Indeed, we like students to teach others once they have some skill precisely because it requires greater mastery of the skill. But is this higher bar useful in assessing math skill?

My approach as a teacher was to shoot for procedural competence and create opportunities for any to soar higher, thinking many simply will not get any higher (and I am sadly awaiting CCSS’s proof of that).

Agreed also: throw them those “extra credit” problems if one wants to push them further, do not take them into the realm of pedagogy/exposition — do we make skilled writers diagram a sentence before admiring it?

The pro-explanation crowd is all in a panic that kids may have memorized something, but I do not think they have thought it through: I can memorize chess positions, but to recognize and execute a pin or a fork requires understanding the moves and weights of pieces regardless of chess position. No math student ever memorized all possible expressions and how to transform them.

Given the technology we have today could a computer based assessment tell you if a student is proficient in grade 8 math?

Of course this depends on the definition of proficiency and if that definition includes being able to explain to another human why a particular solution or method works then it is likely much easier for a human than a computer to assess this.

But taking an alternative tack in Canada the CEMC Gauss competition is used to rank high proficiency in math at grade 8. Is it likely that the top performers on this contest are not proficient at grade 8 math?

Perhaps they are not as proficient at explaining their work as a grade 8 teacher but my guess is the top performing grade 8 students perform well above the average grade 8 teacher on the Gauss competition.

Also despite being simple multiple choice questions the top performers find these to be very satisfying. I am guessing but I would bet many find it more satisfying to get the right answer to a challenging problem than to explain their reasoning used in a simple problem.

Math competitions after grade 8 do evolve to include both multiple choice and written answers. But I wonder if at the early grades some educators are putting an unwarranted emphasis on the math skill they hold most dear, explaining an answer, rather than one kids enjoy, getting the right answer to a challenging problem. At grade 8 which of these do you need to assess?

In summary if you removed all assessment of students ability to explain their solution at grade 8 and below and replaced it with assessment by more challenging problems what would be the result on the students further math education?

(I am being careful here to ask a question about assessment. This says nothing about how much interest students have in understanding math. But just as students very able to read enjoy reading for its own rewards I am betting those very capable of doing challenging math problems enjoy learning about it. Just a guess.)

But instead let’s do this. Let’s imagine we sat down for coffee and wrote down all of our favorite verbs from 8th grade math — the best actions a student should be able to perform at the end of their year. We rank their importance also. We shelve our disagreements about when and how students should learn how to do those verbs. We agree that summatively, to be stamped proficient in 8th grade math, students should be able to do them.

I’m guessing the overlap of our two lists would surprise us. I’m willing to bet that your list would contain more breadth and depth than Khan Academy’s list of skills. I know mine would.

I want students to do more satisfying work than Khan Academy requires. More consequentially than what I want, though, is the fact that this assessment consortium requires that breadth of work. That’s the problem.

It seems the bee in your bonnet, however, has little to do with CC itself — it focusses narrowly on a certain ideology pertaining to the value of assessing what some refer to as “process standards” (a misnomer — they tend to refer only to cognitive processes): things that may be going on in a student’s mind, rather than what they can concretely accomplish on paper with the tools they’ve learned.

All large-scale studies comparing cognitive-focussed instruction in elementary school versus skills-focussed instruction generally come to the same conclusion: cognitive-focussed instruction is a nasty failure, not only at inculcating skills, but also at fostering the understanding upon which it purports to focus.

The epitome of these studies, which happens also to be the largest comparative educational study in history (and also the most ironically-named) is Project Follow Through — for those unfamiliar with this landmark study, Athabasca University has helpfully posted a summary here: http://psych.athabascau.ca/html/387/OpenModules/Engelmann/evidence.shtml

So … if this is the only way in which Khan fails to align with CC I say “hurray for Khan”! Because many of the commercial texts in use fail in other, significantly more damaging ways, such as failing to teach the standard algorithms of arithmetic and/or failing to provide sufficient consolidation practice for the mastery of said procedures. The cumulative effect of such lack of mastery, and the frustration it infuses in children, do much damage.

It is, indeed, important that children learn to demonstrate mathematical results. This must be done with children who have mastery of fundamentals. It is not (I emphasize *not*!) a clever way to *teach* the fundamentals — this is unsupported by any significant and empirically valid research; yet many of said texts blissfully build this presumption into their methodology from the start.

Students who have mastered mechanics and facts in early education have an easy time learning the cognitive content of later years in math education — because they needn’t dwell on trivial details that needlessly clutter up working memory, creating a mental logjam. Students who have not mastered this earlier material are almost universally destined to fail.

That is the beauty of Khan. Sure, there is more to do on the cognitive side in math education. That’s what live teachers are for. That is the bread and butter of those of us who teach university math. That is why high school math teachers are generally expected to have far more mathematics in pre-service training than elementary school children; because this is a highly mathematically challenging thing to do. Students are not helped when teachers with no background in the subject, themselves often struggling with the basics, are asked to teach, and assess, these sophisticated skills. And it does those teachers no service to place them in that situation.

I don’t love KA. I find it’s generally missing higher level concepts and inspiration, so it should fit into the new SAT where the idea is that nothing is “tricky”. But at least it doesn’t promote the idea that those who can learn math do not get too far ahead of those who cannot.

I guess colleges are just going to start looking for other evidence of IQ if the SAT stops providing that.

CCSS is concerned kids might get the right answer for the wrong reason. OK, but more will get the answer for the right reason and not be able to explain it coherently, or their explanation might be fine but elude the examiner. This problem with natural language is one reason we have math!

I am reminded of one of my rules for working on other people’s computer programs: first, delete all the comments. Code is unambiguous, we coders are inarticulate geeks.

btw, do not celebrate too soon on KA vs CC. The CC-aligned SAT now links directly to KA from test results: https://www.khanacademy.org/sat Of course there is no way KA can parse argumentation, so no danger of them going there.

CC makes the mistake, a big one, of asking for wishy-washy argumentation that cannot graded clearly or evaluated. The mathematical form of argumentation is the proof. Traditional mathematics trains students until they are ready to do proofs. Those who don’t make it that far are better off doing their argumentation in other classes and can become lawyers but they won’t become mathematicians. They’ll still be able to calculate their grocery bill and even do their taxes with the math they did learn. Math is not the only place where logical thought is taught.

The CC standards calling for argumentation in English are useless. It seems an effort to make mathematics something different from mathematics. I applaud KA for not mucking up their curriculum with it.