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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68
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11answers
7k views

Whence the “everything is linear” phenomenon, and what can we do about it?

$$ \color{red}{(a+b)^2 = a^2+b^2}$$ $$ \color{red}{\sqrt{x^4+y^4} = x^2+y^2} $$ $$ \color{red}{e^{t^2+C} = e^{t^2}+e^C}$$ I've observed this phenomenon -- wherein, implicitly, students say, "...
71
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6answers
5k views

Issues with “equals”, where does this come from and how do I combat it?

An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=...
36
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24answers
7k views

Good, simple examples of induction?

Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
24
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9answers
2k views

How can mathematics educators encourage innovation and creativity?

Almost by definition, innovation requires that things be done differently than established custom has it, and comes from the young more often than from the old. In a field as old and established as ...
27
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14answers
2k views

Revisiting topics from previous courses

I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.) ...
18
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12answers
1k views

Mathematical problems for preschoolers

What are some mathematical problems that are feasible for preschool children to stimulate their intellectual development? There are multiple natural laws that are not apparent to them, for example: ...
14
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4answers
1k views

When should I say “nothing is as it seems”?

"Intuition" is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true ...
18
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2answers
3k views

Example “bad proofs”?

As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the ...
22
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4answers
2k views

Lesson plan to self-teach real analysis to student with comp-sci background

For my background, I'm a software engineer currently studying for his master's degree in information security. But when that's all done, I plan on going back to mathematics to keep me busy. But with ...
46
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3answers
6k views

How do blind people learn mathematics?

I am interested in how blind people learn mathematics at any level, but particularly before college. Math is often taught using a lot of visualization; how does this work with blind people? My ...
24
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7answers
4k views

When should we first teach variables in school math? And how?

From a pedagogical point of view, when is the "right" moment to introduce letters and variables to school children? Let's say we taught arithmetic, the four operations, did computation exercises, or ...
34
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2answers
2k views

What does math education research know about difficulty vs. effectiveness?

I've asked basically the same question previously on on math.SE, then cogsci.SE without much response, surely here is the place to ask this. As anecdotal evidence is plentiful, but unfortunately ...
35
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7answers
4k views

A Lexicon of Math Mistakes

Neil Postman wrote an interesting (and freely available) article called "The Educationist as Painkiller." I highly recommend you read the article for your own enjoyment and as a background to this ...
35
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23answers
6k views

Imbuing a six year old with a sense of mathematical wonder

My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home ...
12
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3answers
1k views

Moving From Rote Learning To Creative Thinking

My mathematics education was essentially rote, you learned the formulas and applied them almost algorithmically to the problems you were presented with; the teacher dictated a method and you followed ...
19
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6answers
2k views

Is there a good way to explain determinants in an elementary linear algebra class?

Many colleges offer an an elementary linear algebra class for sophomore math, science, and economics majors. Such a class typically covers a chapter on determinants, including the following aspects: ...
15
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5answers
1k views

Is Peer Instruction suited to mathematics classroom?

Peer Instruction is a method developed by Eric Mazur in Harvard, designed with a student-centered approach in mind. In a nutshell, the core of the method is that when presented with a problem, ...
11
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3answers
448 views

How to teach a student algebra who misses too much previous knowledge?

I am now tutoring a student in Grade 9, who falls behind in math study. He lacks the basic understanding of operations and inverse operations, and have trouble dealing with negative numbers and ...
9
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3answers
465 views

How many problems do we have to do as undergraduate mathematicians in order to learn a subject?

I'm wondering how many problems are needed in order to learn a subject, let's say Calculus of Several Variables. We know that the professors often assign us a list of problems to solve as homework, ...
42
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8answers
2k views

Knowing mathematics does not translate to knowing to teach mathematics. Why?

Many brilliant mathematicians seem to make average or even poor classroom teachers. Is this an accurate assessment? Has there been any research to explain the phenomena? What is the difference ...
27
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6answers
790 views

How can we help students who are very anxious about math?

In many parts of the world, the majority of the population is uncomfortable with math. In a few countries this is not the case. We would do well to change our education systems to promote a healthier ...
12
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3answers
7k views

Good examples of proof by contradiction?

In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ...
17
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5answers
727 views

What is a variable?

There are two kinds of answers I'm looking for: What do students think a variable is? What do YOU, the teacher, think a variable is? I'm also interested in why you think a variable is what you think ...
16
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11answers
10k views

Looking for simple “interesting” math problems that cannot be easily solved without algebra

I often find students who dislike algebra. They prefer to work with numbers in solving problems. I believe there are many problems that are hard to solve without algebra. For example: Finding the ...
15
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6answers
2k views

Is there any difference between teaching calculus for math and engineering students?

In our university both math and engineering students attend in the same calculus classes. There are arguments in our department about the possible influences of this approach on students. It seems ...
22
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2answers
981 views

Is Knuth's suggestion on teaching calculus a good idea?

Note: I myself am not a math educator, though I plan to be one someday. In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A ...
15
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5answers
995 views

Cost and benefits of compartmentalization in k-12 curriculum

This is a soft question perhaps not well suited for the format of the site but I'm interested to hear opinions from this community on this topic. K-12 mathematics textbooks (understandably) divide ...
15
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0answers
985 views

Is metacognition ever bad?

Metacognition seems pretty universally positive. I'm wary of viewing it as such. Aside from the obvious criticisms like "you can't learn to ride a bicycle by thinking about and writing a 200 page ...
11
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6answers
1k views

How can I convince students that Fourier series are useful?

Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...
4
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7answers
346 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 ...
14
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3answers
419 views

Calculation versus writing in mathematics

Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas ...
10
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4answers
1k views

Mathematical concepts and techniques that **pay off the most**? [closed]

There is a smart way of learning, and it consists in first finding out what are the most valuable pieces of knowledge to acquire. The ones that will give you the highest value for your investment in ...
6
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3answers
278 views

Should one justify formulae in middle school?

Consider two possible lesson outlines: Check homework. Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279 Calculate the area of a circle as an example. ...
3
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3answers
2k views

Why are most math textbooks so uninspiring; unless you already like Math? [closed]

Why are math textbooks so often boring? It requires some mental discipline to do math; cartoons can help make the principles easier to digest. The Mathematical Association of America has many FUN ...
12
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2answers
779 views

Definitions/proofs that allow “useless” cases?

I often see students confused/mystified by definitions (and proofs) that allow/consider "useless" cases. A case in point is the definition of a DFA (deterministic finite automaton), which allows ...
10
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11answers
2k views

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

How can we break down the complexity of eigenvalues/vectors to something that is more intuitive for students. I feel like the proofy way isn't a good intuitive representation of the mechanism that ...
8
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2answers
2k views

Test Correction Analysis

In hopes of helping my students practice and deepen their understanding of math knowledge, I wanted to bring up the topic of a test correction analysis - I didn't find much on this topic when I ...
5
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8answers
510 views

How to justify single digit answers to long problems?

I have been teaching my student BODMAS and giving him problems like $$4\times4-6+8\div4+10\div2+5-4\times2$$ While solving the problems he asked me that, "So small answers for such big questions?" ...
85
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17answers
15k views

Unique candidate that fails

In the comments to David Speyer's answer here, he points out that "the distinction between 'if there is a formula, it is this one' and 'this formula works' is subtle." Does anyone have any simple, ...
56
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13answers
8k views

How to get past the “mystique” of Maths

This question is primarily discussing maths education for adult learners, on courses teaching engineering mathematics at an undergraduate level. These students generally never set out specifically to ...
28
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9answers
7k views

Can mathematics be learned by ONLY solving problems?

Here is the concept: Student is presented with a problem. He/she may not even understand what is being asked, or may attempt. Students reads a solution to the problem. In it there may be ...
29
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7answers
956 views

How can we help students learn to write about their mathematics?

As a guiding example, imagine an undergraduate Calculus II course where students have to complete a guided "research project" and write a "paper" about their work. This can be a shockingly new ...
30
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14answers
8k views

Justifications for: Why learn mathematics?

I wonder how you teachers walk the line between justifying mathematics because of its many—and sometimes surprising—applications, and justifying it as the study of one of the great ...
20
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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 ...
32
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13answers
4k views

Lecturers “(intentional) mistakes” as a teaching tool

I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting students see how a mathematician's mind ...
29
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13answers
7k views

What do you say to students who want to apply Banach-Tarski theorem in practice?

Once when I was talking about Banach-Traski theorem (paradox) I said: OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can ...
17
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3answers
7k views

Learning Mathematics with the aid of spaced repetition systems

I am currently self-studying, so I learn using a textbook which I work through in a linear fashion. As I am introduced to new concepts I ask myself questions and I attempt to answer these questions ...
36
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12answers
6k views

Is it advisable to avoid teaching “multiplication as repeated addition”?

I've had this discussion with a couple of friends. I argued that teaching multiplication as repeated addition isn't a good idea because it doesn't help children differentiate between the two ...
44
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22answers
17k views

How to explain Monty Hall problem when they just don't get it

Talking to some friends, I was asked to explain the answer to the Monty Hall problem (see also here;) .... they were having some trouble because whoever explained it to them didn't do a very good job. ...
32
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18answers
2k views

How to teach someone that $-3>-4$?

I am trying to teach a teenage person math, but he doesn't seem to be able to grasp the concept of negative numbers and $0$. Again and again he finds $-4$ greater than $-3$ because he has spent ...