Young Mathmaticians 5-7

I think I have mentioned this before, but the fourth and fifth grade classrooms I tutor in are currently working through fractions right now too so it has been interesting for me to think about what I am seeing them struggle with compared to some of the examples raised in the texts. One thing that I think creates struggles for students with fractions is that it seems like the classrooms I observe usually go over one way to add or multiply fractions and it remains a very abstract idea for students. I have seen things like the double number line like they talked about in chapter 5, but this text just demonstrates how many ways you can approach looking at fractions. I looked at the array for quite awhile- I remember being first introduced to it in math 201- and while I can understand the answer I'm not sure I can really make sense of it for myself. I could easily plug the numbers in and make an array but I just can't relate it to anything else. Fractions are difficult for me just because I feel like I always think about them in abstract ways. It is hard for me to really make sense of dealing with complex fraction problems- I can do it- I just feel like I don't always understand exactly Why I'm doing what I'm doing. For some reason with the array in chapter 6 when they put a short story problem with the array it seemed to make more sense to me. I could kind of picture what was happening instead of just seeing tiles shaded in for no real apparant reason. Based on my own experience, making fractions as realistic to students as possible is an important part of teaching multiplying fractions. In chapter 7 I also thought it was advantageous for the teacher to have students work with ideas of multiplying (in this case groups of dots) before she introduced the idea of looking at it as fractions of the whole. Students already explored some math around the problem and found the whole before they started thinking about fractions. This also helps connect what students already know about multiplying and dividing whole numbers. I have heard of strategies like using the clock and money before because it is something very concrete to students. I am looking forward to talking more about this topic just because I feel like it is something I am weak in.

YMaW Ch. 5-7

First and foremost I just want to say that this week's readings were very tough for me to comprehend and understand. Not only is the material extremely heavy in content and intricate processes but it also founded upon one's basic knowledge and understanding of fractions. Because I have not taken a math class in 2 years it has been a struggle for me to understand the strategies students within the text are implementing and the rationale behind them. That being said, I did take away some very knowledgeable ideas and resources that could be used when teaching in the future. Primarily, I learned how important it is to develop a plethora of ways to model mathematical problems and situations so that students can mathematize effectively. Page 73 states, "Mathematizing-the human activity of organizing and itnerpreting reality mathematically-rather than a closed system of content to be transmitted or even discovered, mathematical models become very important." The book then goes on to provide a variety of direct modeling strategies that can be utilized to directly represent equations concerning fractions. I found the Ratio Table to be an extremely efficient way both organize fractions and rations and determine how to adjust those ratios by either making them bigger or smaller. The number line and double number line models, however, proved to be rather confusing to me and I am not sure I thoroughly grasp the concept. Another way to represent the multiplication or division of fractions is through use of an array. Arrays assist students in visually representing the direct action taking place in the equation and allow them to obtain appropriate answers through these means. On page 125 it states, "When children are given the chance to compute in their own ways, to play with relationships and operations, they see themselves as mathematicians and their understanding deepens." I found this quote to exceptionally summarizing of the preceding chapters in that its focus and emphasis remains on allowing children to explore mathematics in order to find what models and/or methods are most appropriate and conducive to their own, specific learning. As future teachers, this is one of the primary goals we should strive for.

April 16 Readings

I realized that this week's readings seemed to focus more on discussion and mathematical thinking than anything else. I found the article in particular to be extremely interesting. After the CGI book, I figured that discussion could be helpful to children's mathematical development, but I never realized the extent. I realized that the discussion process is just as important as the discussion itself. This article taught me about how important it is to have children make mathematical arguments. In order to do this, we as teachers, must set standards and expectations beforehand so that our students have a safe place to share ideas. I found it interesting that from revisiting their own thinking, "The student not only develops a through understanding of the ideas that she is grappling with, but she may also construct new understandings" (Whitenack, 525). I do worry personally about teaching these processes though. I do realize that children naturally question and form their big ideas and strategies, but I always find myself asking "what if?". The teacher in the article, "...focused on different aspects of the students' reasoning and continued conversations with different students in different ways" (Whitenack, 526). From this, I ask myself, what if I miss opportunities with children or are not able to recognize gateways to these conversations. I suppose this is something I will hopefully learn in experience.

Math content and processes

I read the Whitenack and Yackel article first where they were highlighting a classroom where the teacher had set up an environment where children discussed and defended mathematical ideas with each other. The teacher was able to facilitate the discussion to allow this to happen and you could see the process of sense-making in math really unfold for these children. In all of these articles it always seems so effortless in the classroom. As I prepare to teach my own math lesson, I wish I had more resources available to me about how some of these teachers started working to build this sort of environment. I wonder how much of it comes naturally and how much teachers have to help foster to make children comfortable with and eager to share.

While reading about fractions in Young Mathematicians the students had so many different ways to approach the problem. When I read about the "which cat food is cheaper" I immediately solved it the way that Helaina and Lucy did and didn't really think about it much further. The process skills in this classroom are a lot deeper though, these students were really pushing to find different ways to explore the problem. The two boys constructed a ratio table and figured out their answer that way, then two other girls came through and did it the same way with different numbers to find out the answer as well. I wouldn't have thought that this conversation could have led to so many different ways to discuss equivalent fractions but the teacher was able to steer the conversation from method to method to discuss different ways to represent the same idea. This teacher seemed to demonstrate the approach of getting as many students involved in talking about the math as possible through the use of many different solution ideas and the "pair talk" which I'm assuming was where students talked in pairs about a certain math idea so everyone had a turn to express their thoughts.

The big idea discussed in chapter 4- the whole matters- seems so obvious that it feels odd to me that this was never brought up when I have taken prior math classes ever. At some point of course I realized it, but it was never consciously in the front of my mind. Yet in Carol's class the students are free to explore this idea which really will help ground fractions with different denominators in their minds. Fractions are always in relation to one another. Again, in this case I thought it was so critical that the teacher had the students present so many different examples of the problems and were allowed to explore the idea in depth without her just giving an explanation that might not have made sense to them. This seems to be an integral part of having content make sense to students and to allow them to develop the process skills necessary to really "do" math.

Content & Processes in YMAT Ch. 3 & 4

Upon completing Chapters 2 and 3 of Young Mathematicians at Work, I can identify several content and process goals in place. First and foremost, like Carols classroom, Susan was focusing specifically on the content of developing fractions and reducing them to their lowest form. While the word problem she presented her students with did not necessarily constitute a context problem and provide opportunities for the students to engage in a plethora of strategies, they were still able to solve the problem using a few different tools. (After reading the chapter, I now know that the use of a variety of tools does not necessarily suggest multiple processes implemented. In the case of this classroom, tools vs. strategies was what ensued as a result of the provided problem) While some students chose to implement paper and pencil techniques, other were able to perform the necessary computations in their head, while still other implemented unifix cubes. However, this primarily represents a use of diverse tools rather than processes because all students approached the problem from the same angle knowing that it indeed required a fraction of 18/24 to be reduced. Perhaps had Susan framed the problem more similarly to Joel, who provided a context problem regarding the best buy of cat food, the students would have had a better opportunity to approach it from various angles. Looking at the problem that Joel presented to his class, we can certainly see that not only did his students use different tools, but used different processes and strategies in order to obtain the solution. Because he presented them with a context problem, one that had meaning to the students without an implied agenda, they were able to truly explore all of the options for solving. While some students began dividing 15 by 12 and 20 by 23, others began reducing the number of cans and price in half until they reached the number 1. Some students solved for 20 cans, figuring out how costly these would be at each store, while still others solved for one can in order to determine the difference and decide which store was the better buy. In addition, this problem lent itself to rich inquiry and truly engaged the students in the task at hand. Those these two teachers had the same general content goals in mind, the processes that each problem provide were drastically different. While Susan's word problem was primarily teacher-driven with more of a linear model of problem solving, Joel's was student driven and allow the students to explore a variety of feasible processes.

Young Mathematicians at Work

Upon reading the assigned chapters of Young Mathematicians at Work I have made a few discoveries. First, I realized that the ways that students' problem solve may be drastically different from the approach we may personally choose to implement. For instance, when presented with the "submarine sandwich" question in class, my immediate instinct was to whip out my calculator and begin computing division problems in order to obtain percentages. The student's within Carol's 4th/5th grade class, however, used a plethora of strategies such as manipulatives (unifix cubes) and visual representations (drawings, charts, etc.). Because these strategies are not ones that I, myself, was naturally inclined to implement, it was extremely difficult for me to make sense of their thought processes. I struggled while reading, and re-reading, the passages describing the various steps the children took in their solution strategies. As teachers, it is so important that we wrestle with mathematical problems ourselves and find multiple ways of solving it before introducing it to our students. In doing this, we are able to discover a variety approaches that can be taken to problem solve and anticipate responses and methods that students may offer. In addition, I also learned that though this story problem is directly related to fractions, decimals, and percentages, it is not necessarily appropriate or warranted to force these equations and ideas upon students. As Fosnot states, "Asking children to adopt multiplication and division shortcuts too soon may actually impede genuine learning. When introduced at the wrong place or time, good logic may be the worst enemy of good teaching" (5). This serves to say that though developing these concrete ideals is the focus and goal of the lesson, it is best to let children discover these processes on their own through the formation of connections and constant exploration. If this knowledge does not arise naturally, or out of genuine curiosity, then it may not be thoroughly understood or retained. Finally, the text stressed the importance of developing mathematical equations that are both relevant and meaningful to the student's lives. By introducing the "submarine sandwich" problem, the teacher was able to present a real-life dilemma that she encountered with a previous class and sincerely enlisted the help of her students in solving it. Fosnot states, "When children are given trivial word problems, they often just ask themselves what operation is called for; the context becomes irrelevant as they manipulate numbers, applying what they know" (2). In essence, through the development of problems that students can relate to, they will be more willing to invest the time and energy that it takes to truly explore the possible solutions rather than robotically computing meaningless information.

Division

Since we talked in-depth about the sub problem last week I wanted to start off with that. I was amazed at how frustrated I was while working on that problem. It's no wonder so many students do not understand fractions or spend a lot of time trying to understand how they work. I'm still at the point like many children where it is much easier for me to think about lots of fractions while drawing pictures. In the fifth grade classroom that I tutor in last Friday they were working with fractions by having fraction bars with each fraction laid out. I thought these were helpful because I was working with some of the students to discover what equivalent fractions were. We worked on taking things like the 1/3 bar and seeing what other bars would line up exactly equal with it, like 2 1/6 bars, or 1/2 and two 1/4 and four 1/8. Students could also explore how as the denominator got larger the size of the bar got smaller.

While I was reading the DMI chapter on division I was trying to pay attention to the different strategies that were being used. The problem like case 24 when the teacher was trying to get students to divide 134 jelly beans among 6 children was just like the division problems I was working on with a student in the fourth grade classroom that I tutor in. I was really amazed at all the different strategies that April came up with to try to solve the problem. Since it took her so many steps I was left thinking the same thing the teacher was at the end, when should we teach students the standard division algorithm? Or should we at all? Is it better for students to struggle and reason through it and then put it into an algorithm or is the algorithm even necessary? I wish I had read through this before working with the 4th grade student in my class on this subject. The teacher had only provided students with these flat counter chips that were rather difficult to work with because it was hard to create piles of them to keep numbers straight. I first tried to have the student model the division out but as the numbers quickly got a lot larger I realized there must be an easier way. The student seemed to sort of grasp dividing by separating into different piles but I'm not sure he completely grasped what he was doing at this most basic step. I think I was leading him too much and not really allowing him to explore the problem on his own. In retrospect I can see this, at the time I was just celebrating that he managed to sit in his desk and work on something for 30 minutes, which was a very large step for this student. I tried to work with him on trying to estimate for some of the larger numbers like April was doing but without a basic knowledge of his multiplication tables this quickly proved to be an impossible task. I worry about solidifying his division knowledge without a solid understanding of multiplication. In Betty's case she was using base 10 blocks with her students. Maybe that might have made this task somewhat less confused because a major part of each problem was sorting out the counters and then recounting and he was often one counter short which was very confusing when trying to arrive at a correct answer. I understood Betty when she was talking about how it was difficult to see the division when she counted out 10 7 sticks and 2 cubes to illustrate 72 divided by 3. I think just looking at that without seeing each one as a single I would not immediately think of my first step for division. I hope after our discussion in class about this issue I will have some more concrete ideas to take back to my 4th grade student next time I tutor him.