The teaching tag has no wiki summary.
17
votes
16answers
927 views
Examples of mathematical induction
What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
5
votes
1answer
96 views
Who came up with the arrow notation $x \rightarrow y$?
I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it?
Each map needs both an explicit domain and an explicit codomain (not
just a domain, as in ...
12
votes
5answers
193 views
Issues with text problems
When I tutor, I often see people who kind of know the stuff they cover in school at the moment and succeed at straight problems like:
Find the derivative of $f(x) = \frac 12 x^2$
But when it ...
5
votes
2answers
98 views
certain examples of fields of fractions
Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of ...
3
votes
0answers
74 views
Motivating questions for some topics in undergraduate calculus
Being a grad student I'm going to teach a whole class for the first time the coming summer and I'm looking for some motivational problems which I could use to introduce different topics. In other ...
-2
votes
2answers
77 views
“the product of the factors” versus “the factors of the product”
Could somebody please compare and contrast the meanings of the two phrases:
"the product of the factors"
and
"the factors of the product."
In terms of expressing possession. Thank you.
-8
votes
0answers
115 views
Berkeley Ring Theorist Solves 48 ÷ 2(9+3) [closed]
Possible Duplicate:
What is 48÷2(9+3)?
In the youtube video Berkeley Ring Theorist Solves 48 ÷ 2(9+3) he says @1:38 there is not a well known convention about performing the operations ...
1
vote
1answer
66 views
“unexpected” isomorphism between finite posets?
The set of all divisors of a square-free number, partially ordered by divisibility, is trivially isomorphic to the set of all subsets of the set of prime factors, partially ordered by inclusion.
Are ...
10
votes
11answers
245 views
Explaining Horizontal Shifting and Scaling
I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.
For example, ...
1
vote
2answers
144 views
What is an effective way to teach children the Cartesian coordinates?
My nephew is preparing for a $4$-th grade state test. They need to learn topics like reflection about $x$ or $y$-axis of a point( say $(3,5)$ reflected about the $y$-axis).
I tried to explain but ...
-1
votes
2answers
200 views
High school math definition of a variable: the first step from the concrete into the abstract…
variable: A symbol used to represent one or more numbers.
High school students are justifiably confused by the two distinct concepts:
a variable as something that “varies” in an
expression, such ...
0
votes
1answer
129 views
When my teacher gives me a question involving summation notation, do they expect us to calculate it by hand?
Assuming we don't have a calculator that can do summation notation. My class is not up to summation yet, but I'm asking a question involving this concept because I'm not all that experienced using it. ...
6
votes
5answers
167 views
trivial but non-trivial equivalence relations
Define a binary relation $R$ on a set $A$ by saying $xRy$ iff $x$ and $y$ have the same whatever.
"Whatever" is of course some specified function on $A$.
This is a "trivial" equivalence relation: ...
0
votes
1answer
235 views
Why they dont teach Fundamental Theorem of Algebra in High School? [closed]
I am currently in AP Calculus BC and one more year to go, I have heard about Fundamental Theorem of Algebra several times, and with the resources that is out there today I tried to search and study ...
2
votes
2answers
103 views
Opinions on foundational math materials to teach 8th grade, 9th grade kids at a Summer Camp
I have been asked to teach mathematics/physics to a few 8th grade/9th grade kids for a summer camp. I have been thinking about it and I realized that I could go about it in two ways: One of the ways ...
1
vote
1answer
71 views
learning maths for statistics
Apologies if I have posted this in the wrong place first off.
My work has taken me into a unexpectantly large amount of statistics. In order to really understand what I am doing I need to understand ...
6
votes
1answer
116 views
Implicit use of the Implicit Function Theorem when finding tangent lines to polar curves.
Recently I found myself having to teach students how to find the slope of a tangent line to a curve in $\mathbb R^2$ given in polar coordinates by the equation $r = f(\theta)$. The students' calculus ...
8
votes
2answers
231 views
Infinite Series: Fibonacci/ $2^n$
I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)
In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... each term ...
17
votes
8answers
1k views
Why do introductory real analysis courses teach bottom up?
A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
6
votes
3answers
293 views
How to justify small angle approximation for cosine
Everyone knows the picture that explains instantly the small angle approximation to the sine function (as defined by the parametrisation of the unit circle): "what's the length of that arc?" "See how ...
5
votes
0answers
141 views
Connecting finite automata and regular languages in teaching/applications
I am considering giving a presentation to middle schoolers, aged about ten to fourteen, about finite automata and regular languages.
Average American students have no problem with uses of the ...
2
votes
6answers
224 views
Explain for students: Why does 0 mod n equals 0 (zero)?
I told my students that the mod operator basically gives the remainder of division, so upon seeing:
0 mod 10
Some students (apparently) reasoned that, "10 goes into 0 zero times and there are 10 ...
8
votes
4answers
481 views
What is the best way to develop Mathematical intuition?
I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the ...
7
votes
2answers
179 views
A Book of Neat Theorems for Laymen
I'm looking for reading assignment ideas for my students. I'd like them to read up on results in mathematics in layman's terms. For example, the Monty Hall problem, or Borsuk Ulam as the "Ham ...
0
votes
1answer
108 views
Taxonomy for math exercises (K-12 highschool) [closed]
I would like to build an online collection of math exercises that can be indexed and cross-referenced. So I need a way to characterize an exercise somehow.
Example: $\sin^2x + x =0$
Does it ...
18
votes
5answers
437 views
Elementary problems with group theoretic solutions
I am helping a friend develop a course in abstract algebra that is designed for high school students who have no knowledge of abstract algebra or any real exposure to formally rigorous mathematics. To ...
2
votes
0answers
90 views
Books which help to develop logic and brain sharpening for those who fear Maths.
My aim is to get books which help in developing logic, and thus induce brain sharpening.
I am NOT looking for books involving high level theorems, proofs, integral calculus etc.
I saw this thread: ...
8
votes
3answers
235 views
Motivation for solution to constructing a set of 1983 distinct integers such that no three are consecutive terms of an arithmetic progression
Problem:
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $100,000$, no three of which are consecutive terms of an arithmetic progression? (Source: IMO 1983 Q5)
...
1
vote
0answers
138 views
How can I use an abacus to teach concepts to a toddler?
My 18-month old son got a $10\times10$ abacus as a Christmas present, and he enjoys it as a toy. I'm fine with him just playing with it, but I don't want to miss an opportunity to introduce ...
4
votes
3answers
124 views
diversity and teaching
I recently attended a discussion about interviewing for math jobs, and apparently a question that is coming up frequently is something like this:
"We have a culturally diverse student body. How does ...
5
votes
1answer
99 views
Elementary arguments concerning the stereographic projection
How does one give a proof that is
short; and
strictly within the bounds of secondary-school geometry
that the stereographic projection
is conformal; and
maps circles to circles?
6
votes
3answers
314 views
Motivation behind the definition of GCD and LCM
According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the ...
8
votes
2answers
197 views
Why the emphasis on Projective Space in Algebraic Geometry?
I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed.
Why does Miranda (and from what little ...
1
vote
4answers
127 views
Two questions related to probability theory and pedagogy
A housemate of mine and I disagree on the following question:
Let's say that we play a game of yahtzee. Of the five dice you throw, two dice obtain the value 1, two other dice obtain the value 2, ...
1
vote
0answers
75 views
weakly locally one-to-one?
Is there any standard name for this concept that is weaker than local one-to-one-ness?
In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$.
Or, if you like: In ...
2
votes
1answer
113 views
what is teaching kids the rules and exceptions in multiplication called?
I recall reading a website quite some time ago about the rules and exceptions of multiplication with regards to teaching children. For instance: The result of multiplying any number times 9 will have ...
2
votes
2answers
92 views
Advice for Calculus Tutoring
I am tutoring a friend in calculus. Right now, she is working on finding relative maxima and minima as well as Rolle's theorem. While she gets how to find relative maxima and minima she does not get ...
1
vote
0answers
88 views
Lesson plan for teachers
I teach mathematics at school level. Lesson plans are soul of any lesson that a teacher takes in a class. I want to create lesson plan on different mathematics topic as per the level of the syllabus ...
19
votes
6answers
855 views
Do online lecture recordings hurt or help math students at university? [closed]
Continuing with my series of soft questions on teaching practice:
My university uses a system whereby all lectures (given via computer slides or hand-writing on a sort of overhead projector called a ...
13
votes
2answers
402 views
“Best practice” innovative teaching in mathematics
Our department is currently revamping our first-year courses in mathematics, which are huge classes (about 500+ students) that are mostly students who will continue on to Engineering.
The existing ...
5
votes
3answers
170 views
Good lecture optimization problem involving $\ln x$ or $e^x$
I am teaching a Calc 1 of sorts, like a slightly easier version of Calc 1 with no trig. I want a good optimization/practical problem to do in lecture that involves $\ln x$ or $e^x$, to combine review ...
14
votes
6answers
389 views
Exciting games and material to motivate children to math
We are a group of people trying to motivate children, especially living in the countryside, to science and math. We have different activities with children such as doing scientific experiments and ...
5
votes
3answers
214 views
The Power of Taylor Series
I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational ...
4
votes
1answer
139 views
Algorithm for keeping a concrete version of Euclid's argument simple
(My actual question is at the very bottom of this posting.)
Suppose you're teaching a course in mathematics-for-liberal-arts majors and it's the last math course they'll ever take. It has almost no ...
13
votes
4answers
317 views
Fun math for young, bored kids?
For 6 months, I'll be organizing, as part as my volunteer work in an NGO, math classes with small groups (~10 students, aged 16 or 17). These classes are not compulsory, but students willing to stay ...
4
votes
2answers
180 views
Interesting but elementary properties of the Mandelbrot Set
I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
2
votes
0answers
109 views
Have Changes in Applications Made Linear Algebra More Central/Urgent?
In the days when my father taught civil engineering (some decades ago), mathematical applications seemed to be mainly "scientific." (This was the "space age.) Hence the most important branch of ...
8
votes
2answers
256 views
Is there any toy for learning algebraic manipulation of fractions?
Is there any toy for learning algebraic manipulation of fractions? If you don't know of any, how would you design one?
What I'm imagining is something similar to a Rubik's cube whose manipulation ...
31
votes
18answers
2k views
What is your favorite application of the Pigeonhole Principle?
The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item.
I'd like to see your favorite ...
4
votes
2answers
253 views
Simpson's Rule and other Newton-Cotes Formulas
I am curious about the value of Simpson's rule (also called the parabolic rule or the 3-point rule) for approximating integrals. The calculus text I am now teaching from uses this rule any time an ...
