Part of our role as educators is to continually reflect on the teaching methods and evolve them as we adjust to students’ learning. One such adjustment came about while teaching our first graders addition and subtraction for numbers within 20. Specifically, the problem of subtraction with regrouping was a major hurdle the kids had to get through. Let us look at some examples from the textbooks and workbooks, and why we decide to tweak the method for our classes.
Examples from Textbooks/Workbooks
In some Singapore Math textbooks/workbooks, the subtraction problem is solved by first decomposing the minuend to 10. The following are examples of how some of these textbooks/workbooks present subtraction with regroup problems. They are from respectively:
- Math In Focus Workbook 1A page 206
- Shaping Math Coursebook 1A page 88
- My Pals Are Here 1A page 104 (only used in Singapore)
For example, to find the answer for 12 – 7, the steps are as follows:
Step 1: Decompose 12 (minuend) to 10 and 2
12 – 7 = (10 + 2) – 7
Step 2: Subtract 7 (subtrahend) from 10
10 – 7 = 3
Step 3: Add back the number 2 from step 1 into the result of step 2
2 + 3 = 5
What is Wrong?
Well there’s nothing wrong with the approach, we wonder if there is a better way and more importantly, why it might be better.
First let us show how we are teaching our kids now. This example is taken from our worksheet TM107 Morning Work & Review – Addition and Subtraction to 20.
In the above example, 15 – 7, the number bond is drawn below the number 7 (subtrahend), indicating that 7 is the number we need to decompose. Students first ask themselves: “What must I take away from 15 to make 10?” The answer is 5. So, in the number bond diagram, they should decompose 7 into 5 and 2. Now, the students take 5 away from 15 to make ten, then take away the remaining 2 from 10, giving the answer 8.
Diving Deeper – the cost of working memory
We sat down and reflect on the fundamental difference in these two approaches. This is what we realized.
The main difference is the amount of times working memory is involved. Let’s define the following cognitive “costs”:
- Procedural (P) – we assume equal effort for decomposing numbers, adding and subtracting within 10, and answering questions such as “What must I take away from 15 to make 10?”
- Working memory (M) – we assume equal effort for both storing and retrieving single digit numbers.
Due to the prevalence of the base-10 numerical system, it is much less “painful” to remember the number 10 than any other numbers. Hence, we assume the cost of storing the number 10 in working memory is negligible.
Now let’s look at the difference in both methods, using the example 15 – 7 once again.
The textbook methods use the following cognitive resources:
- Decompose 15 to 10 and 5 (P)
- Store 5 into working memory (M)
- Subtract 7 from 10 to get 3 (P)
- Store the result of the subtraction, 3, into working memory (M)
- Retrieve the two numbers, 5 and 3, from working memory (M+M)
- Add the two numbers 5 and 3 to get 8 (P)
The decomposing to 10 method we use involves:
- Decompose 15 to 10 and 5, to answer the question “What must I take away from 15 to make 10?” (P)
- Decompose 7 to the 5 and 2 (P)
- Store the number 2 into working memory (M)
- Take 5 away from 15 to make 10 (negligible – already incurred in step 1)
- Retrieve the number 2 from working memory (M)
- Subtract the retrieved number 2 from 10 to get 8 (P)
Thus, we can see that the procedural cost are about the same (3P), but the cost of storing and retrieving from working memory is twice in the original method (4M vs 2M)!
We are convinced this reduction in working memory is just as important, if not more, than the number of steps (or procedures) required, when it comes to the total effort a child has to undertake to solve one question. What do you think?
For more related resources, please see our Number Sense, Addition and Subtraction page.