Three weeks into the new academic year, it is clear that thirty to forty percent of my algebra 1 math lab students, all 9th graders, misunderstand subtraction, even with whole numbers. Some of them even took algebra 1 last year and received As, Bs, and Cs, while others received Fs, and yet others took algebra readiness or a similar course, also with grades ranging from Bs to Fs.
When faced with an expression like the following:
most students usually find the correct solution, which is 3. However, reverse the order as follows:
and half of my students write their answer as
which is incorrect, as opposed to the correct answer of:
even when the earlier example preceded this one.
When asked how they arrived at positive three instead of negative three, to a one they responded that they subtracted two from five. Why that is so was not as clear. It was difficult to get a clear explanation of why they started with the five in the expression. I asked if they saw the five first, since it was largest, and many nodded that was the reason. Whether that is the case or not is still unclear.
What is clear is that these students have not developed a sense of number, or more specifically a sense of operations with numbers, especially when denoted symbolically. In an attempt to explain various methods to evaluate such an expression, I reminded students to read from left to right, as they did any western-based writing, to include mathematics. While the order of operations dictates the order in which operations are usually carried out within any expression, students should still follow a left to right sequence through any expression to reduce errors with operations.
Additionally, I reminded them to think about the expression, and what it implies. Specifically, in the first expression above, they are asked to take away three (items) from five (items), which obviously leaves two (items). However, when asked to take away five (items) from two (items), they need to recognize that they only have two (items) to subtract from initially. Hence, they need to borrow three (items) to complete the operation. I used money as a context for this class of problem, which they seemed to understand. However, some may not have understood.
For a visual representation, I showed them how to denote two positive items in one color (blue) from which they are to add five negative items (red), which when the positive and negative numbers cancel each other out, leaves three negative items in red. I also used positive (“+”) and negative (“-“) sign symbols to denote the different numbers in a problem, which followed the same logic.
Additionally, I showed them a number line where they start from two and move left, the negative direction, by five arriving at negative three.
Students seemed to appreciate our discussion, and signaled they understood. However, in subsequent, similar expressions, they made the same mistake, likely a result of continued misunderstanding.
The following day, after reviewing the principles we discussed the prior day, students were asked to work on a double-sided worksheet with integer addition and subtraction on one side and multiplication and division on the other. Most exercises involved one and two digit integer operations, mostly with two numbers, sometimes three for addition and subtraction. The following table depicts student results for a sample of six expressions.
As can be seen, students still struggle with subtraction. Except for the the expression -2 + (-3), a sizable number of my 9th grade algebra 1 math lab students have not attained mastery of a grade 5 / grade 6 level standard. Without mastering this concept, and any associated procedure, they will continue to struggle with algebra. That is simply not acceptable to me. So, I will continue to work with these students to ensure they conquer operations with signed numbers.