Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children.

Boaler

I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.

Robinsons1

Robinsons2

“Math Play” Presentation for Early Childhood teachers

Even though my teaching experiences are all middle and high school, as a PreSchool-12 math chair and father of 3 young children, I’m intensely interested in how math is presented to very young people.

As a result, I’m presenting ideas for teaching math through fun and exploration to about 55 Cleveland area pre-school through kindergarten teachers this morning.  My handout is on Scribd and should show below.  Math is  about Play and Curiosity.  Teach it that way.

 

FREE TI-Nspire iPad App Workshop

Nspire_Wshop

On Saturday, 31 May 2014, Texas Instruments (@TICalculators) and @HawkenSchool are hosting a FREE TI-Nspire iPad Workshop at Hawken’s Gries Center in Cleveland’s University Circle.  The workshop is designed for educators who are interested in or are just beginning to use the TI- Nspire App for iPad® (either CAS or numeric). It will cover the basics of getting started and teaching with the Apps.  Tom Reardon will be leading the training!

Sign up for the workshop here.  A pdf flyer for the workshop is here:   iPad App Training.

Which came first: Math Ability or Computational Speed ?

I’ve claimed many times in conversations over the last two weeks that I believe many parents and educators misconstrue the relationship and causality direction between being skilled/fluent at mathematics and being fast at computations.  Read that latter as student accomplishment defined by skill on speed testing as done in many, many schools.  Here is a post from Stanford’s Jo Boaler on math anxiety created by timed testing.

Here’s my thinking:  When we watch someone perform at a very high level in anything, that person appears to perform complex tasks quickly and effortlessly, and indeed, they do.  But . . . they are fast because they are good, and NOT the other way around.  When you learn anything very well and deeply, you get faster.  But if you practice faster and faster, you don’t necessarily get better.

I fear too many educators and parents are confusing what comes first.  From my point of view, understanding must come first.  Playing with ideas in different contexts eventually leads to recognizing that the work one does in earlier, familiar situations eventually informs your understanding in current, less familiar settings.  And you process more quickly in the new environment precisely because you already understood more deeply.

I think many errantly believe they can help young people become more talented in mathematics by requiring them to emulate the actions of those already accomplished in math via rapid problem solving.  I worry this emphasis is placed in exactly the wrong place.  Asking learners to perform quickly tasks which they don’t fully understand instills unnecessary anxiety (according to Boaler’s research) and confuses the deep thinking, pattern recognition, and problem solving of mathematics with rapid arithmetic and symbolic manipulation.

Jo Boaler’s research above clearly addresses the resulting math anxiety in a broad spectrum of students—both weak and accomplished.  My point is that timed testing–especially timed skill testing–at best confuses young students about the nature of mathematics, and at worst convinces them that they can’t be good at it.  No matter what, it scares them.   And what good does that accomplish?

Cover Article

I was pretty excited yesterday when the latest issue of NCTM’s Mathematics Teacher arrived in the mail and the cover story was an article I co-wrote with a former student who’s now at MIT.

The topic was the finding and proof of a cool interconnected property of the foci of hyperbolas and ellipses that I made years ago when setting up my TI-Nspire CAS to model conic sections via the polynomial definition.

Image5a

Image5b

After pitching the idea to teachers at professional conferences for a couple years with no response, I asked one of my 9th grade students if she’d be interested in a challenge.  Her eventual proof paralleled mine, and our work together enhanced and polished each other’s understanding and proofs.

While all of the initial work was done with the TI-Nspire CAS, we wrote the article using GeoGebra so that readers could freely access Web-based documents to explore the mathematics for themselves.

You can access the article on the NCTM site here.

While a few minor changes happened after it was created, here is a pre-publication proof of the article.

Traveling Dots, Parabolas, and Elegant Math

Toward the end of last week, I read a description a variation on a paper-folding strategy to create parabolas.  Paraphrased, it said:

  1. On a piece of wax paper, use a pen to draw a line near one edge.  (I used a Sharpie on regular copy paper and got enough ink bleed that I’m convinced any standard copy or notebook paper will do.  I don’t think the expense of wax paper is required!)
  2. All along the line, place additional dots 0.5 to 1 inch apart.
  3. Finally, draw a point F between 0.5 and 2 inches from the line roughly along the midline of the paper toward the center of the paper.
  4. Fold the paper over so one of the dots on line is on tope of point F.  Crease the paper along the fold and open the paper back up.
  5. Repeat step 4 for every dot you drew in step 2.
  6. All of the creases from steps 4 & 5 outline a curve.  Trace that curve to see a parabola.

parabola1

I’d seen and done this before, I had too passively trusted that the procedure must have been true just because the resulting curve “looked like a parabola.”  I read the proof some time ago, but I consumed it too quickly and didn’t remember it when I was read the above procedure.  I shamefully admitted to myself that I was doing exactly what we insist our students NEVER do–blindly accepting a “truth” based on its appearance.  So I spent part of that afternoon thinking about how to understand completely what was going on here.

What follows is the chronological redevelopment of my chain of reasoning for this activity, hopefully showing others that the prettiest explanations rarely occur without effort, time, and refinement.  At the end of this post, I offer what I think is an even smoother version of the activity, freed from some of what I consider overly structured instructions above.

CONIC DEFINITION AND WHAT WASN’T OBVIOUS TO ME

A parabola is the locus of points equidistant from a given  point (focus) and line (directrix).

parabola2

What makes the parabola interesting, in my opinion, is the interplay between the distance from a line (always perpendicular to some point C on the directrix) and the focus point (theoretically could point in any direction like a radius from a circle center).

What initially bothered me about the paper folding approach last week was that it focused entirely on perpendicular bisectors of the Focus-to-C segment (using the image above).  It was not immediately obvious to me at all that perpendicular bisectors of the Focus-to-C segment were 100% logically equivalent to the parabola’s definition.

SIMILARITY ADVANTAGES AND PEDAGOGY

I think I had two major advantages approaching this.

  1. I knew without a doubt that all parabolas are similar (there is a one-to-one mapping between every single point on any parabola and every single point on any other parabola), so I didn’t need to prove lots of cases.  Instead, I focused on the simplest version of a parabola (from my perspective), knowing that whatever I proved from that example was true for all parabolas.
  2. I am quite comfortable with my algebra, geometry, and technology skills.  Being able to wield a wide range of powerful exploration tools means I’m rarely intimidated by problems–even those I don’t initially understand.  I have the patience to persevere through lots of data and explorations until I find patterns and eventually solutions.

I love to understand ideas from multiple perspectives, so I rarely quit with my initial solution.  Perseverance helps me re-phrase ideas and exploring them from alternative perspectives until I find prettier ways of understanding.

In my opinion, it is precisely this willingness to play, persevere, and explore that formalized education is broadly failing to instill in students and teachers.  “What if?” is the most brilliant question, and the one we sadly forget to ask often enough.

ALGEBRAIC PROOF

While I’m comfortable handling math in almost any representation, my mind most often jumps to algebraic perspectives first.  My first inclination was a coordinate proof.

PROOF 1:  As all parabolas are similar, it was enough to use a single, upward facing parabola with its vertex at the origin.  I placed the focus at (0,f), making the directrix the line y=-f.  If any point on the parabola was (x_0,y_0), then a point C on the directrix was at (x_0,-f).

parabola3

From the parabola’s definition, the distance from the focus to P was identical to the length of CP:

\sqrt{(x_0-0)^2-(y_0-f)^2}=y_0+f

Squaring and combining common terms gives

x_0 ^2+y_0 ^2-2y_0f+f^2=y_0 ^2+2y_0f+f^2
x_0 ^2=4fy

But the construction above made lines (creases) on the perpendicular bisector of the focus-to-C segment.  This segment has midpoint \displaystyle \left( \frac{x_0}{2},0 \right) and slope \displaystyle -\frac{2f}{x_0}, so an equation for its perpendicular bisector is \displaystyle y=\frac{x_0}{2f} \left( x-\frac{x_0}{2} \right).

parabola4

Finding the point of intersection of the perpendicular bisector with the parabola involves solving a system of equations.

\displaystyle y=\frac{x_0}{2f} \left( x-\frac{x_0}{2} \right)=\frac{x^2}{4f}
\displaystyle \frac{1}{4f} \left( x^2-2x_0x+x_0 ^2 \right) =0
\displaystyle \frac{1}{4f} \left( x-x_0 \right) ^2 =0

So the only point where the line and parabola meet is at \displaystyle x=x_0–the very same point named by the parabola’s definition.  QED

Proof 2:  All of this could have been brilliantly handled on a CAS to save time and avoid the manipulations.

parabola5

Notice that the y-coordinate of the final solution line is the same y_0 from above.

MORE ELEGANT GEOMETRIC PROOFS

I had a proof, but the algebra seemed more than necessary.  Surely there was a cleaner approach.

Parabola6

In the image above, F is the focus, and I is a point on the parabola.  If D is the midpoint of \overline{FC}, can I conclude \overline{ID} \perp \overline{FC} , proving that the perpendicular bisector of \overline{FC} always intersects the parabola?

PROOF 3:  The definition of the parabola gives \overline{FI} \cong \overline{IC}, and the midpoint gives \overline{FD} \cong \overline{DC}.  Because \overline{ID} is self-congruent, \Delta IDF \cong \Delta IDC by SSS, and corresponding parts make the supplementary \angle IDF \cong \angle IDC, so both must be right angles.  QED

PROOF 4:  Nice enough, but it still felt a little complicated.  I put the problem away to have dinner with my daughters and when I came back, I was able to see the construction not as two congruent triangles, but as the single isosceles \Delta FIC with base \overline{FC}.  In isosceles triangles, altitudes and medians coincide, automatically making \overline{ID} the perpendicular bisector of \overline{FC} .  QED

Admittedly, Proof 4 ultimately relies on the results of Proof 3, but the higher-level isosceles connection felt much more elegant.  I was satisfied.

TWO DYNAMIC GEOMETRY SOFTWARE VARIATIONS

Thinking how I could prompt students along this path, I first considered a trace on the perpendicular lines from the initial procedure above (actually tangent lines to the parabola) using to trace the parabolas.  A video is below, and the Geogebra file is here.

http://vimeo.com/89759785

It is a lovely approach, and I particularly love the way the parabola appears as a digital form of “string art.”  Still, I think it requires some additional thinking for users to believe the approach really does adhere to the parabola’s definition.

I created a second version allowing users to set the location of the focus on the positive y-axis and using  a slider to determine the distances and constructs the parabola through the definition of the parabola.  [In the GeoGebra worksheet (here), you can turn on the hidden circle and lines to see how I constructed it.]  A video shows the symmetric points traced out as you drag the distance slider.

http://vimeo.com/89861149

A SIMPLIFIED PAPER PROCEDURE

Throughout this process, I realized that the location and spacing of the initial points on the directrix was irrelevant.  Creating the software versions of the problem helped me realize that if I could fold a point on the directrix to the focus, why not reverse the process and fold F to the directrix?  In fact, I could fold the paper so that F touched anywhere on the directrix and it would work.  So, here is the simplest version I could develop for the paper version.

  1. Use a straightedge and a Sharpie or thin marker to draw a line near the edge of a piece of paper.
  2. Place a point F roughly above the middle of the line toward the center of the paper.
  3. Fold the paper over so point F is on the line from step 1 and crease the paper along the fold.
  4. Open the paper back up and repeat step 3 several more times with F touching other parts of the step 1 line.
  5. All of the creases from steps 3 & 4 outline a curve.  Trace that curve to see a parabola.

This procedure works because you can fold the focus onto the directrix anywhere you like and the resulting crease will be tangent to the parabola defined by the directrix and focus.  By allowing the focus to “Travel along the Directrix”, you create the parabola’s locus.  Quite elegant, I thought.

ADDITIONAL POSSIBLE QUESTIONS

As I was playing with the different ways to create the parabola and thinking about the interplay between the two distances in the parabola’s definition, I wondered about the potential positions of the distance segments.

parabola2

  1. What is the shortest length of segment CP and where could it be located at that length?  What is the longest length of segment CP and where could it be located at that length?
  2. Obviously, point C can be anywhere along the directrix.  While the focus-to-P segment is theoretically free to rotate in any direction, the parabola definition makes that seem not practically possible.  So, through what size angle is the focus-to-P segment practically able to rotate?
  3. Assuming a horizontal directrix, what is the maximum slope the focus-to-P segment can achieve?
  4. Can you develop a single solution to questions 2 and 3 that doesn’t require any computations or constructions?

CONCLUSIONS

I fully realize that none of this is new mathematics, but I enjoyed the walk through pure mathematics and the enjoyment of developing ever simpler and more elegant solutions to the problem.  In the end, I now have a deeper and richer understanding of parabolas, and that was certainly worth the journey.

The Value of Counter-Intuition

Numberphile caused quite a stir when it posted a video explaining why

\displaystyle 1+2+3+4+...=- \frac{1}{12}

Doug Kuhlman recently posted a great follow-up Numberphile video explaining a broader perspective behind this sum.

It’s a great reminder that there are often different ways of thinking about problems, and sometimes we have to abandon tradition to discover deeper, more elegant connections.

For those deeply bothered by this summation result, the second video contains a lovely analogy to the “reality” of \sqrt{-1} .  From one perspective, it is absolutely not acceptable to do something like square roots of negative numbers.  But by finding a way to conceptualize what such a thing would mean, we gain a far richer understanding of the very real numbers that forbade \sqrt{-1}  in the first place as well as opening the doors to stunning mathematics far beyond the limitations of real numbers.

On the face of it, \displaystyle 1+2+3+...=-\frac{1}{12} is obviously wrong within the context of real numbers only.  But the strange thing in physics and the Zeta function and other places is that \displaystyle -\frac{1}{12} just happens to work … every time.  Let’s not dismiss this out of hand.  It gives our students the wrong idea about mathematics, discovery, and learning.

There’s very clearly SOMETHING going on here.  It’s time to explore and learn something deeper.  And until then, we can revel in the awe of manipulations that logically shouldn’t work, but somehow they do.

May all of our students feel the awe of mathematical and scientific discovery.  And until the connections and understanding are firmly established, I hope we all can embrace the spirit, boldness, and fearless of Euler.