Chapter 14 discusses algebraic thinking: generalizations, patterns, and functions. I found it very interesting to find out that children start learning algebraic thinking in pre-K! And being an intern in Kindergarten I now see that! I guess I didn't realize what they were actually doing when they were making patterns on the calendar and what not. I knew it was patterning and math but not actual algebraic thinking! That's amazing! I also recently found out that the reason they changed the name of the shape oval to ellipse in order to help get students ready for geometry! I think that it is truly important for students to be introduced to all of this at a young age because it is when they most easily soak up the knowledge and this way they won't feel so overwhelmed later.
Chapter 14 is about algebraic thinking which occurs as soon as students learn about numbers in Kindergarten. I like figure 14.1 with the monkeys. This is great way to begin to paint a picture of the words in the beginning algebra problems. The generalization in the hundreds chart is another good way of connecting the dots in a students minds. Before interning in a first grade classroom I had not heard a teacher say that the equal sign means "the same as," and "is." I like to hear teachers using the same terminology throughout the different grade levels. This chapter also outlines many different ways of connecting the dots for students by integrating within the math curriculum. Such as using money to count or using pictures like the monkeys in the trees. I think it is important as educators we know the common core standards inside and out. Being able to integrate within the curriculum is a critical component of fully understanding each math concept we teach our students. Estimating how much an item costs and knowing number patterns are interrelated but we need to teach how they are interrelated and why it is important in real life. All of this comes back to having a good number sense.
Comment for Kristle C: Amazing that some students don't know what an oval is... I find it weird. I agree, the algebra concepts are starting younger and younger but in the long run I think our students will gain a much deeper understanding of this concept. Learning that happens at an early age and continues on throughout the grade levels gives those 6th grade teachers a much easier job to do when they are expected to master fraction skills. Growing the students number sense over time enables the student to have the answers behind why they are learning all of the math concepts. Relating math to the real world gives the students a reason to learn.
Jena-I find it weird also. I overheard a parent-teacher conference in my intern classroom where the teacher was telling the parent that the student called it an oval and that was incorrect and the parent was like well that is an oval and the teacher informed her of the change and the parent said oh well that is stupid! I can understand where the parent is coming from but they just don't see how the change in something like calling an oval an elipse will help the student later on! I think that is hard for parents to understand!
I think that before teaching anything, a foundation of vocabulary needs to be set. It is important that students understand what shapes are called and what they look like as well as what symbols mean. I am amazed at what students do not remember or do not have knowledge of what a pyramid or cone looks like. As Dr. Stramel has said, the equal sign means "same as". This sign has been overlooked and can confuse students unless taught the real meaning.
Jena, I agree that relating math to the real world gives the students a reason to learn. Capitalizing on their experiences in all areas will help them in their learning process.
I never really realized how dynamic algebraic thinking / concepts are. When I think about algebra I always think of my high school math classes and equations like y = mx + b. It’s hard to think of algebra at an elementary level when you’ve been beyond that level of thinking for years.
As usual there are so many great examples that help me realize the simplest problems are or can be considered as algebra. I think my favorite examples are the open sentences. Those can be used in forms as simple as 5 + ___ = 8, or as complex as 20 + 48 = ___ x 20. These are thing you could use for all grades from Kindergarten to 6th grade by modifying or enriching depending upon the grade level.
I also learned a lot about the five types of representations. These representations include patterns (or the context), tables, verbal descriptions, symbolic equations, and graphs. All of these representations are useful in their own ways. I would imagine though that using multiple representations at one time would help the students tremendously when it comes to grasping the information or concept and retaining it longer.
You bring up a valid point about the equal sign. I’m currently in a 3rd grade classroom, and I can’t recall any time my mentor teacher has used the words “the same as” or “is.” I hear equal a lot though. That makes me wonder what the other teachers in the 3rd grade are using and what teachers in the other grades are using terminology wise.
I also agree with you on needing to know the standards and how we can use those to integrate within and across the content areas / curriculum. There’s a lot we future educators need to step up to, but I think those that are dedicated in this field will have no problems doing so!
When reading through Chapter 14 over Algebraic Thinking: generalizations, patterns and functions I was able to learn some great information that is going to be helpful to me in my future classroom. I liked how the book pointed out that algebra is an established content strand in almost of the grades from K to 12th grade. Up unto this point I have never thought of algebra being presented that much in the lower elementary school setting but after the book gave me a couple of examples I was able to clearly see how algebra is connected throughout all of the grades. I also liked how the book talked about the five different forms of algebraic thinking which was very helpful for me to know. I was able to relate a lot in the area of algebraic thinking when the book talked about how children in kindergarten start to recognize and duplicate simple sequential patterns. This is happening right now in my kindergarten classroom where I am able to show the students a pattern of shapes such as diamond, triangle, diamond, triangle and the students are asked what shape will come next. I liked how the book talked about teachers need to focus more on the meaningful use of symbols. The students need to have a strong sense of what the symbols in the problems mean in order to correctly figure out the solution to the problem. I thought that all of the activities presented throughout chapter 14 were helpful to me as a future teacher.
In response to Kristle C., I was also surprised when reading this chapter that algebraic thinking starts in kindergarten. I also was able to see that in my kindergarten classroom when I am doing the calendar activities each and every day that when the students are asked what shape comes next in the pattern that that is algebra and requires algebraic thinking. Wow! I was wondering the same question about why the students need to know that an oval is also an ellipse so thanks for answering my question and it now makes sense that they are getting the students ready for geometry. Lastly I agree with you that it is important that the students learn the information early that way the students are not overwhelmed down the road.
One of the things that Chapter 14 talked about was conceptualizing the equal sign as a balance. I had never really thought about it till this semester in my math internship. When we discussed solving a problem such as 9+5=__+10 a lot of the students in my internship would put 14 as the answer for the blank. But when I explained to them that we should think of this problem as a balancing scale then it started to make sense to them. The students realized that the goal of the problem wasn't to find the answer just to 9+5 but to make both sides equal the same amount so that if it were on a scale it was balanced.
Brooke M I had never really thought about algebra being presented in early elementary grades either. I always figured it started in jr. high just because that was when I started having classes with the title of Algebra. But looking back it is quite clear to me that I was doing algebra way before jr. high. I think part of the reason that I never really thought about it being a part of the curriculum in early elementary was because I never really had a clear understanding of what algebra was before taking this class. Now that we have discussed it so many times in class it just seems so obvious to me.
Chapter 14 was a big section of reading and had a lot about algebra. Personally I feel very confident with algebra so it was interesting as I read and tried to relate to students who might not understand. Right off the bat I was really surprised how they had categorized concepts that were considered algebra. The fact that we start Pre-K with algebraic thinking is unreal. I was totally unaware of how they classified specific concepts and how much more was considered to be algebraic. Again I really don’t mind doing algebra. I like doing long problems no really problem solving but just long equations that solve for multiple variables. When they began discussing the equal sign I was taken back. I had no idea that a student might be troubled with associating the equal sign with equivalence of both sides not just one. Since we do start students off so young with this concept I do feel the chapter is right as they suggest we begin using variables at a sooner age instead of using boxes. No matter what you are talking about you should be able to label what you solving for like distance or time- D or T in the equation.
In response to Joel Stucky,
I think you make a great point by reminded math teachers they still need to focus on vocabulary. A student could get lost big time if they don’t understand the terminology used to explain the concept. I feel frustrated and overwhelmed when doing something and not fully comprehending the dialog. Math has many vocabulary terms to cover!
Chapter 14 is about algebraic thinking including generalizations, patterns, and functions. One of the topics is generalization through exploring a pattern. This is something that I have seen my mentor teacher work with her students on in pattern boxes. I don't remember doing this in school, but it is a great way to get students to look at the problem and find the pattern. The book discusses how students need to learn to search for patterns and learn how to describe, translate, and extend them. One thing I never really knew, but learned in this chapter is that prediction is an important part of algebraic thinking. The book discusses how predicting real-world contexts is appropriate for upper elementary. For example it discusses finding patterns and predicting when the summer Olympics and when the winter Olympics will be held based on when they have been held in the past. This is a fun way to help students see they will use this in their life.
Unlike you, I am not very confident in my algebra skills, but wish I was. I was also extremely surprised through that students actually do algebra as young as pre-k and kindergarten. There really is so many more topics covered under algebra than I ever knew.
Chapter 14: Chapter 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. discusses using algebra in the classroom which is used more and earlier than people think. According to Van de Walle et al. textbook “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions” ” (Van de Walle et al., 2010, p. 254). Algebra is a concept that can difficult for students but I think once they understand it it can be for students. A piece of information that I learned from this text was all the different activities presented in the textbook. On activity that I liked learning about was 14.8 titled “Balls, Balls, Balls”. When I first read this activity, honestly I had no idea how to go about it but then I discovered how the book did this activity by having the reader really look at the material and really evaluate the information presented. Chapter fourteen really made me think about my time in algebra class. Algebra was not easy for me at first. I had a teacher who did not really put a lot of effort into her teaching which really had an impact on me. Then when I took Algebra over again I had a teacher that really took time and consideration into what we did. Every assignment that we did in her class we could fix and turn in for extra credit which is very important in Algebra. Getting that extra credit made me what to go back and recheck my work to know where I went wrong. I think this is what all teachers should do. I was then able to understand the process of algebra and now I actually like doing it for fun. 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.
Jennifer Pen reply to Elizabeth, I agree with you that prediction (as stated in this chapter of the textbook)is very important. With algebra students can think about what makes sense and what would not make sense. A lot of times students have to even work backwards to figure out problems. There are so many different ways to do math problems that students just have to take a step back and think "does this make sense?"
Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions The meaning of the equal sign on page 258 really does amaze me how many students really miss those problems. I know it really is a struggle for students, even when we break it down. I see the same problem when they put the answer and then the equal sign before the problem, 6=4+2. It’s very hard to determine if something is equal when it does not look alike. That’s probably why they have such a hard time with the scales, cones, and balls and liquids in different types of containers. The mathematics curriculum that we are using at my school is introducing simple algebra in the 1st grade and then it just increases from there. This section is packed with activities which are very helpful in teaching such a hard concept. I think it is important that they understand not just that it is equal but that it is equivalent. I think using real world examples help so much. I watched a teacher used different containers and she filled a wide container with water and then said, Can we fit all this water into this tall skinny container? Many children said no and then she did it. The shape changed but the capacity stayed the same. I think real world examples help.
Lane A. I agree with you. Scales are very inexpensive and now they have nice plastic ones. Bring out the scale and use some kind of manipulative. Take bears for instance, 5+5=_+2. Put 5 bears on the left side and let them see you add 5 more to that side. Now put 2 in on the right side. Then tell the children lets add bears and count until they are back at the same point. Add bears 1,2,3,4,5,6,7,8 now the scale is back at it's starting point. How many did we add, 8 yes, 8 is the number that we needed to put into this spot. This would help them visualize. I thinks some times we forget it's easier when we can see and touch things. Algebra was very hard for me so I should expect that it might be hard for the children. Just try as many examples as possible so that they will get it.
This week we read chapter 14, Algebraic Thinking; Generalizations, Patterns, and Functions. It’s important that we think beyond the algebraic equations that took a whole piece of paper to complete that we did in high school. These problems rarely had anything to do with real world situations and in most cases were completed by following the teacher’s specific direction. Algebraic thinking is much more than that, it starts in pre-K with shapes and is added on to each year. Algebra is used to help students see patterns and generalization in the real world; once they have an understanding of certain patterns they become more successful in math and have an easier time seeing connections between numbers. I love doing Sudoku puzzles, and in many ways this is just many different number patters and problem solving. Another important concept in this chapter is the different properties (associative, distributive…). These help students see that whatever is on one side of the equations is equal to the same as the other side. The next part students have to understand is the meaning of variables, and unknown values. One they understand what the variables are and how to represent them they then can create their own way of solving the equations. In this chapter they also talk about thinking across the curriculum in algebraic, this is why it is important to teach students how to algebraically think so that they can look for patterns and generalization in all areas. This will also help them think beyond the curriculum in both math and the other area creating better reasoning skills.
In response to Tammy M. You mentioned that so many students have trouble with equations that are equal but don’t look the same. For my formal lesson I was teaching about algebraic equations and how to solve them, but one of the main focuses was on that both sides of the equations had to be equal. For my lesson plan we used balance beams, I showed the students how if they are not equal then the balance beam falls towards that way. These manipulative may help your kinetics learners understand the concept better.
Chapter 14 talks about algebraic thinking: generalizations, patterns, and functions. I thought that it was very interesting that children start leaning algebra at the beginning of pre-k. Patterns are one of the very first things that we learn as we are little. This is happening in my kindergarten right now during my internship. I also liked that this chapter gave lots of different activities for me to use in my future classroom.
To Katie C: I myself don't feel as confident in my algebra skills like you. But I also was surprised that students actually do algebra as young as pre-k . There is so much to learn about math!
Chapter 14 was over Algebraic Thinking and I thought that it was very informative. I thought it was interesting that the algebra that we went through is not the algebra that is being taught in todays’ schools. I have always thought that mathematics was the one consistent subject that is taught in schools. For example, in my internship, we just began to introduce the concept of multiplying larger numbers. The students were taught a different method than I was. (I think the teacher said Singapore Math) It was difficult for me to help the students because I would begin to explain it in the way that I was taught. Chapter 14 gave me some great advise on how to teach these new methods.
In response to Rebecca- I agree with you when you say that we should think about algebraic equations differently than when we were in school. The focus of algebra now is to connect it to the real world, and to develop the students’ critical thinking ability. I think algebra is important because it teaches the students critical thinking skills, and how to generalize from experiences that they have had.
Chapter 14- Algebraic Thinking: Generalizations, patters, and functions. When I think Algebra, I am immediately a little scared. Why did we become so scared of something that was intended to be so helpful to us? Plus, we have been learning and being introduced to it since we started school. The text talks about using our basic math fact families – that’s algebra! Early on we are learning patterns and symbols. If we failed to get a good grasp on this concept we can be setting ourselves up for failure down the road. Then the text talked about the equal sign and variables and how misunderstood these symbols really are in our society. This really hit home for me. I am mentoring in a 4th grade math class and the students do struggle with their math facts and after ready this chapter I am almost certain, they think the “=” mean the answer is. The text touches on the importance of conceptual knowledge of patterns. That students should be able to see the XXYXXY is the same pattern as red, red, blue, red, red, blue – wow – I honestly have never really understood the true importance of patterns until now. I am also able to link what is going on in my mentor class with the importance of learning pattern and algebra. This is all part of a major foundation that must be built strong. No pressure on us – right?
@ Joel Stucky – You are right on the money. That vocabulary is so important and knowing that shapes have names and symbols have meaning is something that is critical for us as educators to be getting across to our students. That “=” meaning is a really important one. Dr. Stramel did mention she like to use the term “same as”. I agree that using the best terminology or vocabulary is so important. If we are not doing things in the proper way or manner, our students will never learn it correctly. Great points made –thanks.
Lindsay H @ Brooke M-I agree with you. I also did not really think of algebra being used in early elementary. But now looking back from previous experiences I am seeing it more and more!
I like many other peers found it so interesting that early education teachers are already implementing algebraic thinking into their students. I loved the part about patterns. I think patterns are so important to know when it comes to math. In my internship class we have been working on attributes. It is kind of neat because they teacher says the attributes then the students work on making a pattern based on the attributes that she says. It really gets students thinking and some students really struggle with the process. I really struggled in college algebra so I am a bit worried about teaching it. It was interesting to note about algebraic thinking across the curriculum. I think it so important to integrate and relate different subjects across the curriculum.
I think most people think of high school when the word Algebra is used. My boys are in 4th grade and came home so proud because their teacher told them they were learning Algebra. I believe math needs to be taken seriously from the moment that students start school. In the lower grades we may not use the word Algebra, but everything a child learns in math is important because math concepts continue to build on each other. As teachers we must be careful that students are not falling behind in the lower grades. Learning those math facts is important because in high school it is tough to do math without the basic skills.
To Katie C: In my internship class I have noticed many students who do not understand what certain symbols mean. I think as adults we take it for granted. If I see a student struggling I try to get them to read the problem to me. This way I can see if they even what it is asking them.
To Andrew: I have noticed the same thing. I am amazed that the ways math has been taught. Sometimes I wonder who comes up with these different ways. I am very interested to see the new ways that a concept can be taught. I think each teacher and each school probably has their own preference on how to teach a concept. I don't remember many of my teacher ever using manipulatives, but I am very excited to utilize hands on activities in my classroom.
Lacey Keller In my personal education, mathematics was my strong subject, most likely due to a great set of teachers. However, I must also conclude that I also enjoy algebraic functions and problems. I am a sensible gal who has to make solutions "just add up." Haha! I believe algebra is just real life mathematics. If we can get our students to buy this philosophy, then we will have great success as mathematics instructors.
This chapter discussed in detail functions in algegra. Yes, we see patterns way before we start kindergarten. We introduce our young babies to repetition in nursery rhymes and continue to use algebra long into our adult lives. When I was in junior high, my math teacher was also our computer teacher. Fortunately, the teacher integrated both subjects accordingly. I gained so much algebraic knowledge just by simply using a spreadsheet program, such as Microsoft Excel. Our teacher allowed us to just explore and find the patterns of certain functions. We then graphed several expressions. It was so much fun and I learned so much!
Lacey Keller Shawna, You are right to say that most people get a little scared just by mentioning the word "algebra." Really, it's just patterns! I think of all the things that use algebra... I mean patterns: music, art, money, landscaping, farming, etc. The concept of algebra is found in many professionals and should be thought as helping us find the answer faster!
I have to say this was the most beneficial chapter for me in relation to my internship for this class. I taught unknown variable equations to 5th graders for internship formal observation. I had went through and read this chapter before I created my lessons and I have to say it helped me greatly. When someone says algebra often times students cringe at the thought. In fact I was one of those students until I began actually learning how to do the equations. I actually like to do equations and find it very fulfilling when I am able to complete problems! It is amazing that at the ages of 3, 4, 5 students are already being taught algebraic thinking! In the Kindergarten classroom patterning is the biggest concept taught throughout the year. It is so interesting to know that the entire concept behind Math is patterning. One thing that I am not good at is recognizing number patterns in relation to recursive patterns and formulas. When I am in the 3rd grade classroom I often find myself getting confused trying to show the students how to complete the patterns they are learning! I am not a fan of graphing and linear functions. When I think of either of these items I immediately think of Economics class and that was not my favorite class! It is very interesting all of the information that can be calculated and observed through the use of graphs and linear functions. I enjoyed the section on teaching considerations. When I taught my formal lesson I stressed the importance of knowing the meanings of specific words in relation to the lesson I was teaching. It is very important that the students know the meaning and reasons behind what they are learning. I liked the explanation of the multiple representations of functions. The way the book presented the information is great knowledge for a teacher to know when teaching his/her students. This chapter was very informative not only for giving me information to use in my formal lesson but very useful in helping me teach algebraic thinking to my students I now work with everyday and in the future.
I liked how the book discussed how students comprehend the meaning of the equal sign in different ways. I hadn’t really thought about students not completely understanding the meaning of the equal sign in relation to what is on one side of the equal sign has to equal what is on the other side of the sign no matter how the numbers are presented. This was one concept that I had to stress the importance of when teaching my formal lesson on variable equations because the whole point of teaching my lesson was for students to find the value of the unknown variable. Without this information there is no way that the students could complete the problems!
Certainly the idea that they are teaching algebraic thinking to kindergarteners is an interesting one. I suspect they taught it to us as well, they just did not use the word algebra or any form of the word.
As students and as adults many of us allow the word algebra to scare us. As teachers we need to look at why this is and make sure that is does not happen to our students.
Chapter 14 discussed Algebraic Thinking. I enjoyed this chapter because it allowed me to look at different theories and techniques used in algebra differently. I particularly liked the section on Meaningful use of Symbols. I didn’t realize how important the use of symbols in math really is. The section discussed how some students just don’t have a strong understanding of what the symbols are and how to use them. I definitely agree with this because when I was growing up it all seemed to blend together and I couldn’t tell the difference between the symbols or when to use what. I think that teachers should definitely spend time focusing on explaining and making sure the students have a clear understanding of the meaning of each symbol. In Response to: Angela R. I observe in a Kindergarten class and they are just beginning to work with patterns. Some of the students are getting it right away and some are taking a little longer. I was surprised to see that they do more than just AB patterns at such a young age.
Chapter 14 has a lot of information in it, as the chapters in this book generally do. As teachers the first thing we need to do is make sure that our students never allow the word algebra to scare them.
The first thing that this chapter has to tell us is make sure that we relate the work to our students lives. Which is really quite different from a high school math teacher of mine who actually told us that unless we went into teaching or engineering we would never use these formulas as adults.
There are a number of good strategies in this chapter, but what we need to get from even this is that fact that we must relate what our students are doing to their real life.
This is a major theme in elementary education today, it is a good one, and it is something that historically has not been done.
Chapter 14 discussed Algebraic Thinking and it reminded me of the multiple conversations we have had during class about what exactly algebra is. I have to say that I was surprised to find out that algebra is being taught as early as pre-kindergarten and I have seen this in my internship classroom. Every morning during the calendar routine, the calendar helper will pick out a few different shapes and create a pattern. The rest of the class has to guess if the pattern is an AB, ABA, ABC, etc. Before this class, I didn't realize that this was actually algebra. I'm glad that the text explains that the equal sign actually signifies equivalence and not always "the answer is". The examples given in the text were helpful in understanding this concept and I liked to read how the students were asked to explain their reasoning.
I thought that the Applet presented on page 275 is a great way to help students visualize time and distance. The chapter also explained how algebra can be used in a variety of ways throughout the curriculum such as with measurement and data. The experiments using manipulatives that were listed gave some great ways for teachers to make algebra fun and interactive in the classroom.
I'm glad that you mentioned how many people cringe at the sound of algebra when it really is not as terrifying as people think it is. Math has never really been my strong subject, but when I got into Algebra class and started learning how to find x, I actually found math sort of fun. I realize now that algebra is so much more than simply finding x. I'm glad that students are being taught about algebraic thinking at an early age and it will definitely be beneficial to them in the future. It's great that you are able to apply what you have learned through this text to your internship classroom.
Chapter 14 discusses “algebraic thinking.” Before reading this chapter, I didn’t really think that algebraic thinking began in “prekindergarten and continued through high school.” I also hadn’t thought of algebra as something that involved patterns, but as this chapter explains, algebraic thinking should begin early and one of the earliest forms is recognizing mathematical patterns. This chapter is also full of wonderful activities designed to help students build an understanding of algebraic concepts. I love the “seesaw students” activity which helps students learn that both sides of an equation must be balanced. This is a great activity for younger grades, and even at a very early age students are learning an algebraic concept. Another thing I learned from reading the text was that “patterns are found in all areas of mathematics. Learning to search for patterns and how to describe, translate, and extend them is part of doing mathematics and thinking algebraically.” Patterns should be emphasized continually as children are taught mathematics. Lastly, I liked that the text pointed out that algebraic thinking should be integrated across the curriculum. Overall, a very informative chapter, filled with a lot of great information and activities!
I also enjoyed the section on teaching considerations. I feel it is very important to introduce key vocabulary before teaching a lesson. I also didn't realize before teaching this lesson that student learned algebraic concepts at such a nearly age, but I really enjoyed reading about the different activities that help students develop these key concepts.
Chapter 14 is about algebraic thinking. We are often wondering what exactly algebra is. We have had this discussion in class. In fact, I have learned from Dr. Stramel and did not realize that children of all ages are learning algebra. They start from kindergarten and continue through until high school. Patterns is something widely used in algebra as this chapter points out. I found tons of new activities that will help me teach my students about algebra. My son is in 8th grade is in pre-algebra. Not sure exactly what the difference is between pre-algebra and algebra but he does a lot of problems that are all algebraic. He has a tough time through some of the chapters. I think he is bored because he has a teacher that just assigns homework page by page of the textbook. They never get to do other typo e activities or assignments. She also goes through each chapter and section pretty quick and does not really stop for those that the students are struggling with. I was amazed when I read that patterns are in all aspects of algebra problems. As teachers, we need to remember to use algebra across the curriculum and have it included in other subjects we teach.
In response to Angela R. – I was too one of those students that cringed when I heard algebra. It always seemed to be the toughest math for me. As I have helped my son and I look at the examples in his book, it teaches me that each problem has a correct answer but there is more than one way to do some of the problems. I am still in awe that even kindergarten students are learning algebra, but when we think of the different parts of algebra, it does not seem so bad.
Chapter 14 was over algebraic thinking. I really enjoyed this section because it provided so much useful information that will definitely be helpful in mathematics in the classroom. One thing I found very interesting was that algebraic thinking begins in as early as kindergarten. I knew it started early, but not this early. I do believe it is important to know this concept because it is something you will use all through your math courses. I am not a math person, but I do enjoy algebraic thinking. It makes you think and have to work out the problem to acquire the answer. With technology, there are so much fun, interactive stuff to do with this concept. There are tons of games out there that would be great when teaching this lesson. I also enjoyed the activities provided in this reading as well!
In response to Angela B--I feel for your son when it comes to his pre-algebra homework. This is something I will never do when I have a classroom of my own. I do not believe children learn anything from just working on worksheets or working out of the book every single lesson, every single day. I believe they need to have fun and do interactive things that go along with their lesson. As I stated in my post, with technology these days, I am sure there are tons of games that can be incorporated with this lesson. The book also provided great activities to try in your classroom as well. Thanks for sharing your story, it lets us know what NOT to do when we have classrooms of our own :)
Chapter 14 is about algebraic thinking. I really think the definition of algebra is in the title of this chapter. Algebra really is generalizations, patterns, and functions; they all just happen to increase in skill difficulty as we advance in education. I have always loved algebra. I liked the challenge it set me up for. I liked the section in the book titled “Meaningful Use of Symbols”. I think we forgot that simple problems such as 2+ __ =4 is an algebra problem because we are so used to thinking 5x+76y(-5z+4x) is the only way an algebra problem can look. I think symbols are very important in learning algebra. I also think it is important to have more than one symbol in the lower grades to prepare for future algebra in the upper grades. I liked how the book used pictures to show multiple variables. I actually think using the pictures is confusing. I also liked reading about odd and even relationships. The first graders in my internship are studying odd and even and the other day before break I read them a book about odd and even facts and how the sum of two even numbers is even and the sum of two odd numbers is even and the sum of an even and an odd is always odd. I think these facts would be useful on the assessments if they have to choose between two answers.
In reply to Kristi: I think a lot of people are surprised that algebra starts so early. I also talked about that in my post. Some children really like "finding the missing number" which is a type of algebra. I wonder what turns them off when they advance to upper level grades? Maybe it is the traditional way of thinking we force upon them? I remember my high school math teacher taught only this one way. You had to show your work and every step for you to get all the points possible. If you did one thing out of the ordinary it was automatically wrong. I didn't mind much, because it came easy to me and I loved his method of teaching, but a lot of students in my class struggled.
Chapter 14 is all about algebraic thinking. Until I took this class, I didn’t realize just how early this type of thinking and learning starts. I had been told by a parent educator that dramatic and imaginative play help children visualize algebraic concepts in the future, but I thought this was something that would happen way down the road. I was surprised that when my daughter learns patterns in pre-school she is doing algebra. I really like the idea of using the balance to teach students about the equal sign. It makes a lot of sense to teach them that the equal sign means that there is the same amount on each side. Graphing was something I absolutely hated in high school. I didn’t know how to use the calculator and it was just awful for me. I think starting out by having the students draw and interpret graphs with no numbers or formulas is a great way to help them understand the way graphs represent situations. I feel like I need to do a lot of studying if I am ever going to be able to teach algebra effectively.
Deidre J., I have found in my third grade internship classroom that the use of pictures is extremely helpful. If the students are struggling with a problem, I ask them to draw a picture. They are able to figure things out on their own almost every time this way. Sometimes they need a little direction about what to draw, but this is a strategy that really works. This can also be a way to show and explain their work.
The chapter on Algebraic thinking drives home how early algebraic concepts are introduced. When I observed a kindergarten class for professional practice and observation, there were creating patterns and repeating them. At the time I did not realize this was part of learning algebra. When Dr. Stramel mentioned in class that people say that they hate algebra I thought about how that is true for me, I always say that I hate algebra, but I never realized until this class that algebra is really part of all math. Although my overall view of mathematics is changing from total disdain to more of a cautious curiosity. I realize that as a teacher the only way I can instill a love of learning, including in Math, is to love it myself. This is a changing view for me, and I am thrilled that I can still change my view. The biggest help for me is how students are encouraged to do the math in the way that works for them, and not just one way. I am not very confident about my Math skills yet, but I am realizing that I know more than I give myself credit for. I love the way our text uses student examples, like on p. 259, showing where Latisha drew her explanation of how she understood that the 2 balls were the same weight as 1 cylinder. The examples are what make the chapters make sense to me and help me to realize that every one can do mathematics. One p. 262 there is a sentence that was sort of like a small epiphany for me when it talks about the use of variables. It says: initial work with finding the value of the variable that makes the sentence true should initially rely on relational thinking. It goes on to say "later students will develop specific techniques for solving equations when these relationships are insufficient." I think that the reason this was so exciting to me is because sometimes I want to give students too much information and I forget that they need that initial relationship to build on. At some point the relational thinking may not be enough to solve the problem, but it is an important piece to build on.
Deidra~ I agree that I tend to forget that a problem with a missing piece in it (such as 2+ __ =4) involves algebraic thinking. This chapter is full of reminders and information that help me realize that there are a lot of things that are actually algebraic thinking, like patterns and the odd and even number relations.
Chapter 14 was about algebraic thinking; generalizations, patterns, and functions. Children begin learning about algebra in prekindergarten and it continues through high school. There are five different forms of algebraic reasoning: generalization from arithmetic and from patterns in all of mathematics, meaningful use of symbols, study of structure in the number system, study of patterns and functions and process of mathematical modeling, integrating the first four list items. I struggled with algebra in 7th grade, but then got a teacher that explained it very well and from then on I loved doing algebra. The two weeks before fall break in my internship the students were learning algebra. My mentor teacher did a really good job and breaking it all down and gradually implementing variables. She also involved real world things in which the students were using variables. I feel algebra is a concept that can be confusing for students because letters are being substituted for numbers and that can be confusing. I think it just needs to be explained in a way students can understand. It is important to create word walls with the vocabulary needed for algebra and to teach the students to use the vocabulary when using algebra. It is a good idea for the students to keep a journal of the words.
In response to April B, I remember being in algebra in middle school and I had teachers that actually gave us different types of activities that helped us learn algebra. My internship right now is 6th grade math and they have learned algebra and my mentor teacher did a good job at incorporating real life items. I did my lesson on distance=rate*time and they caught on quickly because I related it to different things they see everyday like the speed limit signs and other things. They also had to plot points on a graph and I made a reference to battleship and it clicked. I think how a student feels about math in general depends on the teacher and I think that's the same with all subjects.
The biggest takeaway I had from Chapter 14 was the importance of symbols and vocabulary. Dr. Stramel does a great job of emphasizing the importance of using different verbiage for =. However, I'm not sure I really got it until I read about in this chapter. The light bulb finally turned on. The equals sign represents relationships. The distributive property helped me make sense of it. 6 x 7 = 5 x 7 + 7.
Then comes vocabulary. If I would have been hearing a lecture or listening to someone talk about the distributive property I would have had no idea what they were talking about. I have been surprised by the terms that are used in the 5th grade class. Like the text states a large part of understanding mathematics is the ability to communicate mathematically. If we constantly use the correct terms and have our students use the correct terminology the students will have an easier time in the future. Brandi Schroeder
@ Lacey I'm so glad you had a good experience with algebra. Using Excel files with Algebra is great. Those are two things that I always struggle with. For one I had a bad experience with algebra and hated it and never really had a class for Excel (mostly taught myself). I love the idea of integrating them. I have observed a lot of lessons with Powerpoints and Word documents, but nothing in Excel.
Chapter 14 was all about algebra. Unlike most people, I happen to have loved algebra. However, after reading this chapter and spending some time talking about the topic in class, I can understand how algebra might be very confusing for some people. There are a lot of terms that are used interchangeably so it is important for students to understand the vocabulary behind algebra. One section that we have discussed in class is the meaning of the equal sign. Many children think the equals sign means what is the answer. They do understand that it means everything on one side equals everything on the other. This is something I have experienced with the students in my internship. When given problems such as 8 x 3 = 6 x ___, they do not understand what to do. I like the idea of drawing a balance with expressions on either side to demonstrate the meaning of the equals sign, I think this would be a good way to gain student understanding. Once again, there were a lot of great activity ideas throughout the text. Hopefully I can get a job teaching middle school math and refer back to this text often!
I agree with you about teaching and displaying vocabulary. Most people don't think to incorporate vocabulary lessons into math class, but it is very important for understanding and shouldn't be overlooked.
Algebraic Thinking begins in prekindergarten and continues through high school and college. This chapter talks about how algebraic thinking and the different forms of it in the classroom. A good part of this chapter is the part about the meaning of the equals sign. It is so important for children to understand the meaning of the equal sign. I think a bad way to teach it is that the answer goes behind the equals sign. There are several tips for teachers to help them understand the meaning of the equals sign. There are many other definitions in this chapter that will be helpful for me in the future. Patterns in this chapter are uniquely talked about. There are so many ways to teach patterns. Actually you can teach patterns in so many areas its crazy. Teaching patterns can’t just be in one area.
I also did not know how important algebraic thinking was in younger years. There are so many ways to incorporate it into algebraic thinking into children's curriculum. I did not even know that algebraic thinking started in kindergarten and it is still surprising to me.
This is very helpful! I now realize just how many algorithms there while tutoring at a local elementary school. I now know what it felt like when I took homework home and my parents had no clue what my assignments were discussing, especially within mathematics. My personal favorite algorithm is the Russian Peasant. This is obviously a mathematics function from Russia and takes some time to work out, but is very interesting and you can always check your work with this algorithm (as long as your procedures are correct). Has anyone else seen this used within schools today or practiced this form yourself? Just learning all of the new ways of algorithms is fun for me, I know it's students have more fun and feel like there are options for them in mathematics-because there are options! As long as students get the correct answers, there are good to go. They can also help each other by showing their peers their favorite algorithms within the classroom or at recess.
I agree, symbols were key and VERY helpful within this chapter. I like this book overall-I love all the references. Knowing all of the symbols is a domino affect to knowing what's down the road... For us as educators, but also are students.
This chapter talked about algebraic thinking. I liked reading this chapter because it related to my internship I am in right now. The chapter talked about understanding equal signs. I know my students are having trouble with equal signs when they are in different positions. Sometimes its in front and sometimes it's at the end. Most of the students understand but there are some that don't. It can be difficult for students to understand that it is asking the same thing no matter where it is in the problem.
I agree I think it is crazy how early children begin to learn algebra, without even knowing it! I think the term "algebra" can be a turn off for some students. Maybe if we as teacher just talked about the lesson instead of algebra the students wouldn't get so worried about it and may enjoy it more.
One day in class we were asked what we thought Algebra meant. I related it to working through mathematics by using order of operations and variables. I thought of it as being used in higher grades but the text states that algebraic thinking begins in prekindergarten. I hadn’t thought about the fact that algebra deals with lots of patterns which prekindergarten students work with a lot. The section in this chapter that discussed patterns on hundreds charts had an example in figure 14.2 that reminded me of the students in my observation class. I hadn’t thought of it as being algebra at the time but I remember that when they would count by multiples on hundreds charts many of them would stop counting after the third or fourth number because they would realize a pattern being formed and just followed it. Algebraic thinking is all about patterns and I don’t think it’s bad that the students used patterns instead of counting each multiple out. Like we’ve learned, mathematics is the pursuit of laziness.
I was also able to relate this chapter to experiences in my internship class right now. Although mine was with the hundreds chart examples I also noticed how students have troubles when the equal sign is moved. It can be confusing to students when they constantly learn that the equal sign goes in the same place over and over. Like the text says, having the equal sign in the same place constantly make students see it as standing for “and the answer is”. It can be confusing to see it as something different when it’s always mean the same thing for so long. I can see why some of the students in your internship class may have trouble at first. Good post!
Chapter 14 had some interesting concepts. I first noticed how it mentioned that we start learning about algebraic thinking in prekindergarten years and then continues through high school. I think we don't realize the importance of the younger years at times but when we put it into perspective we learn SO much from what we learn as young students and these things travel with us through high school. Once again when looking at the part of the chapter about the hundreds chart. Another thing that we learn at a young age that I know i still use this when figure math. At least I use what I learned with the chart for many daily activities like adding by 5's or 10's.
In response to Shannon H: It's funny that you also talked about how learning algebra starts in prekindergarten. I was also amazed by how algebra isn't just something you start in high school but also something we learn in grade school and carry out through out our school years.
I did not realize that algebraic thinking begins in prekindergarten. According to the book, children at this age begin to recognize and duplicate simple sequential patterns. After reading about this, I can now see how kindergartners use algebra and how the teacher goes about teaching it. Kaput describes five forms of algebraic reasoning which are: 1) generalization from arithmetic and from patterns in all of mathematics, 2) meaningful use of symbols, 3) study of structure in the number system, 4) study of patterns and functions, and 5) process of mathematical modeling, integrating the first four list items. I find this list very helpful as it explains how algebraic thinking is not just one idea but several pieces put together.
It is very important to begin algebraic thinking as soon as possible and I too thought it was interesting that we begin before kindergarten. It is amazing to know how young we begin to learn things and that they stay with us all throughout life.
Joel Stucky You mentioned that the equal sign means 'same as'. If you remember it was also mentioned that we will typically tell small children 'take away' for 'subtract' and that this can be confusing when they get into negatives. It is very important that we are sure to use and teach proper vocabulary! Good note.
An interesting thing happened to me on the way to the forum [math]. I have been working with math students for 6 years and it wasn't until the 7th year that I put 2 and 2 together. Our text says that the 'box' "is a precursor of a variable used". Basically, we learn algebra in early grade school years. I was working with a student on a problem somewhat like: 'what' - 212 = 40 when I realized it was algebra in the sense that it uses an unknown number within the equation. We teach the students to add the two known (in this case) to find the unknown, however, we will give the same student the same problem in an upper high school grade and tell them to balance the two sides of the equation! Wow. Imagine my surprise when I realized we learn and teach algebra all through our grades! Imagine the pride my student had when I told him he was doing algebra!
Chapter 14 is about Algebraic Thinking. It seems weird to think about it because we don’t enter into classes called “Algebra” until junior high and high school, but Algebra is happening from the very start of a child’s math career. Students are constantly being asked what the missing number is or how many different ways can different things be put into order.
I have seen Algebra used weekly in my internship and on one of the weekly math flyers the students are required to do, there is even a section named Algebra. It worries me that every week when the flyer comes around the students psych themselves up and get worried about completing the algebra section. Too many times the students think too much about it and expect it to be much harder than it is. When it is time to do fix-it papers, and the problem is explained a little better to them, the students often need to fix the algebra section and are surprised they didn’t get the right answer the first time around.
I think a lot of this chapter is helpful but a lot of it had to focus on older students. It is important to calm students down and let them know that all too often the algebra is easy to figure out.
I never thought about how early algebra is used until I started this course. From the very beginning we are helping students to find "unknown numbers" without them realizing that they are actually doing and learning algebra.
I completely agree with you that we must use the same terminology. When I was doing my formal teaching I remember I said subtract and minus in the same breathe! Just because I know what it means doesn't mean that a classroom of 5 year old will know they mean the same! It is imperative that we stick with the same terms to give the students a better understanding! Great point!
I never realized how early we really start using Algebra. As early as kindergarten sounds crazy, but when you take into consideration of patterns and finding the missing block or number it absolutely is Algebra! It was interesting in class doing some of the activities in our small groups and not even realizing that we were doing Algebra! I think as we get into the actual Algebra class in junior high and high school, we tend to freak out and have some sort of mental block! I did anyway. If I wouldn't have seen the word "Algebra" and just had done the work I would have done better!! I think this chapter gave me great insight on how often we actually use this successfully and don't even know it!
Chapter 14 discusses algebraic thinking: generalizations, patterns, and functions. I found it very interesting to find out that children start learning algebraic thinking in pre-K! And being an intern in Kindergarten I now see that! I guess I didn't realize what they were actually doing when they were making patterns on the calendar and what not. I knew it was patterning and math but not actual algebraic thinking! That's amazing! I also recently found out that the reason they changed the name of the shape oval to ellipse in order to help get students ready for geometry! I think that it is truly important for students to be introduced to all of this at a young age because it is when they most easily soak up the knowledge and this way they won't feel so overwhelmed later.
ReplyDeleteChapter 14 is about algebraic thinking which occurs as soon as students learn about numbers in Kindergarten. I like figure 14.1 with the monkeys. This is great way to begin to paint a picture of the words in the beginning algebra problems. The generalization in the hundreds chart is another good way of connecting the dots in a students minds. Before interning in a first grade classroom I had not heard a teacher say that the equal sign means "the same as," and "is." I like to hear teachers using the same terminology throughout the different grade levels. This chapter also outlines many different ways of connecting the dots for students by integrating within the math curriculum. Such as using money to count or using pictures like the monkeys in the trees. I think it is important as educators we know the common core standards inside and out. Being able to integrate within the curriculum is a critical component of fully understanding each math concept we teach our students. Estimating how much an item costs and knowing number patterns are interrelated but we need to teach how they are interrelated and why it is important in real life. All of this comes back to having a good number sense.
ReplyDeleteComment for Kristle C:
ReplyDeleteAmazing that some students don't know what an oval is... I find it weird. I agree, the algebra concepts are starting younger and younger but in the long run I think our students will gain a much deeper understanding of this concept. Learning that happens at an early age and continues on throughout the grade levels gives those 6th grade teachers a much easier job to do when they are expected to master fraction skills. Growing the students number sense over time enables the student to have the answers behind why they are learning all of the math concepts. Relating math to the real world gives the students a reason to learn.
Jena-I find it weird also. I overheard a parent-teacher conference in my intern classroom where the teacher was telling the parent that the student called it an oval and that was incorrect and the parent was like well that is an oval and the teacher informed her of the change and the parent said oh well that is stupid! I can understand where the parent is coming from but they just don't see how the change in something like calling an oval an elipse will help the student later on! I think that is hard for parents to understand!
ReplyDeleteI think that before teaching anything, a foundation of vocabulary needs to be set. It is important that students understand what shapes are called and what they look like as well as what symbols mean. I am amazed at what students do not remember or do not have knowledge of what a pyramid or cone looks like. As Dr. Stramel has said, the equal sign means "same as". This sign has been overlooked and can confuse students unless taught the real meaning.
ReplyDeleteJena,
ReplyDeleteI agree that relating math to the real world gives the students a reason to learn. Capitalizing on their experiences in all areas will help them in their learning process.
I never really realized how dynamic algebraic thinking / concepts are. When I think about algebra I always think of my high school math classes and equations like y = mx + b. It’s hard to think of algebra at an elementary level when you’ve been beyond that level of thinking for years.
ReplyDeleteAs usual there are so many great examples that help me realize the simplest problems are or can be considered as algebra. I think my favorite examples are the open sentences. Those can be used in forms as simple as 5 + ___ = 8, or as complex as 20 + 48 = ___ x 20. These are thing you could use for all grades from Kindergarten to 6th grade by modifying or enriching depending upon the grade level.
I also learned a lot about the five types of representations. These representations include patterns (or the context), tables, verbal descriptions, symbolic equations, and graphs. All of these representations are useful in their own ways. I would imagine though that using multiple representations at one time would help the students tremendously when it comes to grasping the information or concept and retaining it longer.
@ Jena Simms
ReplyDeleteYou bring up a valid point about the equal sign. I’m currently in a 3rd grade classroom, and I can’t recall any time my mentor teacher has used the words “the same as” or “is.” I hear equal a lot though. That makes me wonder what the other teachers in the 3rd grade are using and what teachers in the other grades are using terminology wise.
I also agree with you on needing to know the standards and how we can use those to integrate within and across the content areas / curriculum. There’s a lot we future educators need to step up to, but I think those that are dedicated in this field will have no problems doing so!
When reading through Chapter 14 over Algebraic Thinking: generalizations, patterns and functions I was able to learn some great information that is going to be helpful to me in my future classroom. I liked how the book pointed out that algebra is an established content strand in almost of the grades from K to 12th grade. Up unto this point I have never thought of algebra being presented that much in the lower elementary school setting but after the book gave me a couple of examples I was able to clearly see how algebra is connected throughout all of the grades. I also liked how the book talked about the five different forms of algebraic thinking which was very helpful for me to know.
ReplyDeleteI was able to relate a lot in the area of algebraic thinking when the book talked about how children in kindergarten start to recognize and duplicate simple sequential patterns. This is happening right now in my kindergarten classroom where I am able to show the students a pattern of shapes such as diamond, triangle, diamond, triangle and the students are asked what shape will come next. I liked how the book talked about teachers need to focus more on the meaningful use of symbols. The students need to have a strong sense of what the symbols in the problems mean in order to correctly figure out the solution to the problem. I thought that all of the activities presented throughout chapter 14 were helpful to me as a future teacher.
In response to Kristle C.,
ReplyDeleteI was also surprised when reading this chapter that algebraic thinking starts in kindergarten. I also was able to see that in my kindergarten classroom when I am doing the calendar activities each and every day that when the students are asked what shape comes next in the pattern that that is algebra and requires algebraic thinking. Wow! I was wondering the same question about why the students need to know that an oval is also an ellipse so thanks for answering my question and it now makes sense that they are getting the students ready for geometry. Lastly I agree with you that it is important that the students learn the information early that way the students are not overwhelmed down the road.
One of the things that Chapter 14 talked about was conceptualizing the equal sign as a balance. I had never really thought about it till this semester in my math internship. When we discussed solving a problem such as 9+5=__+10 a lot of the students in my internship would put 14 as the answer for the blank. But when I explained to them that we should think of this problem as a balancing scale then it started to make sense to them. The students realized that the goal of the problem wasn't to find the answer just to 9+5 but to make both sides equal the same amount so that if it were on a scale it was balanced.
ReplyDeleteBrooke M
ReplyDeleteI had never really thought about algebra being presented in early elementary grades either. I always figured it started in jr. high just because that was when I started having classes with the title of Algebra. But looking back it is quite clear to me that I was doing algebra way before jr. high. I think part of the reason that I never really thought about it being a part of the curriculum in early elementary was because I never really had a clear understanding of what algebra was before taking this class. Now that we have discussed it so many times in class it just seems so obvious to me.
Katie Coulter
ReplyDeleteChapter 14
Chapter 14 was a big section of reading and had a lot about algebra. Personally I feel very confident with algebra so it was interesting as I read and tried to relate to students who might not understand. Right off the bat I was really surprised how they had categorized concepts that were considered algebra. The fact that we start Pre-K with algebraic thinking is unreal. I was totally unaware of how they classified specific concepts and how much more was considered to be algebraic. Again I really don’t mind doing algebra. I like doing long problems no really problem solving but just long equations that solve for multiple variables. When they began discussing the equal sign I was taken back. I had no idea that a student might be troubled with associating the equal sign with equivalence of both sides not just one. Since we do start students off so young with this concept I do feel the chapter is right as they suggest we begin using variables at a sooner age instead of using boxes. No matter what you are talking about you should be able to label what you solving for like distance or time- D or T in the equation.
In response to Joel Stucky,
I think you make a great point by reminded math teachers they still need to focus on vocabulary. A student could get lost big time if they don’t understand the terminology used to explain the concept. I feel frustrated and overwhelmed when doing something and not fully comprehending the dialog. Math has many vocabulary terms to cover!
Chapter 14 is about algebraic thinking including generalizations, patterns, and functions. One of the topics is generalization through exploring a pattern. This is something that I have seen my mentor teacher work with her students on in pattern boxes. I don't remember doing this in school, but it is a great way to get students to look at the problem and find the pattern. The book discusses how students need to learn to search for patterns and learn how to describe, translate, and extend them. One thing I never really knew, but learned in this chapter is that prediction is an important part of algebraic thinking. The book discusses how predicting real-world contexts is appropriate for upper elementary. For example it discusses finding patterns and predicting when the summer Olympics and when the winter Olympics will be held based on when they have been held in the past. This is a fun way to help students see they will use this in their life.
ReplyDeleteIn response to Katie C,
ReplyDeleteUnlike you, I am not very confident in my algebra skills, but wish I was. I was also extremely surprised through that students actually do algebra as young as pre-k and kindergarten. There really is so many more topics covered under algebra than I ever knew.
Chapter 14:
ReplyDeleteChapter 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. discusses using algebra in the classroom which is used more and earlier than people think. According to Van de Walle et al. textbook “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions” ” (Van de Walle et al., 2010, p. 254). Algebra is a concept that can difficult for students but I think once they understand it it can be for students.
A piece of information that I learned from this text was all the different activities presented in the textbook. On activity that I liked learning about was 14.8 titled “Balls, Balls, Balls”. When I first read this activity, honestly I had no idea how to go about it but then I discovered how the book did this activity by having the reader really look at the material and really evaluate the information presented.
Chapter fourteen really made me think about my time in algebra class. Algebra was not easy for me at first. I had a teacher who did not really put a lot of effort into her teaching which really had an impact on me. Then when I took Algebra over again I had a teacher that really took time and consideration into what we did. Every assignment that we did in her class we could fix and turn in for extra credit which is very important in Algebra. Getting that extra credit made me what to go back and recheck my work to know where I went wrong. I think this is what all teachers should do. I was then able to understand the process of algebra and now I actually like doing it for fun.
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.
Jennifer Pen reply to Elizabeth,
ReplyDeleteI agree with you that prediction (as stated in this chapter of the textbook)is very important. With algebra students can think about what makes sense and what would not make sense. A lot of times students have to even work backwards to figure out problems. There are so many different ways to do math problems that students just have to take a step back and think "does this make sense?"
Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions
ReplyDeleteThe meaning of the equal sign on page 258 really does amaze me how many students really miss those problems. I know it really is a struggle for students, even when we break it down. I see the same problem when they put the answer and then the equal sign before the problem, 6=4+2. It’s very hard to determine if something is equal when it does not look alike. That’s probably why they have such a hard time with the scales, cones, and balls and liquids in different types of containers. The mathematics curriculum that we are using at my school is introducing simple algebra in the 1st grade and then it just increases from there. This section is packed with activities which are very helpful in teaching such a hard concept. I think it is important that they understand not just that it is equal but that it is equivalent. I think using real world examples help so much. I watched a teacher used different containers and she filled a wide container with water and then said, Can we fit all this water into this tall skinny container? Many children said no and then she did it. The shape changed but the capacity stayed the same. I think real world examples help.
Lane A. I agree with you. Scales are very inexpensive and now they have nice plastic ones. Bring out the scale and use some kind of manipulative. Take bears for instance, 5+5=_+2. Put 5 bears on the left side and let them see you add 5 more to that side. Now put 2 in on the right side. Then tell the children lets add bears and count until they are back at the same point. Add bears 1,2,3,4,5,6,7,8 now the scale is back at it's starting point. How many did we add, 8 yes, 8 is the number that we needed to put into this spot. This would help them visualize. I thinks some times we forget it's easier when we can see and touch things. Algebra was very hard for me so I should expect that it might be hard for the children. Just try as many examples as possible so that they will get it.
ReplyDeleteThis week we read chapter 14, Algebraic Thinking; Generalizations, Patterns, and Functions. It’s important that we think beyond the algebraic equations that took a whole piece of paper to complete that we did in high school. These problems rarely had anything to do with real world situations and in most cases were completed by following the teacher’s specific direction. Algebraic thinking is much more than that, it starts in pre-K with shapes and is added on to each year. Algebra is used to help students see patterns and generalization in the real world; once they have an understanding of certain patterns they become more successful in math and have an easier time seeing connections between numbers. I love doing Sudoku puzzles, and in many ways this is just many different number patters and problem solving. Another important concept in this chapter is the different properties (associative, distributive…). These help students see that whatever is on one side of the equations is equal to the same as the other side. The next part students have to understand is the meaning of variables, and unknown values. One they understand what the variables are and how to represent them they then can create their own way of solving the equations. In this chapter they also talk about thinking across the curriculum in algebraic, this is why it is important to teach students how to algebraically think so that they can look for patterns and generalization in all areas. This will also help them think beyond the curriculum in both math and the other area creating better reasoning skills.
ReplyDeleteIn response to Tammy M.
ReplyDeleteYou mentioned that so many students have trouble with equations that are equal but don’t look the same. For my formal lesson I was teaching about algebraic equations and how to solve them, but one of the main focuses was on that both sides of the equations had to be equal. For my lesson plan we used balance beams, I showed the students how if they are not equal then the balance beam falls towards that way. These manipulative may help your kinetics learners understand the concept better.
Chapter 14 talks about algebraic thinking: generalizations, patterns, and functions. I thought that it was very interesting that children start leaning algebra at the beginning of pre-k. Patterns are one of the very first things that we learn as we are little. This is happening in my kindergarten right now during my internship. I also liked that this chapter gave lots of different activities for me to use in my future classroom.
ReplyDeleteTo Katie C:
ReplyDeleteI myself don't feel as confident in my algebra skills like you. But I also was surprised that students actually do algebra as young as pre-k . There is so much to learn about math!
Chapter 14 was over Algebraic Thinking and I thought that it was very informative. I thought it was interesting that the algebra that we went through is not the algebra that is being taught in todays’ schools. I have always thought that mathematics was the one consistent subject that is taught in schools. For example, in my internship, we just began to introduce the concept of multiplying larger numbers. The students were taught a different method than I was. (I think the teacher said Singapore Math) It was difficult for me to help the students because I would begin to explain it in the way that I was taught. Chapter 14 gave me some great advise on how to teach these new methods.
ReplyDeleteIn response to Rebecca-
ReplyDeleteI agree with you when you say that we should think about algebraic equations differently than when we were in school. The focus of algebra now is to connect it to the real world, and to develop the students’ critical thinking ability. I think algebra is important because it teaches the students critical thinking skills, and how to generalize from experiences that they have had.
Chapter 14- Algebraic Thinking: Generalizations, patters, and functions.
ReplyDeleteWhen I think Algebra, I am immediately a little scared. Why did we become so scared of something that was intended to be so helpful to us? Plus, we have been learning and being introduced to it since we started school. The text talks about using our basic math fact families – that’s algebra! Early on we are learning patterns and symbols. If we failed to get a good grasp on this concept we can be setting ourselves up for failure down the road. Then the text talked about the equal sign and variables and how misunderstood these symbols really are in our society. This really hit home for me. I am mentoring in a 4th grade math class and the students do struggle with their math facts and after ready this chapter I am almost certain, they think the “=” mean the answer is. The text touches on the importance of conceptual knowledge of patterns. That students should be able to see the XXYXXY is the same pattern as red, red, blue, red, red, blue – wow – I honestly have never really understood the true importance of patterns until now. I am also able to link what is going on in my mentor class with the importance of learning pattern and algebra. This is all part of a major foundation that must be built strong. No pressure on us – right?
@ Joel Stucky –
ReplyDeleteYou are right on the money. That vocabulary is so important and knowing that shapes have names and symbols have meaning is something that is critical for us as educators to be getting across to our students. That “=” meaning is a really important one. Dr. Stramel did mention she like to use the term “same as”. I agree that using the best terminology or vocabulary is so important. If we are not doing things in the proper way or manner, our students will never learn it correctly. Great points made –thanks.
Lindsay H
ReplyDelete@ Brooke M-I agree with you. I also did not really think of algebra being used in early elementary. But now looking back from previous experiences I am seeing it more and more!
I like many other peers found it so interesting that early education teachers are already implementing algebraic thinking into their students. I loved the part about patterns. I think patterns are so important to know when it comes to math. In my internship class we have been working on attributes. It is kind of neat because they teacher says the attributes then the students work on making a pattern based on the attributes that she says. It really gets students thinking and some students really struggle with the process. I really struggled in college algebra so I am a bit worried about teaching it. It was interesting to note about algebraic thinking across the curriculum. I think it so important to integrate and relate different subjects across the curriculum.
I think most people think of high school when the word Algebra is used. My boys are in 4th grade and came home so proud because their teacher told them they were learning Algebra. I believe math needs to be taken seriously from the moment that students start school. In the lower grades we may not use the word Algebra, but everything a child learns in math is important because math concepts continue to build on each other. As teachers we must be careful that students are not falling behind in the lower grades. Learning those math facts is important because in high school it is tough to do math without the basic skills.
ReplyDeleteTo Katie C: In my internship class I have noticed many students who do not understand what certain symbols mean. I think as adults we take it for granted. If I see a student struggling I try to get them to read the problem to me. This way I can see if they even what it is asking them.
ReplyDeleteTo Andrew: I have noticed the same thing. I am amazed that the ways math has been taught. Sometimes I wonder who comes up with these different ways. I am very interested to see the new ways that a concept can be taught. I think each teacher and each school probably has their own preference on how to teach a concept. I don't remember many of my teacher ever using manipulatives, but I am very excited to utilize hands on activities in my classroom.
ReplyDeleteLacey Keller
ReplyDeleteIn my personal education, mathematics was my strong subject, most likely due to a great set of teachers. However, I must also conclude that I also enjoy algebraic functions and problems. I am a sensible gal who has to make solutions "just add up." Haha! I believe algebra is just real life mathematics. If we can get our students to buy this philosophy, then we will have great success as mathematics instructors.
This chapter discussed in detail functions in algegra. Yes, we see patterns way before we start kindergarten. We introduce our young babies to repetition in nursery rhymes and continue to use algebra long into our adult lives. When I was in junior high, my math teacher was also our computer teacher. Fortunately, the teacher integrated both subjects accordingly. I gained so much algebraic knowledge just by simply using a spreadsheet program, such as Microsoft Excel. Our teacher allowed us to just explore and find the patterns of certain functions. We then graphed several expressions. It was so much fun and I learned so much!
Lacey Keller
ReplyDeleteShawna,
You are right to say that most people get a little scared just by mentioning the word "algebra." Really, it's just patterns! I think of all the things that use algebra... I mean patterns: music, art, money, landscaping, farming, etc. The concept of algebra is found in many professionals and should be thought as helping us find the answer faster!
I have to say this was the most beneficial chapter for me in relation to my internship for this class. I taught unknown variable equations to 5th graders for internship formal observation. I had went through and read this chapter before I created my lessons and I have to say it helped me greatly. When someone says algebra often times students cringe at the thought. In fact I was one of those students until I began actually learning how to do the equations. I actually like to do equations and find it very fulfilling when I am able to complete problems!
ReplyDeleteIt is amazing that at the ages of 3, 4, 5 students are already being taught algebraic thinking! In the Kindergarten classroom patterning is the biggest concept taught throughout the year. It is so interesting to know that the entire concept behind Math is patterning. One thing that I am not good at is recognizing number patterns in relation to recursive patterns and formulas. When I am in the 3rd grade classroom I often find myself getting confused trying to show the students how to complete the patterns they are learning!
I am not a fan of graphing and linear functions. When I think of either of these items I immediately think of Economics class and that was not my favorite class! It is very interesting all of the information that can be calculated and observed through the use of graphs and linear functions.
I enjoyed the section on teaching considerations. When I taught my formal lesson I stressed the importance of knowing the meanings of specific words in relation to the lesson I was teaching. It is very important that the students know the meaning and reasons behind what they are learning. I liked the explanation of the multiple representations of functions. The way the book presented the information is great knowledge for a teacher to know when teaching his/her students.
This chapter was very informative not only for giving me information to use in my formal lesson but very useful in helping me teach algebraic thinking to my students I now work with everyday and in the future.
In response to Lane:
ReplyDeleteI liked how the book discussed how students comprehend the meaning of the equal sign in different ways. I hadn’t really thought about students not completely understanding the meaning of the equal sign in relation to what is on one side of the equal sign has to equal what is on the other side of the sign no matter how the numbers are presented. This was one concept that I had to stress the importance of when teaching my formal lesson on variable equations because the whole point of teaching my lesson was for students to find the value of the unknown variable. Without this information there is no way that the students could complete the problems!
In response to Angela R
ReplyDeleteCertainly the idea that they are teaching algebraic thinking to kindergarteners is an interesting one. I suspect they taught it to us as well, they just did not use the word algebra or any form of the word.
As students and as adults many of us allow the word algebra to scare us. As teachers we need to look at why this is and make sure that is does not happen to our students.
Chapter 14 discussed Algebraic Thinking. I enjoyed this chapter because it allowed me to look at different theories and techniques used in algebra differently. I particularly liked the section on Meaningful use of Symbols. I didn’t realize how important the use of symbols in math really is. The section discussed how some students just don’t have a strong understanding of what the symbols are and how to use them. I definitely agree with this because when I was growing up it all seemed to blend together and I couldn’t tell the difference between the symbols or when to use what. I think that teachers should definitely spend time focusing on explaining and making sure the students have a clear understanding of the meaning of each symbol.
ReplyDeleteIn Response to:
Angela R.
I observe in a Kindergarten class and they are just beginning to work with patterns. Some of the students are getting it right away and some are taking a little longer. I was surprised to see that they do more than just AB patterns at such a young age.
Chapter 14 has a lot of information in it, as the chapters in this book generally do. As teachers the first thing we need to do is make sure that our students never allow the word algebra to scare them.
ReplyDeleteThe first thing that this chapter has to tell us is make sure that we relate the work to our students lives. Which is really quite different from a high school math teacher of mine who actually told us that unless we went into teaching or engineering we would never use these formulas as adults.
There are a number of good strategies in this chapter, but what we need to get from even this is that fact that we must relate what our students are doing to their real life.
This is a major theme in elementary education today, it is a good one, and it is something that historically has not been done.
Chapter 14 discussed Algebraic Thinking and it reminded me of the multiple conversations we have had during class about what exactly algebra is. I have to say that I was surprised to find out that algebra is being taught as early as pre-kindergarten and I have seen this in my internship classroom. Every morning during the calendar routine, the calendar helper will pick out a few different shapes and create a pattern. The rest of the class has to guess if the pattern is an AB, ABA, ABC, etc. Before this class, I didn't realize that this was actually algebra. I'm glad that the text explains that the equal sign actually signifies equivalence and not always "the answer is". The examples given in the text were helpful in understanding this concept and I liked to read how the students were asked to explain their reasoning.
ReplyDeleteI thought that the Applet presented on page 275 is a great way to help students visualize time and distance. The chapter also explained how algebra can be used in a variety of ways throughout the curriculum such as with measurement and data. The experiments using manipulatives that were listed gave some great ways for teachers to make algebra fun and interactive in the classroom.
In response to Shawna W:
ReplyDeleteI'm glad that you mentioned how many people cringe at the sound of algebra when it really is not as terrifying as people think it is. Math has never really been my strong subject, but when I got into Algebra class and started learning how to find x, I actually found math sort of fun. I realize now that algebra is so much more than simply finding x. I'm glad that students are being taught about algebraic thinking at an early age and it will definitely be beneficial to them in the future. It's great that you are able to apply what you have learned through this text to your internship classroom.
Chapter 14 discusses “algebraic thinking.” Before reading this chapter, I didn’t really think that algebraic thinking began in “prekindergarten and continued through high school.” I also hadn’t thought of algebra as something that involved patterns, but as this chapter explains, algebraic thinking should begin early and one of the earliest forms is recognizing mathematical patterns. This chapter is also full of wonderful activities designed to help students build an understanding of algebraic concepts. I love the “seesaw students” activity which helps students learn that both sides of an equation must be balanced. This is a great activity for younger grades, and even at a very early age students are learning an algebraic concept. Another thing I learned from reading the text was that “patterns are found in all areas of mathematics. Learning to search for patterns and how to describe, translate, and extend them is part of doing mathematics and thinking algebraically.” Patterns should be emphasized continually as children are taught mathematics. Lastly, I liked that the text pointed out that algebraic thinking should be integrated across the curriculum. Overall, a very informative chapter, filled with a lot of great information and activities!
ReplyDelete@ Angela R
ReplyDeleteI also enjoyed the section on teaching considerations. I feel it is very important to introduce key vocabulary before teaching a lesson. I also didn't realize before teaching this lesson that student learned algebraic concepts at such a nearly age, but I really enjoyed reading about the different activities that help students develop these key concepts.
Chapter 14 is about algebraic thinking. We are often wondering what exactly algebra is. We have had this discussion in class. In fact, I have learned from Dr. Stramel and did not realize that children of all ages are learning algebra. They start from kindergarten and continue through until high school. Patterns is something widely used in algebra as this chapter points out. I found tons of new activities that will help me teach my students about algebra. My son is in 8th grade is in pre-algebra. Not sure exactly what the difference is between pre-algebra and algebra but he does a lot of problems that are all algebraic. He has a tough time through some of the chapters. I think he is bored because he has a teacher that just assigns homework page by page of the textbook. They never get to do other typo e activities or assignments. She also goes through each chapter and section pretty quick and does not really stop for those that the students are struggling with. I was amazed when I read that patterns are in all aspects of algebra problems. As teachers, we need to remember to use algebra across the curriculum and have it included in other subjects we teach.
ReplyDeleteIn response to Angela R. –
ReplyDeleteI was too one of those students that cringed when I heard algebra. It always seemed to be the toughest math for me. As I have helped my son and I look at the examples in his book, it teaches me that each problem has a correct answer but there is more than one way to do some of the problems. I am still in awe that even kindergarten students are learning algebra, but when we think of the different parts of algebra, it does not seem so bad.
Chapter 14 was over algebraic thinking. I really enjoyed this section because it provided so much useful information that will definitely be helpful in mathematics in the classroom. One thing I found very interesting was that algebraic thinking begins in as early as kindergarten. I knew it started early, but not this early. I do believe it is important to know this concept because it is something you will use all through your math courses. I am not a math person, but I do enjoy algebraic thinking. It makes you think and have to work out the problem to acquire the answer. With technology, there are so much fun, interactive stuff to do with this concept. There are tons of games out there that would be great when teaching this lesson. I also enjoyed the activities provided in this reading as well!
ReplyDeleteIn response to Angela B--I feel for your son when it comes to his pre-algebra homework. This is something I will never do when I have a classroom of my own. I do not believe children learn anything from just working on worksheets or working out of the book every single lesson, every single day. I believe they need to have fun and do interactive things that go along with their lesson. As I stated in my post, with technology these days, I am sure there are tons of games that can be incorporated with this lesson. The book also provided great activities to try in your classroom as well. Thanks for sharing your story, it lets us know what NOT to do when we have classrooms of our own :)
ReplyDeleteChapter 14 is about algebraic thinking. I really think the definition of algebra is in the title of this chapter. Algebra really is generalizations, patterns, and functions; they all just happen to increase in skill difficulty as we advance in education. I have always loved algebra. I liked the challenge it set me up for. I liked the section in the book titled “Meaningful Use of Symbols”. I think we forgot that simple problems such as 2+ __ =4 is an algebra problem because we are so used to thinking 5x+76y(-5z+4x) is the only way an algebra problem can look. I think symbols are very important in learning algebra. I also think it is important to have more than one symbol in the lower grades to prepare for future algebra in the upper grades. I liked how the book used pictures to show multiple variables. I actually think using the pictures is confusing. I also liked reading about odd and even relationships. The first graders in my internship are studying odd and even and the other day before break I read them a book about odd and even facts and how the sum of two even numbers is even and the sum of two odd numbers is even and the sum of an even and an odd is always odd. I think these facts would be useful on the assessments if they have to choose between two answers.
ReplyDeleteIn reply to Kristi: I think a lot of people are surprised that algebra starts so early. I also talked about that in my post. Some children really like "finding the missing number" which is a type of algebra. I wonder what turns them off when they advance to upper level grades? Maybe it is the traditional way of thinking we force upon them? I remember my high school math teacher taught only this one way. You had to show your work and every step for you to get all the points possible. If you did one thing out of the ordinary it was automatically wrong. I didn't mind much, because it came easy to me and I loved his method of teaching, but a lot of students in my class struggled.
ReplyDeleteChapter 14 is all about algebraic thinking. Until I took this class, I didn’t realize just how early this type of thinking and learning starts. I had been told by a parent educator that dramatic and imaginative play help children visualize algebraic concepts in the future, but I thought this was something that would happen way down the road. I was surprised that when my daughter learns patterns in pre-school she is doing algebra. I really like the idea of using the balance to teach students about the equal sign. It makes a lot of sense to teach them that the equal sign means that there is the same amount on each side. Graphing was something I absolutely hated in high school. I didn’t know how to use the calculator and it was just awful for me. I think starting out by having the students draw and interpret graphs with no numbers or formulas is a great way to help them understand the way graphs represent situations. I feel like I need to do a lot of studying if I am ever going to be able to teach algebra effectively.
ReplyDeleteDeidre J., I have found in my third grade internship classroom that the use of pictures is extremely helpful. If the students are struggling with a problem, I ask them to draw a picture. They are able to figure things out on their own almost every time this way. Sometimes they need a little direction about what to draw, but this is a strategy that really works. This can also be a way to show and explain their work.
ReplyDeleteThe chapter on Algebraic thinking drives home how early algebraic concepts are introduced. When I observed a kindergarten class for professional practice and observation, there were creating patterns and repeating them. At the time I did not realize this was part of learning algebra. When Dr. Stramel mentioned in class that people say that they hate algebra I thought about how that is true for me, I always say that I hate algebra, but I never realized until this class that algebra is really part of all math. Although my overall view of mathematics is changing from total disdain to more of a cautious curiosity. I realize that as a teacher the only way I can instill a love of learning, including in Math, is to love it myself. This is a changing view for me, and I am thrilled that I can still change my view. The biggest help for me is how students are encouraged to do the math in the way that works for them, and not just one way. I am not very confident about my Math skills yet, but I am realizing that I know more than I give myself credit for. I love the way our text uses student examples, like on p. 259, showing where Latisha drew her explanation of how she understood that the 2 balls were the same weight as 1 cylinder. The examples are what make the chapters make sense to me and help me to realize that every one can do mathematics. One p. 262 there is a sentence that was sort of like a small epiphany for me when it talks about the use of variables. It says: initial work with finding the value of the variable that makes the sentence true should initially rely on relational thinking. It goes on to say "later students will develop specific techniques for solving equations when these relationships are insufficient." I think that the reason this was so exciting to me is because sometimes I want to give students too much information and I forget that they need that initial relationship to build on. At some point the relational thinking may not be enough to solve the problem, but it is an important piece to build on.
ReplyDeleteDeidra~ I agree that I tend to forget that a problem with a missing piece in it (such as 2+ __ =4) involves algebraic thinking. This chapter is full of reminders and information that help me realize that there are a lot of things that are actually algebraic thinking, like patterns and the odd and even number relations.
ReplyDeleteChapter 14 was about algebraic thinking; generalizations, patterns, and functions. Children begin learning about algebra in prekindergarten and it continues through high school. There are five different forms of algebraic reasoning: generalization from arithmetic and from patterns in all of mathematics, meaningful use of symbols, study of structure in the number system, study of patterns and functions and process of mathematical modeling, integrating the first four list items. I struggled with algebra in 7th grade, but then got a teacher that explained it very well and from then on I loved doing algebra. The two weeks before fall break in my internship the students were learning algebra. My mentor teacher did a really good job and breaking it all down and gradually implementing variables. She also involved real world things in which the students were using variables. I feel algebra is a concept that can be confusing for students because letters are being substituted for numbers and that can be confusing. I think it just needs to be explained in a way students can understand. It is important to create word walls with the vocabulary needed for algebra and to teach the students to use the vocabulary when using algebra. It is a good idea for the students to keep a journal of the words.
ReplyDeleteIn response to April B,
ReplyDeleteI remember being in algebra in middle school and I had teachers that actually gave us different types of activities that helped us learn algebra. My internship right now is 6th grade math and they have learned algebra and my mentor teacher did a good job at incorporating real life items. I did my lesson on distance=rate*time and they caught on quickly because I related it to different things they see everyday like the speed limit signs and other things. They also had to plot points on a graph and I made a reference to battleship and it clicked. I think how a student feels about math in general depends on the teacher and I think that's the same with all subjects.
The biggest takeaway I had from Chapter 14 was the importance of symbols and vocabulary. Dr. Stramel does a great job of emphasizing the importance of using different verbiage for =. However, I'm not sure I really got it until I read about in this chapter. The light bulb finally turned on. The equals sign represents relationships. The distributive property helped me make sense of it. 6 x 7 = 5 x 7 + 7.
ReplyDeleteThen comes vocabulary. If I would have been hearing a lecture or listening to someone talk about the distributive property I would have had no idea what they were talking about. I have been surprised by the terms that are used in the 5th grade class. Like the text states a large part of understanding mathematics is the ability to communicate mathematically. If we constantly use the correct terms and have our students use the correct terminology the students will have an easier time in the future.
Brandi Schroeder
@ Lacey
I'm so glad you had a good experience with algebra. Using Excel files with Algebra is great. Those are two things that I always struggle with. For one I had a bad experience with algebra and hated it and never really had a class for Excel (mostly taught myself). I love the idea of integrating them. I have observed a lot of lessons with Powerpoints and Word documents, but nothing in Excel.
Chapter 14 was all about algebra. Unlike most people, I happen to have loved algebra. However, after reading this chapter and spending some time talking about the topic in class, I can understand how algebra might be very confusing for some people. There are a lot of terms that are used interchangeably so it is important for students to understand the vocabulary behind algebra. One section that we have discussed in class is the meaning of the equal sign. Many children think the equals sign means what is the answer. They do understand that it means everything on one side equals everything on the other. This is something I have experienced with the students in my internship. When given problems such as 8 x 3 = 6 x ___, they do not understand what to do. I like the idea of drawing a balance with expressions on either side to demonstrate the meaning of the equals sign, I think this would be a good way to gain student understanding. Once again, there were a lot of great activity ideas throughout the text. Hopefully I can get a job teaching middle school math and refer back to this text often!
ReplyDeleteKayla R -
ReplyDeleteI agree with you about teaching and displaying vocabulary. Most people don't think to incorporate vocabulary lessons into math class, but it is very important for understanding and shouldn't be overlooked.
Algebraic Thinking begins in prekindergarten and continues through high school and college. This chapter talks about how algebraic thinking and the different forms of it in the classroom. A good part of this chapter is the part about the meaning of the equals sign. It is so important for children to understand the meaning of the equal sign. I think a bad way to teach it is that the answer goes behind the equals sign. There are several tips for teachers to help them understand the meaning of the equals sign. There are many other definitions in this chapter that will be helpful for me in the future. Patterns in this chapter are uniquely talked about. There are so many ways to teach patterns. Actually you can teach patterns in so many areas its crazy. Teaching patterns can’t just be in one area.
ReplyDeleteIn response to Elizabeth A.
ReplyDeleteI also did not know how important algebraic thinking was in younger years. There are so many ways to incorporate it into algebraic thinking into children's curriculum. I did not even know that algebraic thinking started in kindergarten and it is still surprising to me.
This is very helpful! I now realize just how many algorithms there while tutoring at a local elementary school. I now know what it felt like when I took homework home and my parents had no clue what my assignments were discussing, especially within mathematics. My personal favorite algorithm is the Russian Peasant. This is obviously a mathematics function from Russia and takes some time to work out, but is very interesting and you can always check your work with this algorithm (as long as your procedures are correct). Has anyone else seen this used within schools today or practiced this form yourself? Just learning all of the new ways of algorithms is fun for me, I know it's students have more fun and feel like there are options for them in mathematics-because there are options! As long as students get the correct answers, there are good to go. They can also help each other by showing their peers their favorite algorithms within the classroom or at recess.
ReplyDeleteBrandi-
ReplyDeleteI agree, symbols were key and VERY helpful within this chapter. I like this book overall-I love all the references. Knowing all of the symbols is a domino affect to knowing what's down the road... For us as educators, but also are students.
This chapter talked about algebraic thinking. I liked reading this chapter because it related to my internship I am in right now. The chapter talked about understanding equal signs. I know my students are having trouble with equal signs when they are in different positions. Sometimes its in front and sometimes it's at the end. Most of the students understand but there are some that don't. It can be difficult for students to understand that it is asking the same thing no matter where it is in the problem.
ReplyDeleteElizabeth Adams,
ReplyDeleteI agree I think it is crazy how early children begin to learn algebra, without even knowing it! I think the term "algebra" can be a turn off for some students. Maybe if we as teacher just talked about the lesson instead of algebra the students wouldn't get so worried about it and may enjoy it more.
One day in class we were asked what we thought Algebra meant. I related it to working through mathematics by using order of operations and variables. I thought of it as being used in higher grades but the text states that algebraic thinking begins in prekindergarten. I hadn’t thought about the fact that algebra deals with lots of patterns which prekindergarten students work with a lot. The section in this chapter that discussed patterns on hundreds charts had an example in figure 14.2 that reminded me of the students in my observation class. I hadn’t thought of it as being algebra at the time but I remember that when they would count by multiples on hundreds charts many of them would stop counting after the third or fourth number because they would realize a pattern being formed and just followed it. Algebraic thinking is all about patterns and I don’t think it’s bad that the students used patterns instead of counting each multiple out. Like we’ve learned, mathematics is the pursuit of laziness.
ReplyDeleteJordan O.,
ReplyDeleteI was also able to relate this chapter to experiences in my internship class right now. Although mine was with the hundreds chart examples I also noticed how students have troubles when the equal sign is moved. It can be confusing to students when they constantly learn that the equal sign goes in the same place over and over. Like the text says, having the equal sign in the same place constantly make students see it as standing for “and the answer is”. It can be confusing to see it as something different when it’s always mean the same thing for so long. I can see why some of the students in your internship class may have trouble at first. Good post!
Chapter 14 had some interesting concepts. I first noticed how it mentioned that we start learning about algebraic thinking in prekindergarten years and then continues through high school. I think we don't realize the importance of the younger years at times but when we put it into perspective we learn SO much from what we learn as young students and these things travel with us through high school. Once again when looking at the part of the chapter about the hundreds chart. Another thing that we learn at a young age that I know i still use this when figure math. At least I use what I learned with the chart for many daily activities like adding by 5's or 10's.
ReplyDeleteIn response to Shannon H:
It's funny that you also talked about how learning algebra starts in prekindergarten. I was also amazed by how algebra isn't just something you start in high school but also something we learn in grade school and carry out through out our school years.
I did not realize that algebraic thinking begins in prekindergarten. According to the book, children at this age begin to recognize and duplicate simple sequential patterns. After reading about this, I can now see how kindergartners use algebra and how the teacher goes about teaching it. Kaput describes five forms of algebraic reasoning which are: 1) generalization from arithmetic and from patterns in all of mathematics, 2) meaningful use of symbols, 3) study of structure in the number system, 4) study of patterns and functions, and 5) process of mathematical modeling, integrating the first four list items. I find this list very helpful as it explains how algebraic thinking is not just one idea but several pieces put together.
ReplyDeleteIn response to Tessa W –
ReplyDeleteIt is very important to begin algebraic thinking as soon as possible and I too thought it was interesting that we begin before kindergarten. It is amazing to know how young we begin to learn things and that they stay with us all throughout life.
Joel Stucky
ReplyDeleteYou mentioned that the equal sign means 'same as'. If you remember it was also mentioned that we will typically tell small children 'take away' for 'subtract' and that this can be confusing when they get into negatives.
It is very important that we are sure to use and teach proper vocabulary!
Good note.
An interesting thing happened to me on the way to the forum [math].
ReplyDeleteI have been working with math students for 6 years and it wasn't until the 7th year that I put 2 and 2 together.
Our text says that the 'box' "is a precursor of a variable used". Basically, we learn algebra in early grade school years.
I was working with a student on a problem somewhat like: 'what' - 212 = 40 when I realized it was algebra in the sense that it uses an unknown number within the equation. We teach the students to add the two known (in this case) to find the unknown, however, we will give the same student the same problem in an upper high school grade and tell them to balance the two sides of the equation!
Wow. Imagine my surprise when I realized we learn and teach algebra all through our grades! Imagine the pride my student had when I told him he was doing algebra!
Carissa Kruse
ReplyDeleteChapter 14-BLOG
Chapter 14 is about Algebraic Thinking. It seems weird to think about it because we don’t enter into classes called “Algebra” until junior high and high school, but Algebra is happening from the very start of a child’s math career. Students are constantly being asked what the missing number is or how many different ways can different things be put into order.
I have seen Algebra used weekly in my internship and on one of the weekly math flyers the students are required to do, there is even a section named Algebra. It worries me that every week when the flyer comes around the students psych themselves up and get worried about completing the algebra section. Too many times the students think too much about it and expect it to be much harder than it is. When it is time to do fix-it papers, and the problem is explained a little better to them, the students often need to fix the algebra section and are surprised they didn’t get the right answer the first time around.
I think a lot of this chapter is helpful but a lot of it had to focus on older students. It is important to calm students down and let them know that all too often the algebra is easy to figure out.
In response to Linda McC...
ReplyDeleteI never thought about how early algebra is used until I started this course. From the very beginning we are helping students to find "unknown numbers" without them realizing that they are actually doing and learning algebra.
In response to Jena,
ReplyDeleteI completely agree with you that we must use the same terminology. When I was doing my formal teaching I remember I said subtract and minus in the same breathe! Just because I know what it means doesn't mean that a classroom of 5 year old will know they mean the same! It is imperative that we stick with the same terms to give the students a better understanding! Great point!
I never realized how early we really start using Algebra. As early as kindergarten sounds crazy, but when you take into consideration of patterns and finding the missing block or number it absolutely is Algebra! It was interesting in class doing some of the activities in our small groups and not even realizing that we were doing Algebra! I think as we get into the actual Algebra class in junior high and high school, we tend to freak out and have some sort of mental block! I did anyway. If I wouldn't have seen the word "Algebra" and just had done the work I would have done better!! I think this chapter gave me great insight on how often we actually use this successfully and don't even know it!
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