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10

I would recommend Python combined with SageMath, as already recommended by Joseph O'Rourke, or rather SageMath and Python comes naturally. Python is a modern, and widely used, interpreted language (no compilation needed) it supports big integers via the bignum type. (But using SageMath I think this is tangential, I mention it for completeness mainly.) ...


8

Check out http://jiblm.org. There are lots of scripts here, some better than others. A nice book in this style is "Distilling Ideas" by Brain Katz and Michael Starbird. I also recommend the following method: Take a reputable text on a topic, and try to prove all the theorems for yourself. If you get stuck for a long time, take a quick peek to get unstuck…...


7

IBL is a really wide umbrella term, nowadays at least. I strongly suggest browsing the Academy of Inquiry Based Learning website. They have videos for many of the talks at the R.L. Moore conferences, such as this one that seems to be about "larger" classes. The canonical text on "Moore method" may have some ideas for your larger classroom situation, but I ...


7

I am going to copy and paste my answer from another question on this site, because I think one would be hard pressed to beat it in terms of the number of suggestions it covers, and the general quality with which it presents these suggestions: You might be interested in the expansive answers that were generated on math.stackexchange by the questions Book ...


6

What computer languages might one recommend for, say, investigations in number theory? I find Mathematica ideal, e.g.: "Mod sequences that seem to become constant; and the number 316" "Does 53 diverge to infinity in this Collatz-like sequence?" But: (a) there is a huge start-up learning curve, and (b) Mathematica is not free. Because of the latter, I ...


6

Because you used the education-research and reference-request tags, I will give my usual recommendation of Powerful Learning: What We Know About Teaching for Understanding for the reason that Schoenfeld's chapter is well-referenced and written for practitioners. The benefit here is that it is a good introduction to methods of "teaching for understanding" ...


5

The linear function component is covered early(ish) in Algebra 1, and quadratic functions are covered towards the end of Algebra 1; so, the former by 7th/8th grade and the latter - if at all - by 8th grade. But, even for students in New York (where I teach) taking Geometry in 9th grade, more general polynomials may not be covered until Algebra 2 in 10th ...


5

If you absolutely don't want to introduce your students to proof methods yet, I think you should set a clear timeline. Give them a day or two to work on their conjectures (preferably in groups) before bringing them to class. You could look at this as an iterative design problem, where the testing-redesign cycle is out of whack. While an engineer ideally ...


5

A simple search give you plenty of reference materials. Here is The Academy of Inquiry Based Learning. And here is The Journal of Inquiry-Based Learning in Mathematics. Enjoy exploring and experiencing.


5

I spent two days sitting in on math classes at Phillips Exeter Academy. It is an outstanding example of inquiry/problem-based mathematics education in practice. Consider their curriculum. Their format is simple: students do 4-10 homework problems per night (all of these are posted online here), come in to class every day and each put their solution to one ...


4

Here are references directed at teachers more than policy-makers, that analyze teaching from the teacher’s point of view, including the decisions that teachers must make as the lesson unfolds, and how their decisions affect the students. Teaching Problems and the Problems of Teaching by Magdalene Lampert Connecting Mathematical Ideas: Middle School Video ...


4

This is only a brief answer, but I hope that it's a useful counterpoint to the others which present a very positive view of discovery learning. The one reference that I have found most useful for discovery- and inquiry-based learning is the paper Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, ...


4

There are numerous little things you can do to instill an IBL atmosphere in your class; most of them can be found on the AIBL website mentioned earlier. I would also recommend watching this TED talk: http://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en I love his examples of getting the students to be the ones to generate the questions....


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

For beginning number theory, Art of Problem Solving has an online course. The textbook used with it, Introduction to Number Theory by Matthew Crawford, can be used alone for self-study. (I have not used the course. I have used the textbook.) I tutored a very advanced 9-year-old using this book, and enjoyed it. I see they also have an intermediate number ...


3

This is a topic I could imagine not being adequately covered in all U.S. schools, although (as Dave L. Renfro pointed out in a comment), it is listed in the Common Core Mathematics standards under high school functions (HSF.BF.A.2). I could imagine it being presented successfully anywhere from the 3rd grade up to an advanced college class for mathematics ...


3

Seconding/complementing other answers: Python (and/or Python as a part of Sage) has a command-line interface (on Linux/Unix and on Mac OS) that does allow defining variables, pre-loading files that set things up, and so on. Python (and, thus, Sage) has built-in large integers that are easier to use than C++ large integers (in my opinion). And freely ...


3

Nice question and I should say Inquiry-based Project is taking my attention. So, I really like these books: Tanton, J.(2001). Solve this: math activities for students and clubs. Cambridge University Press. Cofman, J. (1990). What to Solve?: Problems and Suggestions for Young Mathematicians. Oxford University Press, New York. I hope these ...


3

Student Research Projects in Calculus Cameos For Calculus I particularly like the first one because the authors include with each project a description of how long it may take a student, any issues they've encountered, and (sometimes) how it can be explored further.


2

Combinatorics Through Guided Discovery by the late Kenneth Bogart is a great introduction to combinatorics through a guided set of problems and is freely available for download at the link given above.


2

Not exactly a "study", but you may be interested in a short article I wrote with Deborah Moore-Russo a few years back that touches on issues relating to problem-posing, in particular the last section ("Students as researchers") seems to be pertinent.


1

Let some $k>1$ - since for $k=1$ we have nothing to prove ($x=x$). Starting from the fact that $a<b∧ c<d\Rightarrow a+c<b+d$ we arrive to the fact that for any $x>0$ we have $x+x>0$. Similarly, we can have $x+x+x>0$ and, so on, applying this "trick" $k-1$ times we arrive to: $$\underbrace{x+x+\ldots+x}_{k-1\text{ times}}>0.$$ Now, ...


1

Maybe just do a general unit on solid geometry--it is a bit undercovered in schools. Just have the last class or two be model building and discussion. With the self standing nature of the models being a little extra fun. Can discuss the mensuration formulas or the like a little, but just have fun too. I don't think that the theoretical mechancis issues ...


1

I think the idea of check digits is pretty compelling. A bunch of examples of where check digits are used can be found here, including things like government ID numbers in various countries, UPC and ISBN codes, credit cards, etc. The basic idea is that you're given a number, say an ISBN number on a book. You'd like to know if the number you've been given ...


1

Rotations of regular polygons. Start with an equilateral triangle. Label the vertices one, two, three. Define a rotation to be 60° counterclockwise, for example (or you could go clockwise). Note that the locations of the vertices have shifted. Consider this a new “state“ or “configuration”. Now, what “configuration” does one end up with after n rotations? ...


1

Mandelpad plots the Mandelbrot set and associated Julia sets in the complex plane, and could spark a few interesting discussions (it has in my class), or even a "fractal art" project. In coordination with some other visual processing apps I've used it to create some very beautiful and intriguing images to use as cover photos, backgrounds, and the like on my ...


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