Questions tagged [mathematical-pedagogy]

for questions on general considerations and problems of teaching mathematics, i.e., issues specific to teaching mathematics yet relevant to various contexts and courses.

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0answers
86 views

Let P be a polygon

I've encountered the following misunderstanding. I pose a question (to undergraduates in the U.S.), for example: Let $P$ be a polygon of $n$ vertices. Is it true that every triangulation of $P$ ...
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1answer
163 views

Taxonomy of bad proofs

I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in ...
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0answers
70 views

Is There Book on Collection of Theorems? [closed]

For example, there is a book titled "Synopsis of Elementary Results in Pure and Applied Mathematics: Containing Propositions, Formulae, And Methods Of Analysis, With Abridged Demonstrations" (...
2
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3answers
126 views

How to teach integrals motivated by the work done in moving an object?

I am now teaching Calculus of several variables this semester. In apllications of integrals, the problem of finding the work done in moving an object under a force $F$ is one of the most common ...
2
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1answer
185 views

Is there a numerical base that is in any way “better” for simple mathematical calculations than others?

I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or grade-school level. I know that different numerical bases (i.e. decimal/...
5
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1answer
161 views

Misdirected, Side-Tracked, and Distorted Ramanujan: Problems in Basic Math Education?

I read that- “The tragedy of Ramanujan was not that he died young, but that his genius was misdirected, side-tracked, and to a certain extent : distorted. The years between 18 and 25 are ...
5
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4answers
217 views

How to convince parents that Mathematical puzzles/games help students in their academics too

I write content and conduct workshops for an education firm and also in schools where I try to make them realise how beautifully mathematics and rational thinking complement each other (on elementary ...
6
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3answers
325 views

How do I convince my teachers that a book on maths must focus on conceptual understanding?

I am a senior teacher at this school. We have to select the textbooks for the upcoming session. I am proposing that we have to select books (in maths) that focus more on conceptual understanding and ...
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8answers
3k views

Should high school teachers say “real numbers” before teaching complex numbers?

Before complex numbers are introduced in senior high school courses, should we emphasise that solutions (e.g. to quadratic equations) are real solutions? If we do, then when non-real numbers finally ...
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2answers
180 views

How should one approach the concept of “plus or minus”, such as in the numerator of the quadratic formula?

The numerator is structured like: $$(-b)\pm\sqrt{b^2- 4ac}.$$ Is it confusing or acceptable to distinguish between the following two things? An idiom; and What is or seems to be a compositionally ...
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14answers
7k views

Why is it possible to teach real numbers before even rigorously defining them?

In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach ...
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2answers
101 views

Should the limits of one system of elementary set theory be the limits of a student's mathematical world? [closed]

In teaching elementary set theory, suppose we refrain from emphasizing historical decisions that were made in theory construction. Is there a danger that students may see the mathematical language ...
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5answers
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Teaching asymptotic notations at the beginning of calculus [duplicate]

I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and $\sim$), secondly explaining their "arithmetic" (things like how to sum little-oh's and ...
6
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1answer
845 views

Why is it popular to teach modulus via the example of mod 12 and analogue clocks?

Why is it popular to teach modulus via the example of mod 12 and analogue clocks rather than rectangles or tables that have a finite number of columns in each row, and infinitely many rows? It's ...
6
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3answers
155 views

(Riemann integrability) How do you explain this to a high school student?

The following question was in a high school teacher's guide: Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ defined by $$f(x)=\begin{cases} x & x\in\mathbb{R}\setminus\mathbb{Q}\\ 2x & x\...
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5answers
247 views

Pedagogical considerations behind current order of presentation of trigonometry

A pre-calculus book (Precalculus ed 1 By Miller and Gerken), presents trigonometry in the following order: 1- Angles 2- Trigonometric functions defined on the unit circle 3- Right triangle ...
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2answers
73 views

Retain problems and combat regression in learning

Regressive Learning It's a really stressful situation. I can achieve but not retain expertise in maths problems. History 6 months back, I studied integration in Calculus at college. I learnt it all ...
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2answers
141 views

Finland's performance on international competitions

Why it is said that Finland has a particularly good education system, but Finland's performance on international mathematics competitions is quite often at relatively intermediate level?
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1answer
422 views

Is there research on the efficacy of taking good notes in math class?

I teach at community college, and have often encountered others talking about helping students learn to take good notes. I have never felt that I took good notes as a student. I was too busy thinking ...
8
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4answers
215 views

Should school syllabus include chapters partially?

In my locality, many schools have this tendency to partially include this and that chapter in the syllabus (for almost every subject). For example, (most of the chapters are subdivided in two or more ...
3
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2answers
371 views

Enlighten younger students about the concept of “procedural justice” in mathematics?

I am tutoring a 16-year-old student from my home country (in Asia) in, roughly speaking, precalculus. I would like to give him a feeling of procedural justice, so to speak, in modern mathematics, ...
6
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4answers
328 views

Ideas for the introduction of the derivative?

I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for ...
6
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1answer
258 views

Propositional and predicate logic, with quantifiers: Is there any research when it is ideal to explicitly teach in mathematics education?

In terms of helping students to understand propositional and predicate logic, with quantifiers, is there any research regarding when it is most advantageous for students studying mathematics, to first ...
2
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0answers
103 views

A compelling example of what complex numbers are for, before teaching them [duplicate]

When talking to kids before they are taught complex numbers, I would really like to give some examples of why it will be exciting to learn them. I am comfortable explaining the intellectual ...
4
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7answers
337 views

Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion)

A friend of mine that teaches math has made many times the following experiment : drawing two circles on the blackboard representing two sets A and B such that A and B are disjoint writing on the ...
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3answers
288 views

Solutions to exercises

I am teaching the exercise sessions for a 3rd year algebra course (intro to field theory, Galois theory and Algebraic geometry). The format of the course is as follows: for every 2 hour lecture by the ...
3
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1answer
146 views

Naming arbitrary constants: subscripts, primes, or just more letters?

When choosing names for arbitrary constants either during a lesson or while working with a single student, should one use$\{n_1,n_2,n_3,\dotsc\}$ or $\{n, n', n'', \dotsc\}$ or $\{a,b,c,\dotsc\}$? Is ...
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2answers
224 views

Pros and cons of randomised question generation

I am developing an assessment piece where the content is the same but the particular numbers are different for each student. It involves finding Triangle Centers given points using coordinate geometry....
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0answers
178 views

why don't we do labs in/for math?

(this is in the US and at a high school level) why don't we dedicate a day of the week each week to do a lab for math for exploration? I mean we already do that for Earth Science, Physics, Chemistry ...
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1answer
220 views

How is math taught in elementry school in Finland?

I read on the internet that Finland has the best education system in the world at that in Finland, students are taught to love mistakes and that's how they learn and become smarter. I could not find ...
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2answers
198 views

Collaborative note taking

I have been encouraging my classmates to connect with me on Google Docs to work collaboratively on taking notes. Still, no takers though. I imagine that if I were a professor, I would attempt to get ...
5
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0answers
237 views

Mimic lecturing on blackboard, but facing audience [closed]

I teach mathematics at MSc and PhD levels. My preferred method of teaching is old-fashioned: talking and writing on the blackboard at the same time. Why? Because it has many advantages: Handwriting: ...
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2answers
102 views

Are questions on overlapping solids of revolutions without prior definitions and instructions fair given that there are divided interpretations?

If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the ...
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2answers
145 views

Math Lessons with Two Parts and a Combination

This is fairly open ended, so I understand if people consider this to be off-topic. I'm interested in creating math lessons where two groups each learn how to use a different simple math skill, and ...
4
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3answers
217 views

How to explain the motivation of parentheses in addition, subtraction and multiplication?

My kid, 5 years old, knows addition, subtraction and multiplication now, of course, in a basic level. Also he understands that parentheses means "whichever inside shall be computed first". When I ...
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4answers
419 views

A more rigorous approach to Precalculus

I am a pure mathematics PhD student and graduate teaching assistant at a major state university. During the summers here, teaching assistants are typically appointed to teach an entire course, rather ...
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3answers
315 views

Could students learn a lot more from school if they're only taught number theory until way later?

According to https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html, when students get taught a concept when they're so young, they're ...
6
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0answers
71 views

Long-form, multi-step, skills-integrating applied mathematics problems in calculus I, II, III

When recently teaching Calculus II to college students, I instructed my students to read and be ready to work through the first 8 or so questions of James Walsh's climate modeling differential ...
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3answers
114 views

How can I measure the mathematical computation skills of high school students through a test?

How to analyze the level of difficulty of mathematical computation of a problem on a standard mathematical test designed for high school students? I mean how to choose some indices that can reflect ...
0
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1answer
181 views

Grades in a university course on category theory, curving, and how they reflect on the students and/or teacher [closed]

I originally posted this on the Mathematics Stack Exchange, thinking that the best place to post it, but the question quickly accumulated a bunch of close votes since it was not quite within the scope ...
4
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2answers
83 views

A Markov chain demonstration that doesn't require computers

I have a large probability class and would like to do some memorable demonstrations of Markov chains for them. Does anyone have any recommendations for a "low-tech" demo that doesn't involve ...
4
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6answers
319 views

How to make a student not overlook easy mistakes such as the wrong sign

I am teaching entry calculus to a bunch of students outside class (more like complementary to their math classes, without making much connections) and I can teach on a much more individual level than ...
6
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2answers
281 views

Mainstreaming math student

I'm working one-on-one with a student who is part of a sponsored refugee family. He's bright and a good learner, but has had a lot of interruptions to his education. No indication of any learning ...
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2answers
105 views

Returning Student for STEM - Brush-Up Resources? [closed]

All, I am hoping to wade into an Electrical Engineering or Mechanical Engineering degree, but I have been out of college for almost 10 years. My last major exposure to math was good grades in ...
4
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2answers
846 views

Should young math students be taught an abstract concept of form?

Should a more general concept of the "form" of an equation or expression, be taught to math students as young as elementary school? I'm a fairly new tutor--do more experienced teachers think this ...
6
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0answers
125 views

Flow diagrams and summarizing strategies in proof-computation courses: good or bad for learning? Unsuitable for Inquiry-based learning?

For concreteness lets keep our discussion to calculus courses where there is a balance of proof and computations (computing limits but also doing epsilon-delta proofs) I can understand that in more ...
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2answers
163 views

What are some suggestions fo teaching statistics concepts to struggling college students?

I'm a private math tutor. I'm fairly new at this, and this semester is the first time I've been tutoring for a statistics class at a community college. I enjoy experimenting and learning about ways to ...
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1answer
109 views

Verifying Simple Expression Equivalence in a Spreadsheet

For simple expressions with easily derived canonical forms (eg polynomials and simple rational expressions), is there a way to leverage existing tools to verify that two expressions are equal when ...
4
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1answer
166 views

Is it feasible to expose undergraduates to a “map”-centric point of view early on?

Question: Would it be feasible to teach undergraduate math students a "map"-centric view early on? If so, how early on? Now that I'm preparing for a phd program, I'm also reflecting on my ...
3
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1answer
138 views

Existing Tools for Math Expression Equivalence Logic

Note - this question was posted here and garnered some decent replies before it was closed as off-topic in Stack Overflow. Online systems, such as ALEKS, Cengage's WebAssign, and even Khan Academy ...

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