Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?

Question

I read a paper in my "Research Issues in Mathematical Education" class that I have applied to the Undergraduate Calculus I and Calculus II class that I teach. I take five minutes to explain the history behind the material we learn each day. So far, and similar to the results of the paper, I feel that students are more engaged when we get to the actual lesson. Has anyone else tried this approach? What were your results?

Answer 1

In Germany, there recently was a Project at two universities which tried to strengthen historical connections in analysis courses, especially for teacher education. For them, history can be an important source of motivation if used with real interest. (Bad use: "Even Leibniz made these mistakes" - no learning from history. Good use: "The concept of continuity changed over time. Let's compare definitions and discuss their use.") Unfortunatley, they didn't publish in English. Anyway, for interested persons:

G. Nickel: Vom Nutzen und Nachteil der Mathematikgeschichte für das Lehramtsstudium. In: H. Allmendinger et al. (Hrsgg.): Mathematik verständlich unterrichten. Springer Spektrum, Wiesbaden 2013, 253-266.

Edit: Since vonbrand asked me, here a small summary of the paper, but not the project which included much more measures and cannot be summarised within a few lines:

Generally, the author criticises that mathematics is often taught in a very ahistorical way, whereas he sees history of mathematics as content in its own right. Benefits:

• Anecdotical use to comfort people, which has to be used pointed and not too often

• Genetic use, explaining how definitions, theorems and proofs were worked out: getting a much deeper image of mathematics in the making, connections, etc. Costs time, however. Also good to identify learning obstacles!

• Alienate topics: Can let you experience how hard things are (like the naturals in non-decimal systems: egyption, babylonian, roman numbers).

• Exemplary use: Let students gain there own insight in research processes, which may be easier in historical issues.

However, an anecdotical use can lead to very superficial discussions, rather on persons than on mathematics. That wouldn't be good. Caricatures like mathematicians as "heores" with an enormous mind or as morons making stupid mistakes do neither serve history nor mathematics. And you sometimes should reduce the historical complexity for sake of learning mathematics, not history.

Answer 2

My experience with giving history and chronology and people-involved has mostly (but not entirely) been disheartening: many students, from calculus to grad students, don't count that material (and the perspective it affords) as having much value. Some do value it, but the more-typical response is more-or-less polite waiting for the "real content" to begin.

Some of this is surely due to the eternal "will it be on the final?" criterion for the students' attention, which is not an irrational response in an environment in which they feel overwhelmed, I guess.

Some others' reaction is connected to their affection for mathematics, perhaps extreme among academic subjects, as being allegedly disconnected from the (sordid?) everyday affairs of human beings... and the reminder that mathematics is a human activity is not welcome.

Some view history-of-mathematics as positively antithetical to contemporary mathematics, in a sort of trans-Whiggish attitude, namely, that things are so much improved by now that it is actually destructive to discuss how we arrived at this point.

Edit: but I forgot to say that I still do give some background, despite unenthusiastic response. :)

Answer 3

I would never start with 5-minute historical introduction, main reasons being:

• It's not the content of the subject.
• It's additional information that is not directly relevant and distracts the students.
• Many of historical math results are incomplete if not plain wrong (e.g. proof by example is not a proof, frequently the reasonings were informal, etc.).
• Wrong intuitions are contagious (e.g. I would never start describing solar system with geocentric view).
• The evolution of the idea might be presented without historical context.

On the other hand, I had presented a brief historical context on a few occasions, during the class, that is, once the basics have been laid out (e.g. to explain inconsistencies or backward-compatibility). That technique is especially useful when talking about sequential improvements, like with lower/upper bounds, or time-complexity of algorithms. First, you present basic technique, or naive approach, then gradually introduce more complex ideas and move to cutting-edge research if time allows.

To give an example (what I'm describing wouldn't fit in a single lecture, but other examples were too short and weren't clear enough), in a graph theory course and topic being matchings, one would start with definitions, the bipartite case, then move to naive algorithms, then Edmonds (1965), Hopcrof-Karp (1973), mention Micali-Vazirani (1980,2013), and then move on to algebraic approaches finishing with Mucha-Sankowski (2004) and latest results of Mądry (2013) using linear-programming techniques.

I have seen this technique during a lecture about paradigms of programming languages where the professor would describe programming languages and their features gradually building from machine code up to newest advances in compilers. It worked pretty well.

Yet, I had to admit, I've never seen such an approach during mathematical course.

Finally, at my university it is compulsory for future teachers to attend "history of mathematics" course (fortunately the old professor who teaches it has a gift, the lectures are interesting and fun).

I hope this helps $\ddot\smile$

Answer 4

I use 30 seconds here or there to give brief, colourful, historical notes. It gives something of value to the students who are waiting for me to go on, and gives some extra time for the students struggling to keep up. I have never tried giving planned, longer historical explorations, only spontaneous quick tangents, as my interest sees fit.

Answer 5

In a calculus class, I'd rather start with 5 minutes of an application than with 5 minutes of history.

History in 5-minute increments will probably:

• follow the textbook's ordering (destroying the chronology),
• use modern notation (usually destroying the geometrical context for anything before 1700),
• omit even basic context (e.g. what language did they use? how did they get the resources for this?).

History in 20-minute increments can be more serious.

Many 5-minute applications have some nice historical names attached, and it might help the motivation to say "Archimedes did something similar around 250 B.C.". But I wouldn't call that history, and I'd keep the focus on the application.

Answer 6

My opinion on this topic has vacillated a little through the years, but for the most part, I've always spent time on background information about mathematical and statistical information because I firmly believe that doing so encourages an increase in student understanding and recall.

I believe it also helps students feel less stressed in my classes, because there are brief moments in the middle of difficult classes when they can relax and listen to their teacher tell short, relevant stories.

I believe it also helps them feel an attachment toward me or with me which encourages them to do everything I want them to do to succeed as students.

Sometimes I suspect that an attachment between the students and the teacher is very important. It seems as though some university level and college level teachers I know and speak to don't even think about attachment at all or even know it exists.

Another thing that matters is that background information about topics in math can be presented well or shoddily, and if it's done shoddily, students may feel you're just presenting useless information as part of a checklist, for example. But if you do it well, then maybe you might sit back and let the students know that the following information is to help them understand a new topic.

Lastly, an introduction to the origin of my opinion seems appropriate. I was from a family with no college education in it prior to me. Therefore, there was simply no one in my family nor school who could give me academic guidance back then. However, I just absolutely loved school, especially math and the sciences, and by the time I was eleven or so, I had decided I was going to go into a math-heavy career in adulthood.

It was all working out well, and I was doing well, enjoying my classes and tolerating those I didn't enjoy, until my sophomore year in high school. That year nearly changed the course of my entire life. My interest in math slammed to a complete halt two or three months into a full year geometry course. The teacher believed in a theorem-proof-theorem-proof-theorem-proof approach, I won't expound, but I decided I was wrong about math. Remember, I was a fourteen year old, young for my grade, with no guidance. I believe there are many with little guidance who we teach.

The next year, as a junior, I backed into a slower paced, single semester, half year, precalculus and trigonometry course intending to change my college goals. I decided to do that instead of taking the two semester, full year trigonometry and calculus course my sophomore geometry teacher, the one who taught in a way I hated, had placed me. I did so because I had changed my mind about math.

The half year course was taught by the calculus teacher. I'll never forget him. I'll never forget his name. He died suddenly that Christmas, so he was only my teacher for one semester. He loved math. Here's how he started the course:

He started by talking about Euclid. Well, actually, by talking about the way the Greeks questioned assumptions. We talked about approximating areas and how the Greeks squared circles, then irrational numbers, then the question of when two lines are parallel, about whether two lines always have to cross somewhere, and about whether two lines ever cross, no matter how far out in space you go.

Being 15, I thought his questions were stupid and jumped right in, and he enjoyed my reaction. When I saw his twinkle of smugness, which he was trying to hold back, about five or ten minutes of class had passed, and we got serious about noneuclidian geometry. That night, I was drawing lines on balls, trying to make them straight. Pretty quickly, I had figured it out.

Years later, I was introducing everything that way, right down to number systems in my introductory algebra classes. I teach at Xavier University. I might start a section on Complex Numbers by explaining why numbers were needed by Egyptians at all - first, they needed natural numbers so they could count things: 1, 2, 3, 4, .... But they realized you might have none at all, so they added zero and a new number system, the Whole numbers, was created by building on the Natural numbers: 0, 1, 2, 3, .... And so on.

I feel that it's appropriate to place this reason for my development of my opinion here. It seems highly relevant.