was don't use lengthy timed tests. The reason behind this is that students will often ignore or abandon their strategies that they have learned as a result of the pressure of a timed test. I often still feel that way today. Whenever there is a timed test, even if it is more than long enough, I still have a little trouble with it. I also think that there is a time for students to be placed in an environment where there is a little pressure. Although, I think the level should be somewhat mastered before the student is placed under time restrictions.
ReplyDeleteJeremiah,
ReplyDeleteI think you are right. If we, as teachers, can find the struggles in subjects that are at the heart of it all, we can make the difference of a student who is now successful in math instead of one that struggles.
Chapter 10 talks about children learning basic math facts and different ways for children to learn the facts. After reading the first part of the chapter I realized that my students in my internship are learning basic math facts. The math facts are not being memorized at this point because the students are still learning why 2+2=4. I think that memorization is important, but understanding what you are memorizing is also important. If a student knows that 2+2=4 but doesn’t know why it becomes just a fact not a problem. In every problem the chapter describes that story problems are the best for learning. Whether it is addition, subtraction, multiplication or division, the student still needs to learn the basis of the fact. There are so many activities in the chapter that students can use to learn the facts. I really enjoyed the last part of chapter 10. It talks about students who have not mastered the facts by the 5th or 6th grade. These students need help remembering their facts. Whether it is number touch points or visual aids that help them. There are a lot of tips to remember. I really liked the one that says stay positive and diagnose their strengths along with their weaknesses.
ReplyDeleteIn response to Joel S.
ReplyDeleteI feel the same way about time tests. Usually when I take times tests I don't really use strategies that I learned and I tend to forget a lot. Students I have seen taking time tests are normally counting on their fingers to find the answer instead of using memorization. If you give the students the time they need to complete the activity but still in a reasonable amount of time, I think the students will do better.
I found the Introducing Symbolism section in Chapter 9 to be great information. After observing a 1st grade teacher as a Para for three years now I understand how important this topic is to a teacher. Understanding the terms is the beginning of making sense of the math world. Model based problems are all over in the elementary school classrooms. I hear word problems being modeled as I walk down the school hallways. This seems to be the way more and more teachers are teaching math strategies.
ReplyDeleteI found Chapter 10 to be very useful information. This is the type of information I want to save in a file folder named: Things To Use When Teaching...
For example, up over 10 strategy uses 10 facts to figure out greater than 10 facts. I am a visual learner so the illustrations are helpful for me to understand the strategies.The nifty nines is a wonderful strategy for multiplying nines. I have used this strategy for years with students who are just learning their multiplication tables. I must say that the literature connection at the back of chapter 10 provides some good books to integrate within math lessons.
Just like many of the students that have already posted, chapter 9 contained a lot of material that was new to me. I had never heard of the Four Problem Structures, but after reading this section it made total sense. Properties of addition and subtraction was a good review. Of course my favorite part of each chapter is the activities. In St. Joseph, math is taught through investigations and activities and this book gives me a lot of tools for my toolbox.
ReplyDeleteChapter 10 begins with the developmental nature of basic fact mastery. Figure 10.1 has lots of good information for addition and subtraction basic fact mastery. And then the book discusses approaches to fact mastery and guiding strategy development. I liked the multiple reasoning strategies for addition, subtraction and multiplication. Many of these strategies I had never heard of. I had to learn math the old fashion way, just memorizing facts, which didn't work very well for me. I would have loved to have known other ways to find the same answer when I was younger. I agree with the book that children learn different ways. Towards the end of the chapter, it discusses mastering the basic facts and tells us as teachers what to do and not to do when teaching basic facts. Finally, the chapter has lots of good activities throughout for me to use in the future.
Comment for Joel Stucky:
ReplyDeleteYes, this section was very important and informative. Proceeding through facts from 0 to 9 is a sure fire way of teaching the importance of lower numbers verses the last in line being nines. I think that teaching memorization to fast is not good as well. Learning the meaning of why 6 X 6 = 36 is the goal. It all comes back to number sense...
In response to Amanda Lewallen,
ReplyDeleteI also agree that the student needs to understand what they are doing, not just memorize the facts. I am a very visual learner, so story problems always made more sense to me. Chapter 10 is filled with lots of activities that I believe will be helpful to me in the future.
Lacey Keller
ReplyDeleteI was glad to see that in chapter nine, the author discusses the use of a number line to add and subtract. I say this because in my first grade class, I will introduce the number line to the students as part of my formal observation!
Addition and subtraction go together like multiplication and division. The author provided many ways to teach these operations. Being a special education para presently, one thing that we always teach our (fourth grade and older) students is how to tell the which operation to perform. So many students see numbers and automatically beginning to compute. I think understanding and comprehending the problem is one of the big problems in my department.
Children need to master basic facts. Pure and simple, these skills will be used again and again throughout life. I remember that I worked with one gal in fourth grade who had dyscalculia. Recalling facts was extremely difficult. Her teacher utilized the use of timed tests; every student had one minute to complete forty multiplication facts. However, after visiting with the teacher, she allowed this student three minutes to complete the timed test so that she had an opportunity to complete the test. Without increasing the time, the student would probably have never mastered her facts.
Lacey Keller
ReplyDeleteJena,
I have been using the nifty nines strategy to my special education students for the past couple of years. It is a great lean-to strategy. Of course, you always want your students to be able to recall the facts instantaneously, but this trick is awesome. Not only do my students with special needs love the trick, but the regular education students just think it is the coolest trick in the books. Matter of fact, one of my classes composed of fourth graders have shown their PE, art, and music teacher the trick!
Chapter 10 talked about learning basic facts. I like how the book stated that the students need to know how to work out their basic facts by hand, but a short cut can really help when it comes to solving the problem. This is so true for me. Plus, it's like we say in class, mathematics is the pursuit of laziness. Shortcuts were always beneficial to me, I thought it was so much better than the long way of solving problems, and I'm sure everyone else does too. Using short cuts to learn the basic skill can go a long way and also help in learning down the road. The more you know the basics, the better off you will be.
ReplyDeleteAmanda Lewallen,
ReplyDeleteI think that a lot of students our age were taught with memorization. I know I used flash cards for multiplication and division facts. I now know why these facts equal what they do, but when I was just learning them I feel like I was just memorizing and not really understanding the full meaning. As teachers we need to make sure that our students fully understand the meaning of a problem and how to solve it.
In Chapter 9 the section on teaching multiplication and division really stuck out to me because that is something some of the kids in my mentor classroom seem to really be struggling with. For me I am finding it difficult to help them understand some of the concepts because it has been so long since I have learned them. The problems seem so simple to me, so I am struggling with finding a way to explain what is going on. This chapter was helpful to me because it explains some of the reasons for the struggles as well and some of the verbiage I was taught with that needs to be abandoned. I really like the idea of having the students create word problems. I feel like this is something that may help some of the struggling students to understand word problems and how to handle them. This chapter was helpful to me because it really hit close to home.
ReplyDeleteChapter 10 – Again, this chapter made me stop to think, because I really do take for granted that “I just know it”. I have forgotten how I learned it. It was interesting to me to go back through some of the strategies for learning some basic addition facts. I enjoyed the activities that were provided. I actually use some of the strategies and didn’t even realize it. Funny. I like the idea of “think-addition” with the subtraction strategies. Once kids know their addition, it really makes sense to encourage them to use that knowledge when solving their subtraction questions. Really what this chapter comes down to is relating new facts to existing knowledge.
@Betsy
ReplyDeleteBetsy, you bring us such a great point in the chapter 9 text. The terminology really has changed today for the ways that we were taught. It is amazing that we even learned to add and subtract. I think watching how we say things in our lessons and instruction is important so that we do not further confuse our students, but I also feel that if we slip up here or there, it probably won’t make or break their learning progress.
Your Make 10 strategy is a great one I think. It is funny because I think I use this often when I am doing mental math and I don’t even realize it. I just am all the time trying to see where and how I can make groups of 10 and then add in any remainders I might have. I feel like this is a really important strategy for kids to learn.
Good Luck with your formal lesson this week!!
Chapter 9
ReplyDeleteWithin Chapter 9, I was introduced to the four problem structures. I always feel that I teach, as well as learn, best with hands-on resources and tools. It's always ideal for me to also visually see the problems, or models/examples, of problems shown within textbooks, on the board, worksheets, etc. I feel that the more options and tools I have the better-this textbook seems to do an overall great job with that and I appreciated Chapter 9 for that as well. As stated within Chapter 9, "Cognitively Guided Instruction is not a curriculum program but a professional development program in which teachers learn to use students' thinking to guide instruction." I agree and feel this concept is beneficial to the classroom and student's learning experience. Along with this idea, I feel that scaffolding is similar to this concept. I have noticed within internships that if you show a student how to do a mathematics problem a few times, but then gradually allow the student to do the first step of the problem and then try to do the next step on the next problem. This helps the student not feel overwhelmed, but feel motivated that they can complete the mathematics problem and be successful at the same time. That is learning. I feel this chapter elaborates on this idea.
Chapter 10
Within Chapter 10, it discusses the overall basics and how crucial they are to a students' learning process. I feel especially with mathematics, if you do not understand and are unable to apply the first step to an equation or problem, then how can you go on to the very next step? It's definitely a domino effect of progress. I want to make sure all of my future students understand (whether one learns one way, and another learns differently) how to get the problem down.
I can remember memorizing and applying all of the fac family facts within mathematics. For example, for the multiples of "9", you can add the first digit to the second digit if you line up "0,1,2,3,4,5,6,7,8,9" up then back down. So that 0+9=9(9), 1+8=9 (18), 2+7=9 (27), etc. It was fun for myself and my classmates in elementary school to come up with creative ways to learn our fac families. I still use all of the fac families to this day. For example, buying a 24 pack of Dr.Pepper and knowing how to split it evenly with 3 roommates.
Overall, I feel both Chapters 9 and 10 were informative and helpful. I will be able to reference back to these chapters on how to set up equations...setting up those basics!
In response to Angela,
ReplyDeleteI agree, Chapter 10 was helpful with useful future ideas and equations set-ups. I too feel that there are more and more ways and techniques that are coming about for the mathematics world. This is great for students because it gives them more and more options as how to get the correct answers. Any way is okay, as long as the student understands how he or she got there. I don't know how many times I have heard my parents say while I was growing up,"Wow! This math problem is way advanced and you are only in junior high/high school! We learned these types of problems as a Senior in high school or in college." I think it will interesting to see how it continues to change and develop in the future...especially as teachers. Is it true that every 5 years, curriculum is updated and processed earlier and earlier...meaning to different ages/grade levels?
I liked how the chapter started off with a definition of basic facts. Basic facts are combinations where both addends and factors are less than ten. The book goes on further saying subtraction and division facts correspond with addition and subtraction facts, for example 15-7=8 is a basic fact because seven and eight are less than ten. I also liked the part where the book talks about memorizing facts. It said that if students were to memorize the facts they would have 100 separate addition facts and 100 separate multiplication facts and sometimes separate subtraction and division facts. However, most students do not know their addition facts in upper elementary and in middle school they do not have mastery of multiplication or division. I agree with the book. It is so hard for children to learn these days. Maybe folks thought that when I was in grade school and I just never heard it. But children of today struggle so much in school. My class consisted of 17 students from first to sixth grade. Everyone was an above average student. When we transferred to junior high school, the students over there were below us. Maybe that is the difference between private and public schooling.
ReplyDeleteIn reply to Rachel: Your example of the nine fact family was confusing to me. This is a perfect example of Dr. Stramel's "It may be easy for me this way, but not easy for you." I think the hardest part about teaching mathematics is understanding the different ways of teaching it. I was taught my memorizing and doing naked number problems. But my class always looked at it as a competition so we pushed each other, which was very helpful. We did packs and packs of worksheets and that is how we learned.
ReplyDeleteChapter 9 started off by showing examples of the four problem structures for addition and subtraction: join, separate, part-part-whole and compare.I enjoyed reading the difference in story problems and context problems. I would prefer to do context problems because they are connected as closely as possible to children's lives. When students are doing story problems the focus on getting the answer. The chapter also talked about the four problem structures for multiplication and division: equal-group, comparison,combination, and area and other product-of-measurement.
ReplyDeleteChapter 10 started off by talking about basic facts and that students eventually get to stage of "just knowing." I found myself just knowing how to do a problem and then when my teacher would ask well how do you know that I would say I just know. It also talked about the three different approaches to fact mastery: memorizing, explicit strategy instructions and guided invention. The book also talked about as a teacher you need to have good command of many strategies. That is something I am worried I won't have because math isn't my strongest subject. The chapter went on to talk about strategies for addition and subtraction and gave some activities. Then strategies for multiplication and division with activities to go along as well. the last part of the chapter talked about effective drill. Doing drills is effective when the student has a strategy they understand and can use.
Chapter 9 and 10 were very useful and informative I believe. Chapter 9 talked quite a bit about multiplication and division. I really enjoyed reading this because these are two very important strategies in mathematics. Students are starting to learn these at such young ages now because they are that important. Another thing I found to be very interesting in chapter 9 was the section it talked about different problem structures. Every student learns differently so I think this was great to read about because it really put things into perspective. Some are more visual than others, others like manipulatives while other just like the white board, etc. I know I am a very visual person so taking classes virtually can sometimes be very challenging to me because I do not have someone actually showing me how to write a lesson plan, etc. Chapter 10 was also interesting to me. It mainly talked about the overall basis of mathematics. I enjoyed how it talked about different methods to teach different lessons. This chapter, especially, will come in handy in the near future. I believe having different methods to teach different lessons will only make the students that more involved. Doing the same thing day after day while trying to learn math facts will not be as fun as trying out different ways to learn them, for instance: white board, online games, manipulatives, etc. Overall, I really enjoyed reading these two chapters and I believe they are both useful resources to use in the future.
ReplyDeleteIn response to Rachel--Thanks for sharing your method of learning your multiplication facts of "9" that is a great method. I also remembering using your hands when multiplying "9's" for example if the problem was 9x3, you would hold up both hands out in front of you and put down your third finger from the left. Then you would count you have two fingers on the left of the finger you put down and 7 on the right. Put those numbers together and you got "27". I agree, there is so many fun different ways to teach things. Thanks for sharing!
ReplyDeleteIn response to Deidre,
ReplyDeleteI agree that it is so hard for children to learn these days. I feel like there isn't a sense of urgency. I feel it's also the attitudes of the students and I feel as though children these days are aloud to do what they want when they want. I feel since students are struggling to learn it is important to try and get them hooked on learning and continue to keep learning fun and interesting.
Chapter 10 was over how a teacher can help their students master basic math facts. I found this chapter to be the most beneficial for me thus far in this semester. In my internship class (4th grade), we have been working on how to add and subtract large numbers. It’s amazing how many students have difficulties subtracting large numbers. This is why I think this chapter was great. A student needs these basic math facts in order to achieve harder problems.
ReplyDeleteAt the end of the chapter, the author talks about what one can do to help students who have not mastered their math facts by 5th or 6th grade. One of the suggestions was to provide hope. I think that this in especially important in math because students can give up so quickly. It’s the teachers’ responsibility to encourage the students. It also says to provide new strategies for the student. This will help the student to not become overwhelmed. If something isn’t working, throw it out and try something else.
Response to Andrew Dempewolf:
ReplyDeleteLike you, I also think that it's important to provide hope to students when they are struggling to master their math facts. As teachers, we should provide that hope in any subject a student in struggling with. If teacher's don't encourage their students to not give up then who else will? If the student is not understanding the concepts in one way it's the teachers responsibility to introduce the concept in a way that student will understand.
Chapter 10 was about helping students master their basic math facts. This is such an important skill in math, but is often taught in an ineffective manner. The book mentions that many teachers go straight to memorizing facts after they introduce the concepts. However, it is important to include a strategies stage. Not only does it help chunk math facts together for easier memorization. It helps students learn how to do higher order thinking, which they will need to complete higher level mathematics. The last part of the chapter really caught my eye and I tried to soak up that information. It covered what to do with students who reach the 5th and 6th grade and have not yet memorized their math facts. This was important to me, because I plan on teaching middle school math. I know I will have students that get to me and still need work on basic math facts. Now I have a resource to help me with these students.
ReplyDeleteAndrew --
ReplyDeleteI completely agree with you, hope is probably one of the most important things to give to a student that is struggling with mathematics. Math can be very frustrating and easy to give up on. As teachers, we have to keep our students believing that they can do it, so that they keep trying to learn and don't shut down.
I learned some new ideas in chapter 9. I have always said “take away” to describe subtraction. I think it will be a hard habit to break. I like the example for finding the difference in figure 9.5 that shows how to use the Think-Addition strategy. I like it because I can look at a visual example that makes sense to me. I am not the person that math concepts come to easily, so I need that visual, especially in order to make sense of a way of doing things that is different from how I have been doing them the past 35 years. The reference I am making is using a number line with the arrow hops to show the numbers in the problem and one big hop for the total number.
ReplyDeleteIn chapter 10 the section on Story Problems was most interesting to me because it shows that there is more than one way to solve a problem, which I love, but was never taught as a child. My daughter learned her “doubles” last year in first grade and she uses them for every kind of math that she does! So one day she had the math problem 6 + 8. She told me 14 and then said, you want to know how I got that? I know that double 6 is 12 and 8 is two more than 6 so 2 more is 14. This way of adding completely foreign to me so if I seem overly enthusiastic, maybe I am. To me, the concepts we read about in chapter after chapter drive home the Learning through Problem Solving idea, which I love. I love it because it means that even someone like me can do math! This class gets me excited about teaching math, because I will be teaching children possibilities, not just one way or no way like I learned.
Andrew, I liked your observation about how many children have a difficult time subtracting large numbers. The children in the 6th grade class I intern in know more ways to solve math problems than I ever dreamed of. One day a couple weeks ago, she put a sample problem up and four students demonstrated four different ways that they solved for the same solution. It was fascinating to me!
ReplyDeleteChapter 10 provided many strategies for helping students master their basic facts. At the beginning of the chapter, three different approaches to fact mastery were mentioned. Those three approaches are memorizing facts, explicit strategy instruction, and guided invention. Like other have mentioned, I am most familiar with memorizing facts. In grade school we had math facts up on the wall. We were timed to see how fast we stated and answered the math fact. They were always in the same order so it was easy for people to memorize the math fact instead of actually learning and understanding how to get the answer. The next topic covered in this chapter was guiding strategy development and story problems and reasoning strategies were mentioned. Next, reasoning strategies for addiction, subtraction, multiplication, and division facts were covered. I found the sections on what to do/not do when teaching basic facts very helpful. This chapter was beneficial because knowing basics facts is very important. If students don't know their basic facts then it will be hard for them to learn other math concepts.
ReplyDeleteChapter 10 focused on helping students master the basic facts so that they won’t have to result to using ineffective ways to solve problems. The addition and subtraction facts are extremely important and will continue to be used as the students increase their knowledge of mathematics in order to solve higher level problems. If students don’t have a good grasp of these basic facts, they are going to struggle as they get into more advanced mathematics classes. One of the approaches presented to help students with fact mastery was memorizing and this approach was used frequently in elementary mathematics classes. I also had a bit of trouble learning the multiplication/division facts and my parents used flash cards to help me. I also had trouble with story problems because I had to set up the equations myself and they weren’t already written out. The reasoning strategies presented in the text can be very helpful in assisting students write story problem equations and allow them to become more aware of the relationships between numbers.
ReplyDeleteSome of the strategies such as Using 5 as an Anchor and Up Over 10 seemed a bit difficult at first and I had to really think about them. I don’t remember being taught to do addition/subtraction this way so using these methods seems like it would take more time than simply memorizing the facts. This made me realize the importance of this chapter and the strategies & methods that were discussed. By simply memorizing facts, students don’t really have to think about why the answer to 6+4=10; that will always be the answer. By using these strategies, they will be thinking deeper about why that is the answer. I thought that using a clock to help students learn 5 facts was really cool. In regards to the strategies used to help students with multiplication facts, I did learn how to use Nifty Nines and I thought that it was awesome. The What To Do & What Not To Do sections at the end of this chapter were very helpful and overall I learned a lot of valuable methods and strategies to better teach students how to master the basic facts.
In response to Monse R.:
ReplyDeleteI liked reading your post and I was also in a classroom where we were timed and were expected to answer the math questions as quickly as possible . I liked that the fact that you mentioned how the problems were always presented in the same order. This is the perfect example of how memorizing and drill is not very beneficial. The students are not even required to think about the problems in order to answer them. Once the students have learned the basic facts, they should be given more challenging and meaningful activities to help them practice these skills.
In response to Sarah R.
ReplyDeleteI also think mastering the basic facts in so important. I work in the middle school and I can tell right away if a student knows their multiplication facts or not. They all seem to struggle more, and need more assistance. I have gotten where I just tell them to grab their agenda right away and use the multiplication chart rather than have them guess. My thought is if they keep looking at this chart slowly but surely they will start to pick up on them and it will make their lives a lot easier in high school. I get so upset when I hear that the elementary school teachers are not focusing on them. There are so many activities I have seen done to help students memorize them, when I was in school we had a banana split party at the end of the semester. Each week we focused on one fact family 1x2, 1x5, ect.. And we had a times test at the end of the week and you earned a part of the banana split. If you didn’t pass it you didn’t get that ingredient. I remember students that only had the banana and one scoop of ice cream. The whole goal was to make a complete banana split with a cherry on top…we loved it! I also learned another trick that is very similar to the nines trick but it has to do with 6,7, and 8. You use your fingers to help yourself solve the problems. It is to complicated to try to explain but I bet if you are really interested you could goggle it or YouTube it and find someone demonstrating it.
Chapter 9 and 10 was over helping students understand the operations addition, subtraction, multiplication, and division and to help them master the basic facts of these. Chapter 9 was over helping students have an understanding of the number sense and what it really means to add or subtract rather than just the process. Students first must know how to set the problem up, what symbols to use and how to solve for the answer. Then they must understand the why questions, in order for students to have a good foundation for number sense they have to understand the terminology, subtract means to take a way, the number will be getting smaller, you will have less than what you started with. These are just some points that a student must understand to have a full understanding of the operations. Chapter 10 was over teaching students their basic math facts. I am a strong believe that students need to learn their math facts (addition, multiplication) if they are ever going to be successful and enjoy math. I work in the middle school and I can point out the students that I know don’t know their multiplication facts, their assignments take longer, they get frustrated quicker, there are more mistakes in their answers and not over the material that was being covered. There are so many different ways to teach students their facts I think working on them for a few minutes of the day would help these students dramatically! When I was in school they taught us how to use our hands to find the multiplication facts of 6-9, and I still use them when my brain is working too slowly. When I was in school I had to use my hands a lot but with practice and a lot of time I ended up memorizing them from doing problems with them in it. This is a very easy way to help students learn them rather than just giving them the answer or telling them to look at the multiplication chart. Overall both of these chapters made me take a step back, I have a very high understanding of number sense, I have always had, I could just see why we would do it that way. But I liked how the chapter broke it down so that when I have a student or students that don’t have this sense I can help them develop it.
ReplyDeleteTracyPer
ReplyDeleteChapter 9 learning about the four basic operations, and the newer terms used to help in all to understand the concepts of basic addition, subtraction, multiplication, and division. I had not realized how important the terminology had such a effect on student learning until now.
Chapter 10 really learning and memorizing like the multiplication facts has sure been useful for many years. Especially with the concept of mental math, like being able to count back change is a dieing art. We have become so accustomed to the computer cash register telling us what to do we no longer think for ourselves.
Chapter 10:
ReplyDeleteChapter ten of the text book Elementary and middle school mathematics: Teaching developmentally (7 th ed.) by Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. discussed how teachers assist students in doing math problems that require little thought process in basic math problems (Van de Walle et al., 2010, p. 167). This chapter was packed with new information that made me think of matters differently and information that made me think of my own experience.
A piece of information that I learned from this text was “What not to do when teaching basic facts” (Van de Walle et al., 2010, p. 183). Van de Walle et al. states five different things not to do when teaching basic math, which are “Don’t use lenghthy timed tests”, “don’t use public comparisons of mastery”, “don’t proceed through facts in order from 0 to 9”, “don’t move to memorization too soon” and “don’t use facts as a barrier to good mathematics” (Van de Walle et al., 2010, p. 183-184). This is very good advice for any teacher especially new educators.
A piece of information that made me look at my own experiences was on page 179 of the textbook that mentions a simple way to multiply 9s. Students can do this by putting down one finger on their to show the divide of tens and ones. This is a strategy that I use all the time. I teach students this strategy as much as I can because it will help some students. This strategy can be difficult for some students but it is worth showing the students in hopes it will befit them. If they do this enough times they can gradually start memorizing the answer and not rely on using their fingers.
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.
Tracy Per to Lane A
ReplyDeleteI so agree with your idea of getting the family to help. I know without my families support I could have never gotten this far in my educational experiences. I think there are a lot of parents just waiting to be encouraged to get involved. I am a parent just like that as well.
I really enjoyed reading chapter 10. This chapter gives a lot of great examples and activities for teaching basic facts. In my internship classroom, when I am helping the third graders I sometimes have problems explaining to them the strategies that are second nature to me as an adult. I could definitely use some of the strategies like, near doubles, taking from the ten, and up/down over ten to explain things to the students. I am also realizing just how valuable the ten frames can be.
ReplyDeleteJennifer Pen reply to Rebecca B.
ReplyDeleteGreat post. I agree with you. I had to use my fingers a lot when I was counting and figuring out math problems and I think that is OK because it became easier and easier the more I did it because I would start memorizing them instead. Some teachers that I have met do not like the students to use their fingers but I thin that they students should do what ever it takes to get the correct answer and assist them in problem solving.
Chapter 9:
ReplyDeleteChapter nine of the text book Elementary and middle school mathematics: Teaching developmentally (7 th ed.) by Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. discussed how to teach students ways to add, subtract, divide and multiply.
A piece of information that I learned from this text was there are "separated addition and subtraction problems into categories based on the kinds of relationships involved. These include join problems, separate problems, part-part-whole problems, and computer problems" (Van de Walle et al., 2010, p. 146). I was seen these different ways before in the classroom did not know that is what they were called.
A piece of information that made me look at my own experiences was figure 9.6 on page 153. I remember learning this strategy when I was younger and find it very useful. Students combine numbers together (equal 10) that are easier to add and then add the rest. This strategy is easier for students because the can relate better to adding to 10.
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.
When subtracting, using take away is not really the best word. Minus or subtract are the best words to use for this. Children learn at different rates and at different ages. Educators need to remember this when teaching math facts. They will not all know the basics as soon as others may know them. Knowing the basic facts is very important when learning or teaching mathematics. Teachers have to help students acquire their knowledge over time, not just assume they know that facts. These chapters touched base on four problem structures. It was very interesting to me to read about these. The examples and walkthrough for the different type of problem structures was interesting to read about.
ReplyDeleteIn response to Jennifer P---
ReplyDeleteI think the section of the separated addition and subtraction problems was very interesting. I am using the strategy on page 153 you talked about in my internship class. They want to just count all the time. The teacher is really trying to show them tricks for remembering certain addition and subtraction problems. The one I did not know about was the 9 trick. When adding, you just put a one and one less than the number. So for example: 9+6= 1 and one less than 6 which is 5 so the answer is 15. It was pretty neat to learn this. I wished I would have learned this when I was in school.
I liked the different concepts and strategies chapter 10 talked about. I liked how they showed different ways students might work out a problem. For example, the text had a problem that was 6+8. Some students may not know it off the top of their heads, so they might add 2 onto 8 to make 10, then they might subtract the 2 they added onto 8 from 2. Now it turned into a 10+4 problem. I can relate tot hat because I can remember doing that when I was younger. I found this chapter had many cool strategies that children can use if they're having trouble with their problems.
ReplyDeleteElizabeth,
ReplyDeleteI sometimes have a similar problem like yours. I also struggle explaining problems to students at times. Like you said, it's like second nature to us and we don't even have to think about how we got the answer. We just know it. I feel like this chapter gave me some helpful ideas from this chapter.
I was a little surprised that they separated the four problems into four different structures, Where, we would have just called them word problems or story problems. I liked how the books suggested that story problems should be started as early as kindergarten and pre K. This is something that the children really struggle with so all the practice is very important. I think that the sooner children learn about symbols the better. The one thing I do see is sometimes when you say addition to a child they know it as plus and don’t recognize the difference in the name. The same thing happens with subtraction and take away, and multiplication and times. So I think it’s important to get them use to all the different ways we call the symbol. Remainders really drive me crazy when I’m walking into a classroom because it is hard to tell what they are doing with the remainder; leaving it as r. or making it round up or continue till we get through all the numbers by adding 0’s.
ReplyDeleteMastering math facts has 3 approaches; memorizing, explicit strategy instruction (teacher shows them a strategy and then they practice), guided invention (using number combinations and doing which ones make sense to them). This chapter has tones of activities I can use. I love that. I was trying to think of some good ways to store them for easy access. (on index cards ¬in one of those recipe boxes) or something like that. When I was looking at effective drill, I noticed the, what to do and what not to do when teaching basic facts. These are great and very important. I think making drill enjoyable and involve the family are most important. This is great car ride work. I can do multiplication facts with my children while I’m cooking dinner or doing the dishes. I can do it at the park or waiting in line at the grocery store. Children have to be reminded about this so they can remind the parents. There is not enough time in the day to teach everything so it’s great if they can learn something extra at home. Fact remediation, there are those kids that just don’t get facts no matter how much time they spend on them, they need something else to help them.
Rebecca B. I really agree with you children need to learn to love math if they will be really successful with it. They need to learn their facts. I'm glad the book gives us so many different ways to teach the same thing. These children really don't learn the same way. This book does get you so many opportunities to guide the children.
ReplyDeleteChapter 10 covers the development nature of basic fact mastery. I was excited to read this chapter just by reading the chapter title. My thoughts before I read it were that it would be about memorizing mathematic facts and the different ways to teach that. One of the strategies the text lists is drill. While this strategy works for me, it does not work for everyone. I have seen in my internship class that some students don’t work well under pressure. I have worked with a student one on one and they get very nervous and I have to tell them to stay calm and just think. I definitely think they would benefit from a different strategy.
ReplyDeleteIn Reply to:
Matthew B.
I also thought this chapter had a lot of interesting strategies to use when working with students. I think that there were a lot of different ways to approach teaching facts when you have students that learn differently.
Chapter nine was very informative for me. I think a key to understanding mathematics is being able to connect different meanings and relationships to it. A group of numbers doesn’t make much sense on its own, it has to have some meaning or interpretation supporting it. I feel like I can do so much more in the classroom after reading this chapter. When I sub or get to help teach in my observation classroom I often struggle with math. I can present the facts and problem solving strategies fine but I have trouble when students need it more broken down. Information can’t just be presented to a student. A key to being an effective educator is presenting information in an understandable way. It’s being able to break down those facts and rules so all learners can grasp them. I didn’t realize that when I was showing them a simple addition word problem that I was using the join structure. These broken down structure examples really help me understand what I’m trying to say even better so I know they’ll help the students understand me better as well. I think when he students can see a concrete structure of something the numbers aren’t so hard to manipulate for the correct outcome.
ReplyDeleteChapter ten was also very informative for me because it discussed what my mentor teacher stresses everyday. She works very hard to instill the basic math facts in her students. She uses tons of repetition when it comes to the basic facts. I’ve realized through this chapter and my observation that even the tiniest basic math fact plays a huge role in a student’s mathematical success. If they don’t understand even the smallest fact it could hinder their future progress. You’ve got to get the basics before progress can be made. I really enjoyed all of the activities listed throughout this chapter as well. I can use these even when working with a group of students during down time in my observations.
Matthew B,
ReplyDeleteI also enjoyed learning about the concepts and strategies in chapter 10. Through my observation class I’ve seen students work out problems in so many different ways. It’s amazing how their minds put things together. One thing I noticed in the classroom is something you mentioned. The students were doing a math magician activity and they were just buzzing through it crazy fast. I asked how they did them in their heads so fast and many of them said they would round to ten and then subtract what they needed. I didn’t do that when I was younger but it makes perfect sense and it seems to work for so many. You’re right, there were lots of useful strategies for us to teach our future students in this chapter.
In Chapter 9 where it talks about addition being referred to as "put together" and subtraction as "take away" and using these definitions puts limitations on students and makes it difficult for them to do problems that do not use those exact words. Those words become their key and trigger words and if they are not there than they struggle. I would say I agree with this totally, I have always been good at math but horrible at story problems. I could not figure out what I was supposed to do it less it asked specific questions like all together or what's left over. I like that we are learning strategies to help children work through word problems because I think word problems scare a lot of students. I have also always thought of the equal sign as equal to, but I can see how it would be better to say "the same as" or to think of it as a balance. I think many of us have the traditional mind set of math and this book does a great job of getting us out of the traditional mind set and getting us to think of things differently so we can teach it more effectively.
ReplyDeleteChapter 10 talks about how a teacher can use strategies to understand how to get to the answer. On page 169 it talks about guided strategy and I recognized a strategy that I used this week in my intern class to help a student. The student was trying to figure out 7 x 4, so I asked her what 7 x 3 is, she didn't know that so I asked her 7 x 2 is & then we built on that. She ended up getting the correct answer and it was rewarding to see her understand it. Another strategy that I used was in division, the worksheet looked like this 30/5= ? and one student asked me if their answer was right and I said "what # do you multiply by 5 to get 30" so just because you are trying to find the answer to a division problem doesn't mean you need to only use division, sometimes you can use multiplication to solve a division problem, you can use addition to solve a multiplication problem. I also love the Nifty Nines trick, and I think that will help some students struggling in my intern class. I want to use these strategies and help students learn.
Tammi Whi,
ReplyDeleteI have a son who gets extremely nervous with math drills. He is a smart kid and gets A's all the time in math but give him a time test and he just falls apart. Last year in 2nd grade I think he only got to 5's in addition but he could add or subtract any other time no problem. I practiced one time with him at home and as soon as the timer went on he was shaking like a leaf and almost in tears from the very beginning. When he was done, he had only half the page left and I asked what was wrong and he just said he couldn't do it. He said at school it's frustrating because every slaps their pencils done real loud, and turns the papers over and he just can't think. I went and spoke to his teacher and she knew about the situation but she did not offer any alternative. I think there should be an alternative and for students like that all they are getting is how frustrated they are when doing math. My son says I hate math all the time and he is only in the 3rd grade. I think we need to re think the math time test drills.
There were a few things from Chapter 9 that I didn't quite understand. It says that you shouldn't use the word "take away". When I was in school, that is always what my teacher told us to do when subtracting. I just think it can be very confusing when you are taught one way and then later you result to another way. In my internship class I was somewhat confused at first too because the teacher would ask the class on some problems if they were to add up or subtract down. I understood it because I just knew in my head what to do, but to be honest I probably would have tried a different approach to teach them. Many seemed to be confused on it. Back to subtraction! I do agree that there are better choice of words than "take away."
ReplyDeleteIn response to Shannon H,
ReplyDeleteI agree with you that basic math facts must be repetitive in the classroom. We don't realize that one tiny mistake makes a huge difference. My mentor teacher sounds much like yours. Every day she goes over the basics before beginning a lesson. We all know that you have to have the basics down before advancing up. Good comment!
Chapters 9 and 10 were very informative. I liked how they had teaching strategies for teaching the basics. Like most everyone else, the fact that using the words “take-away” isn’t considered appropriate anymore is strange to me. Take away, to me, simply turns it into more of a words problem and allows the students to immediately relate the term to taking away something. I definitely think it is important to use words such as minus and subtract are important because it is important to know the technical terms. I am confused about this because Dr. Stramel said using words like “8 and 8 is 16” rather than “8 plus 8 equals 16”. Using the words “and” and “is” seems very similar in nature to using the words, “take away”. Other than that, I thought the reading was very informative and helpful for the future.
ReplyDeleteThe four problem structures were really confusing when I first read this section. However, if you really look at the various problems, it really makes sense.
ReplyDeleteI think sometimes we try and make things easier to understand but what happens is it is made more difficult.
The text labels the different 'parts' of the problems "Initial, Change, and Result" These are for the first two problem structures "Join" and "Separate". These two stuctures are simply add and subtract depending on how it is worded and what numbers are provided!
The key word in ALL of these problems is MORE. This mean we add two things to find the result. If we have the result, we will need to subtract. "5 plus WHAT will equal 10?
To Jen Watson
ReplyDeleteThe strategy you mentioned in your post (how much is 7X3 and then 7X2) is, interestingly, exactly what I did today!
Another strategy I use with a student in special ed is "How much is 3 groups of 7" That seems to work well with him. I then slip in the word "times" so he will eventually associate 'groups of' with 'times'.
It is really cool when we find a way the kids understand.