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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3 votes
5 answers
873 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 ...
10 votes
1 answer
109 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 ...
10 votes
3 answers
990 views

What is the pedagogical justification and history for using mnemonics to teach order of operations?

There was previously a question/rant here on MESE about why so many are still using the PEMDAS/BODMAS/BIDMAS/BEDMAS mnemonics to teach order of operations. The question was deleted (still viewable by ...
-2 votes
2 answers
133 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 ...
13 votes
3 answers
3k views

Tips to improve blackboard writing

During the internship I recently finished, I came to realise how important it is to have a good and structured use of the blackboard when teaching mathematics to 12-14 year pupils. Circumstances ...
42 votes
26 answers
11k views

Good, simple examples of induction?

Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
6 votes
3 answers
923 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 ...
7 votes
1 answer
397 views

How is math taught in elementary school in Finland?

I read on the internet that Finland has the best education system in the world and that in Finland, students are taught to love mistakes and that's how they learn and become smarter. I could not find ...
35 votes
23 answers
6k views

Imbuing a six year old with a sense of mathematical wonder

My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home ...
11 votes
3 answers
665 views

How to teach a student algebra who misses too much previous knowledge?

I am now tutoring a student in Grade 9, who falls behind in math study. He lacks the basic understanding of operations and inverse operations, and have trouble dealing with negative numbers and ...
5 votes
1 answer
145 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 (...
18 votes
11 answers
2k views

Books that every aspirant mathematician should read

I am a student and I would love to become a research mathematician one day. So I would like to ask you---experts in mathematics but also in education---what are some influential ($\star$) books that ...
12 votes
8 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 ...
61 votes
13 answers
9k views

How to get past the "mystique" of Maths

This question is primarily discussing maths education for adult learners, on courses teaching engineering mathematics at an undergraduate level. These students generally never set out specifically to ...
32 votes
14 answers
2k views

Revisiting topics from previous courses [closed]

I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.) ...
-4 votes
2 answers
123 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 ...
10 votes
2 answers
155 views

How to incorporate optional higher level mathematical content in an Engineering Maths course?

Our department teaches two very large first-year "Mathematical Methods" courses (600-ish students) to Engineering students. The syllabus is dictated by their (future) needs and covers a huge array of ...
-1 votes
3 answers
583 views

Could students learn a lot more from school if they're only taught number theory until way later?

According to https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html, when students get taught a concept when they're so young, they're ...
0 votes
3 answers
174 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 ...
16 votes
2 answers
828 views

Let P be a polygon

I've encountered the following misunderstanding. I pose a question (to undergraduates in the U.S.), for example: Let $P$ be a polygon of $n$ vertices. Is it true that every triangulation of $P$ ...
3 votes
1 answer
77 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?
93 votes
18 answers
18k views

Unique candidate that fails

In the comments to David Speyer's answer here, he points out that "the distinction between 'if there is a formula, it is this one' and 'this formula works' is subtle." Does anyone have any simple, ...
22 votes
16 answers
8k views

Why is it possible to teach real numbers before even rigorously defining them?

In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach ...
9 votes
1 answer
305 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 ...
9 votes
3 answers
434 views

Obtaining printed copies of the textbook series Unified Modern Mathematics

I'm seeking the textbooks that were released in the 1960's call Unified Modern Mathematics. I'm aware 3 parts of the course exist online but I would like to use them in hard copy form. I find these ...
7 votes
7 answers
3k views

How do you handle the frustration of having to GRADE student exams / homework?

A math student may write very long and detailed answers, just because he or she does not know what to look for, for example in Geometry proofs. Or - a student may just write an arbitrary step without ...
12 votes
8 answers
745 views

How best to explain the logarithm to the mathematically naive?

Suppose you need to explain "What is a logarithm?" to an intelligent but math-phobic adult (or a student well-prior to her introduction to logarithms).1 I have used base-$10$, saying that, essentially,...
4 votes
0 answers
178 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 ...
8 votes
4 answers
664 views

Exponential & logarithm in a high school calculus class

So recently I was teaching high school calculus to a high school class and I was wondering about the pedagogically best way to make students actually understand why the derivatives of the exponential &...
13 votes
4 answers
338 views

Helping students use constructions in geometry

I work as a teaching assistant in a high school and geometry gives most students headaches. I emphasize understanding the problem by constructing lines and using elementary properties to arrive at the ...
6 votes
2 answers
168 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;...
3 votes
3 answers
823 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
235 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 ...
13 votes
14 answers
3k views

What is the best way to intuitively explain what eigenvectors and eigenvalues are, AND their importance?

How can we break down the complexity of eigenvalues/vectors to something that is more intuitive for students. I feel like the proofy way isn't a good intuitive representation of the mechanism that ...
0 votes
6 answers
839 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 ...
10 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 ...
8 votes
1 answer
273 views

How can I improve my concept map?

I've decided to create a concept map of a chapter I covered in a textbook, it's about basic set notation. What I want is suggestions on how to improve the presentation of the map. It seems quite ...
14 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 ...
9 votes
4 answers
436 views

Teaching mathematics and its charms to non-mathematicians

I am teaching English in Japan and I have a student who speaks English well, and to keep up his level, in our weekly lessons would like to learn some subjects related to my degree in mathematics. I ...
6 votes
2 answers
404 views

Do my students know elementary algebra; do they just use online calculators or external help; and is this ok?

Background I know the question in the title is very broad so I will try to explain it as succinctly as I can. Half my time is spent on as a researcher on didactics, while the other half is devoted to ...
13 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 $C(f,g) = f∘g$ ...
24 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 ...
6 votes
4 answers
772 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 ...
5 votes
0 answers
121 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 ...
2 votes
1 answer
188 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. ...
19 votes
8 answers
709 views

Hands on activities for a college history of mathematics course

I will be teaching a course in history of mathematics to juniors/seniors who are math and math education majors, many future school teachers. It should include highlights from antiquity to early 19-th ...
6 votes
0 answers
107 views

Maximize retention

I tutor high school math students. Students often struggle with a problem they had completed few months prior. Like any skill, it's natural to forget what you learned after a while. As high school ...
9 votes
3 answers
679 views

Advantages of using calligraphic or script letters

In which areas of mathematics is it traditional to use calligraphic letters, such as $\cal{ABCDEFG}$, or script letters, such as $\scr{ABCDEDG}$, and is there a pedagogical advantage to doing so? (...
1 vote
0 answers
371 views

Is it more efficacious, productive to jump to perusing full solutions — before and without attempting to solve problems?

Too many students lack the luxuries of time and effort to mull exercises and problems. They must juggle MULTIPLE jobs to pay exorbitant tuition fees. Single parents or adult learners must prioritize ...
46 votes
24 answers
19k views

How to explain Monty Hall problem when they just don't get it

Talking to some friends, I was asked to explain the answer to the Monty Hall problem (see also here;) .... they were having some trouble because whoever explained it to them didn't do a very good job. ...

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