Hoping today’s post finds you doing well! We continue our book study of Teaching Student-Centered Mathematics PreK-2 by Van de Walle, Lovin, Karp, and Bay-Williams with a discussion of chapter 12. To read past posts, simply visit our book study archive!
Chapter 12 – Building Strategies for Whole-Number Computation
“The days of children memorizing computational algorithms are fading fast. Rather than constant reliance on a single method of adding and subtracting, methods can and should change felxibly as the numbers and the context change.” This quote sums up chapter 12 quite well, so I will take it apart in pieces (out of order) and add a bit more detail as presented by the authors.
“Methods can and should change flexibly as the numbers and the context changes.”
One of the best aspects of this chapter is how the authors help the readers understand the three types of computational strategies, the importance of each strategy, and some suggestions for guiding students in the use of each strategy.
In this stage, students use tools, drawings, or fingers along with counting to show meaning. This is an important stage for students to work through. Students should not be pushed beyond this stage too soon, yet they will eventually need to move past this stage to invented strategies.
The authors define invented strategies as those other than the standard algorithm that do not rely on the use of physical materials or counting by ones. It is important to remember here that when a student uses an invented strategy, he/she may still draw a model to show his/her thinking. According to the Common Core Standards for Mathematics, first and second graders need to be able to use their understanding of place value and the properties of operations to solve addition and subtraction problems. It is here where invented strategies camp out. Some examples of invented strategies follow (keep in mind that the numbers added or subtracted would most likely be found in the context of a word problem/problem solving situation):
9 + 8 —> 10 + 8 = 18, 18 – 1 = 17
23 + 56 —> 20 + 50 = 70, 3 + 6 = 9, 70 + 9 = 79
125 + 236 —> 130 + 240 = 370, 370 – 9 = 361
600 – 245 —> 250 + 250 = 500, 500 + 100 = 600, 250 + 100 = 350, 350 + 5 = 355
84 – 35 —> 84 – 30 = 54, 54 – 5 = 49
96 – 78 —> 78 + 2 = 80, 80 + 10 = 90, 90 + 6 = 96, 2 + 10 + 6 = 18
You will see, an understanding of place value and the operations is needed to use any of the above strategies. I am sure you can also see where many of the ways of thinking shown above lend to mental computation. Could all of the above strategies have been used by the same student? Absolutely! The invented strategy a student uses will vary depending on the numbers in the problem. My last year’s students could be found using any of the above strategies, and they were expected to show their thinking. This is where writing equations and/or using an open number line to show thinking become common place in my classroom. As a second grade teacher, this is the stage I want my students to get to when adding and subtracting within 1,000.
The standard algorithm of lining numbers up top-to-bottom and adding or subtracting by “carrying” or “borrowing” was how I learned. It’s funny, though—I can’t even remember the last time I lined up numbers and followed the steps of the process for adding or subtracting something. Why? I have more efficient ways of adding and subtracting mentally. What have I done then? Well, added up to subtract (creating landmark tens or hundreds), estimated, created friendly numbers…
I like the position the authors take on teaching students the standards algorithms for addition and subtraction. Yes, they need to be understood. Yes, they should be delayed until students have developed invented strategies and possess the understanding needed to understand why the standard algorithms work. Yes, they are strategies and should be valued as such. BUT, as a second grade teacher, this should NOT be the main focus. I loved how the authors gave the example of 7000 – 25 to illustrate how the standard algorithm would not be efficient and would lend to errors. I don’t know about you, but I do not want to line up those numbers and subtract across zero. BUT I would love to use what I know about sums to 100 and 1,000 to help me!
“The days of children memorizing computational algorithms are fading fast.” Why is this?
- Algorithms for addition and subtraction do not build on the composing and decomposing of numbers in ways that enhance a child’s number sense.
- Many invented strategies are easier and faster than the standard algorithm.
- Students typically make more errors when using standard algorithms versus alternative methods.
- Invented strategies naturally lead to mental computation and estimation.
- Alternative, or invented strategies, are flexible and instill in students that there is not one right way to compute.
- When ample time is given to students’ use of invented strategies and making meaning, less reteaching is required.
- A reliance on teaching standards algorithms leads to students using a process without understanding why it works. Furthermore, students focus on digits rather than the value of numbers.
“Rather than constant reliance on a single method…”
Establishing a safe environment for students to invent, share, and borrow strategies is essential for making sense of mathematics. There is great power in helping students understand that there isn’t one right way to compute. Students must be encouraged to explore their thinking in a setting that praises risk taking and respects the thinking that is shared. The authors give some great tips for establishing such an environment:
- Don’t immediately identify the correct answer when shared. This gives everyone the opportunity to ponder whether a shared answer is correct.
- Encourage students to ask questions of one another and engage in thoughtful discussions.
- Address right and wrong in a way that is not evaluative or threatening.
- Use coaching and questioning to move less sophisticated thinking to more sophisticated thinking.
- Use familiar contexts and story problems to build background knowledge and connect experiences.
- Instill in students an openness to new ideas and trying new things.
From the very beginning of the year, my efforts begin to establish such an environment for my students. Much time is spent helping students understand that mathematics is more than just getting an answer—there is great value in understanding why a strategy works and that there are multiple ways to arrive at a solution. Being able to talk to others about our thinking is an important skill in our classroom. Students learn how to do this in the whole class setting, in small guided math groups, and independently with a partner or partners. Years ago, when working with fifth graders, I created a discussion aide that I adapted when I began teaching second graders. I created this aide, a discussion fan, because I truly believe that students can and need to have meaningful discussion about their work without me. Yet, students may need a place to begin. Therefore, I introduce discussion fans to my students early on. As time goes on, students use the discussion fans less and less because they naturally begin asking their own questions. Click here to read about the discussion fans my students use and to download a free copy. Last year, Courtney and I also created a performance task/assessment for our students and incorporated discussion by using math talk cards. Students can choose from the cards or draw them randomly to guide discussion. You may also be interested in downloading some thinking prompt posters for your classroom.
At the start of the year, there are always a few students who are quite resistant to “seeing past” their own thinking as the one way, but I love when these students, more times then not, become the encourages of others or get fired up about their ability to solve a problem in more than one way. Furthermore, I love to see a student “borrow” a strategy that is shared by another student and make it their own by using the strategy in a new setting (proving their understanding of the strategy).
As always, the authors included some fabulous activities to help support our students. I especially like Crossing a Decade (p. 212) and How Far to My Number (p. 217).
Well, I have only brushed on the basics of the chapter. We, of course, AGAIN urge you to get yourself a copy of Teaching Student-Centered Mathematics PreK-2, if you haven’t already. If you do not teach PreK-2, and you have been following along with our book study, you will be happy to know there is a version for you, 3 – 5 and 6 – 8.
Please feel free to leave your thoughts about chapter 12 in a comment! We love to hear from you!