All Questions
3,686
questions
6
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2
answers
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views
Any examples of calculus sequence that deemphasizes calculation tricks?
I'm considering creating a series of classes that explore deeper ideas in calculus without overemphasizing the various computational tricks used in integration and differentiation.
My vision is ...
- 251
2
votes
0
answers
110
views
Which Books Comprised The Houghton Mifflin Modern Mathematics Series
Around the 1960s a collection of math books by authors such as Mary P. Dolciani, William Wooton, Edwin F. Beckenbach, Ray C. Jurgensen etc. started to appear which were part of a series of books ...
- 173
1
vote
0
answers
78
views
"Tools" (literarily) for solving linear or quadratic equations
Since a few weeks, I teach as a tutor (not from that school) a support course in a German 9/10 class. I quickly noticed a horrible lack of basics. (Partly based on just different names - I had to ...
- 111
5
votes
5
answers
202
views
Loaning students calculators during exams
Context: I am an associate professor at a small liberal arts institution in the US.
I find in my introductory business math course that students sometimes fail to buy a calculator for the course, ...
- 1,399
7
votes
1
answer
596
views
Bridging the gap between students' intuitive problem-solving abilities and expressing ideas through formal writing
Seeking guidance on how to assist students who possess a solid grasp of problem-solving concepts, allowing them to intuitively arrive at solutions, yet encounter difficulties when it comes to ...
- 71
2
votes
1
answer
179
views
What age group is this puzzle appropriate for?
I have created the following simple tiling puzzle for children.
The puzzle consists of 9 tiles shown below (4 pieces of type A, 4 pieces of type B and 1 piece of type C):
The goal of the puzzle is to ...
2
votes
2
answers
2k
views
Where can I find new types of problems regarding graduate level mathematics?
Though not an undergraduate student , I just wanted to know where can I find hard new types of problems regarding the problems in graduate level mathematics. As per my information , standard books ...
- 159
9
votes
5
answers
2k
views
Requesting a Polynomial System of Equations
I am teaching a course in commutative algebra, and it includes a project where the students research on a particular topic, solve a small problem and present it to the class.
I usually give my ...
2
votes
1
answer
159
views
Help finding a YouTube channel for Linear Algebra
I remember watching a youtube playlist on a video about equation of a plane and about vectors in 3D. The video was fully animated in 3D and had good sound effects too. I feel like it had a clock ...
2
votes
2
answers
245
views
How to motivate ‘unless’ = ‘if not’, with etymology?
My undergrads still reckon Nelson Lande’s 2 motivations for ‘unless’ = ‘if not’ (below) too abstract, formalistic! They crave simpler intuition! Thus I want to proffer a 3rd motivation, this time with ...
-1
votes
1
answer
37
views
Ratio in Fractional Form [closed]
I have 3 puppies and 2 piglets.
Can you that ratio be written in this fractional form 5/2?
If yes, how would you describe this ratio as 5/2?
- 7
6
votes
3
answers
236
views
What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?
So in an advanced mathematics course for engineers, there are often problems of the type:
Prove claim A.
Given equation A, show that you can obtain equation Z.
I am frequently faced with a problem ...
- 737
12
votes
11
answers
4k
views
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 ...
- 771
0
votes
5
answers
250
views
Are logic textbooks wrong to teach ‘unless’ = ‘or’? How to motivate?
Most times in English, it’s ungrammatic to swap ‘unless’ with ‘or’! Aren’t the textbooks below wrong?
My undergrads can master Truth Tables, but still fail to intuit why ‘unless’ = ‘or’. How to ...
10
votes
12
answers
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 ...
- 363
0
votes
2
answers
204
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 ...
- 159
1
vote
1
answer
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 ...
- 159
1
vote
1
answer
101
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
-4
votes
2
answers
205
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 ...
- 1,935
5
votes
4
answers
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,...
- 4,245
8
votes
3
answers
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 ...
- 6,121
-4
votes
1
answer
104
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 ...
- 1,935
3
votes
11
answers
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 ...
- 945
1
vote
1
answer
86
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 ...
- 10.7k
6
votes
11
answers
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 ...
- 89
2
votes
3
answers
164
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 ...
- 1,935
6
votes
3
answers
147
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 ...
- 61
9
votes
6
answers
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
2
answers
72
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 ...
- 957
4
votes
4
answers
327
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.&...
- 6,121
3
votes
0
answers
69
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 ...
- 4,606
0
votes
1
answer
102
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 ...
- 101
13
votes
18
answers
5k
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 "...
- 373
1
vote
1
answer
106
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 ...
- 19
9
votes
8
answers
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, $|\ldots| < \varepsilon+\varepsilon+\varepsilon$ ...
- 139
8
votes
2
answers
150
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 ...
- 1,128
13
votes
5
answers
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 ...
- 1,128
2
votes
1
answer
353
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 ...
- 1,128
1
vote
3
answers
497
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 ...
- 2,102
4
votes
2
answers
133
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-...
- 141
2
votes
4
answers
380
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
1
answer
196
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....
- 945
3
votes
1
answer
93
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 ...
- 1,128
33
votes
17
answers
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 ...
- 1,019
2
votes
3
answers
259
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 ...
- 1,084
28
votes
3
answers
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 ...
- 389
5
votes
13
answers
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 ...
- 419
0
votes
4
answers
273
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 ...
- 419
0
votes
3
answers
282
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 ...
- 419
14
votes
5
answers
13k
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 ...
- 419