Hands on activities for a college history of mathematics course

Question

I will be teaching a course in history of mathematics to juniors/seniors who are math and math education majors, many future school teachers. It should include highlights from antiquity to early 19-th century, but I'd like to supplement it with "hands on" experiences of what it was like doing mathematics in centuries past. Especially since it's a small class, 20-25 students. I don't mean "hands on" literally, like manipulatives, it could be group activities, individual tasks on paper, or even small exploratory projects, but with doing rather than just listening.

What concerns me the most is balance, it is always hard to come up with activities that are meaningful but at the same time what is given in class is enough to reasonably expect students to work out things on their own, I don't want to get them discouraged. Unfortunately, our students have limited background and skills, they can be expected to know (algorithmic) linear algebra, calculus, differential equations, etc., but I expect them to struggle with any kinds of proofs, even in Euclidean geometry and elementary number theory, or with problems not directly analogous to examples.

I was thinking about breaking down some of the classical proofs/solutions into steps and letting them do some of the steps, with hints and nudges, but it is hard to do for the entire course on my own, or to gauge the balance. So I was hoping to find online resources or books that do something like this, or other things, I am open to suggestions. Unfortunately, although books like Eves's or Burton's list a lot of exercises and suggested activities, almost all of them seem to be much too advanced for semi-independent work. And what I saw online is mostly for grade school and/or historical nuggets embedded into other math courses, not a systematic history course.

Any advice and links/references are appreciated.

Answer 1

I had my students in Math for Elementary Teachers doing arithmetic in the Babylonian, Mayan, and Egyptian systems. It's not beyond them at all, and it helps them understand place value more deeply.

I don't have recommendations for a text for the whole course, but you may find a lot of useful material in Count Like an Egyptian. I hope to run a math circle with material from this book. It explains multiplication and division by Egyptian methods, and Egyptian fractions, which we'll be exploring.

I haven't finished it yet, so I just read the reviews on Amazon to see what I might be missing. Apparently, the book is broader than I'd thought: "Despite its apparently limiting title “Count Like an Egyptian”, this book delivers all that its subtitle “A Hands-On Introduction to Ancient Mathematics” promises. Besides presenting a comprehensive overview of the ancient Egyptian computing methods, Professor Reimer also introduces you to the Mesopotamian sexagesimal system, then compares these with the Roman numerals and Mayan counting glyphs, as well as with the modern decimal and even binary ways of expressing numbers."

Answer 2

You might use various proofs of the Pythagorean theorem. Several lend themselves to experimenting with paper cutouts. And they can be connected to historical documents. See the Cut-the-Knot website for over 100 proofs, including this ancient gem:

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Answer 3

The MAA (Math Assoc Amer) maintains a collection of resources at the website for Convergence to support teaching math via its history. They have a link to a "Treasures" list which includes facsimiles of many, many historical documents.

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          ^{The "Witch of Agnesi": Maria Gaetana Agnesi's cubic curve.}

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The publication includes articles that could each form a classroom module, e.g.,

Nicholas A. Scoville, "Connecting Connectedness: A Mini-Primary Source Project for Topology Students." 2016. (MAA link.)

Answer 4

In addition to the good answers already given, here are some additional ideas.

• Joseph's answer predates (or is just at the beginning of?) the official start off the TRIUMPHS project, which has many, many more such mini-projects. They are intended for not history of math classes, but are very useful for those as well. The Chinese linear algebra one is a very good choice, for instance.
• In addition to Sue's excellent examples, I suggest these ones that my students seem to enjoy: Egyptian fractions (explicit computation); Greek numerals (the learned system here)
• Anything Euler: his amicable number formula, finding an Eulerian trail, Euler characteristic ... a lot of these are possible to compute pretty quickly in a classroom setting
• Doing a derivative using Newton's fluxion notation. Any math history book should have this - but it's nice since his approach makes finding slopes of a variety of curves (not just a function graph) doable, analogous to implicit differentiation
• Solving a cubic or quartic polynomial using the various formulas of Cardano's era. I don't personally like this, but students seem to.

Answer 5

It might be difficult to find paper copies these days, but the first edition of A History of Mathematics by Carl Boyer had great exercises at the end of each section. For example:

• Exercises from Chapter 12 on China and India
• Exercises from Chapter 2 on Egypt
• Exercises from Chapter 26 on the Rise of Abstract Algebra

Later editions of the book dropped the exercise sets, unfortunately.

Answer 6

Have them use an abacus. (I prefer to teach using a Chinese abacus, which has 2+5 beads per column, and not using a Japanese abacus, which has 1+4 beads per column.)

I propose the following activity. (I've only done this in undergraduate classes a handful of times, but the students seemed to enjoy it.) Teach them how to use the abacus to add numbers. Do not teach them how to subtract. Then have them add nines in succession, with you starting the process. Start from 0, then add 9, but add 9 as +5+1+1+1+1. Do this three or four times. Then have volunteers do the addition by 9. The students will find this quite difficult at the start. Then (in my experience), a student will discover (on their own) that it is much easier to do the addition by 9 as +10-1. Once a student has done this, point it out to the class. Praise the student for finding (without being prompted) a simpler procedure than the one taught by the teacher. This ends the activity.

Answer 7

Why is this assumption that you should do proofs at all, even in simple fashion for the activities? There are activities that are fun and ability appropriate that have nothing to do with proofs.

Some ideas (on a super slow net connection, so no links):

1. Google Eleanor Dickey on Youtube. Her main interest is education in Roman times (what they actually did). While a lot of her talk is about language instruction or cultural learnings (slavery accepted, cleanliness emphasized), she does have a video showing a little example Roman school with current little English girls taking part. Shows them pushing beads around on the little strange segmented rectangle counting boards. Teachers would like that as it is in some ways similar (at least in appearance) to the new fangled boxes and stuff for math...at least it shows them something to think about. I would even let the troops dress up if they want...might be surprised at the creativity and enjoyment they get out of it...even if they aren't MSE Rudin studs, can be surprised by how some people like amateur arty activities. And it increase the sugar to medicine ratio. More sugar, more sugar, more sugar...help that medicine go down!

2. Finding pi statistically using needles dropped on a board. (There are some math lessons about stats and angles and such...but don't underestimate the fun and the traction from being able to doing something real and physical.) There are some good 'net articles on it and after the exercise, you can teach the history. But I would keep it simple and design the test using a line spacing equal to needle (or toothpick or matchstick)--the Wiki article needlessly complicates the exercise with making spacing not equal to the needle.

3. There is a nice Six Sigma exercise using a catapult...gives learning about angles and statistics and such.

4. Guessing the jelly beans in a jar (average, range, etc.). Leads into a general discussion of Bayesian guessing--please don't get all wrapped up with the conditional math, but talk a little about how this is an important concept nowadays and helps people make money (efficient markets, election and sports betting, etc.) These are topics that excite people since they have impact on their lives ($$). [Always try to have some "motivation" for science or math subjects...usually this will be economic, but it might be social or military. And by motivation, I don't mean a long complicated word problem derivation of a diffyQ...I mean mentioning that it helps people make money or save lives.] Let 'em eat the beans afterwards. Kids love that sort of stuff. Adults are big kids. Leaves them with a good feeling about the experience also.

5. Less formed idea, but you could do something with shooting stars (or sun or moon). Don't teach them celestial nav (right ascension of Ares, blabla) or all the details of every measurement. But just a couple measurements so they get to "learn with fingers" and just play a little. Then a discussion of how development of trig, navigation, time-keeping, etc happened. This was math with both economic and military consequences and very historical. Furthermore in the (probable) event that your students are never able to take all the general science that would benefit them as a high school teacher, at least you have given a little exposure here.

6. Less formed idea, but some simple card game or gambling exercise and the stats insight from it. Don't go super in depth on the stats, but do a little afterwards so they get that it wasn't just a game. Can also discuss some of the history of probability and games of chance (this is fun and excites interest). Also, there is an important cross-cultural component in that anthropologists have seen evidence across many cultures (rain forest to Eskimo, etc.) that gambling is a concept that people understand. [Need to research to find some good articles on this, but I have seen it talked about at times.]

7. Even less formed idea but some simple prob/stats game with pretend farms. Much of probability and stats and even genetics comes out of agriculture. It is also an area that intuitively resonates with people (getting food to eat) and some of the first words and concepts we learn as a child are related to food and animals. Many business frameworks use barnyard terminology since it resonates well with people. [You probably don't need to design this thing from scratch...I bet someone has had same idea and designed an exercise, you could research.]

Answer 8

I saw a You Tube video recently which was shockingly clear on the topic of quadratic and cubic equations and their history. See: How Imaginary Numbers were Invented. In particular, the video shows quite nicely how completing the square was literally completing a square. This could easily be adapted to an exercise for the classroom.

1.) Study $x^2+26x = 27$ by visualizing the $x^2$ as the area of an $x \times x$ square and $26x$ as a $26 \times x$ rectangle. Use colored paper to make it easy to track. Or, have them color a template if you want coloring as part of the exercise.

2.) Cut the $x \times 13$ rectangle in two and arrange one half horizontally with the $x \times x$ piece and the $x \times 13$ piece. Rotate the other half and set it vertically so the $x \times x$ is over a $13 \times x$ piece.

3.) Complete the square with a $13 \times 13$ square.

Notice to be fair we added an area of $169$ to our picture so we must likewise add $169$ to both sides: $$ x^2+13x+13x+169 = 27+169 = 196 $$ The equation above represents the areas of the parts of the square (left hand side) and the total area of $14^2 = 196$ (right hand side). Since the big square pictured has side-length $14$. Thus $14 = x + 13$ and we find $\boxed{x=1}$.

I really like this demonstration for two reasons:

• It gives a reasonable origin story to the terminology completing the square
• It makes the history of ignoring negative solutions to polynomial equations much more reasonable. The mathematicians of that era were not really studying polynomial algebra. Rather, they were asking particular geometric questions for which only geometrically reasonable answers were permitted.

Moving past this particular demonstration, I think it is interesting to note mathematicians were hung up on a visualization. Often students seek visualization as a means of understanding. Sometimes, that is a wrong idea. To truly understand something, you might need to give up on visualization and instead embrace an algebraic approach. For the problem visualized here, $$ x^2+26x = 27 \ \Rightarrow \ (x+13)^2-14^2=0 \ \Rightarrow (x+13-14)(x+13+14)=0 $$ hence $x = 1$ or $x=-27$. A single line of algebra has replaced a page of cumbersome pictures.

Of course, seeking an algebraic understanding is not a universal advice. Just like some screws require a flat-head and others require a star-head. Each problem has tools which are best. Knowing which tool to use, that is an art we're still learning.