25

In regards to "math anxiety", the 1990 paper by Ray Hembree helped me out a lot. It's a large meta-study of about 150 papers and a total of 25,000 students. Summary of the results, as I wrote on my blog previously: Whole-group interventions are not effective (curricular changes, classroom pedagogy structure, in-class psychological treatments). The only ...


23

Perhaps the key-word needed here is not just struggle but productive struggle. Hiebert and Grouws (2007) discuss two key features of mathematical teaching/instruction for "promoting conceptual understanding" (p. 383). Their paper can be found here: Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ ...


12

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....


12

Such an implication as you suggest seems highly far fetched. Firstly, there is a chicken and egg problem here. Suppose that research showed those who study mathematics end up far more likely to have strong characters and be expert at day-to-day problems. How would you tell, without advanced brain imagery techniques, that it is not the predisposition to ...


10

In order to cultivate a greater appreciation for precision in one's mathematical statements, I ask my students not to copy the problem question, but rather to transform it from a question or problem statement into a theorem statement, which they then prove or solve. For example, if the quiz question is Is the ring $\mathbb{Z}_2\oplus \mathbb{Z}_3$ an ...


10

Personally I encourage students to write answers that make sense as a self-contained piece of writing, because I think that that is a valuable skill. Certainly it is required when writing about mathematics in any context other than homework or exams. This generally means that they need to reproduce some or all of the content of the question, possibly ...


9

Here is an NPR article that discusses how teachers' efforts to engage learners in productive struggle (or not) may be culturally situated. (Of note, Benjamin cites Jim Hiebert above, who has written The Teaching Gap with Jim Stigler, interviewed in this article.) Jim and Jim have conducted research on how instructional approaches differ culturally between ...


7

For all of the community college algebra classes I teach, I certainly make proper mathematical writing the number one priority; which is not to say that I have students compose everything in English writing. It's already an overwhelming challenge for students to get the algebraic grammar and syntax right, such that I already feel like there's not enough time ...


6

You have asked three questions, and, in mostly answering the second, my response is already quite long. For now, let me not broach the third question about consequences of the distinction in practice. Yes, I distinguish between the two concepts. As for how I distinguish between them: Let me note first that a rigorous answer would probably come from within ...


6

Responding to Q1: At the grade-school level, there is considerable evidence that music (and also dance) can be used to teach fractions. E.g., "Rhythm and Music Help Math Students." Scientific American. 2012. Article link: "Grade school kids who learned about fractions through a rhythm-and-music-based curriculum outperformed their peers in traditional ...


6

I agree with the premise, but as Ittay's answer suggests, a study sufficient to prove this would be difficult. The issue of correlation vs causation comes into play with the task of separating them to be difficult. In the end, it's fair to say that young children learn more easily. Things like language that are far more difficult to learn say, in high ...


6

I think that you might find useful Harel & Kaput's chapter in Tall's Advanced Mathematical Thinking (1991). You can find it for free download at http://www.math.ucsd.edu/~harel/publications/Downloadable/Conceptual%20Entities.pdf. To summarize, they identify three roles for "conceptual entities" (including reified objects): Alleviating working ...


5

A while back I posted the related question How can we help students who are very anxious about math?, so that I could offer up a few answers of my own. My suggestions include a few good books: My students have had some success in decreasing their anxiety with books like Mind Over Math (Kogelman, Warren), Overcoming Math Anxiety (Tobias), and Managing the ...


5

I recommend this to most people I have helped or tutored over the past years for a few reasons. Problems are complicated when you don't understand them. Basic problems are not as easily digestible to students, especially those who are less confident. I have found people consistently make mistakes based on the problem definitions even if they are ...


4

This is answer is self evaluation of what I think happened to me. I don't know about building characters to be 'stronger'. (What does 'stronger' or 'better' even mean here? What is the partial order?). But I do think it influences character. I recently graduated from my BSc in mathematics and I can guarantee that I'm much more honest now that I graduated ...


4

A small contribuition; This might help discussion going. I do not have a comprehensive direct answer, however, at the moment I am working on something else that allowed me to see some of the potentially relevant information: “…Klinedinst (1991), who also found links between retention in an instrumental music programme and scholastic ability, reading ...


4

Another thought occurred to me regarding this, after having read Reuben Hersh's collection of essays. There is a quote of Bill Thurston, which I paraphrase as "thinking is the same as seeing". In a sense, having the logic of one's proof "at the tip of the tongue" produces a sense of unity in the proof that is analogous to an object one can mentally ...


4

Not a high-quality answer, since I can only refer to some decades' observation, rather than systematic/deliberate study of the phenomenon: Having observed maybe 1,000 pretty-darn-good grad students, some of them very good, I find that it is not typical among mathematically-interested, mathematically-talented people (at least 20-30 yrs.) to have a good ...


4

(This answer has two parts: The first one is about existing research, and probably relevant, but succinct; the second one is about a problem solved in practice, and possibly relevant, but definitely rambling. I will leave the determination of what constitutes a "related" answer to the reader!) Part I As I perceive Polya's (1945) How to Solve It, the chief ...


3

To answer your actual question about guidelines, I would say this. There aren't really any "official" guidelines that I'm aware of. Institutions may have their own policies regarding this, although to me that would be borderline micromanagement if they did. I've never been affiliated with such a place as far as I know, so I've always just gone by this: ...


3

The literature lacks a clear mechanism, that is why theories on the process-object-duality are criticized sometimes. Anna Sfard's reification, Dubinsky's APOS and Tall's procept may help describing the situation, but hardly give specific advice what to do. (Note that Anna Sfard had suddenly stopped her work on reification and moved to commognition, a socio-...


3

I can't really comment on the psychological reasons, but I can say from personal experience that some students have practical reasons for doing so: by copying the problem from the textbook or question sheet onto work paper, this allows directing focus entirely to the paper the problem is being solved on rather than having to switch between the two to verify ...


2

I'm going to hazard a guess - For 2 column geometry proofs, the repetition of every 'given' is required. When working with students on their Geo, it struck me that many times, the 'givens' are 2/3 of the solution, just a couple steps to finish the proof. For algebra, the question is usually brief enough, it makes sense to re-write it on the answer sheet. ...


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