Teaching Student Centered Mathematics Book Study: Chapter 9

All I have to say about chapter 9 is yes, yes, yes, yes, yes!!!

Alright, I have a bit more to say than that but what a great chapter. I found this chapter to be so affirming in terms of why we use word problems and how to teach our student to navigate these tricky, tricky waters.

The chapter discussed students solving word problems in order to learn about and solidify an understanding of operations, to further develop number skills and concepts, and to promote computational fluency. This is to say that a lesson containing word problems could be addressing any number of mathematical goals. This is something the teams at my school talk about a lot. We are using the Engage NY modules but I am sure the conversations we have mirror those of any elementary teacher picking up a program and determining what that means for daily instruction. Questions we often have include "Why are the modules using such BIG numbers! Second grade standards only expect word problems to 100?" or "Why are these word problems so easy? Of course my students can tell how many cookies there are when Ashley brings 3, Sarah brings 2 and Suzy brings 7 cookies... can I skip this day?" Sometimes we are right on the money with our questions and we are able to adjust the lessons as necessary. Sometimes, we are not understanding the purpose of the lesson. Maybe that second grade question isn't really about solving a story problem with sums to 100. Maybe that second grade question is about adding together two three digit numbers but the context of the story allows the students a jumping point for an invented strategy. Maybe the first grade word problem isn't really about adding 3 numbers together, maybe it's about noticing that when combining three numbers, you can be strategic about the order in which you combine them for ease in solving. Ashley's 3 cookies and Suzy's 7 cookies make ten cookies. Add on Sarah's two and we can quickly notice that 10 and 2 makes 12! Whenever introducing a word problem is it important to note the PURPOSE with which you are presenting that learning opportunity.

This chapter also had a small section on avoiding key words. Yes, yes, yes, yes, yes. I wrote a rather long entry on avoiding key words a few months back and this chapter confirmed what I had written... and did it in a much more concise manner :) My favorite quote from this section stated "The key word approach encourages children to ignore the meaning and structure of the problem and look for an easy way out." I could probably go on and on about this quote but isn't it the truth! When we teach our students these shortcuts we are doing them no favors towards helping them to develop their number sense, operation sense, and overall ability to think mathematically. If you are using key words to teach word problems (or rhymes to teach rounding, or "The Butterfly Method" for comparing fractions or any other method that requires no more than a rote application) your students may show success on the lesson today, but will they know it is appropriate to apply this rote process tomorrow? And next year, will they have a strong foundational understanding of operation upon which they can connect to and build future math understandings?

*Stepping off of soap box*

So, if key words are off of the table, how can we help our students to solve word problems? The chapter discussed the methods that students might use to solve an addition or subtraction word problems including direct modeling, counting strategies, and derived facts. Interestingly enough, the authors mentioned that younger children sometimes have an easier time attacking word problems because they think more completely about the context of a problem whereas older students "think that [computation] is what solving story problems means- grab the numbers and compute". A way around this with older students is to require explanations for the way in which a student arrived at their answer. And an equation is NOT an explanation. When students are required to draw a picture or use words to justify their response they can no longer "plug and chug" to solve.

In terms of modeling, the text cautioned students from drawing intricate illustrations in order to solve a word problem. This thought was echoed at a math workshop I was able to attend a few months back. The speaker at the conference really resonated with me in terms of how she recommended dealing with this problem. She said not to tell students that their way of solving the problem is wrong- it's not! She said that if students are drawing a full on illustration that you want to pair their work with a more abstract representation, for example, a student who drew circles to represent bunnies. Ask the student to find the different features in the word problem in their own illustration but then ask them to find the part in the other students abstract representation that means the same thing. Then ask the student which method was more efficient (or faster) when it came to getting the right answer.
The prompting might sound like this:
The story said that first there were 3 bunnies in the field. Where are the three bunnies in your drawing? Oh, yes, that matches the story. Let's look at this model of the story. Can you see the part in this model which shows the three bunnies that were in the field at first? Oh, you see a group of three circles. They used circles to stand for the bunnies! Which way is faster or a more efficient way of showing three bunnies? The next time we solve a problem, why don't you try this more efficient way of drawing.

This conversation does not need to occur 1:1 but could be a full class discussion. Additionally, this method of comparison and conversation can move students from a direct modeling approach to a more efficient bar model or tape diagram as well.

The ends of my posts are beginning to sound like a broken record but, really, truly, this chapter was so dense with information that my reflection is only brushing the surface. I would recommend clicking the links below to read others' reflections on the chapter and/or picking the book up yourself!

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