Milton Chen Event

I attended a FutureofEducation.com event yesterday.  This was a talk with

Milton Chen, the former director of edutopia, the George Lucas foundation enterprise.

He was promoting his new book “Education Nation, Six Leading Edges of Innovation in our Schools”.  I came in late, so I did not get to learn what the 6 leading edges were, but what I heard was still quite interesting.  The book has a picture of a power up button on the cover and Chen is a big proponent of innovation and technology in education.  Chen is highly supportive of the state of Maine’s efforts in using educational technology.  Maine has a program where every student has access to a laptop with broadband access.  The cost of this program is $250 per student which Chen points out is not much more than the average cost of a textbook ($158).

He is also all for common core (national) scholastic standards, and wonders why it took so long for us as a nation to get to the point where we are seriously considering them.  He carries the philosophy that most of the education movements (home school, charter school, etc.) have something useful to offer, and likes to take the best from all of them.  He sees the leading edge of the work of edutopia as the documentary films on the website.  These are extensive and offer lengthy (30 minute) clips of various schools with innovative programs.  There are over 150 different schools filmed and archived.

GeoGebra Applet

I have written an applet for demonstrating reflections and rotations of a function about the x and y axes, and origin.  I was unable to embed the applet in my blog, but it is embedded on the class wiki, here.  The explanation of how to use the applet is with it on the wiki page.

I can see a couple of instructive uses for this applet.  One use is as a demonstration tool during a lecture.  A second use is as an independent study tool for students to explore symmetry and reflection of functions.

Symmetry Assignment

Symmetry occurs when parts of an object are identical to other parts of the same object.  In this lesson, you will explore three types of symmetry.

Try to get as far as you can with the tasks below.  If you get stuck, here are some additional resources that discuss some of the topics below:

http://home.windstream.net/okrebs/page43.html for Parts 1 & 2

http://www.purplemath.com/modules/symmetry3.htm for all parts

http://ed526b.wikispaces.com/Applet+Collection Look at the Symmetry and Functions applet.  This will help with all parts of the assignment.

Also, you may discuss the problems with each other.

Part 1
Using GeoGebra, type in f(x) = 4 – x^2 and g(x) = x^2 – 4.
What do you notice about the two graphs?

Now try f(x) = 1 – x^3 and g(x) = x^3 – 1.  What do you notice?

What do you notice about the combined graphs as an object?

We say that the objects made by the combination of f(x) and g(x) is symmetric about the x-axis.  Do you see why?
We call f and g reflections of each other in the x-axis.

Can you find a rule for creating the reflection in the x-axis, g(x), of a function, f(x)?  Can you give an argument why the rule is true?

Part 2
Now compare f(x) = 1 – x^3 and g(x) = x^3 + 1.  What do you notice?
How about f(x) = x^2 – x and g(x) = x^2 + x?
How about f(x) = x^3 – x and g(x) = x – x^3?  What’s happening?

We say that the object made by the combination of f(x) and g(x) is symmetric about the y-axis.  Do you see why?
We call f and g reflections of each other in the y-axis.

Can you see a rule for creating the reflection in the y-axis, g(x), of a function, f(x)?
Hint: For each value x, what does g(x) need to equal to be the reflection of f?
Can you give an argument why the rule is true?
Can you see a characteristic of functions that are the same as their reflection in the y-axis?
Hint: Such functions, which have y-axis symmetry, are called even functions.
Question: Why (with one exception) aren’t there functions with x-axis symmetry?  What is the exception?

Part 3
Now compare f(x) = 1 – x^3 and g(x) = -x^3 – 1.  What do you notice?
How about f(x) = x^2 – x and g(x) = -x^2 – x?
How about f(x) = x + 2 and g(x) = x – 2?

We say that the object made by the combination of f(x) and g(x) is symmetric about the origin.
We also call f and g 180 degree rotations about the origin of each other.  Do you see why?

Can you see a rule for creating the rotation about the origin, g(x), of a function, f(x)?
Hint: For each value x, what does g(x) need to equal to be the rotation of f?
Can you give an argument why the rule is true?
Can you see a characteristic of functions that are the same as their reflection in the y-axis?
Hint: Such functions, are called odd functions.

Assessment questions:
1.  What does it mean to be reflected in the x-axis?
2.  What is the difficulty in finding functions that have x-axis symmetry?
3.  Give two examples of polynomials that have y-axis symmetry.
4.  Give two examples of polynomials that are symmetric about the origin.
5.  If a graph is symmetric about both the x and y axes, does it have to be symmetric (by    180 degree rotation) about the origin?
6.  Create (using GeoGebra) with your own inputs:
a graph that is symmetric about the x-axis,
a graph that is symmetric about the y-axis,
and a graph that is symmetric about the origin.

Overview of Mathematica

I attended the virtual overview of Mathematica today (at 10 am EST).
Mathematica is among the most powerful computer programs for doing mathematics, and now has many other applications.  I think the presenter, with his great enthusiasm, bit off more than his audience could chew in a one hour session.  Mathematica has some very nice abilities for the teaching of calculus.  It is very handy with making slides – the presenter whipped off a set of integration slides in literally a few seconds.  It also has a tool that makes it easy to create homework assignments, and even easier to grade them.  This feature, however, was not presented since time ran out just as the presenter was getting to it.  What was presented were some of the basics, like syntax, and then a wealth of information that mainly had to do with advanced uses of Mathematica in various fields of research.

We learned, very quickly how to:
Solve systems of equations, do regression and interpolation
How to do integrals with Mathematica
How to use a palette to enter commands as if in longhand
How to do 3D plots, presenter made a nice upside down cone
That Mathematica does very difficult integrals such as Integral[(sqrt( tan x))] or the Integral[sin x/ e ^{-x}]
How to solve a system of partial differential equations.  How to plot 3-D parametric equations.  He plotted a beautiful 3-D butterfly pattern
In 3d plots – how to choose x and y range, color, restrict domain etc..
How to do stream density plot as in plotting an electric or magnetic field.
Visualization of graphs in Mathematica is excellent.
Lots of data is available for manipulation and plotting; plotted finance data for Google and weather data for Sacramento high temps with single commands.  27500 proteins available on data with detailed 3-D rendering
There is a tool that can construct a food chain diagram intuitively
Generate a complicated random walk with one line of code
Three methods of Programming in mathematica
Procedural program which has loops like in C, Pascal or Fortran
Functional programming :  makes use of built in Mathematica functions: for example, you can use Norm or Euclideandistance to find distance.
Rule based approach: This finds a pattern and replaces it everywhere with a calculation
Pattern matching  (genome data) using rule based program to find substrings in a gene
Parallel computation – If you have access to a network or multiprocessor, Mathematica can do
heavy calculations in parallel, which makes them run faster.
Advanced image processing
Lots of external connectivity — to data bases e.g.
Tools for making slide shows
Generating and correcting homework assignments
Mathematica Community:  Math group for posting questions, Web Group

Mathematica is probably too powerful and costly for high school use.  However, it does have many awesome tools for solving math problems, doing graphics, and creating lecture slides.  If I were teaching at a college, I would be excited about using it.

Symmetry Lesson Plan for Middle School

I reviewed a lesson plan I found at

Click to access corporatelogo.pdf

This is actually a lesson plan by PBS | Mathline on using symmetry to create corporate logos.
I thought it was a nice lesson plan.  Don’t be put off by its 14 pages – it is actually three copies of one five page plan.  The objectives are about learning to recognize different types of symmetry and creating figures with those types of symmetry.  The target grades are 6 to 8.  The main activity is the construction of a symmetric object, similar to a corporate logo.

Students learn about reflection, rotation and translation symmetries.  I like the first exercise where the students identify which letters of the alphabet have vertical and horizontal reflective symmetry (or both).  I though using very familiar objects was a nice way to get the students minds thinking about this concept.

For rotational symmetry, they could have done more in providing examples.  They did not provide an example of an object with rotational symmetry, even though so many exist.  They could have used a starfish, sand dollar, hubcap, etc…  They also could have made it clear that you may rotate by any angle (though choosing an angle that divides 360 makes things simpler), not just 90 degrees.  It also would be nice to have an exercise, as there was for the concept of reflective symmetry.  How about showing some common shapes and asking students to identify how many axes of rotational symmetry there are?

For translation, they also could have done much better in providing examples.  For instance, a picture of a traditional Asian rug would have been a good example.

One thing that is missing is a method or evaluation.  For this activity, a rubric would be a really helpful guide for the students, as well as for the teacher to evaluate their work.

I think this is a good plan in terms of active and authentic learning.  Dealing with designing and constructing, it is very active.  Using nature and corporate logos makes it feel authentic (real world).  It is also interesting to think about the lesson in terms of Bloom’s Taxonomy and the Van Hiele model.  It does the first level of Bloom’s well, introducing the three types of symmetry.  There is plenty of opportunity to get to all levels of the taxonomy.  It could have more exercises to help with transforming that information into understanding.  Implicit in the main task is the level of applying and creating.  Analyzing is needed for the alphabet exercise, and more exercises would increase the amount of analyzing and/or evaluating in the lesson.

This exercise takes place mainly at the first two stages of the Van Hiele model.  It has a lot of visual qualities to it.  It also requires students to reflect, rotate and translate geometric shapes, and to join them in various ways with copies of themselves.  This activity, I believe, will help develop higher level geometric reasoning.  However, it may do this intuitively rather than in the verbal sense of the Van Hiele model.   One needs to develop some sense of the properties of geometric objects in order to successfully manipulate them to make constructions.

The Van Hiele Model

I learned about the Van Hiele model of geometric understanding from the site http://www.homeschoolmath.net/teaching/geometry.php.  I liked the version of the model offered by the site because it seemed very practical.  The model is best used as a guide, for elementary teachers to foster development of geometric skills or as a diagnostic tool to help high school teachers to understand what is causing a particular student’s difficulties in high school geometry.

The model was developed by two Dutch researchers, Dina van Hiele-Geldof and Pierre van Hiele as a method for analyzing the difficulties their students were having in higher level geometry.  They believe that all learners must progress through a sequence of stages of mastery in order to reach a point where they can undertake the reasoning needed for high school geometry.  In their model, one must master each stage in turn before proceeding to the next.  The first two levels seem at about a primary school level, the third middle school, the fourth is the level used for high school geometry, and the fifth is the level of upper level undergraduate mathematics.  In a nutshell, here are the levels:

Level 1 – visual. Geometric figures are recognized based on their appearance

Level 2 – descriptive/analytic. Students can identify properties of figures and recognize them by their properties. They cannot tell a difference between the necessary and sufficient defining properties of a shape, and extra properties of a shape.  For example, they would understand that a recangle has four sides and four right angles.

Level 3 – abstract/relational. Students can now understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes.  For example, they would understand that a recangle has four sides and four right angles and also understand that this description is sufficient to describe a rectangle.

Level 4 – formal deduction. Students can reason formally using definitions, axioms, and theorems. They can construct deductive proofs starting from the givens, and producing statements that ultimately justify the statement they are supposed to prove.

Level 5 – rigor/metamathematical. Students can reason formally and compare different axiomatic systems.

The site stresses that
“This theory is not perfect but based on other research, it seems to model the progress of geometrical thinking.”

and
“The important point is that a lot of the geometry taught before high school does NOT foster students into higher level of geometrical thinking.”

I like the van Hiele model.  If for nothing else, it is the first educational model I’ve come across that applies well to mathematical reasoning.  It does seem like a reaonable framework to use to judge someone’s reasoning level.

In my opinion, one danger with theories of education is they end up being used as an inflexible set of rules.  This site seems to have the right idea about using the van Hiele model:  Use it as a tool to construct a curriculum for children that will prepare them for the rigors of axiomatic geometry.

A visit to the PLN ning

Today I joined the site for educators, http://edupln.ning.com.  I enjoyed the site.  The site contains videos, blogs, discussion forums, Social Media Information, and links to educational and tutorial sites.  The tutorials and social media info could be really useful for someone new to social networking.  There are tutorials on things like facebook, jing (on the linked teacher training videos site) and other social networking tools.

My favorite video was Will Richardson’s (that’s right, the one that is already on the class wikipage for unit 4).  I looked at a bunch of other videos as well:

Myths and Opportunities by Alan November:
A video on  globalization video (about 13:00) – It was a beautiful video, a work of art, but a bit too slow paced for me.

Eric Sheninger on Twitter in education:
A Principal discusses his enthusiasm for twitter.  Many commenters liked his endorsement of twitter

Welcome to My PLE!:
This was a well made video of a 7th grader giving a tour of her personal learning environment. It shows how great the role of the internet has become.

There are many more videos than can be exhibited on the home page.  Its worthwhile to check out the full selection of videos.

Since I’m inexperienced in electronic networking, participating in PLN could be a good tool for my professional development.  I found Will Richardson’s video to be persuasive on this topic; being a member of PLN is a good way for educators to practice networking and attempt to catch up to the kids they teach.  This site is also very useful for its links to educational resources.

Review: “A Local Linearity Approach”, by Teague

This is a short review of “A Local Linearity Approach to Calculus” by Dan Teague.  The article is available for download at: http://www.ncssm.edu/courses/math/Talks/PDFS/icme_8.pdf

I enjoyed reading this 8 page paper.  The author discusses an approach to teaching calculus by using linear approximations of functions.  Since this is what the derivative is, this approach is logical.  The reason he likes the approach (and I agree with him) is that it is intuitive and therefore will help students get over the feeling of having to take things on faith.  He gives 3 detailed examples of uses of the approach.

The first is to argue for the weak form of L’hospital’s rule.  Here I found the use of local linearity to be very persuasive in giving intuitive support to L’hospital’s rule.  The approach serves its purpose very well, changing the proof from a technical argument to a visual and intuitive one.

The next example was the product rule in differentiation.  His method is to use linear approximations for the functions f and g at the point a, and calculate the derivative of their product h=(f)(g) based on these approximations.  This works pretty well, but his statement at the bottom of page 5: “at x = a, the approximations are exact” might do with some further clarification: It is clear that the approximation to h(x) is exact at a, but for h'(x) it is less clear.  It would also be helpful if the calculations to go along with figure 3 were included.

The last example was using a sequence of linear approximations to prove the Fundamental Theorem of Calculus.  This technique is known as Euler’s method.  This yields a nice proof of the fundamental theorem.  However, it is a challenge to complete the assertion that the limit of the sum is the values equals f(b) – f(a).  This may leave the intuition of some students unsatisfied.

This paper will benefit teachers of calculus as well as students.  And also has other significance.  Discoveries are often made through intuition, and this is probably the method that the inventors of calculus used.  The analysis that we learn today wasn’t developed till hundreds of years after the discovery of calculus.

Wolfram | Alpha query engine

Wolfram Alpha is a math query engine, designed by the maker of Mathematica, the high powered computer algebra that does some nifty things such as solving advanced math equations and rendering 3D plots and doing derivatives and integrals.

You can type in anything on the command line, and it will try to figure out what you want.
E.g., if you type in: solve linear equations and press equals, you get the following back:

A menu of things you can do with some preset examples.  If go to the line that says:
solve a system of linear equations, and press equals you get the aglebraic solution, plus
a graph of the solution set.  You may change the equations on the command line to anything you want.

This tool can make math more of an exploration.  Students could make various queries, read the answers, then make hypothesis about mathematical rules that explain their discoveries.

It also has the potential to inhibit deeper learning, since students can do their homework without understanding the mechanics behind the solutions they receive from entering their homework problems.

Linearity Research Project

Applied Math Linearity Research Project

Subject:  Calculus, Statistics, or Applied Math

Level:  Advanced

Objectives:  To develop research and independent problem solving skills.
Students will establish ground facts, then form an opinion which is supported by argument and research.

Materials:  Computers with access to the internet.

The Project:

In statistics, a linear approximation is used, called the least squares method, which finds a line, L, that minimizes the distance to L from a set of data.

In calculus, a linear approximation called the derivative is used.

In this research project, you will compare and contrast the benefits of these two methods of linear approximation.

This research project has several stages:

Part 1:  Get ground facts.
Learn about the basics of the two methods.  Research how they are defined, for why, when and by who they were invented, and what they were originally used for.  List reputable references to back up your findings.

Part 2:  Write an opinion.
Write an opinion about what the advantages, and disadvantages of each method are.  Give examples or cite scholarly references (or both) to back up your opinion.  The opinion should be clearly written.  It should have substance.  It should be followed with specific details and examples so it is clear what your meaning is.

Part 3:  Research Project: The U. S. Social Security system.
Nowadays, there is frequent conjecture about when our social security system will run out of money.  Choose one of the linear models you have researched, and use it to analyze the rate of Social Security collections and expenditures (two separate items).  Get your data on expenditures and collections from reputable sources (e.g. the U. S. government).  From your analysis, predict
(a) The year expenditures become greater than collections.
(b) The year Social Security runs out of money.

Evaluation:

Part 1    50 pts

40 pts      Factual information      (20 pts for each method)
Factual information will be graded for accuracy, clarity and completeness.
10 pts      Quality of references
References should be from standard text books, peer reviewed journals, or expert sources (e.g., a scholar in the field).

Part 2    50 pts

10 pts    Opinion is substantial.
10 pts    Opinion is clearly written.
30 pts    Opinion is supported by argument and examples.

Part 3    50 pts

10 pts    Reliability of data
30 pts    Choice and execution of model
10 pts    Conclusion