- Phonics
- Multiplication tables
- Algorithms for addition, multiplication, long division, etc.
- Grammar
- Logic

It's particularly aggravating that several of the issues cut to the very heart of why our writing systems in numbers and words are originally designed as they were: to support simple adding, multiplying, and division procedures by hand; to easily convert spoken sounds to writing by way of phonics. Lose sight of that, and you lose the

*very essence*of those systems of writing.

So doesn't this explain why remedial algebra is the single-most devastating course in the university curriculum, preventing about half of all community college students from ever graduating? After all, it's now the first time in their academic career that students are finally forced, inescapably, it attend to the detail and structure of things. And they've been set up for failure; when our students cannot actually parse the structure of a sentence (don't know what a verb is, what clause connects to what, how to diagram a sentence, etc.), then it's impossible for them to translate word/application problems to math (to say nothing of actually solving them). And it's very sad and heart-wrenching to watch.

What is particularly galling about all this is that this has occurred precisely at the same time as the world around has become more driven by machines, computers, technology, and an increasingly technocratic government structure. All of our young people carry computing devices at all times as a very intimate part of their lifestyle, but their understanding is at about the same level as a cargo-cult. This is why "STEM" academic careers are held out as some kind of bizarre alien life-form that the normals cannot hope or imagine crossing into. To the extent that we have removed the capacity to understand structure from our students, we are making them unavoidably victims of the highly technocratic society that controls their lives, without any hope of understanding it.

Obviously this painful disconnect between prior classes and the introductory algebra course cannot last -- and all signs are that, long term, the algebra class will likely be removed as a requirement even for a college degree (my prediction). And thus a mile-high iron wall will be put in place between the unlearned masses and the elite who are educated in the "real deal" of structure, mathematics, language, and computer skills.

There are different ways to get to the structure.

ReplyDeleteInterestingly, when I introduced my son to arithmetic, I did not really focus on the traditional, "algorithmic" way of calculating sums and products. My focus was almost entirely on developing intuition with numbers.

So addition is simply counting, multiplication is counting by multiples, and division is grouping.

So a problem like 3x6 he may solve by counting 3,6,9,12,15,18. This sounds tedious but has a couple of advantages:

It is easier to extend this process to problems that don't fit in the multiplication table. For example, 20x19 he may solve as 20x20 - 20, it is simply recognizing that you are counting by twenty and can count backwards as well as forwards.

With intuition, it is less likely to make certain types of mistakes with numbers. Although he cannot produce the multiplication table "from memory", he can create it, and through this process will be introduced to certain patterns and properties of numbers which are generally applicable. For example, multiples of 5 always end in either zero or 5. This lets you instantly recognize certain kinds of mistakes.

I believe it would be much harder for him to develop this type of fluency when he is older. At that point he would have no choice but to memorize algorithms. He would not be able to really understand numbers though. Many kids don't have that luxury.

I have met some adults that believe a billion is equal to a hundred million. I am talking about college educated or even graduate level adults. That is not a problem in calculation -- it is a lack of intuition, of basic understanding of numbers. They simply haven't devoted enough time to it.

That's fine -- and for example, I still distinctly remember the similar grade-school exercise that crystallized for me that all numbers divisible by 9 are also divisible by 3, etc.

DeleteBut my usual reply is "you need both" this conceptual intuition, plus the written algorithms for larger problems. Having only one or the other doesn't cut it. Students that never memorized the multiplication table wind up crippled for long division, adding fractions, conversion to decimals, factoring equations, etc.

In fact, understanding that our number system was designed to support certain written addition and multiplication algorithms may be the most important intuition of all.