Learning styles

What skills should tutors have for accommodating visual / auditory / kinesthetic learners? None, I’m afraid; that’s simply not a thing.

And yet, many students’ lived experiences suggests that it is. Why?

I suspect it’s for more or less the same reasons that September babies are overrepresented among elite athletes: a small preference or advantage early on leads to more practice with a particular method, which becomes self-reinforcing.

More valuable than identifying learning styles, I think, is identifying skills that are necessary, but whose absence can go unnoticed.

(Then again, maybe the real moral of the story is that utility can be more important than truth.)

Unknown unknowns

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 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.

Error types

This is a post intended only for members of BATS and other related professionals. Thanks in advance for any light you can shed on this problem.


It’s always nice when you find a way to improve your teaching and your business at the same time. I think a good way to do this is to publish your core methods.  Crucially, though, you have to do it in a way that simple enough that anyone can understand it, but detailed enough that anyone can also see that there’s something to it.

I’d like your help today thinking through one such method, and crucially what to name the pieces of this method.

The system

The basic idea is that math mistakes can be grouped into one of three different categories, and that each category has its own “fixes” that work best.  The marketing power here is in distinguishing my methods from that of others.  The educational power is in helping people see right from the start ways in which they might make better progress if they focused on different things.

Here are the categories into which I claim all math mistakes can be grouped:

  • Category 1 is “I can’t believe I made that mistake!”  In this case, you knew everything you needed to know, and you just didn’t execute correctly. This could be “I misread the question” or “I had a brain fart” or “I was operating at less than full capacity (e.g. I was tired)”, among other things.
  • Category 2 is “I just didn’t know enough math.”  In this case, you either don’t understand what the problem is asking, or else you do, and it’s clear that there’s a method for solving, but you’ve never seen that method and don’t have enough information to figure it out.
  • Category 3 is “I couldn’t put it all together.”  You read the question correctly, you brought your “A-game,” and you know all the math you need.  But you can’t quite bring it together into a solution.  There’s a leap you need to make, and you just can’t figure out which direction to jump in, in order to get there.

So, what should I call these categories?

  • In a related method first developed together with Justin Sigars (of bodsat.com), we called them carelessnessknowledge, and hard questions, respectively.
  • Later, I changed them to thoroughness, knowledge, and synthesis.
  • Most recently I’ve been using attentionmath fluency (or mastery, depending on the day), and creativity.

And how should one decide?

Beside the question of what to call them is the meta-question of how to decide.  What makes a good naming scheme?

Please comment below.

Anything you have to say, I’m eager to hear.  Thanks so much!



See wescarroll.com/unknown-unknowns for the resolution of this discussion. Again, thanks so much for helping to clarity this small but vital issue.

The price and the value

Working with me and my team is more expensive than working with most tutors. That’s because what we do is actually different in a few ways. Here’s how to make sure you’re getting good value for your money.

Does one of these describe you?

  • You need the right answers to math problems
  • You need to know the techniques for solving certain math problems
  • You need to know how to read a math book more easily and effectively
  • You need better study skills
  • You need to feel less anxiety in your class or during tests

If so, then we can help you, but you can probably find good help for less money. (Even if you need many of the above things, it’s probably worth looking around.)

Or are you looking for something more like this?

  • You want your classes to go better, but don’t know what more you could be doing
  • You want to understand your own mind better, and understand why you’re actually having trouble
  • You want to get better at solving problems, especially when you feel underprepared
  • You want to learn cognitive skills that will benefit you for decades
  • You want to excel at math competitions through creative thought, rather than by memorizing obscure math
  • You want an expert co-strategist

If so, then I respectfully suggest that you’re in the right place. The price is high, because it’s difficult to do this work well.

But that’s exactly what we do, and we’re grateful to get to do work we love.

Thank you for allowing us to give these benefits to you and your family.