- Times tables
- Signed operations (add, subtract, multiply, divide)
- Rounding whole numbers & decimals
- Comparing decimals
- Converting between decimal & percent
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)
- 3−(−9) = 3+9 = 12
- 5+(−8) = 5−8 = −3
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)