Emory Math Circle example lesson

Being a part of Emory Math Circle is one of my favorite things about being a grad student. Emory Math Circle is a free math-enrichment program for middle and high school students, that takes place on weekends at Emory. Lessons last around 75 minutes and are geared toward developing logical thinking, creativity, and skepticism in our students. I’ve been an instructor for the math circle since 2017 and have been its director since 2018.

It can be difficult to describe Emory Math Circle, so let me show an example lesson instead. One of my favorite lessons is Unfair Thrones. It’s designed for middle school students with some problem-solving experience.

First, the Story: An empress has two children who will inherit her empire. To prevent future war, she wants to build two thrones for them that are as even as possible (having one throne larger than the other sparks jealousy, and then war). So call down two students to sit in chairs in front of the blackboard. Put a fraction bar above each chair, and have students call out numbers from 1 to 4 to fill in the fractions. So you might get 2/3 over one chair and 1/4 over the other chair. You can’t reuse a number, and if you get an “improper fraction” (say, 3/2) declare, “This throne is upside down!” and flip the fraction. Now, what is the difference between the two fractions? In this example, it’s 5/12. Can we make the thrones more “fair” by shrinking the difference?

This is the main goal of the activity: to find the minimum difference between two proper fractions that use the numbers 1 through 4 exactly once. I love this activity because finding the solution (for this first question, at least) isn’t so hard, but justifying why it’s optimal is harder. Many students will say 1/3 and 2/4 are the closest you can get, but why? Force students to challenge their own answer! Can they enumerate all the possible combinations fractions? This would be one way to prove their answer.

Just as the students are all solving this first questions–and the whole time they are working together, practicing explaining their thinking–the empress has another baby! Joyful news! But now we need another throne. This time, we will have three thrones, three fractions, using the integers from 1 to 6. We want to minimize the maximum difference between each pair of thrones. For example, if we write 1/2, 3/6, and 4/5, then 1/2 and 3/6 have no difference–they’re equal!–but 1/2 and 4/5 differ by 3/10. This is the maximum difference, and we want to minimize it. Mini-max problems are likely new to every student in the room, and will be a struggle to them at first.

Of course, we can continue adding new children as long as the lesson last. Some extension questions we might ask are:

As the number of thrones increases, can we find a set of fractions that is more and more “fair,” or does the difference between thrones sometimes increase when a new child is born? (I.e., are the solutions monotonic decreasing?)
If the empress has n children, how many possible sets of fractions are there? Do we need to check all of these to find the solution, or are some set-ups never optimal?

As a warm-up to this activity, I give students this worksheet as they come in the door. It reminds them how to compare fractions, and how to find their sums and differences. The students should have these skills already, but it’s still good to spend some time discussing the warm-up activity. You may be surprised how many different correct ways students can compare the sizes of two fractions.

Communicating math

What’s the point of solving a math problem if you can’t explain the answer to someone? Nearly every other discipline encourages its students to practice written and oral communication, but math courses often forgo this practice so they can focus on covering more material. This is a mistake.

I am thinking about the importance of math communication since I have just attended the QTM Math Circle poster presentation. At the end of each summer, students enrolled in QTM Math Circle complete a capstone research project and present their work at a poster presentation. For many of these students (who are all high schoolers), this is their first time having to explain math to someone. Of course, many of them can explain steps of a calculation–“then you divide by 2 and take the square root to solve for x,” e.g.–but this was their first time explaining what problem they tried to solve, what their approach was, and why it worked.

The best presentations were those that included helpful visuals. For example, one team studied screensaver patterns. You know the style of screensaver that floats around the screen, bouncing off the walls, until it finally satisfyingly hits a corner? The students on this team studied how many times the box will bounce off a wall between corner-touches, and how far the box will travel, and how many times it will cross its own path. A few clear example pictures on their poster were enough for me to understand the problems they wanted to solve and predict some of the patterns. If an audience member doesn’t clearly understand your problem, you’re wasting their time with everything else you tell them.

Also, many good presentations focused on the idea behind the proof over the step-by-step calculations of the proof. Another team, for example, studied strategies for finding your way out of a forest if you are lost (assuming you know the dimensions of the forest). They wanted to minimize the worst-case escape time. Their solution involved evaluating a few integrals, and these were shown on the poster so anyone who was really interested could see the details, but when they presented their work, they just explained the high-level strategy, then noted, “our calculations show that this is how long you would need to find your way out.”

I have notice many professional mathematicians still struggle with communicating their results. I cringe a little every time I watch a math talk where the presenter shows a slide full of equations, as if the audience members will have time (or energy) to digest each line of calculations. These talks would be so much better with a few helpful images and an outline of the method used, without getting too deep into the weeds. As a graph theorist, I find pictures to be worth a thousand equations, and by using a picture to explain an idea from a proof, the real reason for the result becomes more clear to me.

When working with my Calculus students at Emory, I have started focusing more on communicating math. I have assigned a few take-home assignments (an example is here) that involved interpreting their results. Also, many quiz and exam questions ask them to explain their final answer in terms of the problem/scenario given. When I return quizzes to students, I always have those who have solved problems correctly write their solutions on the board and explain them to the class, but this is mostly about explaining the calculations. I would like to introduce into my classes opportunities for the students to explain math ideas to each other and practice this skill. I will try this in the coming semester of Calc I I’m teaching.

Reflecting on student evaluations

Before I talk about the content of the evaluations, I have one observation. I take student evaluations very seriously, and I suspect my students do as well. At least those students who fill them out. This past semester, only 78% of my class evaluated me. This is the first time evals were available to be completed online, instead of in class on paper. Unsurprisingly, this led to a lower completion rate. This is sad though, because students were are unlikely to fill out evaluations are exactly the students I want to hear from. These are the students who need the most help!

I even offered the entire class bonus points on their final exam if 90% of the class completed the evaluations, hoping they were pressure each other to do it, but alas…

My evaluations were good. It’s easy for me to focus on the negative comments, but the vast majority of comments were positive, and I should absorb this feedback as well, so that I remember to focus on what’s already working in my classes. In particular my students like

The organization/structure of my lessons and lecture notes
How flexible I am with meeting them outside of class, by special appointment
My patience when they’re afraid they’ve asked a “stupid question”
That I’m approachable, that I seem fun, and that I make coming to class not so terrible

I attribute this last point mostly to my wearing a Halloween costume one day, and my bringing in candy/treats a few times. I work hard at the organization of the class, so I’m glad they’ve noticed how helpful it is. The second and third points come naturally to me, so I’m not so worried about falling behind on those, though I am afraid someday I might have less time to offer special office appointments.

I also received some criticisms and suggestions for improvement, which I want to take seriously (though not personally). Many students had the same suggestions, which is telling.

The exam problems are much harder than the problems from the lectures and the homework
Grading criteria is unclear on take-home assignments, and I seem knit-picky on graded work
Group assignments often end up being completed by a single group member.

This isn’t the first time I’ve heard that my lecture problems are too easy compared to exam problems. I purposefully start with the simplest possible example of a problem when I introduce a new topic. Then I work up to more complex problems. In the assigned homework, there are even more complex version of the problem. Many exam problems are variations of homework problems I’ve assigned, so part of me feels like some students are not doing the (optional) homework, are just studying their lecture notes, and are then upset that the exam is harder.

But the other part of me feels that I should show more difficult problems in class, so that the students are already primed for how complex these problems can become. If I do this, suddenly I might start receiving lots of feedback that “Brad’s lecture problems are too hard! He should start off with easier examples,” but it’s worth a try. Eventually I’ll be able to strike a balance.

I think what students call “being knit-picky” is what I would call “expecting them to solve a problem correctly.” That being said, one way I might prepare them for how they will be graded is to make comments about grading as I work a problem at the board. Typically, I try to de-emphasize grading as much as possible in my class, but perhaps as I’m working an example problem I could say something like, “And here, if we forget to at the constant of integration, the answer isn’t complete. On a quiz, I’d have to take off a point for that.” That way, the grading isn’t a surprise.

Since the dawn of time, students have been complaining about group work being unfair. I don’t see how I can fix this problem. In their future jobs, they will all have to work in groups, probably with people they don’t like. I think a valuable skill (for many of them, a skill more valuable than being able to do Calculus) is to learn to work in a group, hold each other accountable, and get along. If anything, I want to shift toward using MORE group work in class.

All in all, this was another set of encouraging student evals. I think I’m doing something right!

Mandatory Office Hours

Here’s a quick tip for the new school year: make every student in your class visit your office hours within the first week. Make this a grade. I have a quiz every Friday in my classes, and to get full credit on the first quiz, a student needs to have visited my office by then.

Why? The reasons are two-fold. First, the students benefit because now they know where my office is, and when they can find me there, and how to make an appointment with me if my office hours don’t fit their schedule. And they’ve learned that coming to office hours isn’t so scary, so maybe they’re more likely to come back later. Second, I benefit because now I’ve met and talked to all my students one-on-one, and I’ve had an extra opportunity to learn their names early on (I struggle with names, but I make a conscious effort to know them all by the end of the second week). Everyone wins, and it’s a very low-effort assignment.

Try it! Let me know how it works for your class.

What is math research?

“This might be a stupid question, but what is math research?” is the most common question I get at social events and family functions, and it’s not a stupid question. Many scientists do their research in a laboratory, or in the field, taking samples or surveys. Humanities researchers read texts or look up primary sources. Mathematics research is unique because it can be done alone in a room with nothing but paper, pencil, and one’s mind.*

(*Math research isn’t usually done this way. Usually mathematicians read many other papers before starting a new research project, reference books as they go, and collaborate with others instead of working alone.)

So what is math research? Math research is the process of discovering truths about numbers, shape, and other mathematical objects. Math research is asking, “Is there a relationship between this and that? Is this always this way? Does that ever exist?” and then trying to answer these questions, for the first time. Let me give an example. Someone once looked at a right triangle (a triangle with one angle measuring 90 degrees) and asked, “Is there a relationship between the side lengths of this triangle?” This person used some of the math they already knew (the area of a triangle, for example) to discover that the sum of the squares of the side was equal to the square of the hypotenuse. Or as you may have heard it: a^2 + b^2 = c^2.

“Well that’s not research. I’ve known that since 6th grade,” you may say. But it was research when the first person discovered it. Now it’s a well-known fact, and most people have applied it many times in school without ever wondering why it’s true. Unfortunately, math is taught in schools as a set of facts, presented to be memorized and applied. Students never have to struggle with discovering the facts; they only struggle with when and how to use them to complete their homework. But this isn’t really math. Math is the process of discovering these equations, these new relationships and patterns, for the first time. Real math is math research!

Math research can involve answering (or at least attempting to answer) all sorts of questions, even ones where the answer is already known. Here are some example questions one might consider:

How many spheres of radius 1cm can fit into a box that is 1m cube?
If 100 people are at a conference and all want to shake hands with each other, how many handshakes will occur?
Are there any whole numbers a, b, and c so that a^3 + b^3 = c^3?
Does 1 + 1/2 + 1/3 + 1/4 + … equal a finite number, or does it grow to infinite? How about 1 + 1/2 + 1/4 + 1/8 + …?

So when I say that I’m working on math research, I mean that I’m trying to ask questions about numbers and mathematical objects (in my case, about graphs) and answer those questions for the first time. I’m trying to be the first person ever to notice and prove that a certain relationship or pattern exists. Which, when it happens, is pretty cool!

Calculus project – real-world data

I recently assigned and graded a take-home project for my Calc I class. Here is the assignment I gave them:

This is a group assignment. You must find a partner in the class to complete this assignment with. If you cannot find a partner, let me know and I will help you find one.
1. Find a real data-set that represents some quantity measured over time. This can be population data, the position of some object/particle, height, weight, temperature, etc. Look at the exercises from section 2.1 for examples of the type of data you’ll need. You should provide the original data and a citation for where you found it.
2. Sketch a plot of the data over time. This can be done by hand or on a computer, but it should be neat and legible.
3. Using the data, pick one of the times. Then find the average rate of change for six different time periods, where the first time you pick is one of the end points for each time period.
For example, if your data uses the times 1900, 1910, 1920, 1930, 1940, 1950, and 1960, you might choose 1920 as your fixed time. Then you’d find the average rate of change for 1900-1920, 1910-1920, 1920-1930, 1920-1940, 1920-1950, and 1920-1960. Show your calculations for at least one of these, and create a table displaying all of the results.
4. Estimate the instantaneous rate of change at your chosen time, and justify where this number came from. You should write a few sentences here explaining what the instantaneous rate of change represents and how you tried to find it.
5. Discuss briefly the difficulties in using discrete data to estimate instantaneous rates of change, and suggest in what ways the data collection could be changed to make your job easier.

If you have any questions about this assignment, please ask me, sooner rather than later.

I think my instructions for part 3 could have been more clear, but the class seemed to understand, and I was happy with most of their submissions. Several groups happened to pick “population of Atlanta” as their data set, but some were more creative, and this helped me get to know their personalities better. For example, one group did “number of YouTube followers PewDiePie has per month, so now I know these two students are gamers.

I was also happy to see different methods for answering part 4. Some students took the left and right estimates and averaged them (e.g. averaging the 1910-1920 and 1920-1930 rates), while others found a middle rate (e.g. calculating the 1910-1930 rate). People who chose an endpoint for their fixed time struggled here, so in the future I might recommend they always choose a middle time to fix, but it was nice seeing how these students justified their answers.

The answers to part 5 were of varied quality, but I generally saw the answer I wanted: having more frequent data points would help get a more accurate instantaneous rate of change. I think that it’s important for students to interpret their answers in words instead of just providing a number, which might be meaningless to them.

I liked this project for several reasons, not least of which being that now every one of my students knows someone in class, has their contact info, and knows where/when is a convenient time to meet with them. Studies show that one of the best indicators of student success in math classes is studying with peers. As I assign more projects throughout the semester, I’ll probably require students to have new partners each time, so they can continue to meet more classmates and hopefully find one they work best with, with whom they can study in the future.

Proof by examples

I have spent the past week helping students at QTM Math Camp complete their final research projects. I have loved introducing students to math research. I was impressed with some of the project ideas they invented, and with some of the methods they used. (I’ll discuss these in another post.) Some of their methods, though, lacked rigor, and this became my greatest obstacle with helping them complete worthwhile projects.

Of the 30-odd students enrolled in the camp this year, 20+ took a “coding” elective, where they learned how to program in Python. I think this is a valuable skill for any young student, and for young STEM-oriented students in particular. However, many of them began to rely too heavily on coding when working on pure-math research projects.

Let me give an example. One research group (most groups had 3 or 4 members) wanted to study the moduli of the Fibonacci sequence. I thought this was a nice, reasonable project for high school students. One of their results is that every third Fibonacci number is equivalent to 0 mod 2, while the others are 1 mod 2. This first result is simple to prove, but the students were not relying on any mathematical proof. Instead, they wrote a program to find the first 1000 Fibonacci numbers mod 2 and plot the results. Then they noticed the pattern and concluded “I guess every third Fibonacci number is even.”

I had to push them to justify this conclusion mathematically. When the started to look at moduli larger than 2, they again tried to rely solely on the “proof by examples” that their program gave, and I had to intervene yet again. My reasons for why 1000 examples don’t constitute a proof were that

A pattern can hold for the first 1000 terms then break, and we don’t know which patterns will break and which will hold.
Noticing a pattern is not very elucidating. It’s much more interesting to know why, e.g., every third Fibonacci number is even. Their computer program cannot explain this.

I think the programs these students (and others in the class) wrote are helpful for noticing patterns, or checking hypotheses. Certainly I’ve written similar programs to aid in my research. Most of them were excited to know Python and wanted to use it immediately, which I understand too. I’m glad though that I got to teach them a little about how pure math research is down.

In my teaching going forward, when demonstrating certain proofs, I will try to emphasize why doing a few (or 1000) examples is not sufficient to prove the statement in general.

A quick note on board-work when teaching

A simple hint to improve your teaching: when you’re writing on a board and you run out of space, you should take time to slowly and completely erase part of the board. It is natural to try to erase the board quickly so you can get back to writing notes, but this is a mistake in my opinion, for two reasons.

First, students need a moment to process your lesson as you teach. When you take twenty seconds to erase the board (rather than ten seconds), you give them a better opportunity to catch up with what you’ve been saying. They can finish copying down the last thing you’ve written, or can think about whether the lesson makes sense so far. Students appreciate this short break just to catch their breath during a lesson.

Second, quick erasing is often sloppy erasing. Do yourself a favor and clear the next section of the board completely of all stray marks. Now you’ll have a clean slate (literally) on which to write. Your students will have an easier time reading what you’ve written if there are no straggling serifs sprinkled throughout nor ghosts of what was previously written haunting the background.

Also, taking a bit more time to clearly erase the board gives you a second to breathe as well. You can gather your thoughts and prepare for the next section of notes you want to write, so you can give the best lesson possible.

Emphasizing intuition over algebra

I just saw this great YouTube video giving an intuitive explanation of why e^(i*pi)=-1. The creator doesn’t use any trigonometry or calculus, or really any algebra. Just basic arithmetic and lots of great visuals. Give it a watch here:

I often find myself relying on rigorous notation and calculation when trying to explain an idea–both in the classroom and when writing papers. Of course, being correct is a priority. But being convincing is nearly as important. If a student or reader walks away thinking, “I guess I buy it. I mean, all the algebra checked out,” then I’ve given them very little basis for growth.

I want to start including more intuitive explanations in my teaching. I can still give the formal proof of a concept, but let me first try to justify the outcome naturally.

Why does sin(x)/x approach 1 as x approaches 0? Not because the squeeze theorem says it does, using two cleverly chosen other functions. But because sin(x) has values very close to x near the origin.

Why does the formula for Simpson’s Rule work? Again, not because we can do a bunch of arithmetic and change the indices on a summation. It’s because it’s a weighted average of two other approximation rules, one that over-estimates and one that under-estimates.

I can also focus on giving intuition when writing my research. Before I define some new variable that seems to come from nowhere, and before I start some algorithm on a graph that decomposes it for some reason, let me invest a couple sentences saying “Here is the goal. Here’s how we’ll reach it.” Now the reader has some idea where we’re headed and may suddenly recognize the route we’re taking to get there, instead of spending the entire proof scratching their head. It’s not enough that they agree that all the variables add up the way I claim; the reader should walk away feeling like they’ve learned a method they can reuse.