Mathematics education consists in the practice of teaching and learning mathematics, along with the associated research. Research in mathematics education concerns the tools, methods and approaches that facilitate the practice of mathematics or the study of this practice.

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$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So ...
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4answers
460 views

Is studying mathematics chronologically a good idea or not and why?

In high school nowadays most mathematics you learn is fairly 'old'. You have your geometry, all of which (taught in high school) was known to the Greeks more than 2 thousand years ago. You have ...
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0answers
103 views

How would you explain the pdf of the normal distribution to high school students (11th/12th graders)

I will be teaching the normal distribution in January and I need to know how to effectively explain the concepts that does not in any way confuse students or make them feel that the material is ...
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2answers
245 views

What is a good example to show high school students why a proof for induction is a reasonable kind of proof?

I teach average-level high school students who have not had much beyond Algebra 1. I want to show them why induction makes sense. I want the sort of problem where it is intuitive that a statement is ...
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1answer
177 views

Explaining probability theory versus statistics

I'm not sure whether this question was asked before, but it's hard to search because of lots and lots non-descriptive titles like "statistics and probability". The context: There is an anecdote I ...
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10answers
1k views

What are the applications of continued fractions?

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
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0answers
73 views

Math for kids with Cuisenaire rods

I work with kids and i am searching some cool stuff to do with Cuisenaire rods. Thinking about an application i thought that i can show to my students what will be the sum of first $N\in\mathbb{N}$ ...
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4answers
303 views

Showing How Prime Factorization Helps Solving Problems

I need some problems conceivable by middle school students, which are not easy to solve unless the prime factorization of some number is known. An example: It's not easy to know wheter $n$ can be ...
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2answers
94 views

Motivated and unmotivated mathematics courses [closed]

The standard calculus course does not acquaint the student with the reasons why calculus has been and continues to be important in the intellectual development of humankind. Rather, it attempts to ...
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1answer
209 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...
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0answers
121 views

Question about limits [closed]

I am quite new on SE. I see a lot of question about integrals, series, limits. I am wondering if there is a limit to teachers (or textbooks) imagination in these areas.
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10answers
494 views

$2\times2$ matrices are not big enough

Olga Tausky-Todd had once said that "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." There are, however, assertions about matrices that are true for ...
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0answers
50 views

Resources for Teaching High School Statistics

I am a student teacher looking for resources to teach high school Probability & Statistics (untracked). The second semester will be inferential statistics and will include these following topics: ...
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1answer
126 views

Planning a mockup maths class for high school related to river reactivation

I have to plan a mockup maths lesson where the "main topic" should be river reactivation. The given suggestion is to focus on computing cross-sectional areas of rivers using basic geometry and for ...
2
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0answers
184 views

Dynamic Geometry Software for Straight-edge and Compass Constructions

Geogebra is a very good dynamic geometry software. It has so many default tools, e.g. parallel line, angle bisector, tangent to the circle, inscribed and circumscribed circles, etc. But I want the ...
3
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1answer
97 views

Fair Division: Making the Differences in Players' Valuations Believable

When teaching basic fair division algorithms, the students always propose some simple and (at the first glance) correct solutions for $n$ players, which unfortunately are not correct! The only way I ...
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4answers
286 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
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1answer
106 views

Are Parabolas similar intuitively?

All parabolas are similar, but are they all similar in that it is just a question of 'zooming in and out' intuitively speaking? It seems that there should therefore be on all parabolas a curve from ...
3
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2answers
339 views

Good at abstractions bad with numbers

Ever since I had an interest in math I was aware that what I'm good at and what really pulled me was the abstract thinking. My intuition for even the simplest number related concepts (modulo ...
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0answers
84 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
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3answers
89 views

Mean Value Theorem Motivation

I am currently practicing presenting mathematics to various audiences and am considering the example of the mean value theorem. I was wondering how would I be able to motivate this theorem to a ...
2
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1answer
169 views

Motivations for Prime Factorizaton

I'm at the beginning of some middle school math sessions on divisors, gcd, lcm, and prime numbers. It's the first place in the curriculum that the students encounter the three latter concepts ...
8
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4answers
519 views

Is $\tan\theta\cos\theta=\sin\theta$ an identity?

A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is ...
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2answers
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Explaining Hypercomplex numbers to Children.

Imagine a highschool freshman walks up to you and asks you what hypercomplex numbers are. Explain to her, in a fair amount of detail, the different types of hypercomplex numbers in a way that any ...
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0answers
393 views

What are some good ideas in teaching combinations and permutations

I am a student teacher trying to brainstorm some effective lesson plans for combinations and permutations for a high school statistics course. My master teacher has decided that he will introduce ...
2
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1answer
104 views

Why do we need primitive roots?

What is the most motivating way to introduce the order of a modulo n? Apart from simplifying powers of residues is there any other use of the order? Are there any examples which have a real impact on ...
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1answer
315 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points $A(3,9)$ and $B(-2,4)$ lie on the parabola $y=x^2.$ The line $y=x+6$ joins $A$ ...
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20answers
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Good math bed-time stories for children?

What are some good references/books/articles from which to derive some good bed-time math stories to pique a child's interest in math? I am fascinated by math (used to hate it as a kid) and want my ...
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3answers
52 views

Commutativity or Distributivity - Which One to Use to DEFINE Multiplication of Negative Numbers?

It's easy to calculate $3 \times (-4)$, using the meaning of multiplication: $3 \times (-4)=(-4)+(-4)+(-4)=-12$. But it's not the case about $(-4)\times 3$! To DEFINE $(-4)\times 3$ we can choose ...
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22answers
7k views

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
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2answers
108 views

A Good Example of an Argument That Cannot Be Reversed

My students are having a hard time with assuming what they must prove. A good example of what they are doing is when one wishes to show that $\lim_{x\to 3}(2x+1)=7$. They will assume that ...
6
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1answer
246 views

Are Base Ten Logarithms Relics?

Just interested in your thoughts regarding the contention that the pre-eminence of base ten logarithms is a relic from pre-calculator days. Firstly I understand that finding the (base-10) ...
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3answers
214 views

What is the best way to solve this high school exercise?

Can you share with me how would you best solve this exersise to a high school student? Show that $f(x)=x^2-6x+2$ , $x\in(-\infty,3]$ is $1-1$ and find its inverse.
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0answers
134 views

Why to Use the Same Sign for Minus and Negative?

Using the same symbol for two different concepts may cause confusion. So if one decides to do so, they should justify this choice by showing its advantages over other choices. What about the minus ...
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1answer
188 views

Is the computer changing the way we teach and learn math in schools?

Back in school, what I got taught during school was labeled 'math', but it was actually 'rote arithmetics.' This seems to also be the case of many other people. Some came to hate it and never came ...
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1answer
152 views

What are the big ideas needed to develop conceptual understanding of fractions?

In order to be able to perform arithmetic on fractions, students need to understand what fractions are and how they operate. Just teaching rules (e.g. "to add fractions you must have common ...
2
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3answers
121 views

Derivative in interesting way

I am supposed to give a 15-20 minutes math lecture, where I am expecting around 20-30 people. The lecture is about derivative. Since this would be my first "class", I would appreciate any suggestions ...
2
votes
1answer
296 views

What is the use of Euler Totient or Phi Function?

What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but ...
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2answers
2k views

Number of Lines Passing Through a Given Point in the Plane

How can one prove that infinite number of lines pass through a given point in plane, using Euclid's axioms (or Hilbert's, if necessary)?
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5answers
288 views

Why $\sqrt {-1}\cdot \sqrt{-1}=-1$ rather than $\sqrt {-1}\cdot \sqrt{-1}=1$. Pre-definition reason!

It is for years that I teach complex numbers following a historical route. I start with the famous problem of Cardano: Find two numbers whose sum is equal to 10 and whose product is equal to ...
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1answer
109 views

interactive training in mathematics olympiad competition for 8th Grade: Ages 13–14.

I'll enjoy your kindness to ask this question, despite that it seems it'snt the right destination. Please show me an url, for training in mathematics olympiad competition for 6th Grade: Ages 11–12. ...
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2answers
85 views

What are some good games for teaching maths to children?

I am due to teach maths to a ten-year-old. I'd like to try out some games such as Nim and Conway's Soldiers. I've found this list on Wikipedia but Googling for more just gives me a load of Flash ...
8
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0answers
233 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and ...
3
votes
4answers
418 views

How to explain Fractional and Negative Exponents

My classmates doesn't understand Fractional and Negative exponents, since I was the top of my class, so they all came to me... Is there any way to explain it clearly to them?
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2answers
87 views

An Integral With an Odd Function That Isn't Contrived

When ever I teach calculus, single or multivariable, there is always the point in the text when the author covers odd functions and then gives an example of an integral to evaluates to $0$ because the ...
4
votes
1answer
739 views

What is the use of the Chinese Remainder Theorem

What is the most tangible way to introduce the Chinese Remainder Theorem? What are the practical and really interesting examples of this theorem. I am looking for examples which have a real impact on ...
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2answers
508 views

What's an induction problem that will be hard to answer with “backwards reasoning?”

I'm currently the teaching assistant for a course that serves as an introduction to rigorous proofs, and I've noticed some of my students have a tendency to try and use a sort of "backwards reasoning" ...
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1answer
374 views

Best applications-oriented introductory calculus textbooks?

Note: I've edited this question on October 9th, after establishing a bounty on it. What are the best introductory calculus textbooks that explain why calculus is important in a broad intellectual ...
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2answers
141 views

Trying to teach supremum and infimum.

I'm helping out my former calculus teacher as a volunteer calculus advisor, and I have under my supervision 5 students. They've already had an exam and... well, they failed. I read their exams and I ...
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6answers
2k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...