Yesterday I had one of those spasms of panic that occasionally afflict those with a relaxed schooling approach, concentrated on our decision to delay formal math. I had been reminded of the placement tests for Singapore Math, and decided to print out the first grade tests to see how the ducklings did with them. (D1 is 6, D2 is 5.)

I presented them merely as a page of math puzzles for them to figure out; I explained that some were easy, some were tricky, and some would use ways of asking questions and writing things down they hadn’t seen yet. I let them ask me for explanations, although I tried to make sure they each thought through each question for themselves.

They did fine–certainly they’re not any behind–although it did highlight a few areas I want to work more on (mostly correctly reading and writing numbers, since we’ve been almost entirely oral).

But working through the tests helped me see a significant divergence of approach in how mathematics can be taught, and I’ve realized that the approach I have instinctively been taking doesn’t really fit the two common math approaches.

The big division between mathematical programs that everyone talks about is mastery vs. spiral–do they stay focused on one thing until it’s finished, or do they jump around and then come back to stuff. Mastery is all the rage right now: first you learn addition, then subtraction, then multiplication, then division. (This is the approach of Math-U-See, and RightStart Math, which I had formerly been very interested in. Four-digit addition before you even move on to subtraction.)

But even spiral programs think in this operational way, they just jump about a bit more. Singapore at least combines addition and subtraction and multiplication and division, but you still have to get through two-digit addition before you move on to multiplication. There is always a rapid move from concrete to abstract, and then back to concrete once you move on to a new topic.

But when I gave D1 the test on the second half of first grade, she skipped ahead to the very end where there was a pictorial depiction of multiplication and division and effortlessly filled out the correct answers. Then she went back and sweated over the multi-digit addition. Neither of them had any trouble with understanding what to do in a word problem, which I understand is a source of terror and loathing for many–they found these easier and more appealing than the bare calculations.

This makes sense to me, because what we have always done is stick with small, concrete numbers, and real-life situations primarily. Our math time is usually around the lunch table and usually consists of questions like, “If you ate five bites and you have seven bites left, how many did you start with?” or “If I gave each of the four of you three slices of apple, how many slices are there?”

Or even, “If I am going to give each of the four of you half of an apple, how many apples will I need?” Technically that’s a question calling for division with fractions–a late elementary concept that most schoolteachers haven’t figured out–but because it’s a real object and an easy to visualize amount, it’s easy for them to do. I don’t have to explain it–they work it out for themselves. And although they have a decent aptitude for numbers, they are not child prodigies.

I never consciously set out to do it this way, but the more I think about it, the more I think the approach I am taking is the right one. After all, 5-7 year old children are still fairly concrete thinkers. And large numbers are very abstract. You cannot meaningfully visualize a four-digit number. (Even with the best manipulatives–which are very pricey–it’s hard to play with effectively.)

But twelve? You can do anything with twelve. You can get out twelve beans or glass beads or poker chips and split them up in unequal groups (addition and subtraction) and in equal groups (multiplication and division). You can test whether it is odd or even, prime or composite. You can apply fractions to it. You can tell which numbers it is larger than and smaller than.

All this time you are gaining a deep instinctive understanding of the relationships numbers have with each other, of the patterns and just of the fun of playing with a number.

And isn’t that what they will most need? After all, if you understand how the operations work on numbers under twenty, and if you understand place value thoroughly, all those dreaded algorithms and lengthy calculations are easy to grasp, once you are developmentally ready to handle numbers as pure abstractions.

So I started thinking through math curriculum because I was sure somewhere I’d seen an approach that stuck with small numbers and used all four operations. Sure enough, Ray’s Primary Arithmetic, that long-dress-and-bonnet companion to McGuffey’s Readers, uses just that approach. Primary children learn to add, subtract, multiply and divide combinations making up every number under twenty. And they do it with manipulatives (the basic kind: counters of whatever is on hand) and with real-life problems. Writing the operations comes after they are thoroughly understood, and larger numbers are saved for later years.

Since this is from the era when algebra and geometry were part of an eighth-grade education and people calculated compound percentage longhand, I think it’s pretty evident that this approach is not going to shortchange anyone in solid mathematical skill. It just surprises me it is not more common anymore.

Then I also remembered to check the free MEP curriculum, an experimental program developed in the UK based on Hungary’s mathematics curriculum. Sure enough, this one also concentrates on multiple operations with small numbers first, one number at a time in the first grade. It also seems to keep things fairly concrete at first, although there are the workbook pages so essential nowadays and more use of symbols early on than I would do.

But I will be reexamining it closely to see if that’s the program I would like us to transition into when we’re ready to start doing more bookwork. (The one big downside I see is that it only teaches metric measurements–but then, measurements are one of those real-life skills that can be learned on the side.)

I’m surprised that this approach is so rare (is there even a modern commercial program using it?) when it makes so much sense–to me, at least. I don’t even know a general name for it. It gets lumped with the “spiral” programs, but there’s a huge difference between deliberately mastering the combinations and relationships that make up each number in turn, and hopping about between topics for the sake of variety.