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Edit 2: This story continues to reappear periodically in mainstream media; this month (April 2019) it was featured in a BBC Feature story here. The article title is: The violent attack that turned a man into a maths genius To reiterate my earlier answer: The pictures created by Padgett are (perhaps) interesting, but the media's intention to depict him as ...


17

I remember being excited about the following at a young age. If you add consecutive numbers you get triangle numbers. Triangle numbers are fun. If you put two consecutive triangle numbers together you get a square number. You can also make a square number by adding the next odd number.


16

How about: Numbers go the other way, too (negative) You can cut numbers in half, forever What if you cut a number into three pieces? 1 million is a thousand thousands (100 is ten tens) If you don't know the number, call it a letter (or name it :) ) You can add letters together too What if you cut triangles in half? Rectangles? Can you make a box from ...


15

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 ...


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 ...


13

Edit (5/24/14): For the reader interested in a somewhat longer answer, I am including the literature review (and all references) from my thesis on conceptions of creativity with regard to problems posed from the multiplication table. A copy of the excerpt can be found here. My original, shorter answer remains un-edited below. The story of mathematical ...


11

I have come to view mathematical proofs as legal briefs, intended to convince a certain audience that the theorem is true. The degree of formality very much depends on the target audience. To echo Jim, the argument should be "coherent and convincing" to that audience. One phrase that encapsulates this view is: mathematical proofs are social constructs. I ...


11

Yes, there is a growing literature at the nexus of mathematics education and creativity. The main name to know is Bharath Sriraman (google scholar) though the classic pieces to read for mathematical creativity are the book by Hadamard (see the answer of Lucas Virgili) as well as the chapter Mathematical Creation by Poincare that served to inspire the ...


9

To supplement the other U.S.-centric answer: yes, the U.S. standard curriculum through high school is not now and has not been in recent years comparable to several western European or former-easter-bloc education, for a variety of reasons which are irrelevant to the question. But, either way, there are some implicit hypotheses that are (in my opinion) worth ...


9

I know I shouldn't add another answer, but I think my other answer went off on a tangent that didn't really address your question. I did not initially read it carefully enough to realize you were tutoring students, not teaching a class. I really don't think the label of "Teaching Critical Thinking Skills" is either relevant or pertinent--teaching is ...


8

This depends to a large extent on the background of your students, which varies a lot from country to country. In the United States, many students who take calculus have not seen any formal proofs before, while others had some limited exposure to two-column proofs in geometry class. In this context, it is nearly impossible to require students to ...


7

I believe you would like Jacques Hadamard, The Mathematician's Mind, which has changed the name since I've read it (it was called The Psychology of Invention in the Mathematical Field). From the synopsis: It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to ...


7

There is quite a bit of evidence that critical thinking can be taught (which I found surprising). College students' gains in critical thinking skills over the course of their education are correlated with high standards set by instructors and greater time spent studying. So I think the basic answer is that you have to assign students tasks that require ...


7

The question is very broad, as you surely realize, since you broke it up into six sub-questions (which are also very broad!). In order to attempt an answer, I will have to send you on a bit of a scavenger hunt: First, see my answer here and note that I have a literature review from my doctoral dissertation, which was entitled Conceptions of Creativity in ...


7

This is tangential, but I'll submit it as an answer because I cannot comment yet. the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way It shouldn't be controversial to want the right answer (to an unambiguous question). The 'right way', however, presents serious problems. The 'right way' ...


7

There is a great old Disney short called Donald Duck in Mathmagic Land. As well as being delightfully drawn in the traditional Disney style, it contains lots of useful and occasionally surprising information about where math can be found in everyday life. Personally I found it informative and inspiring, I imagine there would be plenty of conversations to ...


7

Here is my attempt to answer your question, if not directly, then at least in spirit! I am specifically responding to the following: Many students see no beauty in this subject, only fear and a need to be right and get "the answer". Question: Has an approach to mathematics education been tried that explicitly deemphasizes correctness ... ? What ...


6

Ask students to solve problems, pose their own questions, make their own definitions, etc... This is about creating opportunity for creative work. But it is also crucial that you then take the student's ideas seriously! Give the ideas respect. Encourage other students to do the same. Don't replace student thinking or explanation with your own, instead ask ...


6

The earlier question, "Teaching a very enthusiastic and bright 5 year old" could help. Building and manipulating shapes enhances geometric imagination. Consider polydrons, or snap-cubes:             Learning Resources Snap Cubes


6

Below are some great and inspiring books by an excellent mathematician. (In the Really Big Numbers book, on the page where counting by tens is discussed there is an inspiring error (?)…Big Bird is right, everyone makes mistakes!) Really Big Numbers and You Can Count on Monsters by Richard Evan Schwartz. Let me also add this wonderful activity JDH did with ...


5

I have a distinct memory of my second grade teacher handing out a logic grid puzzle to the class one day near the end of the year. I remember thinking, "This is fun, but what does it have to do with math?" Of course, logical puzzle-solving is an important kind of math, and should be explicitly included in the curriculum. Students of all ages should be ...


5

There are studies indicating that metacognition training can improve problem attack. In the linked article by Schoenfield (which I may have gotten from here, I'm not sure), he discusses the differences in how experienced mathematicians approach a problem compared to inexperienced students, characterizing each approach by the stages "Read, Analyze, Explore, ...


5

Another book, to add to @Jon Bannon's list, that my 4 year old daughter and I can recommend is Introductory Calculus For Infants. We also seem to discuss Graham's number a bit after watching the Numberphile videos


5

Arranging counters into groups (multiplication and division), so arranging 12 counters into 6x2, 3x4 etc, and realising that there are some numbers that cant' be arranged, no matter how hard you try (prime numbers). Then, how many prime numbers are there, can you work out which number is going to be prime?


4

If you are looking for a book you can use to study an effective practice of teaching mathematics in a way that is definitely not intended to leave students with mathematical tunnel vision, can I recommend Teaching Problems and the Problems of Teaching by Magdalene Lampert?


4

It depends on how you have dealt with these proofs during the course. You have written, that you appreciate the approach of describing proofs in sketchy way. In that case, you should accept any coherent proof in exams, regardless of its formality. If formality was part of your course, you should give part of the points to coherent, informal proofs. ...


4

If I had to simplify this problem to one question, then what I would ask myself is:                                         To what extent the student ...


4

It helps getting students interested in some of the more "recreational mathematics" type sites and problems. Solving some not-completely-serious problem that interests them is as much creative and intellectually demanding (and often less demanding in specific background, a impassable roadblock for beginners) than "serious" work.


4

My brief answer would include: interdisciplinary, integration with arts, open-ended problems. Of course, right answers are important as long as we can nurture the idea that not every question has one perfect answer. Instead of What is 2+3? I like asking children to find all dominoes that have total of 5 dots. I ask the same questions future teachers, ...


4

Something that helps, is to create problems where it's harder to see the "right way" to solve it. I try to give problems to my students where it's not obvious what type of method may be best to solve them, and then we discuss whatever solutions that come up. I also ask them to solve problems in more than one way. The idea, being that if they were teaching ...


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