52 votes

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

I couldn't agree more with @Steve's comment. The following response is written with elementary-to-high-school mathematics in mind. A lack of a decent number sense really does encumber making sense of ...
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  • 1,041
24 votes
Accepted

The "water triangle" proportional reasoning task

The instrument pictured was created by Barry Kurtz. He writes by email: I completed my PhD under Bob Karplus at UC Berkeley. I was his last PhD student. My dissertation dealt with teaching for ...
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24 votes

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

I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what ...
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23 votes

What is the rationale for distinguishing between proper and improper fractions?

added Oct 6 The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found ...
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  • 6,171
22 votes

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

There is a middle ground: closed-book, with some notes. The disadvantage to open-book exams is that students will waste time looking for answers in the book. I know this from experience. As I ...
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  • 17k
22 votes

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

Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking. I teach Computer Science freshmen and one of the first things we ...
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  • 419
22 votes
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How do you coach students who often make small errors?

Ask the student to "talk through" their calculations Having a student verbalize their calculation may force them to pay more attention (or a different kind of attention) to their work that ...
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  • 4,362
21 votes
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Is there any evidence about the effectiveness of "table proofs" in pre-college mathematics education?

The two-column proof form has been the dominant mode of presentation for proofs in secondary geometry in the United States for most of the past century. You ask about its effectiveness; unfortunately,...
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  • 16.1k
20 votes

Are women better math teachers for little children?

Most of the research on gender and math education is focused on student gender differences. However, a few references can be found that focus on the what differences there may be based on the gender ...
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  • 7,375
19 votes

Books about elementary mathematics written like a good undergraduate textbook

I think you'll find some of what you want on Berkley mathematician H.H. Wu's homepage. More precisely, see: Pre-Algebra (pdf) and Introduction to School Algebra (pdf). Note: I mentioned the same ...
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19 votes
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Examples of vocabulary that have different meanings in Mathematics compared to "everyday" English

This is a problem for some English language learners: The triangle on the left is also a right triangle.
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18 votes

What is an intercept?

This is a case where you might be looking for a distinction that's pretty subtle. By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered....
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17 votes

Effects of early study of advanced books

I have a bit of anecdotal evidence. I was unfortunately not homeschooled, nor did I have a technical childhood; I spent my childhood painting and writing short stories. I was in gifted classes, but I ...
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17 votes
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Good examples of functions defined as definite integrals of elementary functions?

It seems that the key term here may be the somewhat non-specific-sounding special functions. By googling for a few examples (Erf, Si, Li) I came across a Table of Special Functions and, on the Lists ...
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17 votes

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

The gamma function is very useful in counting problems (among others) and is seen as an extension of the factorial function into the reals. It is defined as: $$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-...
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  • 171
17 votes

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

I taught at the elementary and high school levels. At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a ...
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  • 6,735
16 votes

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

Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such: It appears that even many serious professional ...
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  • 13.4k
16 votes
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Gender and groupwork

Here is one article in PNAS. The final sentence quoted below is a summary: "creating small groups with high proportions of women [...] is one way to keep women engaged [...]" Dasgupta, Nilanjana, ...
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16 votes
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Midterm in Mathematics Courses

"Cheating Lessons" by James M. Lang argues (and has many references to back up) the claim that smaller, more frequent, lower stakes assessment both improves student learning outcomes and decreases the ...
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16 votes

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

Agreeing with comments and other posts: If you want more conceptual answers, give them less details in the set-up. Using your velocity problem, here are a couple of examples of making it more ...
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  • 6,350
16 votes

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

Consider a paper from this year: Setren, et. al., "Effects of the Flipped Classroom: Evidence from a Randomized Trial", Annenberg Institute at Brown University (2019). In their introduction, the ...
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15 votes

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

Brief Remarks: It is difficult to find longitudinal studies on calculator use as specified by the OP. One of the reasons for this is that tracking students from, e.g., high school till college is ...
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15 votes

Effects of early study of advanced books

You asked for anecdotal evidence. I was a "gifted student". The school told me to teach myself 11th grade math (Trigonometry and Algebra 2) in 9th grade. I never formally learned algebra 1, but I ...
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  • 6,735
15 votes

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

Years ago, as an undergraduate student, I experienced something close to your description of a utopian mathematics undergraduate-level curriculum at Sharif University of Technology in Iran. We didn't ...
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  • 4,306
15 votes
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Does the "how old is the shepherd" phenomenon occur for more relatable word problems?

Note (Feb 2018): There is an alleged "Chinese math problem" (see, e.g., WaPo article) going around about the second example problem below (cited to Reusser 1988, but can be found in Reusser 1986, as I'...
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15 votes

Are kindergartners supposed to be steered from squares being rectangles?

Kindergartners are generally at an early stage of geometric development, in which shapes are recognized by how well they resemble prototypical images, rather than by whether or not they conform to a ...
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  • 16.1k
14 votes

Students who know high-level math before going to college

The question you are asking has little to do with the particular subject in which the student excels and everything to do with student motivation. The students have not developed the skills needed to ...
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  • 871
14 votes

How do you coach students who often make small errors?

I used to have this problem. What helps me more than anything is: Solve it two different ways if you can and make sure they agree If you are finding a general formula, test it on some examples If ...
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  • 241
14 votes
Accepted

Are there research studies that attempt to determine the value of a "Growth Mindset?"

Psychologist Carol Dweck's "growth mindset" theory has become a popular solution and intervention technique in (mostly American) schools of all ages. We might say that it's become the new ...
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