http://visual.ly/track.php?q=http://visual.ly/visualizing-pi&slug=visualizing-pi

From Visually.

Sequences of Inscribed Polygons.

Interesting Blog — just click the link and enjoy as Michael Pershan describes the process of exploring (or noodling –as he says). I too seek to provide students with the joyful experience of really “doing” mathematics, often by challenging them with competition level problems and hoping that they will begin exploring on their own. Keep up the great work, Micheal!

I found this fascinating quote today:

Last semester I stumbled upon an approach for teaching the concept of the derivative, and later the integral, that worked surprisingly well with my students. It stems from a realization I had that

much of what students see when they first learn about derivatives has very little to do with understanding what a derivative is.The typical approach to introducing the derivative throws students directly into the trickiest possible case: a smooth nonlinear curve, and we want to calculate the slope of a tangent line to this curve at a point. To do this, we have to bring in a lot of “stuff”: average rates of change, tables of sequences of average rates of change, and in a vague and non-rigorous sort of way the notion of a limit. It’s this “stuff” that confuses students — not because it’s hard, but because maybe it’s not suited for their first contact with the idea of the derivative. Maybe we need to build their intuition first.castingoutnines.wordpress.com, Casting Out Nines, Jan 2010

You should read the whole article.Image via Wikipedia

Maria is a talented math instructor who has got it together with regards to technology and “best practice”. She is also willing to share her knowledge and work for those willing to listen. I am one who has benefited from her work and am grateful. Here is a presentation about online math course design:

[authorSTREAM id= 205652_633813593982923750 pl= player/player by= wyandersen]

Link to presentation here

Also check out Maria’s blog at www.teachingcollegemath.com

Now into my second “school” year teaching mathematics to a young man who is home-schooled, it seemed that I might be able to post some relevant perspective on the advantages and disadvantages of our particular educational situation.

When I was first enlisted to work with Ian on Algebra I, he was already committed to using Saxon materials. I had familiarity with the Saxon method, which has a pedagogical bent toward skill building through continuous review and incremental concept building. Each lesson adds a “nugget” of new material, has a few practice problems, and then a homework set which consists of 30 problems. Perhaps 4-6 of the problems are on the new “nugget” and the rest are review problems. It is imperative for the instructor (in this case me 😉 ) to tie the concepts together during a brief time of direct instruction and to then work alongside the student as they practice solving problems. Feedback and homework corrections are vital to catching and eliminating conceptual errors or misunderstandings, because, misunderstandings can potentially be compounded until review problems are no longer just review, but a large and daunting set, full of menacing confusion. With this last point in mind, an important procedural change was made this school year. I now grade all the homework and provide Ian with a graph of the scores, updated weekly. This simple but important modification over last year, has markedly reduced the number of “bad” homework papers. This was one of the big issues from last year, fighting to maintain an acceptable level of quality and constancy with Ian’s work. One lesson, Ian might get 90% correct and the next it would be <50%. This should not happen with the Saxon method — and is not happening anywhere near as much now that Ian can see and is confronted with actual graphical data about his performance on homework.

In future posts, I’ll discuss the role of faith and religion, the relationship which has developed and my use of supplemental materials to bolster the geometry content in a Saxon Algebra I, Algebra II sequence.

Just thinking “out loud” here 😉 Peace!

The A side and the B side of discovering new tools:

Side A–

I feel like an old man who is running out of time to learn. Having been involved in math education for a number of years in rather small town settings, I have been respected as a math tutor and have had quite solid success teaching K-12 mathematics [through AP Calculus]. My pedagogical knowledge is pretty solid as well as my grasp of K-12 content, especially since I can draw upon a number of applications from physics and business background/experience. In school settings, I have always felt comfortable with available technology. I am fluent with the TI calculators, I can write programs, graph and calculate and display results with the best K-12 teachers around. Today, however, I was able to clearly see how little I really know and how far I have to go in order to grab a hold of some self respect again. Experimenting with Mathematica felt like drowning in information. There is just so much cool mathematics, I have not even scratched the surface! The graphical and interactive features in Mathematica are astounding. Here is a simple graphic of a Knot, which is so easy to generate, it is just ridiculous. And I have never had any opportunity to study these . I have no idea whatsoever, how to describe this mathematically, what it’s significance is, what applications it has, … you get the idea. I feel ignorant.

Side B —

This is one of the most exciting days of my life. I am like a kid in a candy store. I happen to love mathematics and I have struggled to learn LaTeX in order to typeset math symbolically over the last few years. I have struggled to display three dimensional graphs and surfaces encountered when teaching about functions and solids of revolution in calculus, for example. But today, I began learning with a new (for me), amazing tool. Mathematica has opened a world which was previously veiled, scales have dropped away from my crusted mind’s eye, and I am intellectually alive and stimulated. I just don’t know why it has taken me so long to jump on the technology band wagon and embrace this tool warmly and fully, but I am excited that I have seen the light today. How long before I am productive with this and am sharing new discoveries, time will tell.

So when is the last time you discovered a new tool that opened up seemingly limitless potential for learning?

When presenting information or learning new material for a course, I have found this “road map”, based on the Meyers-Briggs personality profile, to be very useful. I generally present information and create learning activities in a cyclical pattern, starting at the top right (SF) and progressing counter clockwise.

Have other instructors or students used this framework?

Would anyone be interested in discussing it’s use in more detail?

I have long thought that math and science homework should emphasize depth and quality over quantity, especially once basic arithmetic-type skills have been established; generally somewhere in those middle school years. I have recently been encouraging students, who I mentor, to create LaTeX templates and to use these to create quality papers for submission to their teachers. physhwktemplate22

I generally find that once the template is created to fit a particular class, it does not take much longer to work homework sets directly in LaTeX than on paper, and the result is much more desirable.

This particular homework example uses a LaTeX template based on the MEMOIR document class and highlights the basic problem solving approach of clearly articulating the *Given* information, the thing one is supposed to *Find*, a *Plan* for finding it, the *Calculations* and finally a clear *Solution* statement which directly answers the question which was asked.

**Here is the /LaTeX code:**