48
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
and parsing word problems, as well as the process of exploring
solution strategies. It is akin to interpreting a passage written in a
not-so-familiar dialect: ...
31
I'm primarily a physicist, but I also teach first-semester freshman calc once in a while. Your characterization of a cultural divide between physicists and mathematicians on this subject does not seem at all accurate to me. If anything, I think the characterizations should be reversed, at least on the average -- but it would only be an average, because ...
24
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 proportional reasoning. I invented the idea of a "water triangle" to teach inverse proportions. These were all made by the workshop at the Lawrence Hall of ...
23
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 your groceries will cost within $10\%$ before they are rung up. To know that, you need to be able to add and multiply in your head. You don't need many ...
22
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 personally have a very bad memory, I wanted to keep that aspect out of it. But I saw many students wasting time during exams, flipping through the book. (Have you not ...
21
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 need to do is introducing base 2 as well as number systems working with modulo (two's complement). We also introduce basic circuitry to do addition/...
21
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 causes them to catch the errors as they make them. This feels very related to rubber duck debugging. At the very least, if you're working with a student one-on-...
20
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, I think that question is ill-posed, because the goal state isn't clearly defined (effective at what?) and so there's no way to measure whatever it is you want ...
20
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 of the teacher.
One thing that appears to be common among some of these studies appears to be that perception of student performance varies based on gender of ...
19
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 homepage (and the two pdf textbooks) in an earlier MESE post here; I would have just re-posted this as a comment, but I believe it is the actual answer to your ...
18
This is a problem for some English language learners: The triangle on the left is also a right triangle.
18
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. I'd be ok with a student's answer to "What is the y-intercept?" to be simply the y value, or the $(0,y_0)$ point.
If a teacher prefers one, you can ask
...
17
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 was not seen as a particularly bright student. Due to bullying I looked for alternatives to the local high schools, and ended up applying to university early ...
17
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 of integrals wikipage, there is a sub-section on Special Functions.
As a related remark, one reason that functions may be presented and/or defined in terms of ...
16
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 mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive ...
16
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:
$$
\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.
$$
(Incidentally, this is the example of how to use MathJax in the help section.)
16
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, Melissa McManus Scircle, and Matthew Hunsinger. "Female peers in small work groups enhance women's motivation, verbal participation, and career aspirations in ...
16
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 tool and they still have to think. Students also need to learn that having a calculator doesn't guarantee that their computation will be correct. Finally ...
15
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 quite complicated. Another reason is that studies on technology use are often fodder for theses, which are completed in too short a timeframe to provide such an ...
15
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 understood it. They gave me a book for 11th grade math and I learned. I don't think it did me any harm - but I do think I understood the concepts and didn't just ...
15
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 mind winning Fields Medal. Indeed,one did: Maryam Mirzakhani. I remember, one of the courses we had was "geometric analysis", quite uncommon as an undergraduate ...
15
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've tweeted here). Interesting that it has gone viral without anyone having sourced it.
This study can be easily replicated, and has been: with multiple ...
15
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 definition. Thus, for example, the shape on the left below is likely to be recognized as a "triangle" (despite the fact that it has four sides), the shape in ...
15
"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 frequency of cheating.
15
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 conceptual:
Suppose that a truck's distance from you in meters at a time $t$ seconds after the big bang is given by the function $p(t)$. What does $p'(19)$ tell you (...
15
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 authors write:
Despite the proliferation of the flipped classroom, little
well-identified evidence exists on its impact on student learning.
In this study a ...
14
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 study and persevere through difficult classes because everything to this point has come naturally for them. They haven't needed to set time aside for a study ...
14
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 neither of the above are possible, re-read every step of your work with an attitude like it's trying to sell you a used car.
Adopt the useful exaggeration that ...
13
In A Mathematician's Lament, Paul Lockhart describes what is wrong with math education in schools. To get a good idea of how he would do things differently, you should read his book titled Measurement. The ideas in it are for slightly older students, but the approach is what you're looking for.
I recommend that you try to work through Measurement yourself, ...
13
What do we understand as mental processes? All of us (Math teachers) dream of entering the brain of our students, see what's happening and adjusting some connections...
However, the thing is that their brains are a kind of black box for us. So the only way we actually have to be sure they have catched some mathematical concept (or argument, or property, etc....
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