c o n c u r r e n c i e s

A Little Inquiry For Teachers

2010-05-26T05:50:00.000-07:00

My summer begins Friday. One of my goals over the summer months is get my thoughts down about the mathematics behind the applet below. It is stunningly simple, but it is the details I want to make clear. So, I present it to you as a sort of summer inquiry activity for teachers. As you move the green points around and as you drag the purple-ish circle around and change its radius, what happens?

Follow Up: After sharing this with some of my students after one of their exams, they expressed a desire to be be able to place points and draw lines, so I updated the applet to reflect their wishes.

I was pleasantly surprised over how much time my students spent with this applet and the quality of their observations.

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Can't Quite Get My Head Around The Numbers

2010-05-21T09:15:00.000-07:00

A lot of BIG numbers is this story.

A month's worth of leaking oil could fill enough gallon milk jugs to stretch more than 11,300 miles.
At worst, it's enough to fill 102 school gymnasiums to the ceiling with oil.
The Gulf has five billion drops of water for every drop of oil.

Random or Real?

2010-05-20T05:42:00.000-07:00

A WCYDWT moment from Mind Your Decisions centering on the online game ARORA. Can you tell the difference between random data an real data?

This Changes My Game

2010-05-16T17:17:00.000-07:00

I have been trying to create moodle questions that get my students to construct something in GeoGebra, get some sort of information from the construction, and answer the question. I have been struggling with the idea that any calculated moodle questions I come up with is ultimately reduced to an algorithm. Nevertheless, if my students are unaware of the algebra behind the algorithm, and need to think of what to construct, then perhaps that is O.K.

One question I have used recently is this:

A circle has a center (17,15) and a radius 59. Write the equation of the circle. Then, find the larger of the two y-values when x=33.

By algorithm, the better algebra students will easily be able to solve this circle equation for the two y-values, but the students in this particular class are not my better algebra students. What I hope would happen is for them to graph the circle in GeoGebra (meaning they must write the equation correctly...#1 goal),

find a good window setting (which is essentially scrolling out until you can see the entire circle...do they know what they are looking for? Goal #2),

graph the line x = 33 (which is difficult to do on my calculator of choice, the Nspire, or other graphing calculators),

constructing the two intersection points, and taking the larger of the two y-values (recognizing that this is the desired answer to the question),

To answer the question correctly, there is still a lot of intuitive mathematical understanding that is being displayed, especially when you ask them why the answer must be 71.79.

I have been encouraging my students to find different web-based resources, lately, and one of my students found a web-based calculator web2.0calc.Imagine my surprise when one of my students did this to solve this problem:

producing this result

Not a CAS, but a pretty nifty numerical solver. And intuitive! When I asked the student what made him try this, he told me that he just wrote the equation, and substituted 33 in for x like the question asked him to do. There were two answers for y, so he used the larger of the two. Intuitive.

THIS changes my game.

Go ahead and try this for yourself on the calculator below.

Function Family Guy (Solution)

2010-05-07T04:45:00.000-07:00

Here is one.

and with the grid and axis hidden

With the "model" for a comparison

What I Wish I Did First

2010-05-03T16:51:00.000-07:00

Though things have worked out fine, I wish I had given this Magic Point first.

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Function Family Guy

2010-05-02T14:24:00.000-07:00

I discussed my approach to function families in an earlier post. The instructions for each of these graphing activities were the same:

Write an equation for each of the following exactly as you would enter them into your Nspire, sketching the graph of each as you go. Only when you have checked your equations and graph with me should you graph them on the Nspire.

Nothing fancy, but it was fun. Kids whined about the numbers (too many decimals), and argued about the finished result (there are two famous football-shaped headed cartoon characters). We did all of these on the Nspire (hmm...might be my first Nspire-related post), but we also used GeoGebra. There are some differences in how you enter the equations which make the Nspire a little better for this activity.

For example, to enter the equation of a parabola from vertex A to B in the graph below, it would look like this on the Nspire

But like this in GeoGebra

I think this might be one area where the Nspire is superior, but even as I write this sentence, I think of how graphing circles and and other equations is so much easier in GeoGebra. You do not need to isolate y in order to graph a function in GeoGebra. You can graph non-functions like x = 4. Ellipses and Hyperbolas are peices of cake (or of cones...you choose).

For example, graphing circles in GeoGebra look like this

but look like this in the Nspire

where you must graph the circle in two parts, top and bottom. Of course, I hope that by now, my students will graph a circle on the Nspire like this,

using function notation and the algebraic representation of geometric transformations, reflecting the top half of the circle in f1 over the x-axis, then translating the reflected circle up six units.

A parabola with vertex A, from A to B.
A parabola with vertex A, from A to C.
A parabola with vertex D, from D to F.
A CUBIC with inflection point D, from D to E.
The LEFT HALF of a circle from B to E.
The RIGHT HALF of a circle from C to F.
A line segment from (1,1) to (1.5, 0.5).
A line segment from (1.5, 0.5) to (1,0).
The LEFT HALF of a circle, centered at (1,-1) and radius of 0.5.
A circle centered at (-3,1) with a radius of 1.25.
A circle centered at (4, 0.5) with a radius of 1.25.
A parabola from the vertex (3,2) and stopping at (4, 2.25).
A parabola from the vertex (-2, 2.5) and stopping at (-3, 2.75).
A parabola with vertex (4,0), opening downward, from (3, -0.25) to (5, -0.25).
A parabola with vertex (-3,0.5), opening downward, from (-4, 0.25) to (-2, 0.25).

Circle centered at (1,3) with radius 0.5.
Square root from vertex (1.5, 3) to (0.5, 3.25).
Square root from vertex (0.5, 3) to (-0.5, 3.25).
Square root from vertex (0.5, 3) to (-2, 2.75).
A cubic from inflection point (-2, 2.75) to (-3,2).
A parabola with vertex (1, 3.75) from (0, 3.5) to (2, 3.5).
A circle with center (-3,1) and radius 1.
A cubic from inflection point (-2, -0.75) to (-3,0).
A square root from vertex (0.25, -1) to (-2, -0.75).
A parabola from vertex (0,-1.25) to (-1, -0.75).
A parabola with vertex (2.25, 3.75) from (1, 3.5) to (3, 2).
The bottom half of a circle centered at (2.25, 2) with radius 0.75.
A parabola with vertex (-0.25, 3.75) from (-1, 2.75) to (0, 3.5).
A square root from vertex (2.75, -1) to (2.5, 1.25).
A parabola with vertex (1, -1.5) from (-1,-1) to (3, -1).

The Magic Point

2010-05-01T11:02:00.000-07:00

I provided this applet to my students a couple of weeks ago. It fit well with where we were at the time, exploring matrix transformations and all. I left it on my class Moodle for a week before we began talking about it in depth. My only question to the students was "What does point D have to do with anything?"

I pose the same question to you.

Or perhaps you will share a different question to ask.

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What If Not - Circle as Directrix

2010-04-14T07:34:00.000-07:00

As another application of What If Not problem posing, let's take a look at the parabola construction. What if the directrix was not a line, but instead, what if the directrix were a circle?

As illustrated in the mathlet below, begin with a circle and a point (Focus) different from the center. Place a point on the circle (Drag Me). As we did with the parabola construction, we are looking for those points equidistant from the Focus and the point Drag Me. This equidistant point will lie on the perpendicular bisector of the Focus and Drag Me, so we construct that next. Following the lead of the parabola construction, we construct a perpendicular to the circle from the point Drag Me; this is simply a radius. The intersection of this radius and the perpendicular bisector is equidistant from the focus and the circle. The locus of this point as Drag Me moves along the circle is an ellipse.

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What is the next logical (at least in my mind) What If Not question to ask? What if the circle was not the directrix, but the ellipse was the directrix? What would happen then?

What If Not?

2010-04-12T17:01:00.000-07:00

I imagine (or I hope) most teachers of Geometry, Algebra 2 and above are familiar with the construction of a parabola. As illustrated in the mathlet below, we begin with a line which serves as the directrix and a point not on the line which is our focus. Place a point on the directrix (labeled as Drag Me!). Since each point on the desired parabola is equidistant from the focus and the directrix, we next construct the perpendicular bisector of the "Drag Me" point and the focus, knowing the equidistant point in question must lie on this line. This bisector happens to be tangent to the parabola (which provides an interesting way to find the equation of a tangent line). We next construct a perpendicular to the directrix at the "Drag Me" point. The point where this line intersects the perpendicular bisector is on the parabola. In fact, the locus of this point as you drag the "Drag Me" point along the directrix is the parabola. If you right-click (or ctrl-click) on this point, you can choose trace the point.

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This is where the Art of Problem Posing and the What If Not strategy comes into play.

I played around with the following idea a couple of years ago in one of my classes. I did this as a lark, not quite certain where it would lead. I am glad I posed the questions!

Applying the What If Not strategy to the parabola construction, I changed the construction question. What if the directrix used in the construction were not the directrix, but instead, the parabola? In other words, what if we mimicked the parabola construction using the parabola as our directrix? What would happen?

Let's begin with a parabola, its focus, and its directrix (we still need this line so we can change the direction the parabola opens). Place a point (Drag Me! of course) on the parabola. To mimic the construction, we are looking for a point that is equidistant from the point Drag Me and the focus. This point must lie on the perpendicular bisector of these two points, so we construct it. Continuing to mimic the construction, we next construct a perpendicular to the parabola at the point Drag Me. To do this, we construct a tangent at the point Drag Me, and then a perpendicular to this. The intersection of the perpendicular bisector and this perpendicular is the desired point equidistant from the parabola and the focus. The locus of this point is known as Tschirnhausen's Cubic.

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What else can you do with this? It is a nice exercise to derive a parametric equation of this cubic. What if you used some other shape, say, a circle for the directrix in this construction? Do you recognize this change in the construction as leading to an ellipse or a hyperbola? What if you repeated this construction using Tschirnhausen's Cubic as the directrix? It's never ending! Applying the What If Not strategy to a common problem is a great way to find something new hiding beneath the surface!

I think I am finally starting to recover...

2010-04-08T12:20:00.000-07:00

I guess I shouldn't complain. I know there are quite a few things worse than spending 11 days in Italy for spring break. I was lucky enough to chaperon my son's Latin Club trip over there. I can't imagine a better trip.

Three days in Venezia and the surrounding Islands
Three days in Firenze with a side trip to Siena
One day in Assisi
Three days in Rome, including Holy Thursday, Good Friday, and Easter Sunday. Actually ate dinner at a place where I could look out the window from my table and see the Colosseum!
One day on the Island of Capri (which has to be the most beautiful place I have been, not including HOME!)

Once I get a chance, I will post a couple of my better math related pictures, like some of the 17 wallpaper groups I found on different manhole covers, or some of the frieze patterns found in some of the tiles at different churches.

The Complete Quadrilateral: Still MORE Points on The Orthocentric Line

2010-03-24T15:14:00.000-07:00

Reflect the Focal Point over each side of the quadrilateral.

Yep...you guessed it.

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The Complete Quadrilateral: The Hervey Point

2010-03-23T15:05:00.000-07:00

Recalling the Four Triangles, the circumcenters of these triangles are concyclic, lying on the Circumcentric Circle, whereas the orthocenters are collinear, lying on the Orthocentric Line. Construct the perpendicular bisector of each circumcenter/orthocenter pair for each triangle. These perpendicular bisectors are concurrent at the Hervey Point.

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The Complete Quadrilateral: Even More Points on The Orthocentric Line

2010-03-22T14:24:00.000-07:00

Are you familiar with the Orthopole? I first ran across this at a presentation by my friend Ray Klein, a T3 National Instructor from the Chicago area. If you are not familiar with it, the following applet will illustrate the construction.

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In the Complete Quadrilateral, construct the Orthopoles of the Four Triangles. The Orthopoles lie on the Orthocentric Line.

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The Complete Quadrilateral: More Points on the Orthocentric Line

2010-03-21T14:13:00.000-07:00

Using pairs of opposite sides, drop perpendiculars from the midpoint of one to the other. These pairs of perpendiculars intersect in six points total, which lie on the Orthocentric Line.

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The Complete Quadrilateral: The Pedal Line, The Orthocentric Line, and The Focal Point

2010-03-20T14:12:00.000-07:00

This makes a pretty picture. Drag the slider slowly to the left to reveal each line or point one at a time. If you do anything with conics and/or their constructions, this is an unexpected conic, which are the best kind!

Draw the Pedal Line. Recall the Pedal Line is the line through the feet of the perpendiculars dropped from the Focal Point to each side. Recall the Focal Point is the intersection point of the four circumcircles of the Four Triangles.

Draw lines through the midpoints of pairs of opposite sides.

These seven lines, along with the four lines of the quadrilateral, are tangent to the parabola whose focus is the Focal Point and whose directix is the Orthocentric Line.

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What Does This Say About Grades and Grading?

2010-03-20T11:38:00.000-07:00

University class swaps grades for experience points

Granted, it is a game design class, but could the same thing be applied to other education settings? Followed a link in the above article to this, conaining the following quote

"The elements of the class are couched in terms they understand, terms that are associated with fun rather than education," [Sheldon] told iTnews. "There will always be a portion of the class who will not be motivated to learn no matter what an instructor may try. Those that are not as involved, one or two out of a class of forty, are pretty much drifting through life anyway thanks to factors the classroom can't really address."

I buy that!

The Complete Quadrilateral: Circles on Diagonals

2010-03-19T04:55:00.000-07:00

Construct the diagonals of the quadrilateral, and then construct circles with the diagonals as diameters. The centers of the circles are the midpoints of the diagonals, and we know they lie on the Midline. The three circles intersect in two distinct points on the Orthocentric Line. Therefore, the Midline and the Orthocentric Line are perpendicular. In circle-talk, these three circles are coaxal, and the Orthocentric Line is the radical axis of the three circles.

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The Complete Quadrilateral: The Orthocentric Line

2010-03-19T04:01:00.000-07:00

Construct the Orthocenters of the four triangles.

Before dragging the slider, make a prediction. What will be true about the four orthocenters?

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More Wallace-Simpson Line

2010-03-18T13:22:00.000-07:00

I guess I should elaborate a bit more on the Wallace-Simpson Line, considering it is used to make a picture that produces the same reaction every time I show it to my kids: THAT is frickin' COOL!

The envelope of every Wallace-Simpson line for every point on the circumcircle is the Steiner deltoid whose area is half the area of the circumcircle. The Steiner deltoid is also tangent to the Nine-Point Circle, the unique circle that passes through the midpoints of the sides of the triangle.

Click on the small animation button in the lower left corner. Click on the reset icon in the upper right corner erase the traces.

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An Integrated Curriculum?

2010-03-18T06:31:00.000-07:00

Might this fall under the category of WCYDWT?

I am the chess sponsor at my school. I play, and I can beat some of the kids in my club some of the times, but generally, I just provide a place to play after school, and I drive the school van to the matches. We actually did OK this past year, finishing 2nd in our division, 2nd in the league tournament, and 1st in the league playoffs.

Anyway, since all the of kids who play are big UFC fans, I had joked with them how if we happened to lose the match, we could win the fight afterwords. And then I am up the other morning watching ESPN and THIS comes on

The Complete Quadrilateral: The Pedal Line

2010-03-17T15:09:00.000-07:00

This theorem reminds me of the Wallace-Simpson line in a triangle. The Wallace-Simpson line is constructed by dropping perpendiculars to the sides (extended) of a triangle from a point on the circumcircle. The feet of these perpendiculars lie on a line. This line is the Wallace-Simpson line.

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In the complete quadrilateral, from the Focus Point, we drop perpendiculars to the sides of the quadrilateral. The feet of these perpendiculars are also collinear. This is known as the Pedal Line. The proof of this follows directly from the Wallace-Simpson line.

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The Complete Quadrilateral: Connecting the Midpoints of Opposite Sides

2010-03-17T03:57:00.000-07:00

When you look at the six pairs of opposite sides, you may notice that the pairs themselves could be grouped into three sets of four that "look alike."

For example, the opposite sides formed by segment A₁₂A₁₄ and segment A₂₃A₃₄ seem to naturally go together with the opposite sides formed by segment A₁₂A₂₃ and segment A₁₄A₃₄.

Likewise, the opposite sides formed by segment A₂₃A₂₄ and segment A₁₃A₁₄ seem to naturally go together with the opposite sides formed by segment A₁₃A₂₃ and segment A₁₄A₂₄.

When you construct segments connecting the midpoints of these three sets of opposite sides, the segments intersect in collinear points.

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The Complete Quadrilateral: The Axis of Mean Distances

2010-03-16T03:47:00.000-07:00

Construct the midline of the Complete Quadrilateral. Drag the slider slowly to the left to reveal the significance of the Axis of Mean Distances.

How would this change if two sides of the quadrilateral are parallel?

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The Complete Quadrilateral: The Mid-Diagonal Line

2010-03-15T04:39:00.000-07:00

The next stop on our Complete Quadrilateral tour is the Mid-Diagonal line, or the Midline. This line goes by many names. Ripert called it the Axis of Mean Distances (which we will see why in the next stop on the tour). It is also known as the Newton-Guass line, or the Newtonian. Whatever it is called, it is one of the earliest known lines associated with the Complete Quadrilateral.

We begin by drawing the diagonals of the quadrilateral, and constructing their midpoints. Two of the diagonals will be familiar (for example, the A₁₂ to A₃₄ diagonal). One diagonal - from A₁₃ to A₂₄ - will seem a bit odd. Nevertheless, their midpoints are collinear, and this line is the Midline.

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Function Families

2010-03-13T11:16:00.000-08:00

During the fourth quarter, I do a "pre-Algebra 2" unit on function families. One of the results I would like my students to see is that any function they will encounter in Algebra 2 can be written in a form similar to the "slope" form I use with my Geometry students.

When we study similar triangles, we go back and visit the equation of a line that they learned in Algebra 1 and look at it through slope eyes.

Consider a line through the points (x₁,y₁) and (x₂,y₂). Place an arbitrary point (x,y) anywhere on the line. I know that the slope triangles shown in the graph are similar, hence, the ratios of the corresponding sides are equal. In other words, the slopes as measured by these two point are equal.

Being this is the case, we can write an equation stating the two slopes are equal

The remarkable thing is we can write the equation of an absolute value graph using a similar form. If the vertex of the graph is at (x₁,y₁) and (x₂,y₂) is another known point on the graph, the equation of the absolute value graph can be written in the form

Likewise, a quadratic with a vertex at (x₁,y₁) and (x₂,y₂) another known point on the graph can be written as

Similarly, the equation of cubic (the transformed parent function y = x³ ) can be written as

Only slight modifications are needed to apply this cubic equation form to any cubic function, or to an exponential function, or even to a periodic function.

In essence, the point-slope form of a line from Algebra 1 can be modified to fit three other function families: Absolute Value, Quadratic, and Cubic.

Try using these equation forms in the applet below. Use point A for the vertex and point B for the other point on the graph. Calculate the rise and the run as you would for a linear equation. For calculating convention, your rise should calculated using y_B - y_A, provided point A is your vertex. Enter your equation in the input line. Click on the RESET ICON in the upper left corner of the applet to create a new pair of points.

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The Complete Quadrilateral: Perpendicular Bisectors of Opposite Sides

2010-03-12T15:34:00.000-08:00

Recall the opposite sides of a complete quadrilateral, and construct the perpendicular bisectors of each pair. Provided the bisectors are not the same line, the perpendicular bisectors of opposite sides will intersect at either a real point or at an ideal point (when the two perpendicular bisectors happen to be parallel). Construct the six pairs of perpendicular bisectors and their intersection points.

Drag the GREEN slider slowly to the left, revealing the bisectors and intersection points one at a time.

I will leave the theorem for you to state.

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The Complete Quadrilateral: The Circumcentric Circle

2010-03-12T04:56:00.000-08:00

Construct the circumcenters of each of the four triangles. The four circumcenters lie on a circle. However, there is more to this circle than that. Slowly drag the PURPLE slider to the left to reveal the centers one by one. As you reveal the circumcenters, can you guess what will happen?

Once again, what will happen when two of the sides are parallel? What about two pairs of sides being parallel?

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The Complete Quadrilateral: The Focal Point

2010-03-11T13:17:00.000-08:00

John Wentworth Clawson calls this point the Focal Point. This point is also known as the Wallace Point and the Miquel Point. For each of the four triangles in the quadrilateral, construct the circumcircle. In the sketch below, the conclusion is rather evident. Slowly drag the PURPLE slider to the left to reveal each circumcircle, one at a time, until all are shown.

Play around with this a little bit. What happens when two sides of the quadrilateral are parallel? What happens when two pairs of sides are parallel?

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The Complete Quadrilateral: Four Triangles

2010-03-11T04:21:00.000-08:00

There are four triangles that will also play a part in our investigation. I am not sure about you, but when I see a whole bunch of triangles related to each other in a systematic way, I begin to think about how their centers might be related.

Drag the RED slider slowly to the left to reveal the four triangles one at a time.

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The Complete Quadrilateral: Opposite Sides

2010-03-10T15:38:00.000-08:00

Here is Part 2 of the Complete Quadrilateral study. Still having to hammer out some definitions, in this post we visually define the opposite sides of a complete quadrilateral. Slide the GREEN slider slowly to the left to reveal the six pairs of opposite sides. Two pairs of sides you are in all likelihood familiar with; four pairs are a bit different and may take a while to wrap your head around.

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Eye Candy

2010-03-10T05:25:00.001-08:00

What is up with THIS?

I sat in on the last 10 minutes (I had crappy luck with my session selection) of a presentation at T³ this past weekend that reminded me of Recursive Explorations With Number Cycles.

The sketch below is based on the iterative rule

which has been modified to be

Under certain conditions, it appears that the points might lie on a hyperbola. At other times, it appears the points might lie on some sort of cubic. Certainly, an investigation is in order!

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The Complete Quadrilateral

2010-03-04T05:24:00.000-08:00

John Wentworth Clawson

Last year, I found this relatively short article from 1919 by John Wentworth Clawson called The Complete Quadrilateral in The Annals of Mathematics, Vol. 20, No. 4. I used this article as a basis for a presentation illustrating some of the amazing things that happen when, essentially, four pieces of spaghetti are thrown onto a table.

What is a complete quadrilateral? Four lines, no three concurrent, intersect in six distinct points, creating what is known as the COMPLETE QUADRILATERAL. Why worry about this? There are a number of concurrency, collinearity, cyclic, and harmonic relationships lurking in this rather ordinary configuration. Over the next couple of posts, we will begin to explore these relationships.

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My First Hit...

2010-02-28T16:26:00.000-08:00

from this South American country.

If You Are Interested...

2010-02-28T16:18:00.000-08:00

The Teachers Teaching with Technology International Conference is this coming weekend in Atlanta. This is one of my favorite conferences...it is relatively small, and I always seem to learn something new about mathematics, the technology (even if handheld technology may be on its last leg), and how to use it.

I am co-presenting with Todd Edwards. You will find information about the topic HERE and HERE. I am also giving one myself. You can find information about that topic HERE.

The gist of my presentation is a unifying theme for transforming functions I found a couple of years ago in Keith Kendig's book Conics. In particular, in Chapter 9, his "Action Reaction Principle" is pretty darned cool...if you search for that phrase below, you will find it.

My Favorite Math Movie

2010-02-27T17:51:00.000-08:00

I could watch this movie every day and never grow tired of it. The low-quality video does not do it justice. Play it with your sound turned up. There is a link to better quality video HERE.

Have You Seen Prezi?

2010-02-27T06:25:00.000-08:00

The following is a Prezi presentation given by my good friend Todd Edwards and I when we had the distinct honor of sharing our work with Metropolitan Mathematics Club of Chicago last autumn. By all accounts, I think it was well received!

Prezi is just another one of those free things that should make powerpoint presentations obsolete. Indeed, Prezi has replaced powerpoints in many or the student presentations at my school.

Nspired Connections from Outside the Box on Prezi

Creating an Applet

2010-02-27T04:12:00.001-08:00

This is weird...a movie of me creating the applet for this post. Pat Ballew asked me how to do this.

The ability to do this easily is what attracted me to GeoGebra almost four years ago. I started out using Cabri on the old TI Voyage 200, then moved to Sketchpad because my school had a site license (which meant my students could actually buy the software foe something like $25). One man's opinion, but GeoGebra is just better. There is no site license to purchase (which means my students can use the software for something like FREE). I look at the new features in the current version of GSP, and it is a good program, but they are playing catch-up. GeoGebra has set the bar HIGH with respect to dynamic mathematics software.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Olympic Analytics or What Would My Sled Weigh?

2010-02-16T10:15:00.000-08:00

Japanese Luger Aya Yasuda was disqualified from her first Olympics due to her sled being overweight. This error is apparently attributed to "accidentally miscalculating a somewhat complex formula," I was curious just how complicated this formula was.

RULES
Pre-Race Checks (Singles)
At the start of all four singles runs, each sled is weighed, each athlete is weighed, and temperatures of sled runners are compared to an official "control" runner.
In singles, the sled cannot weigh more than 23kg (50.6 lbs).

There is no maximum weight for athletes. But men who weigh less than 90kg (198 lbs) may add up to 13kg (28.6 lbs) of additional weight to act as ballast. Women who weigh less than 75kg (165 lbs) are allowed to add up to 10kg (22 lbs) of extra weight. The formulas below determine how much weight may be added:

MEN: ( 198 lbs - body weight ) x .75

WOMEN: ( 165 lbs - body weight) x .75

Also, officials check the temperature of the steel runners on an athlete's sled to make sure they haven't been heated. (Warm runners are not allowed because the heat would decrease the friction between the runners and the ice, making the sled faster.) A "control" runner, located by the start house and shaded from the sun, is used as a basis for the measurement. The competitor's runners must be within 5 degrees Celsius of the control runner's temperature.

I got it...you can add 75% of the difference in your body weight and a predetermined minimum amount in ballast, up to a predetermined maximum amount.

Yeah...that formula sucks. Thank goodness that formula wasn't something like this formula for a Player Efficiency Rating used by those who study basketball analytics:

uPER = 1/MIN * (3PM + [(2/3)*AST] + [(2 – factor * (tmAST/tmFG)) * FG] + [FT * 0.5 * (1 + (1 – (tmAST/tmFG)) + (2/3)*(tmAST/tmFG))] – [VOP * TO] – [VOP * DRBP * (FGA – FG)] – [VOP * 0.44 * (0.44 + (0.56 * DRBP)) * (FTA – FT) + [VOP * (1 – DRBP) * (TRB – ORB)] + [VOP * DRBP * ORB] + [VOP * STL] + [VOP * DRBP * BLK] – [PF * ((lgFT/lgPF) – 0.44 * (lgFTA/lgPF) * VOP))])

where

factor = (2/3) – [(0.5 * (lgAST/lgFG))/(2 * (lgFG/lgFT))]
VOP = lgPTS/(lgFGA – lgORB + lgTO + 0.44 * lgFTA)
DRBP = (lgTRB – lgORB)/lgTRB

Once uPER is calculated, it is adjusted for pace and scalar multiplied so that the league average player-minute is a 15 PER:

PER = uPER * (lgPace/tmPace) * (15/lguPER)

Formative Assessing to the Extreme (A Follow-up)

2010-02-15T16:41:00.000-08:00

I just wanted to take a moment and give an update to an earlier post regarding my attempts to check everything my students do as a means of formative assessment.

There is good and bad.

The good? I know almost immediately what I need to address with the entire class or, by looking at a student's work on three or four problems, what I need to address with that particular student. This alone far outweighs any of the "bads" that follow.

The bad? Some students still show no work, or do no work, regardless of all the "atta-boys" and "atta-girls" I can give. For some of these students, yeah...I know they can do it. For others? I am not so sure, and I need to KNOW.

The good? I keep a tally of how many things each student showed me they could do each day. Show me you can do it, you get a check. Make a mistake, I go over the problem with you, ask you to rework it, and either bring the problem back up to me, or shout out the answer to me, in order to get your check.

The bad? Some students just want a check next to their name. I do not like this aspect of it. "How many checks do I have?" I guess I can't really expect them to get too darned excited about calculating the volume of a square pyramid when you are given an edge of the base and the slant height. This is one area where I would gladly take some suggestions as to how I might handle this better.

The good? Seems to work great for the kids who always do what you ask, and for about half of the kids you normally can't get to do anything.

The bad? Some kids just won't do anything. And for some of the things I must ask the kids to do, I honestly don't blame them. The only job I can think of where you will need to calculate the volume of a square pyramid when you are given an edge of the base and the slant height is in the math teaching industry, and I doubt any of these kids will go in that direction.

The good? Lately, I am checking anywhere from 2 to 6 things a day, maybe 10 to 15 things during a week. It really is taking less time than I thought.

The bad? I am still messing with the number things I am checking for each day. This past week, I looked for things like, "Can you calculate the volume of ______?" or "Can you calculate the surface area of _____?" How many do you need each day? Do I include some of the same types of problems each day, this way, you are showing me you can do this over a number of different times? I do know...

The good? I find it is easy to check over the "synthesis" types of problems, the problems where they need to put it all together. I am also finding that my students are improving at presenting and organizing their work, which adds to the easier part.

The bad? It takes longer than the other problems to check over, so I am tending to get lines. I am thinking of going to each group for these problems.

Will I keep doing it? Without a doubt. I feel I am able to give more specific feedback, better feedback, than I ever could before. I also feel I have a better handle of each kid's tendencies...when I go to grade their summative assessment at the end of the week, I can look for those things. I think the jury is still out on my IEP and 504 kids. Overall, their first attempts at the summative assessments have improved, so that is good. However, they generally do not complete everything for whatever reason (mainly, distractions). I think this is helping my ESL students, too, but that data is kind of sketchy. With my ESL students, I am more aware that ever of language difficulties (slant height vs height, for example). Nevertheless, this opportunity for one-on-one time with them is valuable.

Life After Wolfram|Alpha

2010-02-15T12:46:00.000-08:00

Must be another snow day...A nice W|A article

Pythagorean Theorem Proof Without Words

2010-02-15T11:26:00.000-08:00

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Creativity in the Classroom

2010-02-15T04:49:00.000-08:00

THIS short article made me think.

Our current teaching method is based on the idea that we can teach students to solve problems. It presupposes that we can define the problems in advance, which in turn presupposes that we know the answers. We give the problems to our students and grade them based on the correctness of the answer. But while creativity may help in problem-solving, it’s a fundamentally different activity.

The difference is this: Problem-solving is a repair activity. Inherent in the concept of a problem is the removal of an obstacle or difficulty.

Creativity is not about fixing things that are broken but about bringing new things into being.

Take the example of a broken car. “Fixing the problem” will get you to a working car. It will never get you to a fundamentally different kind of transportation, such as, for example, an airplane, boat or motorcycle.

Problem-solving asks the question “What is wrong?” or “What is broken?” It assumes there is a right answer, and it’s in the teacher’s edition of the textbook. But creativity asks the question “What do you want?” or “What is possible?” It presumes that there are not one but an infinite field of possible solutions.

It is the difference an industrial model of education and an information model of education...

Someone who is trained by our system to be a good industrial worker will only be confused and disoriented by an information-oriented workplace.

I think three recent posts and subsequent discussions on dy/dan best illustrate theses differences between problem-solving and creativity, between an industrial model of education and an information model of education...

Comune Di Trentola (Part 2)

2010-02-14T04:40:00.000-08:00

I used this Comune Di Trentola panorama (as well as some others) earlier in the year when we studied transformations. One construction popular with my students was repeated translations (as shown below).

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Comune Di Trentola

2010-02-13T04:21:00.000-08:00

Take a look around this panoramic view. Pay particular attention to the semi-circular cobblestone design in the street. Try to recreate a drag-testable construction of this street using the GeoGebra applet below.

Comune Di Trentola in Naples

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

More Sports and Statistics

2010-02-10T12:29:00.000-08:00

What would be cooler than to attend a conference that exists at the intersection of statistics and sports?

How about the 2010 MIT Sloan Sports Analytics Conference?

Here is the panel discussion on Basketball Analytics from last year's conference.

Eye Candy

2010-02-09T12:56:00.001-08:00

This is what happens on a snow day...

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

200 Visitors!

2010-02-09T07:28:00.000-08:00

My blog has reached the 200 visitor threshold. I first reached 100 visitors on January 15, a little over one month after I started this blog. The next hundred in a little less.

Thanks to those who have visited and have left comments!

When Will I Ever Use Abstract Algebra?

2010-02-09T07:22:00.000-08:00

What can you do with the symmetry group of an equilateral triangle? I saw this episode of How It's Made a couple of weeks ago. In it, they focused on how they make highlighters. At about the 3 minute mark of this clip, they show how they assemble 3-in-1 highlighters. At about the 4:30 minute mark, they show an application of Abstract Algebra. In particular, an application of the symmetry group of an equilateral triangle, D3.

Recall, the symmetry group of the equilateral triangle contains six elements: The identity transformation, a 120-degree rotation, a 240-degree rotation (they are both either clockwise or counter-clockwise), and three reflections, one in the perpendicular bisector of each side. The result of the composition of any two of these transformations is equivalent to one of the others.

Dri Mark Highlighters - How It's Made - Watch more Videos at Vodpod.

A Few More Super Bowl Stats From A Former Coach

2010-02-09T03:48:00.000-08:00

In light of an earlier post about the role of situational statistics in football, there seems to be a statistical interest in three of New Orleans Saints Coach Sean Payton's decisions during the game: His decision to go for a touchdown on fourth and goal late in the first half, his decision onside kick at the start of the second half, and going for a two-point conversion after a touchdown late in the game. An interesting analysis of all three are found in the post Who 'Dat Gonna Kick Onside to Start the Second Half? at Brian Burke's Advanced NFL Stats.

Three Points and an Orthocenter

2010-02-08T13:55:00.000-08:00

Pat Ballew ran with my earlier WCYDWTRP post over on his blog, so I thought I would return the favor.

Pat's take was an old theorem about the vertices of a triangle lying on a rectangular hyperbola. It turns out the orthocenter of these three points will also lie on that hyperbola, as the dynamic applet below illustrates.

Note: Double clicking on any of the applets that follow will open it in a separate GeoGebra window

All of these following applets were made with the FREE Dynamic Mathematics Software GeoGebra.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

An interesting property of three points and their orthocenter is this: Select any one of these four points. The selected point is the orthocenter of the remaining three. These four points are sometimes called an orthocentric quartet. Drag the slider in the applet below to see what I mean.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Anyway, I began to wonder, what would happen if the vertices of a triangle were restricted to some other conic? An ellipse, perhaps? Drag the red points in the applet below around. Right-click (ctrl-click) in the green orthocenter to turn the trace of the point on.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

What about restricting the vertices of the triangle to a parabola?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

The Goal Should Be Statistics

2010-02-07T13:45:00.000-08:00

A year old, but do you agree?

The Colbert Report

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Political Humor

Economy

It's Super Bowl Sunday, And I Am Just An Old Football Coach At Heart

2010-02-07T04:52:00.000-08:00

Back when I coached, I was a big believer in using statistics to inform my decisions. So, my interest was piqued when I saw THIS article, even though I am not sure what it is getting at. Sounds like some sort of a fantasy league or Super Bowl pool. Who knows.

THIS article begins to get at the heart of the matter, trying to convince the reader that mathematics does matter in football play calling. I was reminded of THIS article, briefly discussing the mathematics behind one of the most controversial coaching calls of this past NFL season, when the Patriots' Bill Belichick decided to go for the first down on forth-and-two from deep in his own territory with a lead late in the game. This lead me to recall THIS article about Pulaski (HS) Academy's Kevin Kelly and his use of statistics to inform his football decisions. Consider Coach Kelly's take on the onside kick:

The onside kicks? According to Kelley's figures, after a kickoff the receiving team, on average, takes over at its own 33-yard line. After a failed onside kick the team assumes possession at its 48. Through the years Pulaski has recovered about a quarter of its onside kicks. "So you're giving up 15 yards for a one-in-four chance to get the ball back," says Kelley. "I'll take that every time!"

However, the article I have enjoyed the most was by the University of California at Berkley professor David Romer, It's Fourth Down and What Does the Bellman Equation Say? A Dynamic Programming Analysis of Football Strategy. In a nutshell, Romer concluded that NFL coaches are far too conservative with their decisions, often going against what statistics suggest is prudent. I actually printed out all twenty-some pages and attached the article to one of our practice plans.

Another interesting article is Stephen Abbott's interview with former Washington Redskins Offensive Coordinator Chris Meidt, an undergraduate mathematics major, in the September 2009 edition of Math Horizons. Meidt's take on statistics:

...in the NFL the data for everything is available. There are a number of statistical firm that capture everything so it’s all there. What’s our run/pass ratio for second and three? What’s our success percentage when we run or when we pass? I used to have to do this myself. I computed it for every down distance in the field. I did it for the red-zone [inside the 20 yard-line] ad nauseam. You’ve got two choices—run or pass—and you have three downs. Once you create your tree, it just becomes a game theory problem—what is the most likely outcome and what has the greatest expected value?

If I wasn't teaching, I think running numbers for a professional or college football team would be an intersting thing to do.

WCYDWAARBS

2010-02-06T13:30:00.000-08:00

What Can You Do With An Arby's Roast Beef Sale?

WCYDWTRP

2010-02-06T03:09:00.000-08:00

My writing has been a bit slow the past two weeks, so I pose the question: What can you do with three random points?

Double-click on the applet to open it in a separate GeoGebra window. From there, you can save it.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Speak With Conviction

2010-01-25T04:51:00.001-08:00

Typography from Ronnie Bruce on Vimeo.

What Teachers Make

2010-01-25T04:47:00.000-08:00

Formative Assessing to the Extreme

2010-01-24T16:32:00.000-08:00

For the past couple of weeks, I have been taking How Math Must Assess to an extreme, literally checking every problem every student does in class, providing constructive feedback to each student who needs it, and placing a check next to their names if they demonstrate to me that they can "do" whatever it was they were doing on that day. And it really is easy.

For example, this past week was a short week, having Monday off. My goal for this week was to have students solve right triangles. I really began last week by reviewing the Pythagorean Theorem the Friday before after a very short quiz, checking off on every student who demonstrated to me in class that they could actually find any side of a right triangle when given any two of them, providing constructive feedback for those students who needed it and checking them off after they could demonstrate to me that they could do it. Tuesday, I introduced the three trig ratios and how to use them to find any side of a triangle. Wednesday, we went from sides to angles. Thursday, we put everything together.

What I do is this: After some sort of a "mini-lesson" of fifteen minutes or less, I provide the remaining thirty minutes or so for students to apply the ideas learned during the mini-lesson on a problem set containing somewhere around fifteen to twenty problems, broken up into groups of four or five problems. I do NOT circulate around the room. I pull up my chair and sit next to an empty student desk somewhere in the room, and I ask my students to get up, walk to where I am sitting, and check their work with me after each group of problems. This seems to provide my students just enough movement-with-a-purpose that they can work rather diligently for the entire class.

When a student comes up to me, I can quickly scan their work and answers and provide specific feedback ("If you are going to use this angle in your calculations, then this side is the adjacent side"). If, after looking over their work for a group of problems, I am satisfied that a student can do things correctly, I place a check next to their name. For some students, this may not occur until after they have received feedback from me and re-worked some of the problems. For other students, if it looks like some sort of a calculation error on their part, I may send them back to their seats and have them shout out the answer to me without making the trip back up and standing around my desk ("It looks like you divided 4.3 by the tan(5) instead of the tan(55). Go back and double check that you end up with something around 4, and shout out your answer to me when you get it.").

I have 28 students in three of my four classes that I have been doing this with (the fourth class has 13 kids). I may have a group of four or five students waiting around me to have their work checked, but I ask them to listen carefully to any feedback given to another, because it may apply to them and their work.

My hope is that by the end of the week, every student will have demonstrated that they can meet the objectives for the week before we ever have a quiz. The results from last week? Fifty-eight students in the four classes combined could solve the right triangles perfectly. Only eleven students out of the four class could not solve the right triangles. Of these eleven, nine of them did not demonstrate to me during the week that they could do the work anyway, so this was not a surprise.

Anyway, I plan on continuing my extreme formative assessment this week as work with special right triangles and their trig ratios. We'll see how it goes.

Median Grading (Part 2)

2010-01-22T09:47:00.000-08:00

Two other short articles about median grading: One Example of Median Grading Another Way for Teachers to Compute Student Grades and Teachers Need to Look at Median Grading: Evaluation and Instruction Suffer When Grades are Averaged.

If you look through the author's other articles, there are some thought-provoking reads.

What if I used the Median to Calculate the Grade?

2010-01-21T15:49:00.000-08:00

I have read a couple of posts on different blogs lately about grading, standards-based grading, and assessment. In particular, Why I Hate Grades, Mathematically Speaking at yofx and Another Standards-Based Grading Question at A Math Teacher's Notebook. Thinking about a system like Dan Meyer's found in his How Math Must Assess, turning student mastery records into end-of-quarter grades using the mean as a measure of central tendency seems difficult. After reading Success for All: The Median is the Key, is the median a viable option?

Euclid and his Modern Rivals

2010-01-21T15:32:00.000-08:00

I picked this book up on a lark the other day, and I find myself wishing I had a copy of Euclid and some of the other geometry texts referenced in this Lewis Carroll (published under his real name, Charles Lutwidge Dodgson) gem. Reminds me a little bit of Imre Lakatos' Proofs and Refutations.

100 Visitors

2010-01-15T03:30:00.000-08:00

One hundred visits in just over one month (started this puppy on December 13, 2009).

I am certainly glad that folks have found my blog. It is still evolving...more than anything, I wanted to start a blog as a means of improving my writing. I want to write and publish more than I have (I have written or co-written a couple of things ranging from articles for my state journal to the Mathematics Teacher to the American Mathematical Monthly) but writing is difficult for me. Hopefully, by regularly blogging, it will make the writing process easier.

That said, thanks to the folk who have contributed to the 100 visits!

More of My Mental Rube Goldberg Machine

2010-01-13T11:20:00.000-08:00

It once again starts with a trip down Farnham Street and ends up with my head spinning.

In the Farnham Street post Can You Measure?, I was led to the article Shape of glass and amount of alcohol poured: comparative study of effect of practice and concentration, with the objective to

To determine whether people pour different amounts into short, wide glasses than into tall, slender ones

leading to the conclusion

To avoid overpouring, use tall, narrow glasses or ones on which the alcohol level is premarked. To avoid underestimating the amount of alcohol consumed, studies using self reports of standard drinks should ask about the shape of the glass.

which reminded me of two posts over at dy/dan, Don’t Forget Answers, Iteration and What Can You Do With This: Club Soda.

I was also led to the article Downsize in 3D, Supersize in 1D: Effects of the Dimensionality of Package and Portion Size Changes on Size Estimations, Consumption, and Quantity Discount Expectations, which found that

changes in size appear smaller when products change in all three dimensions (height, width, and length) than when they change in only one dimension

leading to the conclusions

consumers expect (and marketers offer) steeper quantity discounts when packages and portions are supersized in 3D than when they are supersized in 1D; consumers pour more product into and out of conical containers (in which volume changes in 3D) than cylindrical containers (in which volume changes in 1D); and consumers are more likely to supersize and less likely to downsize when package and portion sizes change in 1D than when they change in 3D.

which reminded me of the blog post Price Realization Through Creative Package Sizes over at Iterative Path.

Not sure what this all means. I would like to find a way to use all of this in my classes some time soon.

My Mental Rube Goldberg Machine

2010-01-12T12:38:00.000-08:00

I was over at Farnham Street reading Starbucks and the Rule of Three, which led me to William Poundstone's blog Priceless at Psychology Today his post Decoding Fast-Food Menus, which led me to the video below, which reminded me of Dan Meyer's recent post How Do You Turn Something Interesting Into Something Challenging? and his series of What Can You Do With This? posts.

Not that I am ready to turn any of this into something challenging, nor am I considering a WCYDWT? idea, but there is something here.

Too Much Time on Your Hands?

2010-01-10T09:33:00.000-08:00

Now, if you would only memorize the digits...

Pi buster: French software whiz claims record

Interesting Fraction Division

2010-01-10T08:54:00.000-08:00

After reading Richard Skemp’s Relational Understanding and Instrumental Understanding at the Republic of Mathematics it got me to thinking about something that happened in class this past week.

I teach a class once a week at a local community college close to my school. I like it...it is a nice change, with an amazing mix of students; some, just out of high school, and others being older adults looking to go back to college or to enter college for the first time. I teach a sequence of three courses - Preparatory Mathematics, Introduction to Algebra (part 1), and Introduction to Algebra (part 2) - none of which count towards any of my student's degree plans because they are not considered to be college-level mathematics.

Teaching these classes are very rewarding. I had an older adult who stood outside the classroom door for five minutes on the first day of class last quarter, afraid to come in because she thought she was going to fail the class. She went on to be one of my better students. To see how she (and the rest) grow to realize all the mathematics they know and can do is special. I typically have about 75% of the students from the Prep Math course take all three classes in the sequence with me.

On a side note...it is funny is to see their reaction to how I let them retake parts of quizzes and the like during my office hour before class (actually, it is really funny to see their reaction to WolframAlpha or to GeoGebra). I explain how some students will get it sooner than others, and how I want their grades to be an accurate assessment of their most current understanding of the content. Everyone in the class has gone through high school, they know the game, so this idea is a shock to them. About halfway through the first quarter, they get it, and start stopping in during my office hour. Of course, there are some institutional constraints that are much different than high school, but as long as I can get everything done that is in the syllabus, I am cool.

Something came up this past week when reviewing fraction operations, in particular, division. We discuss the usual finding-of-common-denominators when adding and subtracting, and how you can, but you do not need to, find a common denominator when multiplying or dividing. Many of the students choose to because it is one less thing to remember, and I am ok with that. I am not going to allow fractions to be a deal breaker for them. Show me you can add, subtract, multiply and divide some everyday fractions, that is good.

When it comes to division, we discuss the normal sorts of mnemonic devices they remember from high school. Same, change, flip. "We do not wonder why, we just flip and multiply." How about THAT for a mnemonic device! The usual question arises: "Why can't you just divide straight across in the same way you multiply straight across?" Since they ask, we talk about it. You can divide fractions by dividing the numerators and denominators straight across, but this method will sometimes leave you with a fraction in the numerator and a fraction in the denominator.

Not a big deal. This just means that you will need to do some additional work to change this fraction into one with integers in the numerator and denominator by rewriting the fractions with common denominators

which of course adds a layer of fraction sophistication, such as multiplying the numerator and denominator by 20.

This past week, someone who liked to find common denominators for multiplication and division problems, noticed that when you found a common denominator first, then divided straight across, you could avoid the messy fraction with fractions in the numerator and denominator.

Thought this was pretty cool. I had never thought about fraction division in this manner, probably because I never found common denominators when I divided. I just always flipped and multiplied!

Things That Make Me Go "Hmmm..."

2010-01-07T05:25:00.000-08:00

From Dan Meyer's Blog How Do You Turn Something Interesting Into Something Challenging? and the idea of lowering mathematics into the discussion s l o w l y. First thought that jumped into my head? I suck at teaching math.

From Kate Nowak's Blog Soldiers in the War on Innumeracy and the link to the Verizon Math video. I watched this video with my 8th grade son last night, and I think he wants to join this War.

The closest I have come to this nirvana is using this video when discussing the Prisoner's Dilemma and other Game Theory ideas with my Mathematical Games class last semester.

Fermat Point

2010-01-03T16:44:00.000-08:00

Sticking with the name of my blog, the Fermat Point was the first point of concurrency to be spotlighted. Not sure why I chose this point...I probably should have followed Clark Kimberling's Encyclopedia of Triangle Centers and chosen the Incenter, which is center X(1) in the ETC. The Gergonne Point is the next to be spotlighted, and it is closely related to the Incenter, so I think that could count.

The Fermat point, also known as the Torricelli point, is the point with the property that the total distance from the three vertices of the triangle to the point is as small as possible. The point takes its name from Fermat, who proposed this problem to Torricelli. More on the Fermat Point can be found at Cut-the-Knot. I use this with my geometry students after we study inscribed angles.

If you double click on the applet below, it will open in a GeoGebra window on your computer. It is then yours to do what you see fit.

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Knights and Knaves

2009-12-31T08:27:00.000-08:00

A big push at my school this year is to get kids to write and to present. You know... those "21st century skills." This has never been a problem for because I am asking my kids to write explanations and present proofs and such all the time (which is probably a reason no one else wants to teach geometry). One of the ways I get my kids to write and present, while at the same time applying some if/then thinking, is through the use of Knights and Knaves logic problems. An example is as follows:

A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie.You meet two inhabitants: Zoey and Mel. Zoey tells you that Mel is a knave. Mel says, `Neither Zoey nor I are knaves.'

So who is a knight and who is a knave?

When discussing a problem like this with my students - two inhabitants, each could be a Knight or a Knave - I like my students to examine the four possible cases. By ruling out those cases that lead to contradicting statements between the inhabitants, what is left must be true (an informal introduction to proof by contradiction?).

There does seem to be somewhat of a history to these types of puzzles. These puzzles, of course, lead some of my students to The Hardest Logic Puzzle Ever, as well as to an interesting variation on the theme which introduces a "Spy" that can tell the truth or tell a lie.

The Discursive Practice of Learning Disability

2009-12-28T12:58:00.000-08:00

One of the things I do when studying a research article is to go through the references of the article and see what catches my eye. When reading the Social Construction of Learning Disabilities, I was drawn to the work of Ray McDermott and Herve Varenne and in particular, their article Culture as Disability. What is culture as disability? The following passage provides a definition:

"The world is not a set of tasks, at least not of the type learned, or systematically not learned, at school, but made to look that way as part of political arrangements that keep people documenting each other as failures. Over the past forty years, school performance has become an exaggerated part of established political arrangements, and, by pitting all against all in a race for measurable academic achievement on arbitrary tasks, school has become a primary site for the reproduction of inequality in access to resources. The use of the term LD to describe, explain, and remediate children caught in a system of everyone having to do better than everyone else is a case in point. Even if used sensitively by people trying to do the right thing for the children apparently disabled, the term has a political life that involves millions of people operating on little information about the consequences of their work."

This, in turn, led me to the essay The Discursive Practice of Learning Disability: Implications for Instruction and Parent-School Relations by D. Kim Reid and Jan Weatherly Valle. The following excerpt describes things perfectly (my emphasis added):

"In current practice, teachers assign academic tasks deemed “grade appropriate” and hold expectations for a specified range of responses that represent mastery. The child who responds consistently outside this specified range eventually will, in most cases, be noticed as “a person with qualities to be discovered by agents of the school” (Varenne & McDermott, 1998, p. 215). The child then becomes the object of intense observation and documentation, a process reserved only for children who perform outside of the anticipated range of response but whose capacity for learning is suspected to be “normal.” In order to confirm or rule out the possibility of LD, a knowledgeable teacher makes a referral so that a psychologist (and perhaps other experts) can administer a battery of psychoeducational tests to the child to generate an individualized psychoeducational report based largely on the results. Soon, a special education committee meets to discuss the test results and to determine the child’s eligibility for individualized special education services. If the child is deemed eligible on the basis of “really being” a special education student, an Individualized Education Program (IEP) is developed. The transformation from ordinary schoolchild to disabled individual is now complete.(page 469)"

Why does it appear that I am so bitter about special education and LD? I don't know. I think it started a dozen or so years ago. During a conference with the parents of one of my students, it was suggested by someone in the room that the parents should explore different medication options for their child. The parents never questioned it! I was dumbfounded. There is obviously something wrong with my school that would require a student to be medicated just to make it through the school day.

It was at roughly the same time I stumbled upon John Taylor Gatto's Dumbing us down: The hidden curriculum of compulsory schooling and in it, The 7-Lesson School Teacher. This sort of started it all, and I have been trying to fight it from the inside ever since.

Different Learning Styles Bogus? What Did The Research Say?

2009-12-23T11:30:00.000-08:00

This is a follow-up to my earlier post on this topic. I read the research article published a year ago (a YEAR ago? What does THAT tell you?) Learning Styles: Concepts and Evidence by Harold Pashler, Mark McDaniel, Doug Rohrer, and Robert Bjork in the journal Psychological Science in the Public Interest(PSPI), Volume 9, Number 3. You can read the article for yourself in a free preview.

As I stated in the aforementioned post, I do not care for the "data" that every administrator is ga-ga about, nor do I care much for educational research that treats students as widget coming off an assembly line. I do my best to find what works for each kid, and if educational research can inform what I do, great. But I feel that data is suspect...just too many variables.

In a nutshell, this was a literature review of studies concerning learning styles. It appears to be in response to the learning-styles industry that produces books, workshops, and learning-styles assessments. The rationale behind this industry is that to teach a student most effectively, you must first be able to accurately diagnose their learning style (hence, the need for the assessments). Once you have accurately diagnosed the learning styles of your students, the teacher can begin to implement a learning-styles approach (hence, the need for books and workshops).

If this is the case, what research is out there that demonstrates the effectiveness of a learning-styles approach? The authors were asked to find if learning-style practices were supported by scientific research. They looked for evidence that supported what they referred to as the meshing hypothesis; an instructional method shown to be most effective for students with a particular learning style is not the most effective instructional method for students with a different learning style. If we find evidence of an instructional method like this, then we have found evidence in support of the meshing hypothesis, and we would conclude that there is something to this learning-styles thing.

The authors found little evidence to support learning-style instructional methods. Some of the studies they reviewed did not use appropriate research methods to test the effectiveness of learning style instructional methods. The authors found that those studies that did use appropriate research methods at times contradicted the meshing hypothesis.

It should be noted that what the authors suggest a valid learning-styles study should entail would be along the lines of a students-as-widgets study. At least, that is my take on it.

The authors did acknowledge the anecdotal things that we all see every day in our classrooms. They did not argue the fact the people prefer to have information presented to them in a certain manner (visually, for example), nor did they argue that different people process information differently. They acknowledged that different disciplines call for different instructional methods (for example, content in Language Arts is presented differently than content in Mathematics). They also acknowledged that students may benefit from having content presented in a particular manner, and the teacher, noticing these patterns in their students, may plan for these student differences. The authors simply dispute that the current models of learning-styles assessments do not offer an instructional prescription for each student (which is what the learning-styles industry would want you to believe).

There were some interesting points made by the authors. One being how the popularity of the notion of learning-styles could be attributed to how it provides a means of sorting people into distinct groups. Schools LOVE this stuff. However, learning-styles does not have the negative connotation of tracking because is it a good thing to find out how I learn best, and that how I learn is different from how someone else learns. I am special, I am unique, and learning-styles tells me that (sarcasm).

Another point made reminds me of the point made in the article referred to in my first post. If a child is not succeeding in school, it must be because the child's learning styles are not being addressed.

This runs counter to the argument made in the Social Construction of Learning Disabilities, in that if a child is not succeeding in school, it must be because the child is imperfect in some way. Schools are perfect institutions, and there could never be something wrong with an institution created by us, for us. It had got to be the kid (more sarcasm). Nevertheless, a point made in this article seems to hold for both Learning Disabilities and Learning Styles. This is probably my favorite paragraph in the whole article. Wherever there is LD, you could replace it with LS (learning style) without changing much of the meaning:

"What is important to understand is that LD is intelligible only in the context of schooling. It is in schools, where children are routinely sorted and evaluated in terms of certain learning behaviors, that LD comes to life. Individual students cannot have LD on their own (McDermott & Varenne, 1999). The performance of LD requires an institutional framework that assigns particular meanings to students’ behaviors that, in other cultural contexts, do not carry the same significance. Learning disabilities defined in terms of a delay in acquiring specific (pre)reading skills, for example, make sense only in a context where these skills are salient. Similarly, a relationship between LD and attention demands a context that requires students to deploy their attention in particular ways. In short, the expression of LD requires the institution we know as school, although it is at least possible to imagine schools where students are not sorted by ability, where learning and not the rate of learning is most important, and where the absence of learning is not such a significant presence (pg 484)."

Different Learning Styles Bogus?

2009-12-22T12:23:00.000-08:00

Found this on Farnam Street, a blog ran across when searching for stuff about the Prisoner's Dilemma for a Mathematical Games class I teach (sort or an applied discrete mathematics course). I am a bit wary of educational research anyway. I just feel that there are too many variables that can't be controlled. Nevertheless, I am looking forward to reading the research.

Different Learning Styles Bogus?

Beauty and Facial Symmetry

2009-12-19T04:01:00.000-08:00

After seeing this article earlier this morning on Yahoo, I was reminded about the article Fostering Mathematical Inquiry with Explorations of Facial Symmetry written by Michael Todd Edwards, a good friend of mine at Miami University up in Oxford, Ohio. If you are an NCTM member who subscribes to Mathematics Teacher, login and check it out.

Poorly Drawn Quadrilaterals

2009-12-18T09:21:00.000-08:00

I got this idea from HERE. The idea is this...you are presented with a quadrilateral that is drawn poorly, but which is marked correctly. Your job is to determine the best possible name for the quadrilateral (if it can be determined) or if it is a quadrilateral at all. Some examples are below...

So...what IS the best name for these quadrilaterals?

I have created a GeoGebra applet that generates up to 24 different poorly drawn quadrilaterals for anyone who is interested. You can find the applet HERE.

Semester Exams

2009-12-18T07:24:00.000-08:00

Finished my semester exams today, all on moodle, which means my grading is done.

I have been a moodle user for the past couple of years, and have used moodle to incorporate a weekly spiral review of algebra and geometry topics (among other things I use moodle for). This year, I decided to use it for my semester exams.

Here's how it worked, along with some background.

These weekly moodle assignments are my student's only homework for me...twenty questions a week. I use the calculated question feature to create a hundred versions of the same problem, so different students will have the same questions, but in a different order, and with different numbers. My hope is that my students will talk about the problems (and they do), and not just copy answers (which they can't).

When they submit an answer, they know immediately if their answer is correct. If it is correct, they receive full credit for the problem. If it is incorrect, they receive a 20% penalty. They can attempt the assignment as many times as they want with the understanding that each attempt will mean a new version of the assignment with the same questions but with different numbers. I simply record their best score for the week.

Grading this way, I would hope that every student gets a perfect score. Even with an unlimited number of attempts each week and regular moodle moots I hold for students to stop in and ask questions, this does not happen. Nevertheless, I feel this is an effective way to get the kids to stay sharp on old content.

Over the course of the semester, I have created almost 200 questions like this, categorized by content (Triangles, Circular Areas, Trigonometry, etc). Many of these questions assume the use of GeoGebra or NspireCAS. Other questions require kids to make a choice: Should I complete this by hand or use GeoGebra?

Their exams were constructed the same way, from the same questions, with just a couple of exceptions.

(1) Every student had a different exam. I finally figured out how to randomly include any number of questions from a question category. For example, every student had three right triangle questions selected from 20 possible right triangle questions.

(2) The students could attempt the exam as many times as they wanted in the 2-hour time period. Just as on the weekly assignments, students knew immediately if their answer was correct or incorrect. However, when they attempted the exam again, their correctly answered questions were on the new attempt, along with their answers.

(3) Students could know their grade at any time. Because they could make repeated attempts on the exam, when they finished one attempt, they knew what their grade was on that attempt. This, of course, meant that no one needed to get a perfect score on the exam, because they only needed to get 90% of the questions correct.

Of course, this is not perfect. To use the calculated questions feature in moodle means I am asking a question that can be solved by applying an algebraic algorithm. I guess this could be good or bad. If I ask a question like

Quadrilateral ABCD is a square. Point A has coordinates (8,19). The diagonals intersect at point M, which has coordinates (1,-3). The vertices are labeled in a counter-clockwise direction. What is the x-coordinate of point B?

students need to know something about a square, how to construct it, that the diagonals are congruent and perpendicular, that is possesses 90 degree rotational symmetry.

I am not concerned that my students had seen the exam questions before on their weekly assignments. I am concerned more about the two hour exam period (we do this, the argument goes, to get our kids ready for college. Why, then, don't we have class just three days a week?) than if the kids have seen the questions. Besides, I'm not sure the kids even realized this until they took the exam, even though I had told them this numerous times before.

Anyway, this will work for now. My grading is done.

Changing the Culture

2009-12-17T07:45:00.000-08:00

In an attempt to change the culture of my school, someone decided it would be good to spend money on large signs with motivational quotations from significant figures in our content areas. The motivational quotations are to go along with our school themes that were created during the summer. You know...things like "respect" or "cooperation." I am not sure what the seven themes are. I looked on our school webpage, but they are not listed any place that I can find them.

It is now the math department's turn to suggest some quotes to be placed on signs that will hang outside out room. Here are my tongue-in-cheek suggestions:

"I don't quite hear what you say, but I beg to differ entirely with you. " (De Morgan)

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." (Klein)

"Mathematicians are born, not made." (Poincare)

"To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls. " (Chebyshev)

"It is rare to find learned men who are clean, do not stink and have a sense of humour." (said about Leibniz)

http://www-history.mcs.st-and.ac.uk/Quotations/index.html

My Favorite Curriculum

2009-12-12T17:08:00.000-08:00

Phillips Exeter Academy's mathematics curriculum. I do not know how they do it. I think if a student were to work their way through this curriculum, starting in Math1 and ending in Math4, they would be able to do some good mathematics.

I have used Math2 with my (so called) honors geometry students for three years. I can't convince anyone else in my department to go along with me.

The Social Construction of Learning Disabilites

2009-12-12T04:42:00.000-08:00

I was reminded of the article The Social Construction of Learning Disabilites by Curt Dudley-Marling after sitting through an IEP re-evaluation meeting the other day. After all, if you have attended an IEP meeting, that is all the regular education teachers do is sit through them.

This student's recognized disability is in the area of mathematics. I call it "recognized" because I am not sure I believe there is a disability. The results of the test administered by the special educators to support this claim looked to me like the results for any one of my students. The test ranked her below average, average, or above average on a dozen or so different content strands.

Let's see...on the handful of strands that concerned mathematical operations, this student was ranked as "below average," estimated to be performing somewhere in the 7th to 9th grade level (this student is an 11th grader). Nothing out of the ordinary there. I would reckon that the majority of my students would fall in this range on these strands. Somewhere earlier in their schooling, someone decided that an appropriate modification for this student would be to allow them to use a calculator. These sections of this test were completed without a calculator. No wonder.

Let's see...on the handful of strands that involved operations with fractions and real numbers, this student scored below average, again estimated performing in the 7th to 9th grade range. Again, par for the course when it come to the majority of my students. For some of these sections, the student was permitted to use a calculator. I guess who ever decided on the calculator modification forgot to teach how to use a calculator. I wonder what calculator they used? Did they use their TI-84? Did they use one of my NspireCAS handhelds? Was it one of the Ohio Graduation Test calculators?

Let's see...on the handful of strands that are actually addressed in my classes - Algebra, Measurement, Geometry, Probability, Shapes - this student was ranked in the average to above average range on all, performing in the 10th grade to 11th grade range. I would imagine and expect that the majority of my students would be here.

I guess I do not understand how this disability works, or how it is constructed.