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16 votes
5 answers
1k views

Is there a canonical name for a polynomial-like expression allowing for negative powers?

When introducing the techniques of differentiation, polynomials come up all the time as great examples to familiarize students with the "power rule" and the linearity of differentiation. A ...
5 votes
3 answers
1k views

How to teach using brackets in sums?

How one should teach using brackets in summation? For example, why is it correct to write $\sum_i a_ib_i$ but $\sum_i a_i+b_i$ should be written as $\sum_i (a_i+b_i)$ But $\sum_i\frac{a_i+b_i}{2}$ is, ...
  • 53
6 votes
3 answers
2k views

Are there examples of central symmetry, without axial symmetry, in nature?

Examples of axial symmetry abound, but I could not find an example of pure central symmetry (that is, without axial symmetry)! Do you know of any? A butterfly shows axial symmetry, what shows point/...
  • 654
8 votes
1 answer
162 views

Research into how students read algebraic expressions

In answering another question What is the justification to teach the (redundant) use of parentheses in multiplications? I was left wondering what we actually know about students' progression in terms ...
  • 3,553
6 votes
1 answer
1k views

What is the justification to teach the (redundant) use of parentheses in multiplications?

Example: 5 x 18 = (5 x 10) + (5 x 8) instead of 5 x 10 + 5 x 8?
  • 61
8 votes
4 answers
2k views

Should math for elementary teachers content be taught under the direction of the math department?

I recently was appointed math department chair at a small university. We have a 3 credit math for elementary teachers content course. Administration told us they will change this course into an ...
  • 81
0 votes
0 answers
28 views

How do I calculate the odds of something not being selected in a sample? [closed]

I have a population that a sample was taken from, and one "group" was not selected in the sample. I am wanting to find the probability of one group not being selected. I will use red/blue ...
27 votes
19 answers
7k views

‘Lies to children’ in mathematics and statistics education

In teaching, we sometimes necessarily oversimplify concepts. Terry Pratchett famously referred to this as Lies to children: A lie-to-children is a statement that is false, but which nevertheless ...
2 votes
2 answers
360 views

When teaching online, how does the teacher teach visual concepts?

What would one suggest for online math teaching to draw diagrams? Is Tikz or Matplotlib too slow for helping to visualize problems during the lesson?
  • 31
2 votes
2 answers
170 views

Common mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
2 votes
2 answers
251 views

Introduction of group action as morphism of groups

The usual definition of a group action is as follows. Let $G$ be a group and $A$ be a set. An action of $G$ on $A$ is defined to be a map $\rho:G\times A\rightarrow A$ satisfying certain conditions. I ...
1 vote
1 answer
61 views

Best Free Direction Field Plotter?

Can you recommend one for a first or second year calculus course? Ideally the website that can plot direction fields: is free is 100% WYSIWYG (does not require any coding or markup or anything of the ...
  • 554
3 votes
3 answers
136 views

Relearning math after long COVID using AoPS or developmental math textbooks?

This is a little bit of a niche topic. I've dealt with a pretty bad dose of long COVID that has caused some serious gaps in my mathematics (basically causing terrible arithmetic skills and a really ...
0 votes
2 answers
157 views

Example of a phenomenon from real life where there is a limit going to infinity

I haven't been able to find examples in real life where we have a function or sequence such that the limit goes to infinity when the independent variable goes to infinity. The only one so far is ...
  • 121
12 votes
9 answers
1k views

Is it meaningful to add a number to itself a fractional number of times?

(Migrated from the math stack exchange, where I received an apt-seeming suggestion to pose the question here, at the math-educators stack exchange) I introduce my young kids to basic math concepts in ...
3 votes
1 answer
155 views

How does learning math differ from learning second foreign languages (L2)?

Peer-reviewed publications analogize learning math to learning a L2. But what are the DisAnalogies and CounterArguments? How can you distinguish learning math from learning L2? Luciana Oliveira, ...
1 vote
2 answers
179 views

A high school level proof that $a/b > 0$ [closed]

Is there a high school level proof of the following? If $a,b > 0$ then $a/b > 0.$
  • 301
11 votes
3 answers
1k views

Arithmetic books for adults

I'm trying to learn arithmetic from scratch again. Even though I can use it, I'm not sure if I can teach it to someone and I believe if you can't teach something properly, there might be loopholes in ...
0 votes
1 answer
174 views

Which are the most used Greek letters in math textbooks?

I am looking for a list of the most frequent Greek letters used in high school and college textbooks or some other corpora. I've realized my students don't know Greek letters and I would like to teach ...
6 votes
4 answers
226 views

Should one include the unit in the variable? E.g. should one write $x^\circ = 30^\circ$ or $x = 30^\circ$?

I sometimes come across equations like $x^\circ = 180^\circ - \frac{360^\circ}{n}$ where I would write $x = 180^\circ - \frac{360^\circ}{n}$, or like $3 \text{ cm} + x \text{ cm} = 5 \text{ cm}$ where ...
  • 1,771
11 votes
6 answers
4k views

Is patching up a student's poor solution better than providing a good solution?

A student benefits from their attempt at a solution or proof being checked by the teacher. My own view is that, if the student's work is poor, it is best just to provide a model solution or proof in ...
5 votes
2 answers
268 views

Proof by Contradiction vs. Proof of Negation

In constructive mathematics we make a distinction between "proof of negation" and "proof by contradiction". You can read a great account of the difference in this blog post of ...
2 votes
2 answers
161 views

How to introduce the use of Greek letters in high school?

I am looking for any hints or experience reports or materials/potential difficulties about how to introduce the use of Greek letters in high school Math/Physics.
3 votes
3 answers
266 views

What is the maximum value of the sum of the digits of the sum of the digits of a three-digit number?

The following is an elementary-level Math Kangaroo multiple choice question: What is the maximum value of the sum of the digits of the sum of the digits of a three-digit number? A. 9 B. 10 C. 11 D....
  • 39
24 votes
6 answers
4k views

Students confusing "object types" in introductory proofs class

In my intro to proofs (and discrete mathematics) class, I see a common mistake where students make nonsensical statements because, for lack of a better term, they confuse the types of the mathematical ...
  • 351
13 votes
7 answers
1k views

Introductory Books easier than Dover

I'm looking to books introducing different topics of math with fun for my son, to give him a taste of different areas. My son roughly understands A-level topics until single variable calculus (of ...
  • 667
0 votes
1 answer
76 views

Where can I find a summary of typical material (e.g., mathematics) taught in US elementaries and high schools?

The reasons for my question are complicated but basically I need information about different subjects, mostly languages (Eng, Spa, Fre), math, and sciences, as taught in different grades in elementary/...
1 vote
0 answers
95 views

Math curricula\programs or any experience using "The Road to Reality" as the\a primary textbook

Primarily a reference request, collaborator search-tips requests, and question-improvement request (including improveent by deletion and re-posting to more appropriate stack, meta, wiki, etc). Rank ...
28 votes
6 answers
2k views

Calculus problems arising from real research problems

I am visiting my in-laws for the holidays. My sister in law is a statistician. She asked me to take a stab at a calculus problem which was coming up in her research. The Lambert $W_0$ function is ...
4 votes
1 answer
141 views

About instructor's manual in math textbooks: principles, design, uses and impact

It is common that school math textbooks are supplemented with instructor's materials (answers, hints and guidelines). Are there scientific studies on how instructor's materials impact math teaching? ...
8 votes
3 answers
2k views

Problem set recommendations for a student who has taken classes through Calc III, but has very weak fundamentals

I know an undergraduate who just finished calc III, and really struggled with it. However, it seems like a lot of the issues are coming from skills which should have been established during algebra, ...
-1 votes
1 answer
112 views

On doing scientific research [closed]

What I want to ask is: can I use the results of others when doing scientific research, whether it is math, physics or other science, without trying to prove them myself or try to verify them? For ...
  • 179
0 votes
1 answer
160 views

Learning, exams, math degree

I have only passed the one fourth of my courses so far, which means like one year of the maths undergraduate studies. I live in a different place from where the university i study at is and try to ...
  • 179
6 votes
10 answers
644 views

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38,\ $ and that $\ 3.4 \times 10^{-6} \times 2.1 \times 10^{-5} = 3.4\times 2.1 ...
15 votes
3 answers
326 views

Pedagogical insights to be gleaned from AI attempts to ‘learn’ mathematics

Some ChatGPT answers have been posted on (and removed from) MathOverflow, and there was a resulting MMO discussion. In that discussion, @darijgrinberg said: When I tell it about its errors, it "...
  • 908
0 votes
1 answer
76 views

Suggestion for IB program Analysis and Approaches SL book?

What is the most suitable book for the IB program Analysis and Approaches SL for a student with significant weaknesses? I had suggested the book from HAESE Mathematics yet he finds it particularly ...
2 votes
2 answers
119 views

Suggestions for Virtual Blackboard?

From time to time I need to teach mathematics to my students remotely. I would like to approximate writing on a blackboard together. The subjects I teach are geometry and pre-calculus. So I would like ...
5 votes
0 answers
60 views

English version of the "Spécialité Mathématiques" French baccalaureate course (for a Ukrainian refugee)?

The highschool where I'm teaching welcomes a Ukrainian refugee who is in "classe Terminale" (ultimate highschool grade in France). The math teacher who has this student in charge is not that ...
5 votes
2 answers
229 views

How to teach the concept of probability distribution?

I observed that my students do not understand what a probability distribution is. We do not treat probability axiomatically on the course, so the required level of understanding is knowing all the ...
  • 5,064
1 vote
3 answers
148 views

Where to learn axiomatic geometry from Hilbert's axioms?

I have found that math is relative easy for me to study but I have problems in geometry. I often omit some obvious steps as I think the problem visually but not from axioms. What would help me for ...
  • 13
4 votes
1 answer
70 views

Research Into Self-Learning at Undergraduate Level and Above

My motivation for this question is personal. I'm a software engineer and I study mathematical logic as a hobby. Subjectively, it feels like I make the most progress after asking a question on the Math ...
7 votes
9 answers
6k views

When did math start to be a hated subject in schools and universities?

Mathematics is currently considered one of the most hated school subjects, (at least in Brazil it is but I think it is a worldwide and cultural phenomenon.) My question is when did this start to ...
1 vote
3 answers
321 views

What are some examples of mathematical errors in computer algebra systems (CAS)?

Computer algebra software is easily and freely available for students nowadays but they still have mathematical errors. For instance, ask them to solve the simple equation $a x - b = 0$ for $x$. The ...
-1 votes
1 answer
180 views

Why do we use polysemy?

From Polysemy - Wikipedia: Polysemy (/pəˈlɪsɪmi/ or /ˈpɒlɪˌsiːmi/; from Ancient Greek πολύ- (polý-) 'many', and σῆμα (sêma) 'sign') is the capacity for a sign (e.g. a symbol, a morpheme, a word, or a ...
1 vote
0 answers
107 views

Quality of a solution in mathematics

How can you determine a quality of a solution for a mathematics problem in general ? If you try to explain too much then solution becomes bit longer and it takes time to go through causing less ...
4 votes
0 answers
184 views

When dealing with sequences, should we teach students to start at 0 or 1?

The reason I prefer starting at 0 is due to a computer science background and also, I think it helps to start at 0 because there are certain reasons that demand it (in particular, combinatorics) and I ...
  • 825
5 votes
4 answers
2k views

Point Deductions - Exams and Quizzes

With regard to an undergraduate statistics course, I am developing a standardized list of point deductions with the TAs (doctoral students) so that graders are consistent in what they are taking off ...
6 votes
0 answers
147 views

Online math quiz: students make short video explaining solution

I am teaching high school math. My students are generally hardworking and competent. Class sizes are about 35. We were recently forced online due to the pandemic, and I have been searching for a way ...
  • 519
1 vote
0 answers
124 views

What is the text for "the other second-term course in analysis at MIT?"

My question comes from first few paragraphs of preface of "Analysis on Manifolds" by James R. Munkres, as excerpted below: A year-long course in real analysis is an essential part of the ...
  • 11
3 votes
6 answers
966 views

Is this motivation for the concept of a limit a good one?

tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...

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