by Wes Carroll | Dec 10, 2019 | Blog, education advocacy, math education, show me the steps, teaching style
“Dr. Po-Shen Loh has discovered a new way to solve quadratics.”
Well, yes and no. Dr. Loh is a great coach, educator, and evangelist, and I admire and respect him. If he says he was “dumbfounded,” then there’s something there.
The thing is, though, that the press is making a big thing about the “new formula he’s discovered”. That’s just plain incorrect: the interesting part here isn’t the formula. That formula is just shoehorning a simple idea into the language of math, and in this instance the language is almost as cumbersome as with the original, better-known formula. So, not an improvement.
No, the key idea here is in putting together two facts:
- that the roots of a quadratic are equidistant from the centerline of its graph
- that that allows one to systematically work out the roots of a quadratic without either guessing or an explicit formula
Taught well, this new method will relieve students of the need to memorize any formula per se. Instead, students who understand this will follow the method intuitively, and will wonder why quadratics get so much careful attention in math texts: instead, they’ll just be obvious.
(Now that is a development worth writing about.)
by Wes Carroll | Jan 12, 2019 | Blog, data-driven excellence, low-hanging fruit, quality, show me the steps
You probably remember the quote:
There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don’t know. But there are also unknown unknowns. There are things we don’t know we don’t know.
When it comes to STEM tutoring, test preparation, and contest preparation for especially strong students, this is a shockingly important concept. After all, strong students know a lot; they know what they know; and they are aware of things that they ought to know but don’t yet.
But typically there are things that they are missing that they don’t even know that they are missing. And this is where most of the real trouble lies. I help these students recognize these “unknown unknowns” in their academic lives.
The most common unknown unknown is a deficit in one of three qualities (which some colleagues helped me identify in a previous post):
- Fluent: the successful student knows the material and how it all interconnects.
Otherwise, success is necessarily limited (of course). This category includes not only “I need to study more” but also “I had memorized that fact, but didn’t know how it was relevant to this question.”
- Present: the successful student is fully focused when engaging with the material.
Otherwise, knowledge doesn’t matter; you’ll still flub it, e.g. by misreading the question, answering a different but related question, making an arithmetic error, doing too much in one’s head rather than on paper… in essence, a forehead-slapper. This is often missing in students who are so fluent that they aren’t used to having to focus 100% of their attention.
- Bold: the successful student is willing and able to make progress with incomplete information.
It’s often called creativity, critical reasoning, or problem-solving. But at its core, it’s about reasoning successfully even when some pieces of the puzzle appear to be missing. This is often missing in students who are so fluent that they aren’t used to having anything less than complete information in the first place.
That’s it in a nutshell: to be extremely successful academically, you should aim to be fluent, present, and bold. But most strong students consider any academic issue to be a failure only of fluency, which means they often use the wrong tools for solving their problems.
This can cause extreme frustration, and can threaten both morale and identity.
My diagnostic systems identify gaps in these categories, and my interventions help students build the new habits that bridge these gaps. This eliminates these frustrating “unknown unknowns” for most students.
I’m glad to finally have a way to easily discuss these issues with students and parents, so that we can all help the student as a cohesive team.
by Wes Carroll | Mar 27, 2017 | Blog, show me the steps, Uncategorized
“Math punk” Tom Henderson has written a brilliant essay that I have co-opted and edited here.
In a nutshell, what is the problem with math education in the US? It’s that students are mostly trying to minimize feeling stupid rather than trying to maximize their ability to solve problems.
This manifests as “show me The Steps.”
Many students want a sequence of steps that they can perform that will give them an answer. This is not unreasonable; they know that their performance on exams, and therefore their performance on the All-Important Grade Point Average, is largely determined by being able to Do The Steps. So they want to know the formulas, so that they can float them on top of their short-term memory, ace the exam, and then skim them off.
For their entire mathematical careers, math has been a sequence of Steps, and if they get them wrong, they get red pen, bad grades, No No No Look What You Did. Plus, bonus, there is no apparent relevance of these algorithms other than To Get The Answer.
But that’s crap. “The Steps” aren’t math, and what’s more, The Steps aren’t generally useful in life. What’s useful is the ability to deconstruct thorny problems and figure out a way to tackle each of the pieces.
The Steps are seeing the sorts of symbols that count as “right”, and trying to replicate that dance of steps. It turns out that the easiest thing in the world is to look at a student’s work, and tell the difference between “Knows what’s going on, made mistakes and dozed off” vs. “Can memorize steps, has no idea what’s going on.”
Now, a better way to explain mathematics sort of looks crazy at first. It’s handwaving. It’s referring to certain groupings of symbols as “alphabet soup” and writing it down as a wild scribble with one or two symbols around it. It looks nothing like standard “math class” from the outside.
That’s because the better way avoids showing The Steps and instead shows enough of The Idea that the student can reconstruct what the steps MUST be.
And that brings us to a better way to learn mathematics: you get a fear-free zone, you check your ego at the door, you try a bunch of things that will wind up not working, you ask a pile of dumb questions, and before long, you figure out some crazy way to get the problem solved. And only then do you realize that your crazy, lame-brained, that-can’t-possibly-work solution is in point of fact the official method nine times out of ten. Because math is, at its core, just a collection of “the best way we could figure it out” stories, organized semi-sensibly, and with a specialized vocabulary and language on top.
So, what’s wrong with math education in the US? What’s wrong is: whatever it is that makes students uninterested in learning any more math than is required to minimize feeling stupid. My solution? Provide that safe space; find the genius in every question; and provide interesting problems to solve.
Ultimately, that’s all it takes to get students to say “Oh. That’s not nearly as hard as I thought it was going to be.”
