All Questions
3,849
questions
1
vote
1
answer
143
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
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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
165
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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 ...
- 1,283
13
votes
5
answers
4k
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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 ...
- 1,283
2
votes
1
answer
389
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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 ...
- 1,283
1
vote
3
answers
524
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 ...
- 2,248
4
votes
2
answers
330
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
410
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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
214
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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....
- 1,144
3
votes
1
answer
352
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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 ...
- 1,283
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 ...
- 1,102
2
votes
3
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303
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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
31
votes
4
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 ...
- 419
6
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 ...
- 439
0
votes
4
answers
294
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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 ...
- 439
0
votes
3
answers
304
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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 ...
- 439
15
votes
5
answers
14k
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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 ...
- 439
1
vote
1
answer
190
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 ...
27
votes
7
answers
17k
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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
13
votes
6
answers
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/...
- 2,038
25
votes
14
answers
17k
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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
5
votes
4
answers
305
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 ...
- 1,144
1
vote
2
answers
651
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 ...
- 352
2
votes
3
answers
960
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 ...
- 1,144
15
votes
15
answers
7k
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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 ...
- 291
1
vote
1
answer
239
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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 ...
- 113
8
votes
3
answers
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 ...
- 2,384
1
vote
2
answers
401
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 ...
- 1,068
5
votes
4
answers
2k
views
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. $\frac{3}{4}$ ...
- 415
11
votes
1
answer
1k
views
Does there exist a (statistical) topology induced by students on the space of algebraic formulas? :)
It's kind of a serious question even if the title seems silly.
As math educators, we all know that students link together different algebraic expressions thinking that they mean the same thing, e.g.
\...
- 805
0
votes
2
answers
147
views
Sourcing and verifying calculus applications
There are many questions on this site about specific (or not-so-specific) applications of calculus to the "real world". However, one issue I've noticed in using textbooks for this purpose ...
- 5,996
5
votes
4
answers
426
views
Antiderivative of $1/x$, with or without absolute value?
Many textbooks include $\int \frac{1}{x} dx = \ln |x| + c$ in their list of antiderivative formulas, with the absolute value. Correspondingly, they do the same with the antiderivative of $\tan x$ or ...
- 51
1
vote
3
answers
268
views
Plainly by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{dodgerblue}{|\vec{b}| - |\vec{r}|}$?
Yearly, I teach 16 year olds this diagram beneath (improvement of this) that reappears on standardized tests IN BLACK AND WHITE below with different lengths, letters, and orientation. Tests require ...
user22729
9
votes
7
answers
2k
views
Is there a resource for learning to read mathematical notation/equations/formulae?
Ideally, I am looking for an online resource. But a book or any other would help already.
Background: I am a senior teaching assistant in the field of business and statistics. Most of my students have ...
1
vote
2
answers
170
views
What can be considered as common knowledge in an online Mathematics course?
As I prepare to instruct an online Mathematics course next year, I'm currently writing the syllabus. Right now, I'm writing about student participation in the Learning Management System, which takes ...
- 11
2
votes
5
answers
995
views
Geometrical verifications for Algebraic formulae
What is the importance of using approaches related to Geometric Algebra in teaching,is it only useful when introducing Algebra to the students or can it be helpful to improve creative skills in ...
- 1,144
-1
votes
2
answers
165
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I'm in dilemma while solving arithmetic problems [closed]
I'm competitive exam student learning Quantative aptitude what should i choose over solving more questions and skipping the one i can't solve or spending hours on one question till i solve it and then ...
- 7
6
votes
5
answers
6k
views
What benefit is there to obfuscate the geometry with algebra?
Consider:
In a right triangle:
sin(2x + 4) = cos (46)
What is the value of x?
The question above is from standardized tests for a geometry course. If my goal is to have students understand ...
- 1,068
1
vote
4
answers
222
views
Naming the procedure of converting the place values of digits
Let's say I have the numeral 2.263,3 thousands, and convert it to 2.263.300 units.
How do we describe what I have done to the numeral regarding units ?
I know it has to do with the place values of the ...
- 147
-5
votes
1
answer
123
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Does the "Middle School Mathematics domains" refer to (I) through (V) topics?
Does the "Middle School Mathematics domains" on page 3 of https://www.ets.org/content/dam/ets-org/pdfs/praxis/5164.pdf refer to the the following 5 topics/categories?
(I) Numbers and ...
- 93
4
votes
2
answers
163
views
The key didactical ideas on mathematical modelling?
This question concerns teaching teachers who often already teach mathematics, but are now studying to get a formal qualification for it, and hopefully some more competency as well.
What are the key ...
- 7,475
-1
votes
1
answer
127
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Why do problems should be solved by pen and paper before coding? [closed]
I heard that before you can handle data automatically, you have to know how to handle it manually.
Why is it impossible to find a problem and build some machine learning algorithm to find a proper ...
- 17
0
votes
1
answer
71
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Seeking References on Deterministic and Stochastic Phenomena Suitable for High School Students
Can anyone recommend good and didactic references that delve into the dualism between deterministic and stochastic phenomena? Ideally, I'm seeking materials that provide a conceptual explanation along ...
- 2,384
7
votes
6
answers
2k
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Is 'For all $x$' an abuse of language in math?
I chose to ask this question on MESE because I think it's not about mathematics per se but more about how it should be communicated.
Quantified statements in mathematics are often written for ...
- 447
4
votes
3
answers
430
views
Graphing lines by finding integer points
I always say that the most difficult part of graphing or plotting points is labelling your axis/es. In the case of plotting the graph of a linear equation with integer coefficients in 2 variables it ...
- 967
3
votes
2
answers
109
views
Utillizing Lakatos' "Proofs and Refutations" in Secondary Education
These days I am reading Imre Lakatos's Proofs and Refutations and I can't stop thinking how one could utilize it in the classroom (mostly high school). Some stray half-baked ideas I have had so far ...
- 1,120
9
votes
2
answers
2k
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Explaining Sigma-Notation
I attempted to introduce the summation notation $\Sigma$ to my students. The notation was unfamiliar to the students beforehand. I worked through many examples with them, but for most of them, working ...
- 193
1
vote
1
answer
509
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Is there any university or college in any country where failure and dropout rates in Calculus are not so high?
Calculus is a foundational mathematics course that is often seen as a bottleneck for STEM majors. However, it is also a course that is notorious for its high dropout rates. In the United States, for ...
- 2,384
4
votes
3
answers
290
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Looking for web app resources for symbolic Gaussian elimination
I am looking for a web app software that takes step-by-step directions from a student to perform the linear combination operation on a matrix with symbolic coefficients (as opposed to just numbers). ...
- 1,410
0
votes
4
answers
386
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Why are negative numbers introduced before quotients in the real number subsets?
This is a question regarding why the order of the real number subsets commonly used in the mathematics community is such:
$$ \mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R} $$
Here ...
- 29