Welcome back to our book study of * Teaching Student-Centered Mathematics* by Van de Walle, Lovin, Karp, and Bay-Williams. To read past posts, visit our

**book study archive**. Today we discuss Chapter 9.

**Chapter 9 – Developing Meaning for the Operations**

Growing up I struggled with math. I hated timed fact tests, I hated the “tricks” that were taught to me because they didn’t make sense to me, and I hated word problems! I was one of those kids who would reread a word problem 3-4 times trying to make sense of it and eventually just take a leap into what I thought it was saying.

Fast forward several years and I am a teacher in a 2^{nd} grade classroom. Our district just adopted Saxon Math. As a first year teacher, this program was great because it told me what to do and say when teaching. I did very little with conceptual understanding, but being a person not strong in math, I didn’t realize this. I became that teacher that was trying to teach the “tricks” and was frustrated when they weren’t getting it….I just couldn’t understand why they were writing fact families:

3 + 8 = 11

8 + 3 = 11

11 – 3 = 8

8 – 3 = 11

I emphasized that when adding the biggest number was always last and when subtracting, the biggest number was always first! I mean why were they not getting it?!?!

Chapter 9 focuses on developing meanings for the operations. Children need to be aware that the operations are all related to one another. They will learn that models can be used to solve contextual problems for all operations, and figure out what operation (or operations) are involved in problems. I also love how the mathematical practices are referenced throughout the chapter.

Fast forward to a few years ago and my understanding of math has COMPLETELY changed! I love how the authors in this chapter say, *“through research, we are aware that children can solve contextual or story problems with appropriate numbers by reasoning through the relationships in the problems. We also know that different problems have different structures that can affect the difficulty level of the problem. When teachers are familiar with these structures, they are better able to plan and differentiate instruction”* (p. 127).

Table 9.1 on p. 128 gives a wonderful illustration of addition and subtraction problem types using the number families (oh, those ole fact families again!) After reading this table, I thought about those signs I had seen in classroom and on Pinterest with the large plus sign and minus sign. Inside the signs were phrases students could reference to know if they added or subtracted in a story problem. We shouldn’t teach those “tricks”, but we should teach the problem types so students see the relationship between addition and subtraction and solve using strategies that work for them (this is talked about more on p. 147). It is important that children experience all the problem types to ensure they are developing a broader understanding of addition and subtraction.

Oh, those pesky word problems! Students like to solve them in chronological order. As a teacher I see this all the time. They also find the result unknown problems to be the easiest. *“Many children have difficulty with problems in which the start is unknown because they try to model the problem in chronological order and they cannot make the set that represents the beginning of the problem. Write a question mark on an index card and have children use the card to represent the unknown amount”* (p. 131).

I LOVE the section on Introducing Symbolism. I completely agree that the minus sign should be read as “minus” or “subtract”, but not “take away”, and the plus sign should be read as “plus” and many times as “and”. I have also read the equal sign as “the same as” in my own classroom because, just as the authors mentioned, children begin thinking that it means the answer. First grade is when students learn about the equal sign, and this needs to be reinforced in later grades. The * True or False *activity on p. 134 is a great activity to review equal.

Another great piece to this chapter is how important modeling is when solving problems. We have said before that the conceptual learning and understanding is huge at this level. Students need the opportunities to see and prove how to add parts and subtract parts. This chapter gives many great models and activities for problem solving.

When students understand the relationship of addition and subtraction, they will easily move right into multiplication and division. We are laying the foundation for multiplication and division. Second graders begin to work with equal groups. Equal group problems involve 3 quantities: the number of groups, the size of each group, and the total. For example:

*Jill has 4 bags of crayons. There are 3 crayons in each bag. All together Jill has 12 crayons.*

Students will look for and make use of structure just as they did with addition and subtraction. They will also make sense of problems and persevere in solving them…I just love those mathematical practices.

They say that solving word problems should be done on a regular basis and should be a significant part of math curriculum. In our classrooms, we problem solve with word problems daily, whether in math journals, guided math groups, whole group discussions, or math warm-ups. The authors point out that there are many interrelated objectives that you should have in mind when you pose story problems:

- Understanding the various meanings of the four operations
- Development of number skills and concepts
- Computational fluency

Finally, I love that they referenced the fact that we need to teach students not to spend a lot of time when drawing models for problem solving. I know this is a discussion I have in my classroom every year. The little ones like to try and draw exactly what the problem is talking about, but they can lose sight of the mathematics. Let them know that, in mathematics, drawings should be simple and only include information that helps them solve the problem.

I only hit on the surface of this chapter. Problem solving can be a struggle for students, but learning what the authors have presented will allow for students to become more comfortable with problem solving and give teachers more guidance with creating problem solvers!

Join us again on July 8th as we continue our book study with Chapter 10 – *Helping Children Master the Basic Facts*. AND, if you haven’t already, **visit Chapter 8’s post for a chance to win two WONDERFUL number sense resources!**

The Math Spot says

I really identify with your opening statements about not understanding math in school because of the “tricks”. I could never remember the rote steps but, when I had a conceptual understanding, I was a strong student! This book is so great about providing practical examples of how to go about laying this conceptual foundation.