# All Questions

2,445 questions
52 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, ...
70 views

### Non-US polynomial division notation

I am looking for examples of how polynomial division is presented to secondary students outside of the US. Wikipedia has a nice presentation of this for integer division here. Do you know of anything ...
23 views

### Student-friendly / efficient approach to computing Taylor coefficients of infinite binomial series expansions?

I’m working on a section of a course covering Taylor expansions, and have found that, although there is great notation for simplifying the formula for the coefficients of a general infinite binomial ...
91 views

### Corequisite remediation for “Mathematics for Future Elementary Teachers”

My university is eliminating its developmental math courses, and moving to a system using corequisite remediation. I am trying to develop a coreq for the first course in our "Mathematics for ...
73 views

### How to intuitively convince the students that a strip with two full twists is homeomorphic to the standard annulus?

Intuitively speaking, one space is homemorphic to another if one can be deformed continuously to another without tearing and gluing. It is more or less easy to convince the students that a square is ...
228 views

### What's the point of exercises without answers?

What is the point of exercises for which answers aren't provided? (That is to say, what is the pedagogical justification for such exercises? - Edit by someone other than original poster.) Commentary ...
194 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 ...
117 views

### Interpretation of how to define “bigger” and “smaller” real numbers

This is a variant on the question small real numbers. I have a disagreement with someone about the meaning of "bigger" real numbers. Say we have the real number $-1.$ Is $0$ "bigger" or "smaller" ...
89 views

### math norms for all kind of ruls [closed]

Want to ask if someone knows a official site where all kind of rules like $\infty-\infty$ or $\infty^0$ are classified. Ment an paper rule collector for that kind of definitions which has an certified ...
138 views

### MacLane-Birkhoff's “Algebra” vs Jacobson's “Basic Algebra I,II” vs Lang's “Algebra”

(Cross-posted at Math.Stackexchange) I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. ...
62 views

### A KinderGarten of binomial coefficients [closed]

I tell the beginners that $(-n)!= \pm \infty$, if $n \in N$. Next, I tell them that the definition of ${\nu \choose k}$ as $$\frac{\nu (\nu-1) (\nu-2) (\nu -3) ...(\nu-k+1)}{k!}~~~~(1)$$ is the most ...
250 views

### How important is making definitions plausible?

During my studies I observed that while most lecturers try to explain theorems and their proofs, only very few of them try to explain definitions. However, in my opinion, definitions are the base of ...
2k views

### Are soroban (Japanese abacus) classes worth doing?

The companies that run these expensive abacus programs for children claim it has all kinds of benefits for their mathematics abilities and speed. Apparently it starts with a child learning the ...
123 views

### Practical applications of integration by substitution where integrand is unknown

I posted this question on the Mathematics Stack Exchange a while ago, and got no responses, so I thought I would ask it here. I'm looking for any real-life applications of integration by substitution ...
374 views

### Different Kinds of Variables

Students sometimes ask whether the $x$ in the expression $$2x$$ the same kind of thing as the $x$ in the equation $$2x = 4.$$ In the expression $2x, \;x$ can be any real value. However, in the ...
181 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 ...
170 views

### Scientific results on the usefulness of physical units in secondary education?

When we encounter "real world problems" in math, one can chose different levels of detail with regard to units: from leaving them out completely up to using them everywhere. I'd argue that both ...
100 views

### Online open-course-ware that uses Maclane's book “Algebra”

I am struggling with that book which I find to be more of second-guessing type than a book for self-study: it has cryptically written sections, no examples (and those given, and rarely, are even more ...
184 views

### How many hours / school years does it take for the average child to memorize the $10\times 10$ addition and multiplication tables?

How many hours does it take for the average child to memorize the $10\times 10$ addition table? How many school years does it take for the average child to memorize the $10 \times 10$ addition table? ...
287 views

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### Mobile phone apps that would allow for students to anonymously “buzz” me in real time?

Are there any mobile phone apps that would allow for students to anonymously "buzz" me in real time? What I have visualized is a free app students can download that features a button they can press ...
146 views

### How to come up with a Leslie matrix with convenient eigenvalues?

A three by three Leslie matrix looks like $$\begin{bmatrix} f_0 & f_1 & f_2 \\ s_0 & 0 & 0 \\ 0 & s_1 & 0 \end{bmatrix},$$ where $f_0 \ge 0$ and everything else is ...
90 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 ...
304 views

I am reading the paper Effects of game‐based learning on students' mathematics achievement: A meta‐analysis and can't find a definition for the term "PreK‐12th‐grade students". While I know that "K-...
294 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 ...
7k views

### Is there a simple example that empirical evidence is misleading?

Suppose that I want to show a student that empirical evidence in mathematics is not enough and we do need proofs, what kind of examples can I use? By empirical evidence, I mean that (most of the time)...
223 views

### How to justify that students should come to class?

Nowadays, a student should be able to learn the course material at home through reading the textbook or follow one of the many free online courses. Some universities record video or audio of lectures ...
168 views

### What are the benefits of an expertly curated learning pathway?

What are the benefits of an expertly curated learning pathway? Like that provided by a major publisher's textbook - CPM, a school district's mandated curriculum - IM's Open Up Resources or a ...
427 views

### Real-life exceptions to PEMDAS?

What are some real-life exceptions to the PEMDAS rule? I am looking for examples from "real" mathematical language --- conventions that are held in modern mathematics practice, such as those appearing ...
122 views

### What books are good for drawing an intersecting plane?

I am looking for textbooks pertaining to drawing intersecting planes, intersecting point (among others) as follows. I am not sure this topic is taught in high school around the globe. That is why I ...
177 views

### Are there textbooks that cover most etymological aspects of mathematics?

In most of the science textbooks I read, I observed that most of them contain the terms, definitions and etymology too. But nowadays, the mathematics textbooks are becoming more formal and contain ...
257 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 ...
129 views

### The art of designing of problem sets

For proof-based math courses, the gist of the learning happens in problem sets and so it is essential to design them well. We would appreciate responses containing references (eg. from active learning)...
248 views

### What is an interesting high-school level topic to discuss using Mathematica or Geogebra?

I have to choose a topic to give a presentation. The topic should be high-school level or at most Linear Algebra 1 and Calculus 1. Conics and polygons in the Euclidean geometry are some fine topics ...
131 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 ...
144 views

### Introducing derivative concept and definition

I need to give a short presentation on introducing a class of engineering students to the concept and definition of the derivative. I'm to assume that the students are currently at the appropriate ...
182 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....
86 views

### How to formulate this type of arcsin problem?

Reading and commenting on What are some common ways students get confused about finding an inverse of a function? I was kindly set straight that the use of $\sin^{^{-1}}(x)$ to mean $\arcsin(x)$ has ...
183 views

### What are some common ways students get confused about finding an inverse of a function?

What are some common ways students get confused about finding an inverse of a function? One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. ...
187 views

### Use of Lockhart's *Measurement* in a course?

I greatly admire Paul Lockhart's Measurement (Harvard Press). Many of you know him through A Mathematician's Lament. One review of Measurement said, “Here Lockhart offers the positive side of the ...
139 views

### Is a clear distinction made between signs and operators?

This question about FOIL, comments and answers made me think about the two roles of $-$: as a sign and as an operator. This struck me because the title "Why in the FOIL Method the terms are taken ...
91 views

### Explicit Cross Method

When factoring quadratic expressions $ax^2+bx+c$ it is common to the guess and check factors (AKA the cross method). This would involve factoring $a$ and $c$ and considering particular combinations ...
213 views

### How to overcome the frustration in teaching and doing research at the same time

I understand that this website is designated particularly for teaching-related topics. However as many of us do agree that being a lecturer, we are supposed to run both teaching and research duties at ...
195 views

### Why in the FOIL Method the terms are taken with their signs?

That was the most boring title I could choose but in all honesty, it is what the question is. Here is a reminder of the FOIL method that is used for multiplying two binomials. For example, to multiply ...
114 views

Next year I volunteered to lead the math class for a group of 6th graders (ages 11 - 12). Here are the pertinent details: About 5 - 8 (U.S.) students, for about 45 minutes, 3 days a week (they'll ...
104 views

### What is the notation for polynomial long division in Norway?

I will be teaching a calculus-type course in Norwegian. Our textbook is unfortunately in English (the curse of a small language), but any custom exercises should be and all exams have to be in ...