Charades is probably my favorite game. On the surface it's about communication, sharing an idea through a limited medium (gestures).

But it's really about empathy and reading your audience. What does the other person know? Are they getting lost? Are they having fun? (Am *I* having fun?)

What looks like a communication obstacle to an alien observer is an *enjoyable experience* for the human participants. Sure, there's an idea to convey, but maybe there's a clever, funny, or astoundingly simple way to convey it. Aha!

Math teaching should be the same: convey ideas with empathy for your audience.

## Math Empathy Checklist

I use the ADEPT method to remind me of what helps me learn: an Analogy, Diagram, Example, Plain English, Technical Definition.

But when *sharing* a math idea, I have a different mental checklist. No convenient acronym, just a list of questions to ponder:

## How long did the idea take to be discovered? Accepted?

If an idea was debated for centuries before being accepted, shouldn't that be taught?

Sure. Ok. How many of you know that negative numbers were called *numeri absurdi*? Only accepted (in the West) in the 1800s?

When we have struggles with new concepts (like imaginary numbers, also considered absurd), reference similar struggles in the past. *Hey, you're confused? Good. So was everyone else, and here's how we resolved it.*

## How long did *you* take to internalize the idea?

Hey you (yes you, the teacher) -- what struggles did you have when learning?

Did imaginary numbers click instantly, without doubt? Did the Fourier Transform just snap into place on your first reading?

(You'd think so, given the unblinking, matter-of-fact treatment in most lessons. Argh!)

If you, the teacher, struggled with an idea, don't hide it: what tripped you up, how did you resolve it, and what issues do you still have?

I needed simulations before I understood the Fourier Transform: playing around with them made it click. Instead of writing down the definition, share the "behind-the-scenes" of what helped.

## How comfortable are students asking questions?

Learning is a back-and-forth process. If students don't have questions, they either understood it perfectly, or they are scared/uninterested.

In charades, we can easily see if the other player is confused or having a good time.

## Are we trying to be defensive, or helpful?

Academic writing is a bomb shelter, built to be defended from critics. Stable, rock-solid, but not welcoming.

I'd prefer to make a beach bungalow you look forward to visiting. Yeah, the banana-leaf roof is leaky, and no, Dwight, it cannot withstand an aerial assault from AGM-114 Hellfire missiles. But we'll have a great time all the same.

Lessons barricaded with prefaces and caveats indicate you are protecting yourself, not trying to be helpful. (*If students began Calculus without a month studying limits, they might (gasp) not have a rigorously defensible understanding on Day 1!*)

At some point you reinforce the bungalow, don't start there.

## Goal: Are we making students awesome?

Make your students awesome. I want readers to learn things in minutes that took me a decade to untangle. (Kathy Sierra has a great talk about making users awesome.)

Giving impressively rigorous definitions on day 1 doesn't make students awesome. Ignoring historical and personal confusion doesn't make students awesome. Organized chapters of theorem/proof/exercise doesn't make students awesome.

Share what actually worked, in a way you would have liked to see it.

Happy math.

## Other Posts In This Series

- Developing Your Intuition For Math
- Why Do We Learn Math?
- How to Develop a Mindset for Math
- Learning math? Think like a cartoonist.
- Math As Language: Understanding the Equals Sign
- Avoiding The Adjective Fallacy
- Finding Unity in the Math Wars
- Brevity Is Beautiful
- Learn Difficult Concepts with the ADEPT Method
- Intuition, Details and the Bow/Arrow Metaphor
- Learning To Learn: Intuition Isn't Optional
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Learning to Learn: Math Abstraction
- Learning Tip: Fix the Limiting Factor
- Honest and Realistic Guides for Learning
- Empathy-Driven Mathematics
- Studying a Course (Machine Learning) with the ADEPT Method

## Leave a Reply

19 Comments on "Empathy-Driven Mathematics"

Definitions arise naturally when you have seen enough, different or similar things, and then you figure out a pattern, looking for connections, differences, to eventually arrive a general definition covering a large group of things falling into the same category. So often now things are first taught by throwing a rigid definition at you, while it might be true that it is complete, rigorous and accurate, it simply lies beyond a regular people’s comprehension!

Thanks Shirley – I agree. We start by trying to decipher the rigorous definition instead of starting with a softer version and sharpening it up.

I wish to teach after completing my higher studies in maths and I really think this idea is going to work on the students.Primitive teaching should be discouraged to a certain degree.

Thanks Kalid

Glad you enjoyed it Arun, it’ll be great if it informs your teaching methods!

Kalid,

Another great article. You are the first person I’ve found who feels the same way I do about teaching limits first in calculus. Personally I think calculus should be taught in a way that shows the types of problems Newton was trying to solve, shows how he solved them, and then expands from there and shows how to use derivatives and integrals to do fun real-world problems. Let the students minds expand and have fun in the process, then explain limits. Just like a programming book would give you a real-world project first and then a deep dive, so I think some math books would benefit from a simulated but complex enough real-world problem and walkthrough of how to think through the problems and “discover” basic differentials and integrals, then dive deep into the rules.

For this reason I often recommend people pick up Sylvanus Thompson’s text “Calculus Made Easy” from 100 years ago before they learn from a textbook. He teaches using infinitesimals not limits, and while not “mathematically pure” by today’s standards the calculus methods reach the exact same answers and are FAR more intuitive and give you a gut feel for what is happening. He does a great job explaining what calculus textbooks gloss over.

Also, if you haven’t read it you might like Polya’s “How to Solve It.” The first half is a scenario-based walkthrough of how a math teacher should teach students, and the second half is essentially an encyclopedia containing a sort of pattern language for solving math problems.

Thanks Dave! I like Calculus Made Easy as well, and plan on getting How To Solve It :).

This is how I teach Calculus. The first week we look at two ideas: finding a tangent line / slope of a curve, and finding an area under the curve.

From there we build the idea of limits.

Kalid…Welcome back!…I developed my math empathy with Flowers, Birds & nature’s empathy in Construction. i.e. Bird nests, Rabbit burrows, Worm tunnels. I used not only “Discovery” & the Wonders of nature but, also, “Ordination & Cardination”. I learned these concepts and applied them before the age of 7yrs. old. Have a great Summer! Jamieson (Peter) PS:… Good easy on the Hot Dogs! Hamburgers are better with Potato Salad & String beans!

Thanks Jamieson :).

Hi Kalid, great read. You have a very natural writing style.

Do you believe that concepts in math should always relate to something in real life? It’s a question that has always tickled me. Wanted to know your thoughts on it.

I totally agree. Humans are realistic, humans would not waste too much effort doing things without a good reason. Unfortunately there are too many levels of abstraction in developing math theories that we simply forget what the real problems we are intending to solve. The worst part is, centuries old education style get people so comfortable in teaching in this abstract way so detached from reality.

Thanks Sachin! So, I don’t think math *has* to relate to things in real life, but it’s certainly more understandable and fun that way. Any math taught in school (K-12) is likely to have a real-world application, so if we haven’t found one, we haven’t looked hard enough :).

True, I wonder if our ability to perceive the world around us hasn’t kept up with our ability to conjure up mathematical devices. Your example of negative numbers is perfect in that sense. Another example is the concept of infinity. It’s as if the evolution of our physical senses has fallen behind our mathematical capabilities. If we accept this view then we may never be able to find perceptible analogs for some of our math theories.

At the end of the day, any theories should be used to solved a real-life problem.

Yes, the vast majority of math taught in school has real-world scenarios it can apply to. However, most math is developed/discovered ahead of its application (like number theory being used for cryptography centuries later). We want to understand math as a pattern, which can often show up in the real world (but is not *required* to).

Loved the video ” the great talk”

One of my favorites as well.

I don’t understand why negative numbers were not accepted in the West until the 1800s, as you say. After all, the concept of “owing” or “debt”/”indebtedness” has been around forever. So yes, you *can* have “negative sheep” – they are sheep you owe someone!

One of the most influential books in my math tutoring has been this one. Keep in mind that I am not a math major. I have not read dozens of books on math. Frankly, it’s hard to find good books on helping students understand math. Most are way above me like Kevin Devlin, or they don’t teach the concept-relationship model. Your book is good in theory, but way too ‘mathy’. The examples, etc., need to come down to an algebra level. I got lost when you went into using calculus for examples.

https://www.amazon.com/Problem-Math-English-Language-Focused-Understanding/dp/1118095707/ref=sr_1_1?s=books&ie=UTF8&qid=1473103796&sr=1-1&keywords=the+problem+with+math+is+english