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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How to maintain enthusiasm and joy in teaching when the material grows stale

I recently finished my third semester of teaching calculus to freshman college students. This means I was drawing the same pictures, solving the same example problems, and discussing the same ...
12
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5answers
502 views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
2
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2answers
226 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...
11
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1answer
324 views

Teaching engineers mathematical thinking skills

In my experience, many introductory engineering mathematics textbooks these days tend to skip proofs and discuss logic only in the context of digital electronics. On the other hand, I can imagine that ...
6
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2answers
120 views

Teaching algebra in a culturally relevant way while fitting Common Core standards

I've been assigning algebra textbook and worksheet problems (from the publishers and my own) that look like this: Simplify the following expressions. $x^{- 3} y^2$ $c^2 d^{-5}$ ...
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1answer
15 views

Standardizing a normal curve and getting equivalent areas

For example, a pediatrician finds that his 3 year old patients are normal with mean 38.72 inches and SD of 3.17 inches. The probability that a randomly selected 3 year old patient is between 35 to ...
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2answers
351 views

Examples of open ended calculus “class project” ideas

I have instructed calculus I an II, each once, at the college level and would like to emphasize that math is not just about memorizing formulas and concepts for a test and that applied math is not a ...
114
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31answers
11k views

Stopping the “Will I need this for the test” question [closed]

I am a college professor in the American education system and find that the major concern of my students is trying to determine the specific techniques or problems which I will ask on the exam. This ...
6
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1answer
135 views

How to combat memorization

As a student in high school, I never bothered to memorize equations or methods of solving, rather I would try to identify the logic behind the operations and apply them. However, now that I've begun ...
11
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3answers
524 views

Why study metric spaces?

Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most ...
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2answers
96 views

Explaining the concept of $z$-scores in high school statistics

The students have so far studied the uniform probability distribution and have a working familiarity with relative frequency histograms and the 68-95-99.7 empirical rule. They still have trouble with ...
208
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32answers
30k views

Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} ...
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0answers
53 views

Studies on how the wording employed on the explanation of mathematical concepts helps students to learn?

I remember that I had to learn division in my childhood, I could handle all the other mathematical concepts that were presented until then but division was a real pain to learn, somehow the idea of ...
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23answers
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How would you explain to a 9th grader the negative exponent rule?

Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = ...
4
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2answers
202 views

How can I explain my 9 years old brother that $8a\cdot4a \neq 64a$

My youngest brother had a pre-algebra test yesterday and he was asked to tell if two expressions are equal or not. We agreed on most of the things but on this one I find it hard to make him accept my ...
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2answers
70 views

Why is Cauchy condition for convergence not formulated in a simpler way?

The standard definition of a Cauchy sequence (e.g. it's given in Wikipedia and most textbooks I remember; admittedly those are mostly older ones) is: for every positive real $ε > 0$ there is a ...
2
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4answers
280 views

How would you create a math class that centers on the cultural experiences of African American and Latino students [closed]

I need to write a paper on "Ethnocentric Mathematics" and I have no idea what kind of effective teaching strategies are available. We read an article from this scholar named Tate who explained that in ...
5
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6answers
290 views

What is the point of quadratic residues?

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most ...
11
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2answers
130 views

Is the maximal path through a math book necessarily linear?

I'm studying with two main math books (Munkres and D&F) these couple of months. My method so far is just going through the book page by page constructing everything in it (independently if I can) ...
11
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3answers
140 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
5
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2answers
152 views

Is it better to teach or to grade?

As a graduate student at my university, I have the option many times of deciding what type of work I do for support. The two basic options are to teach either a calculus or college algebra course, or ...
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1answer
144 views

New twist on: Does a negative number really exist? [closed]

[I read the thread from back in 2011 and would have commented there but it is closed.] I think the answer is clearly no, because -3^2 is -9. Likewise, the only way to get 9 is to put -3 in parens ...
3
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1answer
57 views

Topic for teaching assessment

I'm in the position to have a teaching assessment with a tutoring agency next week. This assessment will include me teaching the assessor a topic of my choice in 15 minutes, demonstrating the Socratic ...
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0answers
57 views

(Actual) applications of basic differential and integral methods

If this isn't the place, I apologize: At the end of my calculus class, we asked the students (among other things) what some applications of calculus methods are. Disappointingly, many focused on the ...
4
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0answers
347 views

Soft question : First year student and confused

I hope I won't tire the fellow mathematicians with this question but I am very, very confused... I am a first year undergraduate student of Mathematics. I can't say I am a prodigy, maybe having an ...
5
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5answers
155 views

$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: 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
227 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
59 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 ...
4
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2answers
178 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 ...
2
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1answer
86 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 ...
8
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1answer
95 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?
3
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0answers
61 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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0answers
76 views

Software/Applet to Draw Tree Diagrams (for Enumeration Problems)

I need a software/applet/flash file which easily draws tree diagrams for simple enumeration problems: I want to give number of the vertices in each layer, and it draws the diagram which shows all the ...
4
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4answers
130 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 ...
3
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2answers
75 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 ...
3
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1answer
127 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
92 views

Question about limits

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.
10
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8answers
298 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 ...
2
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0answers
44 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: ...
4
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1answer
69 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
97 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 ...
2
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1answer
58 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 ...
3
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4answers
206 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
76 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
votes
2answers
109 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 ...
3
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0answers
74 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 ...
0
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3answers
56 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
86 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
399 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 ...
7
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2answers
377 views

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 ...