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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3answers
457 views

Mathematical difficulty

There exist a large number of reasons why "mathematics is difficult". If one exclude "subjective reasons" such as: "math anxiety, math fear,..." and education factors ...
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3answers
358 views

How to invite humanities students to study mathematics?

This question comes from the perspective of an undergraduate math major who feels that much (although not all) of the mathematical discipline is a liberal art, rather than a science, and should be ...
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4answers
956 views

The Undergraduate Responsibility Gradient

We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they ...
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10answers
5k views

Complex numbers in high school

Are complex numbers taught in high school in other countries? I am from Germany and complex numbers are next to never touched in high school with the exception of extra-curricular activities, for ...
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1answer
277 views

Why many people believe that: $\displaystyle c>0\implies \frac{1}{c}<0$?

I came across many people who believe the below false implication. I don't know why people believe it true in high school and middle school and also students in university level. Really I would like ...
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1answer
239 views

How to ask a student a question to get the answer '...integer not continuous...'

Context: a very basic level statistics package computer lab. A scatter plot is produced for one integer variable versus another integer variable. The students are asked why the points form a grid ...
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2answers
378 views

Obtaining printed copies of the textbook series Unified Modern Mathematics

I'm seeking the textbooks that were released in the 1960's call Unified Modern Mathematics. I'm aware 3 parts of the course exist online but I would like to use them in hard copy form. I find these ...
7
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2answers
255 views

Ethics of looking at other proofs before submitting work

I am in my third year of undergraduate math, and now that classes are becoming more proof-based, many of my homework questions are proofs of relatively basic concepts that can be found with a quick ...
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1answer
93 views

How to incorporate optional higher level mathematical content in an Engineering Maths course?

Our department teaches two very large first-year "Mathematical Methods" courses (600-ish students) to Engineering students. The syllabus is dictated by their (future) needs and covers a huge array of ...
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3answers
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Is proof-based exercise-oriented math course without solution an effective way to teach pure math?

In recent years I have seen several courses in pure math in the undergrad level (year 2, 3, 4) such as real analysis and topology where the entire course consists of: notes written during the lecture ...
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1answer
442 views

When are partial fractions taught? [closed]

Recently I had taken the SATs, and a question came up that involved partial fractions decomposition. $$\frac{x^2-4x+5}{x-3}$$ This is not the exact problem but a similar one. If the SAT math is ...
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4answers
760 views

Why are proofs by contradiction counterintuitive?

And how to make them intuitive? We are tasked to prove $P \implies Q$. So we assume $P$ and are trying to prove $Q$. We assume not-$Q$ ($\neg Q$) and derive a contradiction, establishing $Q$. There ...
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5answers
671 views

How to get through the boring stuff?

It frequently happens that there's some material I have to cover which is, frankly, boring. The subject itself may be boring, or it may be the particular exercises, but in any case I have to get ...
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2answers
432 views

Common Core Question: What is included and excluded in high school mathematics?

I took pre-calculus in high school, and I did not get to learn about matrices, and conic sections, vectors law of sines and cosines, and etc. I took geometry as well and found that matrices were also ...
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3answers
309 views

Monty Hall challenge

Thinking about the counterintuitive Monty Hall Problem (stick or switch?), revisited in this ME question, I thought I would issue a challenge: Give in one (perhaps long) sentence a convincing ...
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1answer
655 views

Is the current education system as bad as most critics and famous pure mathematicians try to convey? [closed]

Throughout elementary, middle and high school mathematics is quite merely about memorizing concepts and formulas, understanding the theorems (without their proofs) and applying acquired knowledge in ...
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2answers
221 views

Course-based undergraduate research experiences in math

"Course-based undergraduate research experiences" (CUREs, or CBEs) are being explored in various STEM fields, especially biology, chemistry, geology. Here is one geology link that gives a flavor: "...
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1answer
273 views

Native language, writing, and mathematical problem solving

This question is meant to explore the intuition that mathematical thought does not most naturally proceed from writing in one's native language. The hackneyed and not entirely satisfying slogan that ...
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3answers
1k views

Group theory for high schoolers, want the opinion of other educators

So I am going to be teaching the basics of group theory to high schoolers in a few weeks, and I want to hear what the Stack Exchange network has to say on the matter. What are the applications and ...
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4answers
338 views

request for evidence about class perspectives in math word problems

A recent publication: Anita Bright. Education for Whom? Word Problems as Carriers of Cultural Values. Taboo: The Journal of Culture and Education. (Spring 2016). pp. 6-22. Link No paywall. observes ...
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2answers
351 views

Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers

I am teaching an introduction to number theory for high schoolers right now, and there seems to be quite a bit of confusion on what the difference between the natural numbers, the integers, the ...
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3answers
805 views

A good antonym for reducing/simplifying equivalent fractions

I am looking for a good antonym for reducing/simplifying equivalent fractions: 'reduce' and 'simplify' both make sense to me when dividing, but I'm struggling to name what it is we do when we multiply ...
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3answers
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Should I teach Laplace Transforms? How much?

My question is in the title. Let me elaborate and give some context: I'm teaching a first differential equations course, essentially for engineers, at the university. I'm developing the syllabus ...
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3answers
598 views

In what ways can educators introduce polynomials in grades 7 to 9?

Q: Is there a way we can teach polynomials, avoiding the "watch me do it & now you do it" training method, that will allow students to anticipate and predict the existence or formulation of ...
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2answers
232 views

Is there any research on the value of extra credit in the college mathematics classroom?

After teaching mathematics for a year, where in each class I had opportunities for my students to earn extra credit, I am reflecting on whether this has any value. The reason why I am questioning ...
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3answers
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Tips to improve blackboard writing

During the internship I recently finished I came to realise how important it is to have a good and structured use of the blackboard when teaching mathematics to 12-14 year pupils. Circumstances forced ...
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4answers
561 views

Is there research for or against such an approach in teaching calculus?

Copying from Calculus Made Easy by Silvanus Thompson (2nd ed., 1914): CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS The preliminary terror, which chokes off most fifth-form boys from ...
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1answer
218 views

Reference request for studies on gender in math examples, homework problems, or math exams

I am looking for a study or reference on gender in math problems given in mathematics. In math texts or even on math exams, if there is a word problem involving people, these people or "characters" ...
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1answer
1k 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 ...
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3answers
766 views

Summary of the mechanism of reification

The concept of "reification" in mathematics education is interesting. Roughly, if I understand this correctly, one "reifies" processes into mathematical objects. Very recently, it occurred to me that ...
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4answers
2k 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 ...
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1answer
437 views

Polya's "Nearby Problem" Heuristic and Inquiry Based Learning

I've often wondered about the "devise a plan" part of Polya's "How to solve it" outline. What we call "problem solving" can be thought of as what to do when you have no idea what to do. From this ...
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3answers
234 views

Do iPhones help students in their math class?

While the question is stated with reference to the iPhone, my actual question is about phones in general. Just as there was much talk about the use of Computers in the classroom over the past fifty ...
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4answers
2k views

How are the basic trigonometric functions introduced to students?

The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is ...
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15answers
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What books are like Knuth's Surreal Numbers?

I'm looking to find more examples of books which bridge the gap between "story" and "mathematics" using narrative and all those other wonderful features we might find in Harry Potter or some other ...
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2answers
1k views

What methods successfully identify and eliminate severe math anxiety?

What methods are effective in identifying and eliminating severe math anxiety, this most terrible and unfortunate part of modern mathematics education? This question is not about ordinary math anxiety ...
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2answers
153 views

How are geometric proofs related to geometric pictures?

When teaching geometry it is common to use pictures/figures to "show" the problem and its solution. It's also common to say things like "more than one figure can be shown which demonstrates the same ...
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5answers
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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 ...
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3answers
227 views

Pedagogical Purpose in Making Students Do Problems in A Less Efficient Way First

Let's assume that a group of students need to learn to solve a certain type of mathematical problem for which there is two general methods of solving it, $X$ and $Y$. We also assume that $Y$ is more ...
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2answers
180 views

Rigor in secondary mathematics

If rigor doesn't mean more challenging problems, then what does it mean? There is a big push for rigor in common core mathematics, but I'm not sure exactly what rigor means (I'm pretty sure it has to ...
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2answers
558 views

Developing mathematical stories

In a comment on a recent post, Steven Gubkin pointed out that in doing mathematics he likes to develop stories. This motivation for mathematics is perhaps familiar to many practicing mathematicians. ...
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3answers
311 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. ...
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6answers
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Are precise drawings important in geometry?

In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also ...
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0answers
174 views

Research on the use of outlined / structured proofs in instruction

Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs? For the ...
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1answer
172 views

In Text Exercises

For an undergraduate mathematics textbook, what are the pitfalls of inserting all of the exercises in the text? (As opposed to grouping every exercise at the end of the section). IMO I feel it is ...
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0answers
328 views

Was math education following a western trend?

After some research on the recent history of math education in the U.S., from the new math movement to the beginning of the 21st century, I understood that the historic flow of the math education ...
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2answers
293 views

Difference in difficulty between removing and adding structures?

In Fantasy math, Peter Saveliev remarks: Removing structures from consideration is hard; adding is much easier. Compare how you start with point-set topology -- by removing the geometry from the ...
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3answers
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Mathematical education slang

Amir Asghari recently asked a question about mathematical slang. He was "looking for "non-mathematical" terms or phrases that are used to refer to mathematical objects (of any kind) mainly for ...
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4answers
591 views

Elementary physics course for pure math student

Are there any mathematical departments which present the course "elementary physics" for pure math undergraduate students, separately? Is there a way to present this course with the most pure ...
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4answers
2k views

Why do students only see the last term of a sum abbreviated with an ellipsis?

It's very common in learning mathematical induction to prove statements like $$ 0^2+1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.$$ I've found that very frequently, on this sort of problem, when ...

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