39

Such an approach seems designed to force (or at least, strongly encourage) students to learn by pattern-matching from examples. This is one of three modes of student learning in mathematics described in this article by Frank Quinn; it is the least powerful, most fragile, and most error-prone of the three. To quote the relevant passage: There are (roughly) ...


27

Do NOT give exam questions that are intentionally more challenging than homework or in-class problems. I would recommend precisely the opposite. The point of the exam is really a spot-check that students know the basics and aren't just faking their way all through the class. If there is a time-limit, then that is already a lot more pressure/high-stakes ...


14

I have been teaching students for the past 6½ years- in all levels of college undergraduate math (decent bit of physics too). I have found that analyzing learning and all the ways to understand mathematics, in particular, is a very necessary first step. (See above posts!) To the OP: I believe your conclusion is correct, the process you outlined alone ...


12

Daniel Hast's answer is great, but I want to add one thing: What kind of mathematical ability do you want your students to learn? Are you measuring that ability or something else? I have seen way too many students who can do some formal manipulations (solving equations, differentiating functions, or the like) but who do not understand what any of it means. ...


11

First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a while because there's no one to talk in it with. There are two types of mentalities about intelligence and success in life. Studies have shown that children who'...


11

The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as trying a special case. I do not know of a more specific term for the context that you have put forth, but this is often how one approaches a problem if you are initially unsure as to how to solve it in generality. It is a good way to get an initial foot-...


10

I am an alumni of Fazekas Mihály Gimnázium (Budapest) and I can attest to the fact that we were educated in a problem solving manner -- although not exactly as OP describes. For four years, all we did we solved problems. We did nothing else. There were no lectures as such. The teacher provided guidance or stated definitions and theorems based on the ...


9

Anecdotally, based on self-observation and observation of many faculty and grad students: "if it's not in your head in some form, you can't think about it". A funny point here is that it seems not strictly necessary to "completely understand" something, if one can keep it in one's mind. Indeed, I don't see how to make the transition from not-understanding ...


9

I always make homework (from the textbook and online in WeBWorK) and written assignments MORE difficult than exam questions. I tell my students this, with the reason being “if you can run 10 miles in training running 5 miles on race day is easy”. Keep the course learning goals in mind. Your exam questions should be a chance for your students to demonstrate ...


8

Nice question! Let me add one reference to your list: Silver, Edward A. "On mathematical problem posing." For the learning of mathematics (1994): 14(1) 19-28. (PDF download link.) Silver cites Hadamard's famous book, The Psychology of Invention in the Mathematical Field, as recognizing that isolating key research questions is a sign of exceptional ...


8

Part 1: Do they really understand? My first thought is that you are running into the limits of working memory. As students try hard to understand step 5, they are pushing previous thoughts about step 1 and 2 out of their working memory, before having really processed that information. That ~guarantees they will forget steps 1 and 2 almost immediately, ...


8

Problems columns I found (in 2009) useful to challenge undergraduates. Some of them may no longer be current. But even those may be interesting to consult past issues in your school library! Mathematics Magazine https://www.maa.org/press/periodicals/mathematics-magazine Mathematical Mayhem https://www.facebook.com/pg/Mathematical-Mayhem-...


7

You might want to learn more about a learning disorder called dyscalculia. Students (or people in general), who are diagnosed with this learning disorder, are not "thick", nor "stupid". An analogy you might be more familiar with is dyslexia, in which students struggle with reading and writing, but are in no way inherently "thick", nor "stupid". You seem ...


7

(1) Here is Margie Hale's tree: (2) And here is Gaspard Sagot's hierarchy:


6

In light of your edited "more general" question, I thought I would make a few remarks. Historically, an early treatment of the subject of reasoning by analogy can be found in work on Gestalt psychology. One of the better known authors/books in this area (where mathematical examples are plentiful) is Max Wertheimer/Productive Thinking. For a somewhat more ...


5

For me, the process is as follows: Do the exercise. Do the exercise again. This is probably faster than the first time, since I have a vague feeling of what I should be doing, and maybe remember some dead ends that I can avoid. Repeat step number 2. At some point, if I have to do the exercise often enough, I learn it by heart. This is rare. In practice, ...


5

I will take a stab at an answer though clarifying what level of education we are talking about would help. I have never created problems for things like Qual exams (essentially masters exams) so I can't really speak to those. For almost everything else I would tend to start with a very well formed idea of what the problem would be. You always know what ...


5

The strongest students often tell me that they like taking my courses because they learn so much in doing the work for my course. I lecture and give traditional, human graded, homework. One thing I do which I see being done less in other courses is to ask some difficult open-ended problems along-side the "standard working knowledge" component. If I had a ...


5

The answer to your question is yes. Check out the recent textbook: Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. Springer. Google Books. In particular, Part III, which contains 10 chapters, is described as: Here is the beginning of the first chapter in that section: See ...


5

This approach would be fundamentally a violation of the entire axiomatic idiom of mathematical understanding and proof. In particular: Mathematics starts with careful definitions of terms. The proposed learning process contains no mention of definitions. I tell my students all the time that one of the essential strengths and advantages of the mathematics ...


5

What looks to be missing is teacher interaction. The student is interacting with a workbook. So where is the learning occurring? The student may learn something while exploring the problem. However, learning the difficult things this way is pretty difficult. Few people "discover calculus" on their own because some problems were put in their way. So ...


5

Our MAA section (North Central) has an annual team math competition for undergraduates. It has proven more attractive to the students than the Putnams (which we haven't tried for many years). The reasons for this are: (1) It's just a Saturday morning (9:00-12:00), rather than the 6-hour Putnam format. (2) The problems are more accessible, although some ...


5

I had a similar interaction with a student this weekend. I tend to walk it back to talk about these objects are just ways we invented to talk about number and quantity. They translate into a language. So many students get focussed on the machine process of manipulation of quantity to get results. In that they lose the idea these expressions represent ...


5

As written, this seems overly caught up in your personal experience, but I think there's a real question here. A lot of people make it to various stages of education having concluded that they're incapable of learning math, and teachers do need to figure out how to handle students who believe that they just can't learn to do math. More than most subjects, ...


5

An interesting example is Sicherman dice: A pair of 6-sided dice, with positive number of eyes on each face that are not the classic 1..6 ones; if you throw them, the distribution of total eyes is the same as for two regular dice. To construct them, one die is represented by: $\begin{align*} D(z) &= z + z^2 + z^3 + z^4 + z^5 + z^6 \\ &=...


5

I would love to get an answer by a teacher who is trying to teach prime numbers to elementary school students about what's happening with their attempt to teach them prime numbers. I would like them to tell me what teaching style they're using. This may not be exactly what you're looking for but just the other day I introduced prime numbers to my first ...


4

'Thinking Mathematically' by Mason, Burton and Stacey sounds like a good match. It has a large collection of problems/investigations using high-school level maths, and discusses how to go about the thinking process of doing maths. It isn't about contest maths, but my understanding is that contest mathematics is not the same thing as university maths (some ...


4

I've just happened to come across this free online course on logical and critical thinking. Excerpt from the site's "About the course": We are constantly being given reasons to do and believe things: to believe that we should buy a product, support a cause, accept a job, judge someone innocent or guilty, that fairness requires us to do some household ...


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 ...


4

Any exercise is just one example of a family of similar exercises. You can explore those similar exercises by writing and solving your own variations of the problem. To be more concrete, I assume that you have had at least first semester calculus so have seen something like the sliding ladder problem as an exercise. Suppose that you had the following ...


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