# All Questions

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### Does a proof by induction have to explicitly refer to the principle of mathematical induction?

I teach high school math. Some of my colleagues insist that a proof by induction should explicitly refer to the principle of mathematical induction, i.e. it must include the words "by the ...
• 1,245
3k views

### Fixing wrong ideas about coefficients (e.g. subtract 3 from 3x to isolate x)

I tutor a student (9th grade, United States) who is in an algebra class. She consistently makes mistakes when dealing with coefficients. The most common one is attempting to subtract away a ...
• 375
321 views

### What will happen if someone doesn’t make it to the IMO team?

I am highly interested in mathematics, and solving Olympiad maths problems has been a type of hobby for me. But due to my age, I will never be able to give the olympiads a go again. I want to know ...
• 201
2k views

### How does a math Olympian fare in undergraduate maths courses?

I am a maths enthusiast and have been exposed to maths olympiad problems for some time. I wanted to know how does a math Olympian do in undergraduate courses of mathematics, statistics or computer ...
• 201
1 vote
117 views

### Speed math appropriate for middle-school students

There are many "rules" for speed arithmetic. List of some reference links showing speed methods or rules: https://ofpad.com/multiplication-tricks-for-mental-math/ https://ofpad.com/mental-...
• 111
278 views

### Order of operations pemdas

Why was the order of operations established in mathematics with multiplication taking precedence over addition, as dictated by the PEMDAS rule? What historical or practical factors influenced this ...
2k views

### How can we explain intuitively the convergence and divergence of these two series?

It is known that $\displaystyle\sum_1^{\infty} \frac{1}{n^{1.000001}}$ converges while $\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$ diverges. Though we can logically prove these results,...
• 4,275
1k views

### Is there a standard convention for interpreting ambiguous absolute value expressions?

Consider the expression $$|x + 2|x + 3|x + 4|.$$ One way to interpret this is that there are two products being added together: $$|x+2|x \hspace{1cm} + \hspace{1cm} 3|x+4|$$ But you could also ...
• 12.1k
128 views

### PROBABILITY QUESTION [closed]

How to solve this probability question?f a, b, and c are in the range [0, 1], what is the probability that the quadratic equation ax^2 + bx + c has real solutions? justify your answer! This question ...
3k views

### Importance of complex numbers knowledge in real roots

Many students question the importance of complex numbers in real life. We can find many important applications of imaginary numbers in Engineering field and physics. This question is not related to ...
• 1,154
1 vote
111 views

### Summer or Semester Programs which bridge to Graduate Mathematics

A few years back I had a student attend the MASS semester at Penn State. It was a fantastic experience for my student and it certainly helped him find a place in graduate school and I would wager it ...
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5k views

### How can I teach intuition why ‘If P then Q’ and ‘P only if Q’ mean the same, to first year undergraduates?

In every of my ten years as course instructor introducing first year undergraduates to proofs, I always project this quotation on the lecture screen, then dictate the whole caboodle word for word! But ...
211 views

### Engaging Mathematical Riddles for Classroom Enrichment: examples and impact studies references

Hello fellow educators and math enthusiasts, I am on the hunt for entertaining and thought-provoking mathematical riddles suitable for a classroom setting. My objective is to find riddles that not ...
202 views

### Game project using linear algebra

Throughout this year, I have been in charge of a linear algebra course aimed at engineering school. In this course, I asked my students to work on a final project involving finding a winning strategy ...
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1k views

### Intuition for order of operations in compound transformations

This is a close cousin of the previous question asked here about transformations inside and outside a function and how they switch things around. I think some of the perspectives there will help here, ...
1 vote
220 views

### Online Probability Simulation for Compound Events

I'm teaching Grade 9 Probability. We need to compare Experiment Probabilities (Relative Frequencies) with Theoretical Probabilities for Simple and Compound Events. I want to come up with and Online ...
• 967
372 views

### Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

Back in 2018, I wrote a post about asymptotes of rational functions in which I addressed not only horizontal and slant/oblique asymptotes, but also the general case of "polynomial asymptotes.&...
• 12.1k
76 views

### Examples of Financial Institutions that Compute Interest Atypically?

Are there examples of financial institutions that compound their interest more frequently than once-a-month? Are there examples of financial institutions that consider continually compounded interest ...
• 4,845
1 vote
163 views

### Chinese and Japanese most important high school textbooks

I would like to know the best high school math books from Japan and China. Can you suggest some books or free resources? I would like to compare the different approach betweeen China and Japan and ...
• 131
6k views

### What are some "deep" questions to explore in elementary school math?

My first grader is very advanced in math. Rather than doing more and more math and making school math even more boring for him, I recently decided to start going "deeper" rather than "...
• 383
1 vote
137 views

### Idea of using LLMs to help communicate ideas in math

What do you guys think about the ideas presented in this short text: https://github.com/yougetyourmanwww/AI-for-math/blob/main/AI.md The text is about how LLMs like chatGPT can be used for when doing ...
4k views

### Why do most Analysis textbooks overlook, and fail to teach delta-epsilon proofs, using the K-ε principle?

When writing $\delta$-$\varepsilon$ proofs, it's common that the ''natural'' choice of $\delta$ leads to the final inequality in the form, say, $|\ldots| < \varepsilon+\varepsilon+\varepsilon$ ...
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163 views

### Cognitive activation vs cognitive load reduction

During my teaching training in Germany and in many professional development sessions, there has been repeated emphasis on the importance of cognitive activation and challenging tasks for effective ...
• 1,275
4k views

### Role of human math teachers in the century of AI learning tools

In view of recent developments of AI-based adaptive math learning systems like Squirrel, Aleks, Knewton Alta, or Math Academy: Does a human math teacher play any reasonable role, or will those systems ...
• 1,275
383 views

### Feedback from AI based learning to human based learning in math?

There are several AI based adaptive learning environments out there like Squirrel, KNewton or MathAcademy. Are there any studies which try to extract from the (big) data of those learning systems ...
• 1,275
1 vote
522 views

### Basic skill requirement suspension

Oregon appears to have suspended the "basic skills" requirement for graduation; see this. What will be the effect of this on the mathematical proficiency of the graduating class? Follow-up ...
• 2,238
264 views

### What are some decent apps for Hasse diagrams?

What options are out there for software that supports interactively constructing, editing, and manipulating Hasse diagrams? Their semantics is significantly constrained, so garden-variety Let’s-draw-...
406 views

### How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

I always showcase separate pictures of Triangle Inequality, and Reverse, to 16-years-old students in 1st class. I reshow pictures in 2nd class. I preachify Please remember these 4 inequalities. ...
• 139
1 vote
212 views

### Identifying Trigonometrical proofs

How can we identify trigonometrical proofs from geometrical proofs, do we have purely trigonometrical proof of Pythagoras theorem as claimed by two high school students ? https://www....
• 1,154
307 views

### Alternative to Quizlet live that supports latex formulas

With Quizlet Live you can play a flashcard game in classroom, where teams compete against each other. Since Quizlet doesn't seem to support rendering latex formulas, I am looking for a alternative ...
• 1,275
7k views

### Natural origins or learned habit: Why do students skip concepts before applications?

When teaching elementary mathematics, it takes a lot of time and effort to teach students that our goal is not to learn the examples, but to learn the concepts first, and then apply them to specific ...
302 views

### Is this a viable Calculus 1 question?

A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon rises straight into the air at a rate of 12 ft/sec. Is the ...
• 1,410
7k views

### When is it appropriate to warn about the difficulty of a subject?

I've been a TA across every class in the calculus sequence, under the assignment of professors with different teaching styles and curricula. It's often clear to me ahead of time when a certain subject ...
• 419
17k views

### To 17 year olds, how can I explain that two numbers with arbitrarily small difference are equal?

$|a – b| < ε, \forall ε > 0 \iff a = b$ resurfaces on standardized tests to 17 year old (y.o.) students, who can memorize and regurgitate the proof to earn full marks. But the glut of duplicates ...
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291 views

### How to motivate $x^n−y^n ≡ (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$, to 13 year olds?

You can safely presuppose that 13 year old (y.o.) students learned the Difference of Squares and of Cubes identities, before tackling this Difference of Powers Identity ($x^n – y^n$). The glut of ...
• 429
302 views

### How to explain why we can’t factor $x^n + y^n$ for all natural numbers n, to 13 year olds?

Michael Spivak, Calculus (4th edn 2008), p 13. I know this monograph is aimed at undergraduates (not middle schoolers), but this kind of multiple-part question resurfaces on standardized tests for 13 ...
• 429
14k views

### Do undergraduates struggle with δ-ε definitions because they lack a habit of careful use of their native language?

I transcribed this excerpt starting at the 22-minute mark, of Okinawa Institute of Science and Technology’s May 19 2020 podcast with Professor Tadashi Tokieda: For example, this is a bit too ...
• 429
1 vote
183 views

### Proof that volume of cone is 1/3 that of a cylinder [closed]

I am trying to verfy the formula for "cone volume" calculation. It is not clear why cone volume is 1/3 of a cylinder volume with the same bottom size and height. Is there any proof of the ...
17k views

### Why not think of derivatives as fractions?

Back in high school—back in the 1900s, as my sons say—when our calculus teacher was introducing the chain rule... $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$ ...he made a special point of ...
• 401
5k views

### "Real life" examples of limits of functions at finite points

This is more specific than this similar question on math.SE, since I'm not satisfied with the answers there. Question: Can you provide an interesting, natural and simple example of some physical/...
• 2,039
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, ...
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292 views

### Role of history of mathematics in contextual teaching and learning

To get a deeper understanding of mathematics conceptual teaching and learning is supposed to be a much better approach than factual teaching and learning processes. Since the conceptual approach is ...
• 1,154
1 vote
582 views

### How should an educator answer a student who asks "Can this theorem be deduced in other systems of set theory?"

If the educator decides to handle the situation by declaring that the question is beyond the scope of the course, then would it be fair to ensure that the course description and course syllabus ...
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957 views

### Better proof for a proposition when a proof is already available [closed]

What is a much better proof in mathematics, is it need to be a much more advanced one compared with the proof already available or a much simpler one? I think you can challenge a proof in two ...
• 1,154
7k views

### Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs

Background I'm informally helping a few students with college Calc 1. This isn't the first time I've aided people with calculus, and so they've sought me for help, though I don't consider myself to ...
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1 vote
236 views

### What would be a good pacing for teaching this calculus 2 course?

Next semester I'm going to lecture calculus 2 in an institution I just joined. However, when I had calculus 2 back then the syllabus was very different, it mainly covered several variable calculus up ...
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4k views

### What are some common errors and misconceptions about the Pythagorean Theorem?

I'm teaching a geometry class and want to ensure my students understand the most common errors and misconceptions related to the Pythagorean Theorem and its applications. I attempted an initial Google ...
1 vote
335 views

### Is there a particular reason why segment addition postulate and partition postulate are two different things?

I could be wrong but those two ideas sound the same, just that the partition postulate is more general. There is also the angle addition postulate. The segment addition postulate states that if three ...
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### Is there a way to extend the analogy that fractions means "x out of y" to show that fractions are also dividing?

When explaining fractions to my kids, I've used the analogy that $\frac{a}{b}$ means "you want $a$ out of every group of $b$ (of the thing you're finding a fraction of)." E.g. $\frac{3}{4}$ ...
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