Automatic Negatives

In my developmental (remedial) courses, I have been thinking more lately about where and how to communicate which skills need to be automatic -- that is, instantaneous and always correct. As I wrote earlier (link), these are skills which we take for granted in higher level courses, but they frequently get lost in the underbrush of all the other math topics, and students expect that struggling with them for several minutes is acceptable and normal behavior. A list of these remedial automatic skills that students often lack includes:

• Times tables
• Signed operations (add, subtract, multiply, divide)
• Rounding whole numbers & decimals
• Comparing decimals
• Converting between decimal & percent
• Etc.

These are the kinds of things that seem "obvious" to properly-educated people, but if no one ever communicated them to a student in that way, they are opaque. For example, in each of the last few  semesters in my sophomore statistics class, I've had "A" students, otherwise doing very well, but who were completely mystified at how we were converting decimal probabilities to percent on the fly. Notice that these are all one-step, immediate mental tasks: I wanted to include order-of-operations knowledge here (it's so critically important), but the truth is those tasks are inherently multi-step problems, so they don't really belong on the list above.

But let me focus more on the issue of negative numbers (signed operations). I find these to be the greatest stumbling block in students getting through the bottleneck remedial algebra course -- I can pull up any test, including finals, and see that usually at least half of the errors are simply signed-number mistakes. A student can know everything in the algebra course, but if they routinely trip over negatives even after I've begged them to practice it for a whole semester, then they have practically no chance of passing the final.

In June, I had the opportunity to teach an immersive one-week workshop for students who narrowly missed passing our department's prealgebra final (basic arithmetic with different types of numbers: integers, fractions, decimals, percent). This was a great experience, the students were hard-working and highly appreciative, and it gave me a chance to further focus on this issue. I was trying to do frequent one-minute speed drills on things like negative operations, and some students were having what seemed like an inordinately difficult time with them -- particularly the subtractions. So that night I sat down at the bus stop and tried to think through really carefully what we really do in practice as proficient math people.

Here's the thing: Not all negative operations are single-step. In particular, consider subtracting a negative number, written inside parentheses. I find that a lot of students are taught this bumfungled "keep change change" methodology: they will transform expressions as follows:

• 3−(−9) = 3+(+9)
• 5+(−8) = 5−(+8)
• 1−(+7) = 1+(−7)
• 4+(+6) = 4−(−6)

Now, all of these are true statements. But only some of them are helpful. It's not like the students' prior instructors were lying to them, except in regards to when this fact is useful in simplifying an expression with signs (namely, the 1st and 3rd cases above). Let's look at how a math professional would really do it. These are two-step problems; really, we would follow the order-of-operations and get rid of the parentheses in what's really a multiplying step, then combine like terms in an addition step.

• 3−(−9) = 3+9 = 12
• 5+(−8) = 5−8 = −3

Etc. So the lesson is that if an instructor shows students how to mangle signs in & around parentheses, they are really missing the point; when simplifying (evaluating), we will remove the parentheses entirely in the multiply step, and then always perform add/subtracts without any parentheses in the picture.

So in the current discussion, this informs us as to what we should be drilling students for "automaticity" in terms of negative number operations: namely, combining terms with no parentheses has to be the automatic one-step skill. If you want to explain this as effectively adding terms that's fine; but don't fail to clearly communicate that this is expected to be instantaneous and immediate, in one mental step, in practice.

• 2−6 ← Automatic drill ok
• −7+1 ← Automatic drill ok
•  6−7+2 ← Automatic drill ok
• 5−(−2) ← Not automatic drill ok (2-step problem)

I feel like this was an important lesson I got to learn from this summer's immersion workshop. A speed drill can include automatically multiplying or dividing integers, or combining terms with no parentheses -- but add/subtracts with parentheses don't belong in the same category, because you really do need to apply two separate simplifying steps for them. And perhaps most important of all: clearly communicate that the one plan that always succeeds is the standard order-of-operations, not a score of different random manipulations to memorize for different situations.