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0answers
17 views

What were the applications of conic sections before Newton?

As I was being observed today, an administrator asked me for a practical application of parabolas. I responded by talking about objects in free-fall. Afterwards as I was re-thinking this conversation ...
4
votes
5answers
674 views

Algorithmic thinking problems

In Norway we will have a new national mathematics curriculum for all ages including high school beginning august 2020. A fundamental change is the new focus on so called algorithmic thinking. In ...
1
vote
1answer
47 views

How to address opportunities for improvement with a teacher

My daughter’s 5th grade teacher gives some pretty impossible questions and I don’t think she understands the material she’s teaching. For example, this question has no context from previous questions ...
0
votes
0answers
71 views

generalization of the derivative of accumulation function: 1st fundamental theorem of calculus

The 1st fundamental theorem of calculus: $$\frac{d}{dx}\int_a^xf(t)\,dt = f(x),\quad x\in (a,b).$$ No textbook says that this can be generalized into the scenario where x is less than a and where f ...
8
votes
0answers
56 views

Good Source for German Tank Problem

I would like to teach the German Tank Problem to bright students with little prerequisites. They are 16 year old high school students with a lot of maths competition experience, but no prior knowledge ...
9
votes
0answers
50 views

toys/manipulatives for exploring graph theory

I was talking recently with my daughters about non-planar graphs, like $K_{3,3}$, $K_5$, and the 7 bridges of Königsberg. They got pretty interested in it, and seemed to catch on to the core ideas. ...
1
vote
2answers
108 views

Quadratic equations using complex math which have no imaginary roots

Many years ago when learning complex maths we used complex maths as an example in the quadratic equation to find real roots. My nephew is struggling to deal with complex maths, as his teacher is ...
-1
votes
0answers
70 views

Refactoring college maths [on hold]

So this is a somewhat neat idea I stumbled onto late in thinking. It involved refactoring the pedagogy of analysis maths. The typical discourse or dialectic is analysis versus synthesis. Here, I ...
2
votes
2answers
154 views

Introductory exercise for the addition of large natural numbers

I'm starting a repetition with my students in 5th grade after they learned in elementary school how to sum up larger natural numbers (also 5- to 6-digits) by writing down that calculation. As ...
33
votes
9answers
3k views

Beautiful planar geometry theorems not encountered in high school

I would like to impress college students (undergraduates in the U.S.) that there is more to planar geometry beyond what they learned in high school. I would like to show them beautiful theorems they ...
10
votes
4answers
1k views

Co-curricular lessons between geometry and chemistry?

My school is hyped about the promise of co-curricular education and they are giving the math and science teachers paid days off to develop lesson plans that synergize our learning goals. I'm on ...
6
votes
4answers
512 views

Favorite secondary math manipulatives?

I read this is the mathematics educators stack exchange so hopefully this is the right place for this question. I was curious what is your favorite math toys, manipulatives, math games, or tools to ...
8
votes
6answers
368 views

Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?

There are a lot of "proofs" of the identity $e^{ix}=\cos x+i\sin x$ in textbooks, using either differential equations or power series. However, I find those proofs often misleading, because it appears ...
3
votes
5answers
279 views

Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?

In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial $$f(x)=a_nx^n+\cdots a_1x+a_0$$ with $a_n\neq 0$ is defined to be $n$. But I ...
4
votes
2answers
234 views

The trick didn't like me (teaching Fourier transform)

I was teaching Fourier transform for engineering students. Since I didn't want to go into rigourous proofs during class, I often use intuition, just give students an idea to persuade them with the ...
19
votes
13answers
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 ...
6
votes
6answers
391 views

How to answer a three-year-old the question “Why is $2+6$ the same as $4+4$”?

I am teaching my daughter, who is currently about $46$ months old, additions. She is very curious and asks a lot of good questions. For example, when I told her that $2+6=8$ and $4+4=8$, she asked me ...
4
votes
2answers
204 views

The hardest case of integration by partial fractions

The context is explaining to calculus students how to integrate rational functions by using partial fractions decomposition. As we all know, partial fractions decomposition is a method to write every ...
3
votes
3answers
142 views

What kind of problem do you learn most from (hard problems or easy problems)?

I am curious if it is true that one can learn more from hard problems, or from easy problems. I heard that we all learn a lot more from doing harder problems, because we have to learn all of the ...
2
votes
3answers
189 views

Why not write “or” inequalities as $a>x>b$? [closed]

This seems like a stupid question . I just don’t understand why the algebra textbooks I see don’t really address this with students. I boy that I am tutoring brought it up and I was slightly ...
-2
votes
2answers
93 views

Should the limits of one system of elementary set theory be the limits of a student's mathematical world? [closed]

In teaching elementary set theory, suppose we refrain from emphasizing historical decisions that were made in theory construction. Is there a danger that students may see the mathematical language ...
12
votes
5answers
615 views

Teaching a student who refuses to learn

How to deal with a student who refuses to learn? I've met a few of those over the years as a a private-class math teacher. They don't want to learn anything about the subject. Some of them are just ...
-2
votes
1answer
49 views

Where can I find the partial order relation of prerequisites of undergraduate courses in the United States?

Let $A$ be the set of all undergraduate mathematical courses in the US and define a binary relation $\leq$ on $A$ such that for elements $a,b\in A$ (that is, $a$, $b$ are undergraduate mathematical ...
-3
votes
1answer
136 views

A role for a non-symmetric equality relation in teaching mathematics? [closed]

First, I will simply observe that it seems to be standard practice, in elementary set theory, to define relations to be sets of ordered pairs. If we had the option of introducing a "symmetric ...
4
votes
2answers
106 views

Road map to teach undergrads a first course in real analysis that concludes with convergence of fourier series

I am planning to teach (unofficially, I am a Grad student) a course in real analysis. Aim of the course is to understand the convergence of Fourier series. I want to start with the notion of ...
3
votes
1answer
85 views

Are there any textbooks on multivariable calculus that introduce all non-trivial definitions and all non-trivial proofs with a “first draft”?

The idea is that a student who is reading from beginning to end will always be provided with a train of thought. The student will have no motivation to stop, and memorize an unexplained product of ...
2
votes
2answers
129 views

Which textbooks on College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE are written by world-class mathematicians?

For example, Trigonometry was written by Wolf-Prize winner Israel Gelfand, one of the top mathematicians in the 20th century. I am wondering if other world-class mathematicians have written textbooks ...
21
votes
15answers
6k views

Explaining why (or whether) zero and one are prime, composite or neither to younger children

There are lots of discussions out there about whether $1$ is a prime number (such as this one) and even about zero (such as this question, though note zero does generate a prime ideal in $\mathbb{Z}$ ...
8
votes
2answers
124 views

Exercise database

edit: Thanks for all your answers so far. I have decided to develop my own solution, both because it is fun and I can then form it exactly as I want to. Once I am done (which might take some time, ...
4
votes
1answer
69 views

In preparation for exams: question bank or questions with omitted particulars?

I have been doing a little bit of experimenting when it comes time to review with the class in preparation for the final exam. The last handout I have been giving my students has usually been a ...
14
votes
8answers
992 views

How do you attract more math majors at a liberal arts college math department?

It seems to me that we all might benefit from an answer to this question, since math departments must defend their performance within their institutions. I imagine there will be standard answers like:...
8
votes
2answers
1k views

Take-Home Examination on Ordinary Differential Equations?

I am planning to give my students a take-home examination on ODE. The main topic that I would like to cover is Linear Differential Equations of Order Greater than One. For example, I will give my ...
6
votes
1answer
127 views

Symmetry in polar functions - how to explain

In the precalculus curriculum I am teaching (using Stewart's book Precalculus: Mathematics for Calculus, 7th ed.), we do a bit of polar graphing, which includes discussion of symmetry on polar graphs. ...
7
votes
5answers
2k views

Teaching asymptotic notations at the beginning of calculus [duplicate]

I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and $\sim$), secondly explaining their "arithmetic" (things like how to sum little-oh's and ...
3
votes
0answers
80 views

The propagation of the wave equation in even versus odd dimension

I am about to teach a second year undergraduate class on applied differential equation (first time) and, while I won't have time to go into the details, I wanted to show my students the difference ...
17
votes
8answers
5k views

Prisoner's dilemma formulation for children

I am preparing an introductory course on Game Theory for children (between 10 and 17 years old). In the course description, I want to include a prisoner's dilemma in order to catch children's ...
19
votes
8answers
2k views

How can I learn to write better questions to test for conceptual understanding?

I'm worried that I'm bad at realizing when a question I've written requires little or no conceptual understanding to answer. Like, when I'm writing a question for a homework assignment or exam, I'll ...
0
votes
4answers
208 views

Is “Volume of Solids of Revolution” a part of Cal I or Cac II

I took Calc I in preparation for the CLEP. I did not learn about solids of revolution. Is this something that is normally part of Calc I and that most Calc II classes will expect that I can do already?...
4
votes
3answers
550 views

“Table” method for expanding brackets vs “each term in the first bracket gets multiplied by each term in the second bracket”

Hi I've just discovered mathseducators stackexchange. As a maths tutor in the UK, I am irritated that some of my students - particularly GCSE and sometimes below - use the table method for expanding ...
6
votes
4answers
257 views

Examples of informal explanations that cause misconceptions

Every good teacher knows that giving an informal explanation (that is, an explanation not based on strict definitions and maybe making use of metaphors, intuition, or everyday language) of a ...
3
votes
3answers
111 views

What is the name of the form of the line equation $y = m(x-x0)+y0$

I have been looking everywhere for the name of this form of equation of a line $y=m(x-x_0)+y_0$. It's not quite point-slope. It's the same form you would write in the linear approximation of a ...
6
votes
1answer
823 views

Why is it popular to teach modulus via the example of mod 12 and analogue clocks?

Why is it popular to teach modulus via the example of mod 12 and analogue clocks rather than rectangles or tables that have a finite number of columns in each row, and infinitely many rows? It's ...
6
votes
2answers
139 views

Why is there an emphasis on analysis courses in undergrad progams?

In undergraduate maths study, there are three main areas: analysis, algebra, and geometry. (There are of course other small topics as well, but they don't have to be learnt by every student.) I have ...
0
votes
2answers
68 views

Examples for environmental topics in the context of terms or linear inequalities

I want to emphasize the aspect of environmental education in my math class. Now I'm reasoning whether to do that with linear inequalities or terms with two variables - these are our next topics. The ...
0
votes
1answer
91 views

How helpful are university subject rankings when choosing a place to study math?

There are many university ranking consultancies trying to compare one leading maths department with another and to conclude which one is better. Although this doesn't seem to be a very reasonable ...
4
votes
5answers
123 views

How to teach sketching a parametric curve?

I feel very confused when teaching students how to sketch a parametric curve like $$x(t)=e^{-t}t; y(t)=t^{2}+t.$$ Here students are supposed to know derivatives, increasing-decreasing,... In ...
5
votes
1answer
117 views

How to explain even higher moments

In prob/stats, students mostly get the idea of mean/expected value, and variance/s.d. slowly sinks in. Even the third and fourth moments (about the mean) have "standard" interpretations - skewness ...
3
votes
3answers
126 views

(use of de L'Hospital's rule) How would you explain this limit to high school students?

Someone asks the following question: Determine the values for the real numbers $a$ and $b$ such that $$\lim_{x\to0}\frac{ae^x+b\cos x}{x}=2$$ The students directly apply de LHospital's rule for the ...
5
votes
2answers
179 views

Are there mathematical proof info-graphics?

I am teaching mathematical proof to kids (10th grade) and am of the opinion that proofs of theorems are a good place to start, where almost all of mathematics' important players come together. On ...
8
votes
0answers
77 views

tutorial active learning

This is a question I asked on [Academia.se]. It did not get an answer, so I am re-posting it here. In the country where I live, university students studying mathematics usually attend lectures, ...

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