Questions tagged [reference-request]

A reference request is a request to be provided with (links to) documentation, official papers, and specs related to one or more specific algorithms or mathematical procedures, to provide a trusted base for what's being said or written.

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56
votes
15answers
7k views

Is there a virtue to learning how to compute by hand?

I have been professionally tutoring a wide range of students (from elementary school through graduate school) for many years. Most of them are from the United States. I generally focus on helping my ...
38
votes
5answers
3k views

Effects of early study of advanced books

Context: There was recently a question on Math.SE: Inferior to Other Younger and Brighter Kids which starts as follows: I'm a high school student (Junior/Grade 11) and I'm currently studying ...
34
votes
26answers
2k views

What are some great books for exploring mathematics? (not kids' books and not textbooks)

People often think of math as facts and procedure - dry stuff. But it is so much more, even at basic levels. What books about mathematics have you been inspired by? There are some real treasures out ...
30
votes
7answers
15k views

What are the comparative advantages of open-book versus closed-book exams?

I would like to know the advantages and disadvantages of open-book exams as compared to closed-book exams, particularly in standard undergraduate courses like calculus or linear algebra. My practice ...
29
votes
4answers
3k views

Books about elementary mathematics written like a good undergraduate textbook

I've never seen any really good expositions of elementary mathematics (middle school or earlier). A good college-level textbook, written for people with an interest in mathematics, reads like a novel ...
25
votes
6answers
2k views

What is the quantitative data on effectiveness of "modern" teaching methods?

What research has been done on how much and in what circumstances various non-lecture types of teching are effective with regards to student knowledge and performance? Meta/review studies preferred ...
25
votes
2answers
1k views

Students who know high-level math before going to college

There is a high school in the city I live in which has some high-level math courses in their curriculum. It's a special math class mentored by some university lecturers, and the children basically do ...
24
votes
7answers
4k views

When should we first teach variables in school math? And how?

From a pedagogical point of view, when is the "right" moment to introduce letters and variables to school children? Let's say we taught arithmetic, the four operations, did computation exercises, or ...
24
votes
5answers
2k views

Should we "program" calculus students, like the physicists seem to want us to?

If it is true that we first learn formalism...how to do things that we don't understand, should we regard teaching students mathematics as programming dumb machines with formal rules (to the greatest ...
24
votes
2answers
1k views

Can students tell the difference between the "definition if" and the "theorem if"?

The word "if" is used in two meanings in mathematics: Definition. A topological space is compact if every open cover has a finite subcover. Theorem. A topological space is compact if it is ...
24
votes
3answers
2k views

The impact of dyslexia on learning mathematics, and available resources

I have always loved the beauty of mathematics and physics. However I'm severely dyslexic and find it hard to keep numbers in my head, any more than 4 numbers at a time and they melt together and lose ...
23
votes
5answers
520 views

Any support for mathematical "learning types?"

Back when I was an undergrad calculus TA, I participated in a general TA training class. We were taught to be mindful of different "learning types," such as visual learners, audio learners, and ...
22
votes
7answers
3k views

How do you coach students who often make small errors?

Some students are prone to making small calculation errors. Not errors in understanding, but errors like adding or multiplying integers incorrectly, or dropping a negative sign. Unsystematic errors in ...
22
votes
3answers
382 views

Mathematic reasoning in nonEnglish/non Western languages

I am teaching in an Eastern Asian environment (precisely, teaching Mathematics using English in Korea, with Asian students) and I figured out that my reasoning is a lot based on my language ...
21
votes
7answers
2k views

Good examples of functions defined as definite integrals of elementary functions?

I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions. Two examples of such functions are $$ ...
21
votes
6answers
701 views

Becoming a better instructor: where to start?

I just finished a PhD in math at a top department, but not one that placed much emphasis on graduate student teaching. Grad students here teach only as TAs, and the training is minimal. I got great ...
21
votes
2answers
2k views

Gender and groupwork

What does current evidence suggest: doing group work in mixed/balanced gender groups or doing group work in single gender groups? Setting College level mathematics/science course Group size ...
21
votes
2answers
638 views

Separating the roles of "teacher" and "assessor"

Teachers at the university level (at least what I've seen in the US) are responsible for both teaching students the material for a course, and assessing the students' understanding of the course ...
21
votes
3answers
1k views

Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
20
votes
8answers
3k views

How can I learn to write better questions to test for conceptual understanding?

I'm worried that I'm bad at realizing when a question I've written requires little or no conceptual understanding to answer. Like, when I'm writing a question for a homework assignment or exam, I'll ...
19
votes
1answer
447 views

The "water triangle" proportional reasoning task

(I previously asked this at The Mathematics Teaching Community, but I'm hoping it would attract further answers here.) The Wikipedia page on proportional reasoning mentions a "water triangle"...
19
votes
2answers
420 views

Literature on learning from errors in mathematics

In teaching undergraduate mathematics, I implemented some strategies to encourage the students to look at errors they made or at "typical errors" in the current topic. One attempt was to compile a ...
18
votes
3answers
399 views

Critiquing Proof Style During Class

I would like to spend a day with my students analyzing mathematical writing. One way I might accomplish this is to offer multiple proofs (some good, some poor) of the same simple statement and ask ...
18
votes
2answers
575 views

Comparison of different concepts of integral

As the following math stack exchange question (and answers) show: https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy There are a lot of different ways to ...
18
votes
3answers
536 views

Evidence for or against the claim that some students are "algebra people" and others are "geometry people"

Where I live and work, there is a widely-accepted and often-repeated claim that there are two kinds of students: "algebra people" and "geometry people". This claim sometimes gets expressed in ...
17
votes
2answers
1k views

A study comparing effects of calculator usage on later math skills?

Each year my university tries to decide whether or not it will have calculator and CAS based introductory math courses (the calculus sequence, linear algebra, and ODE) or not. Other than some hearsay ...
16
votes
5answers
7k views

Best textbooks to introduce measure theory and Lebesgue integration?

What are the best textbooks to introduce measure theory and Lebesgue integration to undergraduate math majors? Many students in such a class will go on to graduate school and be required to take a ...
16
votes
5answers
807 views

Discovery-based and inquiry-based learning

In general, I think (and I am told by students) that I am good as a tutor. However, I would like to become more rigorously familiar with the discovery-based and inquiry-based learning applied to ...
16
votes
1answer
799 views

Is there any evidence about the effectiveness of "table proofs" in pre-college mathematics education?

I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol ...
16
votes
1answer
498 views

Research supporting "recipe-style" calculus in senior high school?

Anecdotally, I've heard it said that in (Australian) grades 11 and 12 calculus needs to be taught in a procedural way involving merely recipes for doing calculus, rather than teaching for ...
16
votes
1answer
331 views

How to assign grades to proofs: what do(es) the literature/experts suggest?

I am teaching an introductory course on proofs in mathematics in a mid-size American public university, and trying to develop some kind of consistent grading meta-scheme for grading proofs. I am ...
16
votes
1answer
1k views

The general and particular in the psychology of mathematics education

Many students I have spoken with who are drawn to becoming mathematics teachers chose their mathematics major because they enjoyed doing routine exercises in high school. The comfort of a definite and ...
15
votes
4answers
549 views

Thought experiment: Utopian college-level math curriculum without external constraints

An old favourite topic of mine to daydream about on pleasant afternoons is this: If you could completely redesign the university-level mathematics curriculum from the ground up to be as good as it ...
15
votes
3answers
450 views

Calculation versus writing in mathematics

Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas ...
15
votes
3answers
1k views

Does the "how old is the shepherd" phenomenon occur for more relatable word problems?

A friend of mine just showed me this article about the "how old is the shepherd" problem: Link Of course, I'm shocked by the finding that 75 percent of kids give an answer other than "there isn't ...
15
votes
2answers
1k views

"Always/Sometimes/Never" vs. "True/False" questions for mathematical reasoning

Has anyone performed a study on the differences between student interpretations of these words? Background: When I taught high school geometry and later undergraduate precalculus, I noticed that ...
15
votes
2answers
799 views

The Fundamental Theorem of Calculus and Vegetables

When I was an undergraduate, someone presented to me a proof of the Fundamental Theorem of Calculus using entirely vegetables. I found this incredibly fun at the time, but I can't remember who ...
15
votes
1answer
212 views

Reference request for studies on gender in math examples, homework problems, or math exams

I am looking for a study or reference on gender in math problems given in mathematics. In math texts or even on math exams, if there is a word problem involving people, these people or "characters" ...
15
votes
2answers
423 views

Logic in symbols or words

Making precise logical statements is an important part of teaching and learning mathematics. There are many ways to write such statements, and let me divide them into two main types1: writing in ...
15
votes
0answers
167 views

Research on the use of outlined / structured proofs in instruction

Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs? For the ...
14
votes
10answers
1k views

What are some great books for inspiring children to explore mathematics?

Starting from a young age, children can explore deep mathematical questions and enjoy thinking about basic math within the context of a story. There are some real treasures out there. Parents often ...
14
votes
6answers
222 views

Books/(auto)biographies/references on how mathematicians study/studied (as students)?

As Geoff Pointer commented: [...] As a composer I've learnt a lot from studying famous composers why wouldn't that also apply to studying maths and mathematicians of note as well? [...] Are there ...
14
votes
5answers
1k views

Soft questions for 8 - 12 year olds

I am concluding my second year in mathematics at the university of Milan. I also happen to be an educator for 8 - 12 year old children (as a Scout). Recently I have tried to fill some dead time by ...
14
votes
4answers
428 views

Why is polynomial factorization over the integers part of secondary school curricula?

By "polynomial factorization over the integers", I mean problems and solutions like the following: Problem: Find a factorization into irreducible polynomials for $24x^2 +x - 10$ and ...
14
votes
3answers
1k views

Recommendations for inquiry based/aided discovery textbooks

I've recently dipped my toes into the world of number theory; and I've bought a book that to me is quite unconventional: R. P. Burn, A Pathway into Number Theory. I've yet to put the book through its ...
14
votes
4answers
6k views

What is a good book to learn all of prealgebra?

I am an old man trying to learn math, starting off with prealgebra and need a good comprehensive book for it. The book should NOT contain annoying images like in most American textbooks or anything ...
14
votes
4answers
2k views

Teaching a very enthusiastic and bright 5 year old

I was asked to give extra lessons to a 5 year old boy who loves math (he says he likes 3 sports: football, swimming and math). However, I have never tought at this age and I am unfamiliar with the ...
14
votes
2answers
819 views

How does a reliance on calculators affect student performance?

Overheard in the Math Office while another Professor was helping a student with Statistics: Always use a calculator when doing decimal arithmetic because you'll eventually make a mistake if you do ...
14
votes
3answers
276 views

Resources for teaching Riemann integration in higher dimensions and on submanifolds, with view toward Stokes' theorem?

Question I am looking for suggestions of good resources (textbooks or lecture notes preferably) for teaching Riemann integration in $\mathbb{R}^d$ with $d\geq 2$ and also for Riemann integration ...
14
votes
1answer
4k views

What experimental studies have been done on the Kumon method of teaching and learning mathematics?

My dissertation involved, among other things, the East Asian way of teaching and learning mathematics. (See, for example, Leung (2001).) I was particularly interested in the Kumon method. Although I ...

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