All Questions

0
votes
2answers
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, ...
3
votes
0answers
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 ...
1
vote
0answers
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 ...
6
votes
1answer
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 ...
1
vote
1answer
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 ...
5
votes
3answers
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 ...
2
votes
2answers
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 ...
-2
votes
2answers
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" ...
-7
votes
1answer
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 ...
6
votes
2answers
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. ...
-5
votes
0answers
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 ...
7
votes
3answers
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 ...
4
votes
3answers
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 ...
-1
votes
1answer
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 ...
7
votes
5answers
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 ...
4
votes
1answer
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 ...
7
votes
1answer
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 ...
0
votes
0answers
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 ...
2
votes
2answers
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? ...
6
votes
5answers
287 views

Writing up a proof that assumes what is to be proven?

I was working on this question on math, where (among other things), the OP was asked to prove that $$x \oplus y=\sqrt[3]{x^3+y^3}$$ is associative. After some prompting, the offered proof was $$\...
3
votes
1answer
192 views

Resources for teaching calc III

I was very unsatisfied with how I taught Calc III a couple years ago, and this summer I have to do it again. Are there any general resources for teaching this course? It seems like there should be, ...
2
votes
5answers
337 views

How can I explain why numerical integration is easy, but symbolic integration is hard?

I'm asking about definite integrals that can effortlessly be found numerically by high schoolers using software. For example, $$\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\...
3
votes
1answer
125 views

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 ...
5
votes
2answers
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 ...
1
vote
0answers
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 ...
1
vote
1answer
304 views

What are “PreK‐12th‐grade students”?

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-...
3
votes
7answers
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 ...
16
votes
12answers
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)...
2
votes
3answers
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 ...
3
votes
2answers
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 ...
4
votes
4answers
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 ...
4
votes
3answers
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 ...
5
votes
1answer
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 ...
7
votes
4answers
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 ...
3
votes
0answers
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)...
3
votes
6answers
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 ...
2
votes
1answer
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 ...
3
votes
2answers
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 ...
7
votes
2answers
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....
1
vote
1answer
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 ...
5
votes
3answers
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. ...
10
votes
0answers
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 ...
5
votes
1answer
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 ...
1
vote
1answer
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 ...
6
votes
3answers
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 ...
2
votes
3answers
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 ...
5
votes
2answers
114 views

Curriculum for Advanced 6th Graders

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 ...
4
votes
1answer
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 ...
5
votes
0answers
163 views

Mathematical undergraduate education in Syria

I'd like to learn some things about undergraduate mathematical education in Syria EDIT: In particular I'm interested in students 15-16 years old. What are the main differences from the European ...
6
votes
2answers
146 views

What are some tips for framing a graph in the most readable, relevant, and aesthetic way, for secondary education mathematics?

When I say "framing," I mean things like choosing zoom, x-axis/y-axis step, horizontal/vertical shift from the origin, choosing how/when to number steps, labeling axes, as well as, purely aesthetic ...

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