My school started using a blended curriculum model (which I won't name here) recently, and I have a lot of thoughts about it, some pro- and some anti-, none of which would be totally appropriate for me to share here yet. The one thing that it has undoubtedly done, though, is dramatically improve our math curriculum.

In the pre-blended days, our long-suffering but talented and enthusiastic math teacher would randomly pull from a grab bag of mathematical concepts and subdisciplines – geometry, trigonometry, algebra, repeat – trying to patch together a satisfactory sampler of mathematics to students who very often had significant gaps in even their most basic math knowledge. "You have to get 'em when they’re young," she says, sadly, so often.

Now we have an integrated math curriculum, where students re-frame some of their understanding of math to tools that will be more broadly applicable to different mathematical problems: identifying patterns, creating spreadsheet tables, graphing data. These are all skills that I use, more or less, in my day to day work and life &ndash you could use these tools to find significance in a lab experiment or organize your taxes, say.

So that's good. And yet there's still a kind of fog of specialization running through this integrated math; students still learn concepts they'll soon forget, and I find myself dumbing down my "so what" talks so that they'll just remember the difference between a domain and a range, or a mnemonic device to help them find the slope of a line.

It seems to me that there is something very deeply wrong with the whole concept of mathematics at a pedagogical level, and I think the problem with it is the way in which teaching math outside of meaningful applications of that math doesn't even provide some of the basic building blocks of what you'd need to apply it later. Some of the ways we teach reading suffer from this, but at some level the pay-off of knowing how to put letters into words and words into sentences at least has some obvious, immediate value. The higher level meaning-making of literacy is a lifelong project, and subject to constant, frustrating backpedaling, sometimes among very smart people for no reason other than general cognitive biases. (It's amazing, for instance, how your reading comprehension suffers when you really don't want to change the ideas you had coming in.)

But at some level, I understand at least the connection between learning the basics and applying the basics to some other thing – reading a newspaper, scrolling through a Facebook feed, even. I have a harder time seeing that much "higher" a level in mathematics than learning some baseline of competencies.

I think of the non-fiction books I've read on innumeracy, probabilistic thinking, other "mathematical errors" that so many people make, especially when interpreting or conveying their understanding of data. But in many of these cases, what's being described is the breakdown I mentioned above, a way of tactically, if perhaps subconsciously, not understanding something because you don'twantto, not (just) because you can't. I think of someone like Nate Silver, lamenting the inability of journalists to convey his site's probabilities in a way that is true to how they're actually presented. But such inability doesn't necessarily speak to some breakdown in the way in which that person learnedmath. That person can likely do their taxes, use a spreadsheet, figure out the slope of a line if they really needed to. The breakdowns that I most commonly read about are of a kind with reading incomprehension more than they are of innumeracy.

This is a meaningful distinction for me as I work with lots of kids with genuine innumeracy &ndash it just doesn’t look like “dumb analysis” or shallow thinking. It looks like students who really can't calculate numbers, are unable to estimate or make mathematical inferences at a level far beyond misunderstanding or mischaracterization. It is closer to what it looks like when a student can't read a sentence, can't put the words together, than what it looks like when a student uses a poor argument to make their point.

The thing that makes this frustrating for me is that I genuinely have no idea what mathshouldlook like. I have at least some inkling, some model, some research, somesomethingfor how I understand and imagine almost every other school subject and its relevance to student outcomes. I know what I know, I know what I don't know, I have some provisional thoughts about where to go with it, where it works and where it doesn't. But math continues to totally perplex me. (And I now know, as someone who has developed about as much understanding as you’d need to teach all of the math at my school, that the problem isn't just that I suck at math.)

This is the best post I've read all year (at least, best post in the category Something Positive And Just Plain Fun To Think About Even If The Underlying Problem Is Frustrating And Depressing). I may take a while to give this the response it deserves. I wonder if there's a way to get Duncan Watts and Brad DeLong to read it. I think you're completely right that the problem in a lot of what’s called "innumeracy" is bad reading or listening comprehension rather than an incapacity with

*numbers*.

What I'd hope to explore further, though, is that a possible way to connect two areas of intellectual breakdown (or for that matter intellectual success) is through the idea of "logical inference": you say X, someone correctly infers that you must also believe Y but that you haven't implied anything one way or another about Z. Whereas you say X and some

*others*— Maura Johnston or Simon Reynolds, say — incorrectly infer Z, which they've spent the last ten years happily refuting, and BOOM BOOM BOOM. It isn't just that they don't want to understand you; there's actually something askew in their cognitive apparatus. Of course, if they really do want to understand, or one of their friends wants them to understand, they might eventually get it right — bad logical inference isn't an on-off switch, it's more like a dimmer.

But isn't logic something of a bridge between at least some of math and some of the verbal? That's what I'd like to think more about.

I'd put bad logical inference as the core of communication breakdown and bad critical thinking. But logical inference can be taught, at least somewhat. People get better with practice.

This entry was originally posted at https://koganbot.dreamwidth.org/369787.html. Comments still welcome here, there, and anywhere.

## Eurobeat revival continued

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