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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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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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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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 ...
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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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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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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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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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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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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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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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) ...
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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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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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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 ...
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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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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
user155
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Why's math way more puzzling, abstruse than law and medicine? [closed]
Most students find math unfathomable, labyrinthine by the time of univariate integration (Reddit). Even overachievers – who ace undergraduate math without studying – will eventually be convoluted by ...
user155
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How to balance between teaching to the standardized test vs understanding?
For people teaching high school standardized curriculums such as AP, IB, or A-Levels, how do you find the balance between preparing students for the standardized test compared to ensuring they ...
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Teaching Solving Linear Equations before teaching evaluating expressions
Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have ...
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Why the fear of polynomial long division?
Why do people think long division of polynomials is complicated ?
I heard this expressed recently and it seems like an odd sentiment. For me, synthetic division is complicated and totally adhoc ...
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Which function is more elementary: Constant function or identity function? [closed]
I am not a mathematics teacher, I just want to know which function is more elementary and is more reasonable to be learnt first:
Constant function (f(x)=1)
...
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How can I encourage students to show up for exercise classes?
I am doing a maths PhD and naturally that involves leading exercise classes for undergraduate students. The idea is that the students just show up and work through the problem set and I'm there to ...
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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 &...
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Can you talk about (the rest of the) field axioms when the operations are not closed? [closed]
Note: Updated based on this.
In my course, my instructor posed the following exercise:
Let $S$ be the subset of $\mathbb R^n$, $S=\{(a_1,a_2,a_3...a_n) | a_2 = \pm a_1, a_3=...=a_n=0 \}$. Define ...
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The value of making students copy your derivations exactly
I have a hypothesis that I have not yet implemented, and am seeking guidance before I do. I have always taught algebra (and above) in such a way that the students understand the derivations. I check ...
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What studies exist, comparing the efficacy of exercise sheets with or without worked solutions?
I've been tutoring mathematics at university level for over 10 years, and one of the more common requests from students is worked solutions for sheets of exercises. Most educators I've worked with ...
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Why are math test scores dropping in America? Lack of student responsibility movement [closed]
Someone asked me this question today. Why do you honestly believe that high school math test scores in America (particularly the United States) are low compared to other countries? I thought about ...
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Is there a book or Web site that explains how to use and teach the Singapore math block model?
I have been examining sites that illustrate the use of the block model for solving problems. I am an adult with a degree in math and I have an interest in seeing how math can be taught.
From what I ...
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Making the leap from Pre-Calculus to Calculus
This question is targeted at teachers who taught both low and high level mathematics. I have a group of students that I'm currently teaching precalculus and they seem to be doing really well in all ...
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Creating an Engaging Class Atmosphere
I would like to start out next year by creating a classroom community. This year I noticed a lot of burnout towards the second half of the year. Basically, I want to see what kinds of games/activities ...
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Textbooks with solutions and catering to different circumstances
Questions: Are we really taking students into account FULLY when writing textbooks for various areas? Also, are we being unintentionally elitist or dismissive when neglecting to take a more humble ...
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Objectives for group work in undergraduate pure maths
Whether we are preparing undergraduates for research in industry or academia effective collaboration is an important higher skill. I think there are two aspects to this in mathematics - thinking ...
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How many of "The Seven Laws of Teaching" are still relevant for teaching maths today?
Wikipedia shows that in 1886 John Milton Gregory outlined his "The Seven Laws of Teaching"; asserting that a teacher should:
Know thoroughly and familiarly the lesson you wish to teach; ...
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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 ...
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Why are most college level math textbooks black and white only?
Why are higher level math textbooks almost completely black and white? I can't think of any math textbooks on a subject more advanced than calculus that uses colors. Edited to add that the comments by ...
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Which works better for learning 6 * 7 = 42: saying "six sevens are forty-two", or "six times seven equals forty-two"?
When memorizing and recalling the times table, I learned to say "six sevens are forty-two", and always wondered what it would be like to learn to say "six times seven equals forty-two&...
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Why do we typically only teach high-school students affine transformations of elementary functions?
A standard pre-calculus curriculum consists of the study of elementary functions: Polynomials, rational functions, (circular and hyperbolic) trigonometric functions, exponential functions, their ...
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Worth introducing a mandatory short module on $\LaTeX$ into a mathematics degree?
On Mathematics StackExchange for a particular instance, it is highly recommended that questioners mark up their questions using $\LaTeX$. A surprising number of mathematicians and student ...
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