# The Big Ideas of Algebra, Part Two

This post is a follow-up to my post of November 30, where I brought up two points that can illuminate one’s views on the big ideas of algebra:

…we discussed the assignment of partial credit for work in solving a problem — more on this later, but it definitely reflects one’s views on what the big ideas are — and whether the study of algebra is distinct from (and prior to) the study of functions…

Partial credit doesn’t sound like a deep issue, but it really is. All you have to do is gather a group of math teachers, give them a student’s solution to a problem, and ask how many points should be assigned. Regardless of whether it’s out of four (where there are only three partial-credit possibilities) or out of ten (where there are nine), there will be significant disagreements; I reached this conclusion from having participated in such activities many, many times, with various groups of colleagues. And I don’t just mean that one teacher will give two out of four and another will give three, or that one will give seven out of ten and one will give five. No, I mean that one teacher will give nine points and one will give zero! And this is in mathematics, which is supposed to be an objective discipline — unlike English, where such disparities might not be surprising.

So, what does it mean when major disagreements surface in this area?

It usually means one (or both) of the following types of differences:

• differences of opinion about what the big ideas are
• differences in what one values

For instance, consider these three examples of student work that we discussed in the seminar in which I participated in November:

1. The problem read, “The sum of three consecutive odd integers is 81. Find the integers.” One student’s solution was like this:

Let x = 1st odd integer
x+1 = 2nd odd integer
x+2 = 3rd odd integer
x + x + 1 + x + 2 = 81
3x + 3 = 81
3x = 78
x = 26
The integers are 26, 27, 28

How many points (out of ten) is this worth? If a big idea is that odd numbers differ by 2, not by 1, then the setup at the beginning of the solution represents a significant error — especially since the student wrote the word “odd” each time, thus showing that s/he didn’t merely skip over the word “odd” in the problem statement. On the other hand, the solution is otherwise correct, the work is clearly shown, and the answers will even check, being three consecutive integers adding up to 81 (ignoring the word “odd” again). If you highly value all the skills of combining like terms, backtracking to solve an equation, and recording a solution, then the solution is worth a lot of points. But if the idea of consecutive odd integers is important, it may be worth very few points. My colleagues rated the solution all the way from one to nine; I gave it a six.

2. Next we have a different student’s solution to the same problem: 3. The grades on this one ranged all the way from zero to ten! Some teachers gave it only a few points — or even zero points — because no algebra was used. But I was one of those who gave it a ten, because not only was the solution correct but it also showed a thorough understanding of what the problem was asking for, and of course the answer was checked. If an algebraic solution was meant to be required, that requirement should have been specified.

4. Finally, here’s a different problem, along with one student’s solution: “Solve 2(– 10) – (12x – 4) = 20.” The issue here is that the student couldn’t read his own handwriting (possibly her own handwriting, but the odds are against that): “20” got transcribed sloppily and then read as 26. I gave it a nine out of ten, since that error struck me as a very minor one. But other teachers’ scores ranged all the way down to zero, on the theory that the student had nobviously never checked his answer. My own values are that checking one’s answer is a big idea of algebra, if we mean that it’s important to understand that what we mean by a solution is a number that will satisfy the equation. But failure to check, especially in a time-sensitive situation where there are no instructions to check, strikes me as a very minor offense.