# Questions tagged [mathematical-pedagogy]

for questions on general considerations and problems of teaching mathematics, i.e., issues specific to teaching mathematics yet relevant to various contexts and courses.

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### Comparison of two ways to introduce translation to 12-14 year olds

I consider pupils 12-14 years old, who are new to translation. On the other hand, they have been accustomed to placing points in a coordinate system, especially when studying relative numbers. In ...
4 votes
4 answers
360 views

### Why use the vague notion of "vector" when you have $\mathbb R^2,\mathbb R^3,\mathbb R^4,\ldots$?

I'm reading an introductory course on groups. In this course, the author illustrates concepts using the vectors of the plane. For example, "the set of vectors in the plane(or in space) is a group ...
2 votes
1 answer
205 views

### About a difficult exercise for 12 years pupils

You have to go from a point $A$ (start) to a point $B$ (arrival) by crossing a river $(d)$ and traveling as little distance as possible. Pupils first do a search by trying several paths $1,2,3,4$ and ...
2 votes
0 answers
133 views

### When Interpreting "If A, then B" as "A coupled with B" is rational?

It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. As we know, some ones interpret "if A, then B" as "A coupled ...
14 votes
6 answers
2k views

### A good book about mathematical thinking

I am a qualified mathematics teacher but I have left teaching because I could not tolerate the behaviour of students. Now I am a mathematics tutor and I love that I get to teach students who are eager ...
5 votes
7 answers
2k views

### Alternative approaches in basic mathematical operations

Sometimes awareness of alternative approaches of different origins may be helpful for students to improve their creative skills in mathematics. What would you suggest as alternative methods in basic ...
• 1,071
19 votes
4 answers
3k views

### Whether to tell students how difficult (you think) a problem is

Background: Most textbooks end a section with a set of questions ranked either by topic or by difficulty. A distinction is often made between "exercises", which are for directly practicing a ...
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9 votes
3 answers
562 views

### Best practices for Proof Revision/ Proof Portfolio?

I'm teaching a class small enough that I'm considering encouraging proof revisions (i.e. students taking a second try on proof based homework problems after getting feedback) for the first time. I'd ...
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2 votes
0 answers
123 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 ...
7 votes
1 answer
671 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 ...
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 ...
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5 votes
4 answers
346 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.&...
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1 vote
1 answer
204 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....
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2 votes
3 answers
267 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 ...
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24 votes
14 answers
16k 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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5 votes
4 answers
248 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 ...
• 1,071
2 votes
3 answers
951 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 ...
• 1,071
15 votes
15 answers
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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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 ...
13 votes
7 answers
1k views

### What are some good books on mathematical pedagogy?

I suspect that; just as one must "do" mathematics to learn mathematics, one must have practice teaching mathematics to become a great mathematics instructor. Still, a good book on ...
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26 votes
14 answers
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 ...
• 361
4 votes
0 answers
118 views

### Activities that encourage students to create or evaluate mathematical notations

I'm looking for references about activities that encourage elementary school students to create or evaluate mathematical notations. do you know any?
4 votes
0 answers
167 views

### Studies on the change in effectiveness of pedagogical practices over time

Are there any studies that have investigated this question? Why certain pedagogical practices that used to be effective up to a few years ago, may suddenly become less or even no longer effective? I ...
2 votes
4 answers
532 views

### A better example of a logical implication

(Updated) An example of a logical (material) implication that is commonly used is: "If it is raining outside, then the ground is wet." The problem with this example is that it could be ...
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8 votes
5 answers
880 views

### What should I call the "important" values of x?

When analyzing the functions $f(x) = \sqrt{x-5}$ $g(x) = \frac{1}{x-5}$ $h(x) = 2^{x-5}$ we know that it is useful to think about what happens at $x = 5$. For the function $f$, this logic will ...
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1 vote
0 answers
173 views

### What to cover on a first ordinary differential equations module?

I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books, have Laplace Transforms which is fine but I would not use LT to solve ...
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5 votes
0 answers
128 views

### Is it possible to learn some basic mathematics using an app?

I am interested in developing an app for students that are starting a grade career involving mathematics. It is a real problem that they start with almost no knowgladge of basic mathematics and there ...
4 votes
0 answers
207 views

### When dealing with sequences, should we teach students to start at 0 or 1?

The reason I prefer starting at 0 is due to a computer science background and also, I think it helps to start at 0 because there are certain reasons that demand it (in particular, combinatorics) and I ...
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3 votes
6 answers
1k views

### Is this motivation for the concept of a limit a good one?

tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
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10 votes
1 answer
288 views

### How to explain the concept "Without loss of generality" (through examples)?

This is not a precise question. I am curious to know how do you present to your students the (imprecise) concept of "without loss of generality", and how to use it correctly/incorrectly. I ...
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6 votes
3 answers
1k views

### Geometry in the Community College Curriculum

As many Americans know, the “traditional” high school sequence is: Algebra 1 Geometry Algebra 2 PreCalculus Calculus For those who take developmental education at the community college level, it ...
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4 votes
1 answer
222 views

### Elementary examples for non-reversible logical steps

While listening to recordings of Calculus $I$ lectures, I noticed that some students get confused between showing that "some object $x$ is a solution", and showing that "every (...
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12 votes
7 answers
3k views

### Does induction really avoid proving an infinite number of claims?

I am teaching calculus $1$ this semester, and I saw the following motivation for using induction by another teacher: Since we can't go over "manually proving" all claims $1,2,\ldots$ and ...
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-4 votes
2 answers
169 views

### Mathematics and love [closed]

This might seem a bit misplaced, but, is very relevant to mathematics education. The question is, how can I love someone, and teach students to love, or attempt and complete actions of love, through ...
-3 votes
2 answers
152 views

### Best natural language(s) for conveying, conceptualizing, teaching, understanding, and learning Probabilistic & Statistical concepts & theory?

English can be precise but it is rather 'flowery' and easily gets in its' own way. East-Asian natural languages like Mandarin, Cantonese, Korean, and Japanese are notorious for permitting the ...
0 votes
3 answers
191 views

### The two paradigms of seeing a functions

When we are first taught functions , we are typically taught of them as maps between real numbers and we taught to think of them mainly as a mapping between elements. It seems intuitive to take this ...
3 votes
1 answer
86 views

### Fitch Style Deduction in Non-Logic Classes

Has anyone experimented with using Fitch-style proofs as a teaching aid in courses outside of logic specifically and if so, how was the technique received by students?
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9 votes
1 answer
356 views

### The Interleaving Effect: How widely is this used?

I came across the idea of mixed up practice in Benedict Carey's book, How We Learn, in a chapter on the benefits of interleaving, particularly for learning Maths. For instance, in "blocked ...
4 votes
0 answers
647 views

### What are your experiences with Buck’s Advanced Calculus?

I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more ...
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7 votes
2 answers
175 views

### Are there pre-printed wall images that might engender understanding in a very young child?

I just read Moebius Noodles. (Thanks for the recommendation Sue). Part of the book talks about keeping images about math around the house. My child's 18 months. But I figure, why not now. It's passive;...
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3 votes
3 answers
847 views

### Walter Warwick Sawyer: How has reading his works changed your learning or teaching? [closed]

I recently worked my way through Walter Warwick Sawyer's book, Mathematician's Delight, which has opened my eyes to Maths. I used to fear maths, feeling I was incapable. Sawyer (among other authors) ...
4 votes
1 answer
253 views

### Is there a widely respected early childhood math curriculum?

If it's a good idea to work on reading and language from early childhood, I'd bet that it's a good idea to work on math and quantities too. I have an 18 month old. I've pretty much been winging it ...
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0 votes
6 answers
883 views

### Finding an analogy to explain the function of a binary adder

I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of adding something by ...
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11 votes
6 answers
4k views

### What should be memorized in math and what should be reference table?

I can never figure out what should be a memorization concept and what should be in a reference table. For example, in calculus, you are expected to memorize all the derivatives and integrals but in ...
• 945
13 votes
5 answers
2k views

### Use of language: "perfect square". is this useful or a hindrance? [closed]

I have recently been noticing the tendency to use the term "perfect square" when "square number" is really what is meant. Usually I have seen it at elementary level: introductory ...
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15 votes
4 answers
2k views

### Why do we use functional composition in the order we do?

Function composition means, roughly, taking the output of a function and applying it to the input of another function. If we define an object C to represent this operation, we could say $C(f,g) = f∘g$ ...
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5 votes
0 answers
323 views

### Word problems written in past tense, present tense, or future tense

Does anyone have extensive classroom experience regarding the best verb tense to use when writing word problems at an elementary or middle school level? I have been writing some lessons recently and I ...
• 379
6 votes
4 answers
845 views

### What is the best way to introduce Laplace transforms on an Engineering Mathematics course?

Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if ...
• 1,245
25 votes
2 answers
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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2 votes
1 answer
199 views

### How and what to teach on a second year Engineering Mathematics?

In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things in calculus also changed with the advantage of technology. ...
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