*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.

*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)

__. 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__

*But only some of them are helpful**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

__, and then always__

*remove the parentheses entirely in the multiply step*__in the picture.__

*perform add/subtracts without any parentheses*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)