Starting Monday, June 28, 2021, at 9:00 AM ET, our community will hold an asynchronous discussion on this thread about "Expanding our view of math history and culture. This two-day forum will have one guest discussant, Eric Appleton, who is the current president of the Adult Numeracy Network. He will highlight areas for teacher learning and curriculum development. It is often assumed that all mathematics comes from Europe and the ancient Greeks. Studying the history of mathematics can help teachers and students learn about the contributions of many cultures to our current mathematical understandings. Our number system comes from India. Algebra was developed most notably in Islamic learning centers. Mayan mathematicians independently developed 0 and a base-60 number system. Chinese mathematicians proved the Pythagorean Theorem long before Pythagoras. He will also discuss, what can math teachers do to develop knowledge of the history of mathematics in non-Western cultures and share this history with their students?
So mark your calendars for June 28 - 29 and please participate in this discussion. You may post any questions, ponderings, or musings you may have, go ahead and post them below. I look forward to growing as a math practitioner from this community discussion.
Should be interesting ;)
I want to thank Brooke for inviting me and give appreciation to Pam for the discussion last month on culturally responsive teaching in mathematics. I really liked how Pam shared ideas from the books she is reading and the webinars she has joined. She provides us a model for teaching while continuing to learn. Thank you also to Mark for sharing so many resources in the last discussion and continuing to collect resources at the Adult Numeracy Network’s web site.
A bit of an introduction: I have been teaching in adult education for a little more than 20 years. I work as a staff developer for the CUNY Adult Literacy Program in New York City and am the current president of the Adult Numeracy Network. I taught computer classes and reading/writing for a few years before I attempted to teach math. I saw my colleagues teaching math and it seemed like they were having a great time. I was curious and nervous, but caught the bug fairly quickly and have since tried to build my understanding of basic math while learning to teach math effectively. I still have a lot to learn on both counts.
And I am no expert on the topic of math history. In fact, I consider myself a complete novice. I hope that we can work together to brainstorm ideas and resources to improve our use of math history in adult education. Over the next couple days, I’m hoping we can talk about math history, but also try out some possible activities together.
In that spirit, take a look at this image:
What do you notice? What do you wonder?
Please share your noticings and wonderings in a reply.
I notice that the four corners have 2, 4, 6, and 8 colored dots, and that all the other dots are not colored in. It looks like it might be a game--I wonder if it is.
And now I will go see what everyone else noticed and wondered about...
I am coming in taward the end of this discussion... but I am going to read and catch up!
It is the only one that is not connected to another circle in any way.
Notice- There are open dots and closed or filled in dots. The closed dots are only in the corners and in even amounts.
Wonder- I'm naturally a questioner and have about 12 questions but will just list one for this post. Why are the open circles in lines but the closed or filled in dots represented as groups?
I notice that there are lines/dashes that connect most of the dots and that one dot stands out as being all by itself on the bottom of the image, disconnected from any other group of dots.
I wonder why the grouping of eight dots in the bottom right corner has those little dashed lines between only some of the dots in the group whilst other dots in the group of eight are not connected this way.
Some things I notice:
- I notice dots in groups that can work as pairs adding up to 10. The top left has 4 dots and the bottom right has 6. The bottom left has 8 dots and the top right has 2. The horizontal row across the top has 9 dots and the bottom has 1 dot. The vertical column on the left has 3 dots and the vertical column on the right has 7 dots. And there are 5 dots in the middle, 5 being the number you double to get 10.
- Another possibility is that the number here is 15, since each of the pairs I listed above goes "through" the 5 in the middle, except the 5+5.
- When there are an even number of dots in a group, they are arranged in an array of two rows of half the number of total dots (so for example, the group of 6 dots is arranged in two rows of three, the group of 8 dots is in two rows of four).
- The group of 5 dots in the middle is the only one with an odd number of dots not arranged in a straight line.
Some things I'm wondering:
- Is this 10 or 15?
- What would another number look like?
- This figure has dots in the corner and on each sides of a "square". Could we have a figure that has the sides of another regular polygon and its corners - like say a pentagon?
- How are groups of large even numbers arranged? Would 26 still be 2 rows of 13? Would 100 be two rows of 50?
- Are the 5 dots arranged the way they are because 5 is in the middle or because of some special characteristic of 5?
Let's keep noticing and wondering! Thank you for the ideas so far. In the meantime...
As I said, I don’t know a lot about math history. I would say I’m just getting started. But I’m interested in the topic and I want to know more about how scholarship and awareness can change the accepted views of what math history is. There are so many recent examples of how the teaching of history can reshape our understanding of our country. Caste, for example, the book by Isabel Wilkerson that Pam mentioned, is a good example. With a reframing of how race operates in the United States and tireless scholarship supporting her ideas, Wilkerson shifts our idea of what racism in the United States is, not just the prejudice of individual people, but the infrastructure of a system of hierarchy. The History of White People, by Nell Painter, affected me in a similar way. It helped me realize that racism was constructed as a way to justify the economic institution of slavery and was built piece by piece, through specific events in European and American history. The telling of historical events helped me realize that I had the ideas backwards. Racism didn’t cause slavery. Justification for slavery created racism. I think the uncovering and retelling of our actual history is a powerful force for understanding the world. It’s not just about retelling history to include excluded perspectives. It’s about understanding what actually happened, learning the real history.
I’m wondering now about unexamined ideas about the history of math that need to be (maybe already are being) upended by the actual history of how mathematics developed around the world. I love the resource Pam shared in the discussion last month for researching non-white mathematicians. Let’s tell a fuller story of who does math by bringing examples of non-white mathematicians into our classrooms. How can we tell a fuller history of math as well?
Question for you all: Do you teach any examples from the history of math? What stories do you tell? What resources do you use?
Is part of the study of the history of math revisiting what gets valued as math?
Annie Perkins, the Minnesota math teacher who created the Mathematicians Project mentioned above (and in the discussion thread that Pam lead) has also created/curated some resources on Math Art, which she defines as something which inspires curiosity about structure. If you are looking for activities to teach your students math through the making of beautiful art, check out her Math Art Challenges. I bring that up here because in exploring Sona drawings of the Tchokwe of south-central Africa, Mola textiles of the Kuna in Panama, Islamic Geometry, Japanese Hitomezashi stitching, Celtic knots, mandalas, rangoli, etc you can engage with mathematics that came out of a particular cultural context and history.
Professor Filiberto Barajas-Lopez argues all cultures have rich mathematical traditions that we need to draw upon, asking two powerful questions: (1) Do our students feel like their own culture is reflected in the math they are learning? (2) What opportunities exist for them to develop lasting mathematical identities?
His work focuses on ways that teachers can amplify opportunities for immigrant and non-immigrant students of color to productively participate in school math and learn ways to expand the way we understand what kinds of math count.
Here's a 5 minute video of a talk he gave called What Exactly is School Math for?
This conversation reminds me of an Amplify webinar I attended last year, during which I bookmarked... this article, among others:
Thank you, Lakshmi! What a useful article. There are springboards to lots of great stuff to explore. A couple that stood out:
- The Mathigon Timeline of Mathematics - I've just started using Polypad, which is fantastic. Another great tool.
- The Crest of the Peacock, a book about the non-European roots of mathematics.
I love the reminder that teaching math history is about storytelling.
The question is who do we have doing the storytelling and framing what is noticed and important?
I appreciate your raising the historical truth that slavery was not caused by racism, but that racism was developed in order to justify slavery. It makes me think about all of the knowledge in general (and science & mathematics in particular) that was devalued, ignored and/or destroyed in order to justify slavery and colonization.
I remember adapting a story from Rethinking Globalization (Bob Peterson) titled Burning Books and Destroying Peoples and reading it with ABE students - it was about how in 1562 a Spanish priest ordered that 800 years of Mayan books be gathered and burned. At the time we looked at images of Mayan numerals and played around with how they worked and talked about how only 200 or so pages remain of the Mayan codices. I am grateful to those students from continents all over the world who discussed this 440 year old moment of colonial violence in Central America and connected it immediately and deeply with their present. We talked about the colonial mindset of dehumanizing people and targeting language and culture to establish power and control. We talked about what was lost for the Mayan people, but looking back, we did not talk enough about what is lost to the world when systems of knowing are erased and/or hidden.
If I were to teach it again, I'd try to do more with the numerals (what I could have done with notice/wonder back then) with the goal of making connections between math being cultural in that it is a human endeavor of making sense of things that is given shape by the environment we are in and the people who are in that environment with us.
I've only dipped my toes into weaving in math history to provide some context for content. There are some examples in geometry in some student geometry materials I wrote - Lines, Angels, and Shapes: Measuring Our World
- history of the Babylonians, angles, and 360 degrees (see pages 40-45)
- attributing the idea of using square units to measure area to the Egyptians (pages 80-81)
One thing I'm thinking after thinking about your question is the difference (and the overlap) between teaching math history and using math to explore history. I've done lessons on area and percentages looking at Thaddeus Stevens' 40 acres and a mule plan, lessons on area and the land grab of the Mexican-American War, all of which are using math as a tool to study history. Part of the history of math is all the untold history and mathematics. Another part of the history of math is why those histories and mathematics are untold.
I have so many more questions than answers about any of this - Does it matter? If it does, why does it matter? If it doesn't, why doesn't it matter? Who gets to decide? How and where have untold mathematics resisted erasure and survived? What can we learn from that survival and what can we learn from that mathematics? - and I am grateful for this thread.
yours in productive struggle,
Thank you for the story of Mayan books from Rethinking Globalization. I found a pdf online (p. 43). Did you read an adapted version of the Eduardo Galeano story with your students? Or is there a longer section in the book on the conquistadors in Central America?
And thank you for the idea of doing a notice/wonder with Mayan Numbers!
Everyone, maybe we could try it together? I made a Mayan number notice/wonder with a Google Doc. You can edit the document to add your noticings and wonderings or respond in the message board.
Thanks for creating this notice wonder activity with the Mayan numerals! I appreciate the numbers you choose to share as examples to encourage the noticing and wondering.
The image I shared earlier today is called the Lo Shu magic square. Each vertical, horizontal, and diagonal line adds up to 15 (including the 5 in the center). It was created more than 2500 years ago in China and is sometimes depicted on the back of a turtle. Many kinds of magic squares were explored in China, Japan, the Middle East, India, North Africa, and Europe.
For a warm-up activity to use with students: http://www.collectedny.org/frameworkposts/lo-shu-magic-square/
An introduction to the history of Chinese mathematics: https://www.storyofmathematics.com/chinese.html
How might you follow-up with this notice/wonder activity?
Reflecting back on my math education, I don’t know that I was ever taught explicitly that mathematics has been primarily a European field of knowledge, but it was certainly the implicit message. The only figures from math history that I encountered in my schooling were European (Fibonacci, Galileo, Newton) or Greek (Pythagorus, Euclid). I never learned about other mathematical traditions until I started teaching math in adult education. I remember a fellow teacher pointed some of the numbers used in measurement of time are arbitrary (or a cultural invention). A year is the time it takes for Earth to go around the sun. A day is the time it takes for Earth to rotate. A month is about the time it takes for the Moon to orbit Earth. But what is an hour? A minute? A second? And why are there 60 minutes in an hour and 60 seconds in a minute? It never occurred to me that our conventions with these measurements are the result of an ancient civilization centered around modern-day Iraq.
My colleague at CUNY, Mark Trushkowsky, and I spent a few years writing math curriculum for high school equivalency, focusing on the TASC exam (http://www.collectedny.org/ftgmp). As Mark said, we have tried to incorporate math history with examples from different cultures. Mark gave the example of Babylonian mathematics in the packet, Lines, Angles, and Shapes. See pages 11 and 40-45 of Part 1:
360 degrees in a circle. 60 minutes in an hour. 60 seconds in a minute. 12 or 24 hours in a day (the Babylonians counted 12 hours). Why these numbers?
Using this jumping off-point, we could get into factoring numbers. If we are measuring fractions of a circle, why would 360 degrees be nicer than 100? Why would 60 minutes in an hour be better than 100 minutes (something the French tried to do during their revolution.)
There must be hundreds of other examples of mathematics in our daily lives that can be traced back to different cultures. What examples have you used or could you imagine using in our math classes?
The following is an example of an introduction Mark and I wrote to the packet, Expressions, Equations, and Inequalities. We wanted to show that the ideas we’re building on have a rich and varied history around the world. At the same time, we have to be economical in how much math history we bring into the classroom. The readings need to be accessible and most of our time should be spent doing math, not just reading about it. It’s a hard balance to strike.
History of Algebra (excerpt)
Algebra was developed by different civilizations many, many years ago. More than 3000 years ago, Egyptians in North Africa used algebra when building the pyramids, conducting business, and measuring how stars and planets moved across the sky. Around the same time, the Babylonian civilization in present-day Iraq used algebra to measure land and do business. The ancient Greeks used geometry and drawing to further develop algebra. In India, mathematicians were solving algebraic equations more than 1500 years ago. 1000 years ago in China, mathematicians explored different kinds of algebra. They also invented some of the earliest magic squares...
... Living about 1200 years ago in what is now Iraq, a famous Islamic mathematician and astronomer named Al-Khwarizmi (pronounced al-kwar-is-me) wrote a book of instructions for solving equations. The full title of the book could be translated as “The big book of finding solutions by balancing and completing equations.” The book was written as a practical guide for using algebra to answer questions in business, land ownership, inheritance, and construction. The title of Al-Khwarizmi’s book included the Arabic word, al-jabr, which is where the word algebra comes from. Al-jabr referred to how a value from one side of an equation can be moved to the other side, finding a solution while keeping the equation in balance.
Al-Khwarizmi also advocated the use of Hindu-Arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, and 9, which are the same numerals we use today. He learned these symbols from Indian mathematics. At the time, Islamic mathematicians didn’t use letters or symbols in algebra. All algebra was done with regular words. A mathematician at the time might have written, “A number multiplied by two and added to five,” instead of writing 2x + 5.
Original: Pages 5-7 of Expressions, Equations, and Inequalities (http://www.collectedny.org/wp-content/uploads/2019/05/Tools-of-Algebra-Expressions-Equations-and-Inequalities-Fast-Track-GRASP-Math-Packet-Part-1-v1.8-05.06.2021.pdf#page=5)
When I try to understand a bit of algebra of Al-Khwarizmi, I am amazed by Islamic scholars’ ability to reason about generalized quantities, even though they didn’t have symbols for variables and operations. Take a look at this description of a problem and an excerpt of its solution:
If some one says: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less a thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one...
Can you translate this into a modern number sentence with variables and other symbols?
I was hoping that students could look at excerpts of the actual text, but it’s obvious that setting up and solving quadratics in this way is just too complicated for our classrooms. I can’t get my head around it.
I wonder if a next step with students might be about expressing mathematics with everyday language instead of symbols, using simpler examples. Our students could definitely use practice understanding variables as something other than “x.” We might ask students to find some mystery numbers:
When I double my mystery number, I get 26. What is my number?
If I add 5 to my number, I get 18. What is my number?
When I subtract 10 from my number, I get 35. What is my number?
We might also ask them to connect number sentences with mystery number descriptions:
A. Twice my number plus 5 is 10.
B. When I subtract 10 from half my number, I get 5.
C. My number multiplied by 2 is 5 more than 10.
1) ? - 5 = 10
2) ? ÷ 2 - 10 = 5
3) 2 * ? + 5 = 10
Does this seem useful to you? What ideas do you have for connecting math history with the skills that students need to learn?
This would be a great way to connect verbal algebra (expressions we want students to understand) with math history-- because that's where it started!
All number work has verbal, quantity and symbolic parts-- we should really keep making those connections!
Thinking about the colonial mindset of denying and erasing mathematics that looks different from Eurocentric mathematics (as part of the dehumanization that served as a justification for stealing land and enslaving and exploiting people) I'm curious if other people see a connection between that mindset and the inhibiting conception people have that there is one right way to approach math problems. And if so, what effects correcting that narrative could have on all adult learners?
In my experience, "the right way" is usually a symptom of the teacher really only knowing that way, or perhaps choosing to enforce a "right way" because of experience with that being effective in helping students construct concepts in their minds. That said, in higher level maths the power games are almost certainly more pervasive.
There is so much more we could explore together. For example, I was recently introduced to Egyptian fractions, multiplication, and division. I'm dipping my toes into Count Like an Egyptian, by David Reimer. And the book Crest of the Peacock has a chapter on Egyptian mathematics. The way they did multiplication is ingenious. If you can double, halve, and add, you can multiply any two numbers. No need to memorize multiplication tables.
Here's an example:
Can you create your own example? Can you explain how this works?
Thank you all for your participation this week. I hope we can continue these threads. Maybe we could drop in when we hear of a new resource or learn a bit of history.
"If you can double, halve, and add, you can multiply any two numbers. No need to memorize multiplication tables."
Eric, I would love a demonstration of how this works. I couldn't make enough out from the image. The could be the kernel of a very interesting webinar.
Thanks, Lakshmi. Does this explanation help? A webinar or a teachers circle meeting on Egyptian multiplication would be fun. We had a teachers circle meeting on multiple ways of multiplying a while back. It's interesting to compare methods to each other.
Also, I misspoke. I don't think you need to halve. Doubling and adding is enough to do multiplication the Egyptian way.
Thank you, Eric, for leading this discussion. The Math and Numeracy Community appreciate you leading this forum and we thank all those who participated and added to our knowledge. Please continue to discuss and add to this robust discussion. Eric is a member and he may choose to contribute periodically. Many of the participants are also leaders in math and numeracy, so we can all learn together.
Thank you - and let's continue to engage each other!