No study of mathematics can be considered

Junaid Mubeen – Mathematics. Education. Innovation.

complete without attention to its history.

In *Bill & Ted’s Excellent Adventure*, Bill & Ted are confronted by Mr. Ryan, their history teacher, about their not being able to graduate because they are failing his class.

**Mr. Ryan:** Listen guys, don’t forget, tomorrow. Final Reports, 1:30-3:30, okay? (to Bill & Ted as they leave) Hey guys.**Bill:** Mr. Ryan, before you say anything, my distinguished colleague Ted and I wish to express to you our thanks for all the things we have learned in your class.**Mr. Ryan:** And what have you learned?**Bill:** We have, uh…we’ve learned that the world has a great history.**Ted:** Yes, and that thanks to leaders such as Genghis Khan, Joan of Arc, and Socratic Method, ** the world is full of history**.

**Mr. Ryan:**It seems to me that the only thing you have learned is that Caesar was a salad dressing dude.

I became interested in mathematics history when I attended a seminar where the lecturer was Dr. V. Frederick Rickey (see Additional Reading below). His topic was the Mercator map and tangents. Since then I have been delving into mathematics history, and reading about teaching mathematics history, articles, books, and scholarly papers. I would advocate that mathematics history be taught; however, here are my thoughts.

- Mathematics history will add some historical context to what the students are being taught. This will provide a reference as to 5W1H.
- No testing of the history presented!
- I believe teaching mathematics history would be best accomplished within mathematics courses – simply because understanding history of mathematics requires understanding, and good understanding, of mathematics itself.
- I see the following are barriers to implementing mathematics history.
- How it should be taught will create much debate as everyone has an opinion from legislators, consultants, school boards, superintendents, principals, teachers and parents.
- Book publishers will be pushing their books to be used in the schools.
- Consultants will be pushing their ideas, for a price, on how best to implement mathematics history in K-12.
- Teachers will need to be trained but I am sure that schools will not be able to fund teacher education. They already have a funding crisis.
- Some may say that I am suggesting another course be dropped to allow for the introduction of a new course. The answer is NO. I am advocating adding some historical context to math classes, where appropriate.

- “Is there any evidence showing that including the history of mathematics is effective in the teaching of mathematics?” How can there be when it has not been documented as far as I know. Personally, the answer is yes as it stimulated me to be interested in mathematics. I do not think it helped me with the actual math, but it made math class enjoyable.
- When I was taught history in high school it was BORING!! Memorizing dates, names and places without context of how important they were makes it BORING!!

Below are the views of some individuals I believe have some merit as to the benefits of incorporating mathematics history in school curricula.

I propose five reasons for using the history of mathematics in school curricula:^{1}

- History can help increase motivation and helps develop a positive attitude toward learning.
- Past obstacles in the development of mathematics can help explain what today’s students find difficult.
- Historical problems can help develop students’ mathematical thinking.
- History reveals the humanistic facets of mathematical knowledge.
- History gives teachers a guide for teaching.

But sharing history of mathematics will give them three advantages that others will not.^{3}

- When one studies history, one sees connections and motivations. Studying the history of mathematics, one sees what motivated and interconnects key results in math, that can easily be missed in a typical class.
- Understanding the life, the times, the motivations of mathematicians and how they did research will give them a role models and prepare then to do research as well.
- Having such an understanding puts a human face to mathematics giving students in STEM an appreciation for the human condition and endeavors which they take for granted. I believe this will give them a richer appreciation of all areas not just STEM.

Yes. It is important from motivational point of view and also from the historical perspective. Students know just the names of the mathematicians in the context of their theorems, but *not *their lives. It is also important to know the historical development of mathematical ideas to appreciate the beauty and truth of mathematics. On Quora, I have written some answers highlighting some interesting aspects of the lives of some mathematicians like Leibniz, Gauss, Cantor, Ramanujan, Kronecker, Lefschetz, Hilbert, Pontryagin, Gödel, etc.^{4}

Very often, when students learn mathematics they are unmotivated simply because they do not know why the mathematics was developed in the first place. To them, it is just a difficult subject without a purpose.

Teaching history tells the story of mathematics, why did people ever care about it in the first place, and why we should still care about it. For example, Calculus was developed by Newton to answer very practical questions in Physics, and without those answers, something like the Apollo Moon missions wouldn’t have been possible.

Further, while sometimes Mathematics gets ahead of application, it later finds a home. A perfect example of that is Boolean Algebra, for decades it sat as a theoretical novelty, but then found a home in building modern computers. Or fractal geometry, it was conceived and then later found to be of great value in creating realistic computer graphics.

So I discovered when I used to teach Mathematics, teaching history brought the subject alive, provided motivation, and sparked student interest. And once that interest was awakened, students found the work required by the subject was no longer quite as onerous.^{5}

Humans are story-telling animals and the context provides motivation, humanization, and, often, aids in learning and creative problem-solving with the learned material.

I am firmly in the camp that believes that mathematics are created and not discovered. Humans make mathematical models based on real stuff, but the models are human-made things created in particular contexts to solve particular problems – and the models we have aren’t the be-all to end-all of the ways we can use patterns, numbers, and logic to solve problems.

For example, there are multiple geometries and they make sense to solve different problems. On small scales, Euclidean Geometry where we assume the existence of parallel lines can be quite useful to build a house that doesn’t fall down. On large scales, non-Euclidean Geometry can explain the shortest distance between points on a sphere (you know, like the planet we live on?).

Let’s talk about why and how we developed a concept of 0 and the problems that it lets us solve. Why did the Greeks not use irrational numbers and how did it help or hinder them? What factors were at play so that, after thousands of years of people doing math all over the world, two men independently developed calculus at almost the same moment? Why was is primarily social scientists and not pure mathematicians that developed statistics when they tie so closely to probability?

Humanizing mathematics makes it something that is interesting to humans and that humans can do.

Teaching the history of mathematics also clears the air and dispels the prevalent myth that math is something done only by ivory-tower genius white men. Women invented a lot of math, so did people of color. It is very motivating and affirming to see that someone like you succeeded at and contributed to the area you are studying. Like it or not, the default setting for people who think about the development of math is still that it’s a bunch of dead white guys. Not mentioning any humans is letting that status quo remain.^{6}

Many people snoozed through their history classes growing up, and that’s a shame. The more I study it, the more I realize that history is not something boring but rather it’s profound, insightful, and incredibly useful.

Here are a few reasons why you should pick up a history book or two in the not so distant future: ^{7}

- It helps us understand why things are the way they are
- It helps us predict the future by seeing recurring patterns
- It differentiates you

Don’t know much about history

Sam Cooke – “Wonderful World”

## Do teachers need to incorporate the history of mathematics in their teaching?

Po-Hung Liu^{1}, in his paper *Do teachers need to incorporate the history of mathematics in their teaching?*, made the following statements.

The merits of incorporating history into mathematics education have received considerable attention and have been discussed for decades.

Many mathematics education researchers and mathematics teachers believe that mathematics can be made more interesting by revealing mathematicians’ personalities and that historical problems may awaken and maintain interest in the subject.

The idea of using historical mathematics problems in teaching has recently received considerable attention among scholars. In contrast to telling stories to attract students’ interest and improve their attitudes, using historical problems in class has the advantage of improving students’ attitudes about mathematics, as well as improving their understanding of mathematics.

In responding to the question of whether history is important in mathematics teaching, Morris Kline indicates,

“I definitely believe that the historical sequence is an excellent guide to pedagogy. . . . Every teacher of secondary and college mathematics should know the history of mathematics. *There are many reasons, but perhaps the most important is that it is a guide to pedagogy.*” [italics added]

Following the panel’s (International Conference on the Teaching of Mathematics, July 2002) reports, an American mathematics educator raised a critical question: “Is there any evidence showing that including the history of mathematics is effective in the teaching of mathematics?”

We have to clarify one critical conception before answering this question. Namely, what is meant by “effective in the teaching of mathematics”? If it means improving students’ performance on standardized examinations, my attitude would be reserved.

Yet if effectiveness means developing students’ views of thinking and further improving their learning behavior, I am convinced that including the history of mathematics in the curriculum can help.

What may be result if the history of mathematics was incorporated into the teaching of mathematics in a manner that would inspire students? Watch the following video.

## References

^{1} Liu, Po-Hung. “Do teachers need to incorporate the history of mathematics in their teaching?” 2003. *ResearchGate*. https://www.researchgate.net/publication/281223989_Do_teachers_need_to_incorporate_the_history_of_mathematics_in_their_teaching.

^{2} “History Of Mathematics Project | Mathematics Education”. 2022. *history-of-mathematics.org*. https://history-of-mathematics.org/MathematicsEducation.html.

^{3} Koltha, R. “Should A History Of Mathematics Be Taught In High School?”. 2022. *Quora*. https://www.quora.com/Should-a-history-of-mathematics-be-taught-in-high-school/answer/R-Koltha?ch=10&oid=130749703&share=735df7d3&srid=uLYie&target_type=answer.

^{4} Adsul, Rajratna. “Should The History Of Mathematics Be Taught In Addition To Mathematics? Why Or Why Not?”. 2022. *Quora*. https://www.quora.com/Should-the-history-of-mathematics-be-taught-in-addition-to-mathematics-Why-or-why-not/answer/Rajratna-Adsul?ch=10&oid=298544633&share=b16fed7d&srid=uLYie&target_type=answer.

^{5} Schuck, Christopher. “Should The History Of Mathematics Be Taught In Addition To Mathematics? Why Or Why Not?”. 2022. *Quora*. https://www.quora.com/Should-the-history-of-mathematics-be-taught-in-addition-to-mathematics-Why-or-why-not/answer/Christopher-Schuck?ch=10&oid=294357439&share=bdce5b10&srid=uLYie&target_type=answer.

^{6} Bjorkman, Katie. “Should The History Of Mathematics Be Taught In Addition To Mathematics? Why Or Why Not?”. 2022. *Quora*. https://www.quora.com/Should-the-history-of-mathematics-be-taught-in-addition-to-mathematics-Why-or-why-not/answer/Katie-Bjorkman?ch=10&oid=297278525&share=c4bb3d2f&srid=uLYie&target_type=answer.

^{7} Yiu, Tony. “Why You Should Study History”. 2022. *Medium*. https://medium.com/alpha-beta-blog/why-you-should-study-history-c84644a3ec67.

## Additional Reading

“History Of Mathematics Project | Home”. 2022. *history-of-mathematics.org*. https://history-of-mathematics.org/.

Mubeen, Junaid. “Mathematics Without History Is Soulless”. 2019. *Medium*. https://medium.com/hackernoon/mathematics-without-history-is-soulless-978436602fa4.

Shell-Gellasch, Amy (editor). *Hands on History: A Resource for Teaching Mathematics*. Washington, DC: The Mathematical Association of America, 2007.

Research shows that students learn best when they actively participate in their learning. In particular, hands-on activities provide the greatest opportunities for gaining understanding and promoting retention. Apart from simple manipulatives, the mathematics classroom offers few options for hands-on activities. However, the history of mathematics offers many ways to incorporate hands-on learning. By bringing this material culture of mathematics into the classroom, students can experience historical applications and uses of mathematics in a setting rich in discovery and intellectual interest. This volume is a compilation of articles from researchers and educators who use the history of mathematics to facilitate active learning in the classroom. The contributions range from simple devices, such as the rectangular protractor, to elaborate models of descriptive geometry. Other chapters provide detailed descriptions on how to build and use historical models in the high school or collegiate classroom.

“Should A History Of Mathematics Be Taught In High School?”. 2022. *Quora*. https://www.quora.com/Should-a-history-of-mathematics-be-taught-in-high-school.

“V. Frederick Rickey – Wikipedia”. 2022. *en.wikipedia.org*. https://en.wikipedia.org/wiki/V._Frederick_Rickey.

Vincent Frederick Rickey is an American logician and historian of mathematics. Rickey received his B.S., M.S., and Ph.D. from the University of Notre Dame in South Bend, Indiana. His Ph.D. was entitled *An Axiomatic Theory of Syntax*. He joined the academic staff of Ohio’s Bowling Green State University in 1968, became there a full professor in 1979, and retired there in 1998. He was then a mathematics professor at the United States Military Academy from 1998 until his retirement in 2011. He was a visiting professor at the University of Notre Dame, Indiana University at South Bend, the University of Vermont, and the United States Military Academy. He was a Visiting Mathematician at the Mathematical Association of America headquarters in Washington, D.C. and while on this sabbatical he was involved in the founding of the undergraduate magazine **Math Horizons**.

## Videos

“History of Mathematics”. 2022. y*outube.com*. https://www.youtube.com/playlist?list=PLne67oXvVr1L31fDqK7fjVO4dSnApLHS8.

By the 17th century, Europe had taken over from the Middle East as the world’s powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was now on to discover the mathematics to describe objects in motion. Marcus explores the work of Rene Descartes and Pierre Fermat, whose famous Last Theorem would puzzle mathematicians for more than 350 years. He also examines Isaac Newton’s development of the calculus, and goes in search of Leonard Euler, the father of topology or ‘bendy geometry’, and Carl Friedrich Gauss who, at the age of 24, was responsible for inventing a new way of handling equations – modular arithmetic.

Marcus du Sautoy concludes his investigation into the history of mathematics with a look at some of the great unsolved problems that confronted mathematicians in the 20th century. After exploring Georg Cantor’s work on infinity and Henri Poincare’s work on chaos theory, he looks at how mathematics was itself thrown into chaos by the discoveries of Kurt Godel, who showed that the unknowable is an integral part of maths, and Paul Cohen, who established that there were several different sorts of mathematics in which conflicting answers to the same question were possible. He concludes his journey by considering the great unsolved problems of mathematics today, including the Riemann Hypothesis, a conjecture about the distribution of prime numbers. A million-dollar prize and a place in the history books await anyone who can prove Riemann’s theorem.