# All Questions

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### What do you do when you realize mid-lecture that your lesson plan is not working?

I had this happen today, and several times before. I had had difficulty preparing the material, because I knew my students strengths and it didn't seem suited to them. However, it seemed the right ...
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### What is a good motivation/showcase for a student for the study of eigenvalues?

Courses about linear algebra make great demands on looking for eigenvalues and transforming matrices to diagonal matrices (or, at least, to Jordan normal form). This is somehow a technical, recipe-...
• 9,448
4k views

### A Non-Unique Factorization of Integers!

I'm going to introduce my students to the fundamental theorem of arithmetic (uniqueness of integer factorization to prime factors), and I don't want them to take the uniqueness for granted! To make my ...
• 2,363
9k views

### How to teach pure mathematics to a well-educated adult who did badly in maths at school

My partner is a PhD student in philosophy and has recently developed a keen interest in learning pure mathematics. I am doing my best to teach her (I'm a pure maths PhD student myself) and it is ...
• 369
4k views

### Why are we so careful in saying that dy/dx is not a fraction?

Calculus instructors are mostly very careful to explain that $\frac{\mathrm{d}y}{\mathrm{d}x}$ is not a fraction, and multiplying both sides of an equation by $\mathrm{d}x$ is nonsense, wrong, or evil....
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### How to present $\Bbb Z/n\Bbb Z$ to highschool level audience

I have the oportunity to talk to a highschool class about mathematics, the topic I got to present are the integers modulo $n$, ie, $\Bbb Z/n\Bbb Z$ , however I don't want to be very heavy and formal, ...
• 835
3k views

### Common Core, threat or menace? Or maybe ok after all?

I have a background in math but no contact with secondary education or kids. I hear all sorts of stories ... horror stories mostly ... about the Common Core math curriculum in the USA. Then I hear ...
• 361
2k views

### "We already passed that course!" How to overcome this?

I've heard the phrase "we passed that course already!" too many times when asking for e.g. the derivative of a simple rational function or a simple integral, getting blank stares, and digging deeper. ...
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### What is the quantitative data on effectiveness of "modern" teaching methods?

What research has been done on how much and in what circumstances various non-lecture types of teching are effective with regards to student knowledge and performance? Meta/review studies preferred ...
• 757
4k views

### How do you go about writing your own lecture notes for a new course?

In advanced graduate courses, there are often no textbooks on the material being covered, and sometimes no good introductory material at all. Even when there are textbooks, many professors prefer to ...
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### How can I convince someone to use a calculator and not worry about the mechanics too much?

I'm trying to help someone pass their final exam (analysis of functions) so they can graduate high school and move on to college. (Not a teacher, just another student, currently in high level math) ...
• 363
2k views

### How to write like a mathematician?

I learned to do math proofs in college. But recently I have begun studying more advanced math books and I've noticed some mathematicians frequently make assumptions that I don't. When I write proofs, ...
• 1,076
2k views

### How can mathematics educators encourage innovation and creativity?

Almost by definition, innovation requires that things be done differently than established custom has it, and comes from the young more often than from the old. In a field as old and established as ...
• 1,765
1k views

### Too much motivation?

This is something that I felt like was difficult for me in some classes, especially lower division differential equations and linear algebra classes. I know professors want to motivate certain topics ...
2k views

### Should we "program" calculus students, like the physicists seem to want us to?

If it is true that we first learn formalism...how to do things that we don't understand, should we regard teaching students mathematics as programming dumb machines with formal rules (to the greatest ...
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### Lesson plan to self-teach real analysis to student with comp-sci background

For my background, I'm a software engineer currently studying for his master's degree in information security. But when that's all done, I plan on going back to mathematics to keep me busy. But with ...
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7k views

### How is teaching calculus in high school different from teaching calculus in college?

I've taught calculus in college for five years, and it's always interesting to see students coming in who already had calculus in high school. Many of them do very well, and don't even seem like they ...
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6k views

### Pi or Tau? How should the circle constant be taught?

Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent ...
• 519
5k views

### How can I explain why we need proofs to someone who has no experience in mathematical thinking?

I know someone I really like, but sadly, that person has absolutely no experience in math or mathematical thinking above third grade mathematics (+, - are fine, but division already makes problems). ...
• 351
7k views

### Why would you teach Calculus before teaching Real Analysis?

Let's assume our students are actual aspiring mathematicians. Why would we introduce our students to Calculus rather than Real Analysis? After all, "Calculus is a subset of Real Analysis". He will ...
6k views

### Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?

The cross product is an important vector operation in in any serious multivariable calculus course. In most textbooks that I'm aware of, right after the definition, we always introduce the ...
3k views

### Redundant zeros

How to convince a middle school student that $0.50=0.5=0.500=\cdots$? I used the fact that $0.50=\frac{5}{10}+\frac{0}{100}=\frac{5}{10}=0.5$ but that far from intuitive. Then I tried to explain ...
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17k views

### What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

I have helped to TA and taught several courses with mixtures of advanced undergraduate and early graduate students in engineering/STEM. These courses are the classics: signal processing, control, ...
• 749
5k views

### What is the proper verb for "doing" an integral?

It's time to write exams, and when writing in committee we often discover differences in usage between various instructors. Here's an example I noticed today. What is the proper verb to use in a ...
20k views

### Ideas for high school pure maths projects

I am thinking of giving my high school students some pure maths projects to do. It is a lot easier to think of some interesting stats projects but not in pure maths. The students' maths background are ...
• 359
4k views

### Are there disadvantages to teaching complex numbers as purely geometrical objects?

Complex numbers are, or at least were to me, generally introduced like this: There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare ...
• 1,357
4k views

### Why do we care about multiple proofs of the same theorem?

I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
• 553
2k views

### Practical experience with teaching differentials in freshman calc?

There is a well known essay by Dray and Manogue which argues that differentials should be brought back into freshman calculus, and that we shouldn't worry too much about choosing a specific way of ...
1k views

### A Series of Unfortunate Examples!

All of us know, when teaching, the "right" choice of examples is important. Though, making the "right" choice is one of those things that are easier said than done. Here is the ...
• 4,438
3k views

### How do you coach students who often make small errors?

Some students are prone to making small calculation errors. Not errors in understanding, but errors like adding or multiplying integers incorrectly, or dropping a negative sign. Unsystematic errors in ...
• 4,845
4k views

### Are there science-backed effective teaching strategies?

As a math teacher, I am always trying to self-assess my teaching methods. I am trying a lot of different methods but I would like to organize my study on the subject without weighing too much on the ...
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2k views

### What are the historical reasons for the hostility against standardized testing in the US?

NB. Some answers appear to be for a question I did not ask, namely, "Why is standardized testing bad?" Indeed, these answers tend to underscore the premise of my actual question, which can ...
1k views

### How can you explain to students that they should not use the same variable in an integrand and in the limits of integration simultaneously?

When teaching Calculus, one thing that many teachers emphasize is that the variable of integration is a `dummy variable' that is unimportant. Around the same time, we introduce integrals with ...
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### When did US mathematics programs start failing to prepare incoming students for books like "Baby" Rudin?

I've seen in a lot of questions about "which textbook to use for intro analysis", and inevitably Rudin's Principles of Mathematical Analysis comes up, with the (almost cliche) rejoinder that "today's ...
2k views

### Counterintuitive consequences of standard definitions

Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and ...
• 4,239
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### How should one tutor a student in undergraduate real analysis?

I am an undergraduate. Other undergraduates sometimes ask me to tutor them in an introductory real analysis course that covers the equivalent of the first half-dozen chapters of Rudin's Principles of ...
• 539
2k views

### The impact of dyslexia on learning mathematics, and available resources

I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose ...
• 351
1k views

### Student Poisoned Experience with Math

I’d like to start off with saying I am not a teacher so I don’t know how much of this is already trying to be addressed in math education throughout the world. In talking to and tutoring fellow ...
• 530
829 views

### "Function" vs "Function of ...": how much does it contribute to students difficulties?

Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions. For example, when we write $y=f(x)$ we all know that $f$ ...
• 2,039
706 views

### Tutoring elementary student who reverses left and right

I am a volunteer math tutor. My student is in fifth grade. He was evaluated in August and found to be at the first grade level in math. (His reading is very close to grade level.) He has epilepsy, ...
• 755
774 views

### Is there a Piagetian age at which proofs can be comprehended?

I am wondering if there is literature on the developmental age (pre-adolescent?, adolescent?) at which the notion of a "proof" can be understood? I am less interested in mathematical proofs and more ...
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### Is 'estimating' still considered a valuable skill?

I was with a 2nd year high school class, preparing for our (US) state's standardized test. I asked the class how they would solve this, and they flipped through the sheets to find V=\frac{1}{3}\pi ...
8k views

### How does a teacher come up with plausible wrong answers for multiple choice tests?

When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But ...
• 351
4k views

### How to encourage young student to think in unusual ways?

I tutor a young girl aged 11 (grade 4). She is doing OK for her age, but I have observed that she has a tendency for rigid ways of thinking. She is usually more inclined to follow rules and stick to ...
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For example, I gave an exam earlier today with a problem that ended in the sentence Use the chain rule to find $(f\circ g)'(3)$. During the exam, one of the students asked me what the circle ...