59
votes
Is there a virtue to learning how to compute by hand?
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
and parsing word problems, as well as the ...
- 1,802
28
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 ...
- 487
25
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 ...
- 449
25
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 ...
- 7,369
22
votes
Accepted
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 ...
- 4,606
21
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 ...
- 17.3k
21
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 ...
- 7,999
21
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 ...
- 24.4k
19
votes
Accepted
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.
- 10.8k
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....
17
votes
Accepted
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 ...
- 18.4k
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^{-...
- 171
17
votes
Accepted
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 ...
- 24.4k
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 ...
- 14.2k
16
votes
Accepted
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, ...
- 29.5k
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 ...
- 7,500
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 ...
- 24.4k
15
votes
Accepted
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'...
- 18.4k
14
votes
When should we first teach variables in school math? And how?
You don't need 1-letter variable names to do algebra. Basically, as soon as you start giving story problems to children, you need to start teaching algebra techniques. You can teach them as "easier ...
- 3,148
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 ...
- 241
13
votes
Examples of vocabulary that have different meanings in Mathematics compared to "everyday" English
"In general" or "generally"
In mathematics, if we say a specific result holds in general, we mean there are no exceptions to the result.
In every-day non mathematical discussions, if someone makes a ...
- 131
13
votes
Accepted
What is the correlation between students' contentment and educational quality?
From Clark, Richard, Paul A. Kirschner, and John Sweller. "Putting students on the path to learning: The case for fully guided instruction." (2012):
Even more disturbing is evidence that when ...
- 24.4k
13
votes
Accepted
Separating the roles of "teacher" and "assessor"
A question that occurs with a project like this (broader than one department, as you put it) would be: Who is qualified to make those assessments? Probably not any other department at a particular ...
- 24.4k
12
votes
Good examples of functions defined as definite integrals of elementary functions?
My first take is
$$
\ln(x) = \int_1^x\frac1t dt.
$$
Granted, some texts introduce the natural log of the inverse of $\exp$ but other texts define $\ln$ as above and the $\exp$ as the inverse. If I ...
- 2,992
12
votes
What are some good books on mathematical pedagogy?
In The Teaching Gap, James Stigler and James Hiebert analyze research from the TIMSS studies. Part of their conclusion is that much of our experiences of classroom culture are deeply imprinted in us, ...
- 20.1k
11
votes
Good examples of functions defined as definite integrals of elementary functions?
The function $\displaystyle \text{Li}(x) = \int_2^x \frac{1}{\log t} \,\, dt$ comes up in the study of the distribution of primes. Specifically, the number of prime numbers less than $x$ is ...
- 2,601
11
votes
Is there a virtue to learning how to compute by hand?
I attended the Computer based math educational summit back in 2016 and found their ideas interesting. I agree with some of their points and disagree with others, but it is certainly interesting to ...
- 1,228
11
votes
What is the rationale for distinguishing between proper and improper fractions?
I do not know of any relevant research.
Here are my own not-research-informed ideas.
Most people refer to fractions as parts of a whole. If someone says "I lost a fraction of a pound on my diet&...
- 24.4k
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