The actual title for this video is “RSA Animate – Changing Education Paradigms”.

Related posts:

1. Friday math movie: OK Go This Too Shall Pass
2. Friday math movie: Spirals, Fibonacci, and Being a Plant
3. Friday math movie: Salman Khan at TED

She had a massive stroke at age 37, and had the unenviable ability to analyze what was going on in her own brain during the traumatic event.

She wrote a book about it all, aptly named My Stroke of Insight: A Brain Scientist’s Personal Journey. I really enjoyed the book on several levels. When your language abilities are severely impaired, other things take on greater importance. For example, she reported how much the body language of a person spoke very loudly, and small kindnesses were hugely important.

She described one scene where the doctors and nurses treated her as a piece of meat, while one nurse actually looked into Taylor’s eyes and spoke gently. I thought this has so many implications for teaching. Our “aura” in the classroom has profound effects on many students, and can make such a difference to the atmosphere of the room.

Another interesting aspect of the book was how the meaning of symbols (letters and numbers) became messed up after her stroke. She wasn’t even able to call the ambulance, but did manage to call her office, most likely due to a “kinesthetic” memory. That is, we can remember where numbers are on a keypad, without actually remembering the actual numbers, or what they mean.

Once again I thought that has interesting implications for learning math, especially for those students who really struggle with the symbolism.

There are all sorts of insights we get about brain hemispheres, learning – and forgetting – in this story.

Here’s Taylor speaking at TED. At the time of writing, it’s been viewed by 1.7 million people.

She also appeared (I guess inevitably) on Oprah. There’s a ballet based on the story and a movie in the works, too.

Related posts:

1. Friday math movie – Daniel Tammet – The Boy With The Incredible Brain
2. Friday Math Movie – Deep Brain Stimulation
3. Friday math movie: Right Brain Math – Times Table

From the TED summary:

With his team at SENSEable City Lab, MIT’s Carlo Ratti makes cool things by sensing the data we create. He pulls from passive data sets — like the calls we make, the garbage we throw away — to create surprising visualizations of city life. And he and his team create dazzling interactive environments from moving water and flying light, powered by simple gestures caught through sensors.

There are some interesting data visualizations here, too.

Related posts:

1. Friday math movie: Artfully visualizing our humanity
2. Friday Math Movie – Imagine Leadership
3. Friday math movie – The mathematics of war

Autograph claims to be a "3rd generation graph plotter for schools and colleges". It’s clear as soon as you start using Autograph it’s been designed by a math educator, for the purpose of math education.

A teacher can produce objects that students can explore, but even better, it is simple enough to use so students can create their own graphs. This is an essential requirement for such software.

At first glance, Autograph shares features with two other applications I’ve used:

• GeoGebra, a free community based offering, which is strong on geometry but lacks 3-D support (although this is under development for version 5). GeoGebra is available for PC, Mac and Linux, since it is Java-basesd, but it won’t work on tablets; and
• Livemath, a commercial offering which is no longer under development. Livemath has algebra features (e.g. factoring, solving equations, matrix operations and calculus) that Autograph and GeoGebra don’t. Both Livemath and Autograph provide a Web page embed option, so you can view the graph and interact with it using a free plugin, without having to install the full application. Available for PC, Mac and Linux, but not tablets.

However, Autograph is still worth a look becuase it is a user-friendly, integrated package. (It’s around US$100 for a single user license.)

It’s available for PC and Mac, but like the other 2 products, it won’t work on iPad or other tablets.

For this article, I attempted to do similar things I did earlier in my GeoGebra review article, to see how well Autograph performed in comparison.

Disclaimer: The developers of Autograph sent me a review copy, but this is otherwise an independent review.

2-D graphs

You can enter your equation in familiar y = f(x) form, and Autograph will plot it with a minimum of fuss.

In this first example, I’ve plotted a cubic curve (in magenta) and a tangent to the curve. There’s also the derivative curve (the blue dotted parabolic curve).

You can also easily animate each plot, by clicking on the turtle icon:

This is useful for making things more dynamic (a point moves along the x-axis, while the graph is traced out at the same time), and for helping students understand how a particular graph "works".

Geometry

Like GeoGebra, Autograph allows you to draw lines between 2 points (and triangles, rectangles, etc), find mid-points, perpendicular lines, intersections between lines, and so on:

With 3 points selected, you can right click to get the following options.

There are similar options (and the context menu changes appropriately) for 2, 4, or more points.

Statistics, too

Apart from histograms, scatter diagrams and binomial distributions, the statistics options allow you to create best fit curves, like this one through 7 data points:

3-D plots

3-D plots is one area where Autograph is a stronger product than GeoGebra. Geogebra does not have native 3-D graph support (athough they are working on this for version 5 of GeoGebra as mentioned before).

It was very easy to produce the following conic section plot in Autograph, following along with their video tutorial. You can include several "variable constants" when creating your plot, and then change the value of the constants while exploring the resulting curve.

The pinkish plane in the plot was graphed using z = ax + b. You can then change the value of a or b to move the plane around (and thus see the various conic sections).

You then use the "Constant Controller" to vary the values of the constants.

You can animate the changing values and see the effect on the graph. This feature is also available in Livemath.

Calculus

You can use Riemann Sums (using either rectangles, trapezoids or Simpson’s Rule) to approximate the area under a curve:

You can also draw an integral function from a given function (you click on the graph to provide an initial value (the point through which the integral curve passes).

Volume of Solid of Revolution

It was easy to set up a demonstration of volume of solid of revolution, especially when following the provided tutorial.

Modelling a Human Cannonball

This example is from their video tutorial page and shows what can be achieved when modelling using Autograph.

THe problem: A person gets shot out of the end of a cannon. What is the trajectory?

You can model the trajectory and modify parameters until your model lines up with the image, like this:

The dreaded y = arccot(x) function

This has become a standard test for me when exploring new graphing software. Different math software designers use 2 different possible graphs for y = arccot(x). See Which is the correct graph of y = arccot(x)?

Autograph produces the "second" interpretation, which assumes an original domain of −π/2 to π/2:

Output formats

You can output Autograph files as either:

1. A BMP or WMF image of the graph
2. A copy of the equations used
3. A copy of the status bar (for areas, and other output values)
4. An HTML page

Th HTML output option is interesting. The file sizes are nice and small (9 kB for the proprietry .AGG Autograph file, and 8 kB for the HTML file) and so it loads quickly. It even works in Internet Explorer!

The minus point about the HMTL output is the user needs to download the Autograph plugin, and there is always resistance (and confusion) associated with this process.

Example HTML output: Here is one of the Autograph files in HTML output:

Riemann Sum – Autograph file

You can interact with this graph by moving it around, selecting parts of it, sketching on top of it, and so on.

In many ways the Autograph HTML output is better than GeoGebra’s huge and slow-loading Java files.

Gripe: unexplained levels

Most people don’t dig into manuals when using new software (well, I certainly don’t) – they just play with it and see what it can do.

The first time you open Autograph, you are given this dialog box:

There is no explanation on the radio buttons about what these levels mean or the implications of choosing one over the other.

So I happily started using the product (having accepted the default value of "Standard"), and found it quite limited. I couldn’t even change from degrees (the default) to radians (which I almost exclusively use these days).

Some graphs didn’t seem to plot properly and I was getting pretty frustrated.

After digging around in the documentation, I found out what "Standard" and "Advanced" level were for (the former gives a simpler subset of tools for newbies, the latter gives you full options).

There’s nothing wrong with that as a design approach, but it should be explained in that very first dialog box! A simple list of features you’ll get if you choose "Advanced" would have made a lot of difference to my first 30 minutes using the product.

Video tutorials

The short videos on the following page show you some of the capabilities of Autograph, and how to achieve them:

Autograph video tutorials

Conclusion

Autograph is a respectable graphing package which is designed by educators for educational use.

The 2 modes ("advanced" and "basic") allow beginners to explore 2-D graphs with few complications, while the "advanced" option gives you many more ("grown up") things to play with.

This package is suitable for grade 6 to early college level math course.

The price point is probably on the high side, especially considering GeoGebra’s upcoming 3-D developments, but Autograph is certainly worth checking out.

Related posts:

1. Online graph plotter using JSXGraph
2. GeoGebra math software – a review
3. How to find the equation of a quadratic function from its graph

This is part 1 of 3 videos on this topic.

As a follow up, blogger MathHombre constructed a GeoGebra file where you can explore the concept of logarithmic spirals. (It takes forever to load, but worth waiting. Move the sliders or click the tiny right arrow at the bottom left of the file to run the animation.)

Logarithmic Spirals in GeoGebra

See also my article on Golden Spirals, a special case of Logarithmic Spirals.

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3. Friday math Movie – Math Rock and Fibonacci

Biorhythms are interesting, but not at all scientific. They are a quirky example of Composite Trigonometric Graphs, since they involve 3 sine curves starting at (0,0) on your birthday, and having 3 different periods.

They work on the principle that our bodies, minds and emotions go through various cycles throughout the month, leading us to have pre-determined "good days" and "not-so-good" days. I have assumed the "best" days (colored green on my graph) will be the days when all 3 cycles add to give us the largest amplitude, and the "worst" days (colored red) will be when all 3 cycles push us into the depths of despair.

According to the Skeptik’s Dictionary, biorhythms were first "observed" by the Berlin physician Wilhelm Fliess (a good friend of psychologist Sigmund Freud’s) in the 19th century.

Fliess originally proposed a 23-day "male" cycle and a 28-day "female" cycle. An Austrian engineering teacher proposed a 33-day "mind" cycle, based on his observations of how well his students performed over time.

They are now known as the physical, emotional and intellectual cycles. Apparently:

There is the 38-day intuitional cycle, the 43-day aesthetic cycle, and the 53-day spiritual cycle.

Head on over and put in your dates here:

Biorhythm Graphs

Math behind the graphs

Periods: To get graphs having wavelengths of 23, 28 and 33 days, I used these sine curves:

• Physical: y = sin(2πx/23)
• Emotional: y = sin(2πx/28)
• Intellectual: y = sin(2πx/33)

The period of the general graph y = sin(bx) is 2π/b. In the first graph, b = 2π/23, so the period is 2π/[2π/23] = 23.

Composite graph: A "composite" graph is the result of adding the y-values for 2 or more given graphs.

Bars: For the green and red bars used for the "good days" and "not-so-good" days, I used Reimann Sum rectangles. These are used to approximate the area under a curve. (See also this Riemann Sums java applet.)

Conclusion

Don’t take biorhythms seriously! However, hopefully you’ve learned some math from this example.

The link again: Interactive Biorhythm Graphs

Related posts:

1. Trig graphs – how do you feel today?
2. IntMath Newsletter: Biorhythm and parabola graphs
3. The IntMath Newsletter – April 2007

One of my readers, Dave Fashenpour, kindly agreed to do an interview to answer the question – why would a twice-retired space scientist become a math teacher?

The US is keen to recruit work-experienced people like Dave to teach math.

Dave used mathematics as a tool throughout his career in the military then in NASA, and continued to develop a love for the subject. This led to his recent career choice as a math teacher.

He writes that teaching math was the "hardest thing I have ever done in my life".

Dave wrote an interesting research paper about ways to teach mathematics in order to reduce "math fear" during his re-training college days. We talk about it in the interview.

Tell us about your previous careers in the Military & Aerospace industry

Dave: With a couple years of college studying liberal arts behind me, I joined the US Air Force and was sent to a 1 year technical school and learned how to repair secret code (crypto) computers utilizing binary and hexadecimal number systems and even some high level number sequencing methods. 

Six years went by and I was a Staff Sergeant, working all day and taking math courses by night. I was finally selected to an education and commissioning program; sent to Colorado State University to get a BS in Math; and then off to OTS (Officer Training School) to receive a commission as a Second Lieutenant. During the remaining 14 years as an officer, I taught communications-electronics courses to new officers entering the Air Force and received a Master Instructor Award. I also was selected to return to school to get an MS in Computer Science from the University of Southern Mississippi.

International Space Station

Retiring as Captain with twenty years of service, a Top Secret security clearance, experience with modern communication-electronics systems, and an MS in Computer Science, I found a job rather easily with NASA in Houston as a Software Engineer for the Boeing Company. I was assigned to the International Space Station Program (ISS), involved in the design of the communication bus protocol that would “web” through the entire station. Mathematics came to the forefront again, in the creation of deterministic timing methods and data-bottleneck analysis schemes.

Math was also prominent in the design & verification of the Commands and Telemetry signals that would be communicated back and forth to the Space Station’s orbiting laboratories; all in four-digit hexadecimal groups, processed & translated into human readable displays.

I was fortunate enough to see all of these designs and plans become operational in the ISS and participated in the final missions to complete the last construction phase of the ISS. That took another twenty years and resulted in my retirement from Boeing.

How much math did you use in your work?

Dave: As stated above, mathematics played an integral role throughout my first two careers, as a result of computers, communications, and electronic systems.

It wasn’t until my second retirement, that math became a bigger deal.

I entered the University of Houston to take their graduate level series of courses in Mathematics Education, which was to lead to a certification as a Secondary Level Math Teacher. Math suddenly took the front-seat of my attention − mainly because my Math degree was from 1970. I made a couple attempts to pass the Math Teacher Certification examine and finally squeezed by with a passing score.

I got a job at an inner-city high school (Jones HS), and was assigned to teach 11th & 12th grade math. Jumping right into Algebra II, AP Algebra II, and Calculus for six hours a day (averaging 25+ students in each class) was the hardest thing I have ever done in my life and I didn’t even last to the Winter Break! I decided to take a vacation and to visit my terminally-ill sister in Florida and that was my most memorable vacation.

The next year I responded to an ad for a “Math Fellow” in a new program out of Boston, transplanted to Houston, Texas – called the Apollo-20 Program. I was assigned to teach/tutor mathematics to two (2) kids each class period, for six hours a day (totaling 12 students). I was hired and was sent to the sixth grade Math Lab at Dowling Middle school.

There were “GT” and special-education students, which we tried to group accordingly. This program is integrated into the regular curriculum and each of these students also has a traditional 6th grade math class, in addition to the hour that we spend tutoring them. Sometimes we prepare them for upcoming subjects and sometimes we re-visit problematic topics. The kids love their Apollo time!

I am half-way through my second year. The results of year #1 were fantastic – so there wasn’t even a question of not continuing the program (which is funded by local business leaders). It is a great job for a double-retiree, but the money is just slightly more than a burger-flipper.  The photo shows two of my sixth grade students at Dowling Middle School, in Houston, Texas.

Here’s more information about the Apollo 20 Math Fellow Program:

Apollo-20 Math Fellows Program, which is an overview of the program. They are "seeking dynamic candidates to commit to improving the academic achievement of inner-city students."

Apollo 20 tutoring program seems to be making strides, which is a Houston Chronicle article about Dave’s work.

What inspired you to become a math teacher, as your 3rd career?

Dave: The experience of teaching in the military gave me a “warm-fuzzy” feeling when I thought back on it; so going into teaching seemed like a no-brainer (turned out I would need my brain).

The single, most satisfying event for me is the “awww” or the famous “oooo” responses – seeing the eyes light-up and verbal acknowledgment that we just cut through the hard-stuff and arrived at understanding. One should hear at least one of those sounds on a daily basis to keep your “motor running”.

Tell us about your research paper

Dave: Here’s the paper (PDF, 13 pages):

Taking Away the Fear of Solving Mathematical Problems

The paper addressed the emotions behind success in math. I have confirmed that there is an emotional hang-up with learning mathematics over and over. From the original research for the subject paper, to the classroom of Jones HS, to the 3-man desks of the Dowling MS Math Lab; many people have a predisposition to hate or not like certain aspects of math. Some just don’t like word, e.g. Algebra, Probability, etc.; others have had a bad taste from past experiences in math.

So, what does one do about it when you encounter an “I HATE THIS.” I have learned from on-the-job to start simple, repeat, critique & praise, and slightly increment the difficulty – and repeat.

Just today a 6th grade girl said I hate this kind of problem. I asked her if she hated falling off her bike when was learning to ride (she said yes). Then I asked if she still hated riding her bike, after she learned how to ride it (she said no). So then I explain that she will be learning how to solve this problem, just like she learned how to ride her bike. You won’t hate it then. She smiled.

I wrote the paper before I was a teacher. True I was inexperienced in what I was writing about and have since been able to experience the other side. Surprisingly, I did attempt to use many of the described teaching strategies. From the basketball court, showing how the talented basket-maker actually is computing the path, speed, and destination of the ball in his/her head. That connection caused some follow-on discussions and questions. The spatially adept learners love manipulatives and the non-verbal types enjoy computing silently in their head — surprising everyone with the correct answer.  Everyone is different and have a variety of needs and desires.

I really was not prepared to conduct differentiated instruction to my large HS classes of juniors and seniors – but I tried. I found myself walking the room and spending an inordinate amount of time with a special needs senior and having to skim by other needs (the special needs student still emails today, telling me that I was his most best teacher ever).  When you just have to deal with two students at a time (eyeball to eyeball), as we do in the Apollo-20 Program, it is VERY easy to meet the students where they are and to serve up the exact portion they desire.

My view has changed. Teaching is hard – even just doing the minimal tasks required each day. Any extra icing on the cake takes extra effort – analyzing the optimum cognitive strategy for each student can only be done as time and situations permit.

In this my 3rd career, I work harder than I ever worked in the first 20 years or the 2nd 20 years of my previous careers and I think I may not make the full 20 years in this job!

Which activities (in the paper) have you tried out in a real classroom? How did they go?

Dave: The role curiosity plays in the affectivity of problem solving (Goldin, 2000). While the concept of internally generated interest is sound, finding out the student’s special interests and/or skills & abilities is not easy to do. It is time-consuming and requires an informal environment – then when you finally determine some specific interests (i.e. rap music, TV, opposite sex, soccer, etc.); it is hard to connect those to the curriculum topics. But one must be ready to react if the opportunity presents itself – like “I want to design a video game”, then to be ready with–  let’s talk about object oriented software programming and what someone would need to do, in order to be able to design, create, and program a video game.

The affect of interest and excitement in getting started solving mathematical problems (Ormrod, 2008). Reading popular magazines or checking the web for the best widget available, certainly captures their interest; but making these activities relevant and mathematically meaningful is tough. I have found that ‘shopping examples’ play well with money exchange topics, as well as percentage bargain dollars-off the original price, and for determining taxes. Recipe measurement conversions convey some interest to the future cooks and geometric measurements sometimes appeal to the wood-worker or car enthusiast. But beyond those low-hanging fruit, excitement seems hard to come-by.

The Credit Card project changed moans to laughter and enthusiasm (Marks, 2000). Teaching annual percentage rates and monthly finance charges, the credit card type exercise would be fantastic. Lesson plans do not get that sophisticated in 6th grade math and I am not sure I could fit a full-blown Credit Card project into my limited time. But the concept is a good one. Groups designing the art-work for their own Credit Card design are practicing creativity and graphic design. Groups deciding on a specific set of on-line purchases that meet their needs and at the same time keeping the credit card’s monthly payments as low as possible, is an exercise in maturity.  Plus there certainly is a fun factor – which is always a good thing.

The set of cognitive abilities that makeup the theory of Multiple Intelligences (Gardner, 1983). This is a very important topic – understanding why people think differently and what can be done to facilitate those native abilities. I have used insights into different cognitive abilities successfully in three areas: the hands-on learner playing with fraction-sticks for example; the physical athlete pacing off feet on the basketball court; and the most common being a visual learner who can easily understand a picture or a graphic display. One other aspect of a unique cognitive ability was with a special needs student that was lost until class instruction was translated into procedures: step 1, step 2, and etc., all written down and performed repeatedly. Once mastered, there was no stopping him — with the downside being he did not want to leave a subject that he had mastered.

The use of Multiple Representations, which illustrate four different approaches to teaching the same mathematical concept (Jones & Swan, 2006). This theme has already been covered in the above paragraph, because once you understand multiple intelligences, you understand that problems need to be presented and represented in a variety of different approaches, according to the individual needs of the student. This article simply identifies four different ways to solve the same problem, which is very useful – especially in a tutoring environment.

Multiple Strategies, which presents a single problem solved with five different techniques (Meyer, 1999). Multiple Strategies such as: draw a picture, use a model, make a list, eliminate possibilities, use symmetry, solve an equation, look for a pattern, work backward, and guess a & check; are a set of techniques or strategies that are used successfully to solve mathematical problems. These remain in our bag of tricks to be utilized when the individual need arise.

If you had the power to change one thing in math education – what would it be?

Dave: The kids that come to the US from other countries almost all know their basics: addition, subtraction, multiplication, and division. That is extremely useful and easy to build upon. The struggling kids that were educated here just do not have a firm confidence in the basics. Case in point, at a HS in Houston, a senior girl was called to the board to work a problem and she could not solve a simple multiplication problem without a calculator. She was graduated.

If only memorization of the Multiplication Tables from 1 through 12 was mandatory and required for entering Grade 6, a new level of sophistication could be realized. It would not surprise me if we could cover double the material than we currently have to wade through; without stopping to count on their fingers, or draw hash marks on the paper, or write all of the multiples of a number down, before deciding how many are required.

In the meantime, we are here to “give a year” and to “change a life”.

David R. Fashenpour
Houston, Texas
10 Jan 2012

Questions to: dfashenpour@gmail.com

Thank you Dave, and all the best with your third career!

Related posts:

1. What is a good teacher?
2. What is a math teacher worth?
3. A good teacher…

[Image source]

On the left is a square with sides 8 units. This gives us 64 small squares. After cutting the square into 4 pieces as shown, we can re-arrange them to give a rectangle with dimensions 13 × 5 units. This gives us 65 small squares.

Where did the extra square come from?

Try to solve it before watching the video!

This week’s movie gives us some insight into this problem.

Video by: Math Pickle – Million Dollar Unsolved Problems.

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3. Friday math movie – the Rain Man in each of us

I totally agree with this one. Studio Schools was developed by the same people who gave us the Open University concept. They asked a simple question – what would make teenagers fight to get into a school, rather than fight to get out?

It’s fairly close to the apprenticeship model. I’ve always felt it’s a shame that model disappeared.

The Studio School is not perfect, of course – but I’m not sure that any school system is perfect for everyone. The concept is certainly worth investigating, since it involves students in potentially more meaningful learning experiences than what happens in many conventional schools.

Related posts:

1. Friday math movie: Intel Schools of Distinction winner
2. Friday math movie – string theory
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If you have studied math, the word “calculus” brings back memories of differentiation and integration, finding maximums, and finding areas under curves.

But calculus can also mean “calculation” (as in “by my calculus, it doesn’t make sense to own a car”), or a “system or arrangement of intricate or interrelated parts” (source: Webster’s Dictionary).

A final meaning for calculus is something I face each time I go to the dentist – a build-up of mineral salts on the teeth. It’s also called “tartar”.

And this is the connection with math. The Greek word “calcis” means various stones, giving rise to the word “calculate”, or using stones for mathematical purposes.

So before you think I’ve got rocks in my head, on with the show.

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2. Friday Math Movie – Archimedes
3. Calculus Made Easy (Free book)