Teaching

Math and Dinner — NYMC

I’ll be giving a Math and Dinner talk for the New York Math Circle on Monday, August 4. Math and Dinner starts with some interesting mathematics at NYU’s Courant Institute, and after the talk the conversation continues over dinner at a nearby restaurant. The series provides a fun and casual way for NYMC participants to chat about mathematics, teaching, and learning.

My talk is titled An Array of Matrix Explorations. Here is the description.

The world of matrices is rich and diverse, connecting ideas across disparate disciplines and extending familiar mathematical notions into unfamiliar territory. In this talk, participants will explore some common concepts in the high school curriculum–algebra, geometry, and trigonometry–through the mathematics of matrices, where their depth and connections can be seen in exciting ways.

The talk is free, but participants pay their own way for dinner. You can find out more about the series, and register, here.

By MrHonner, 10 years10 years ago

Resources Teaching

My Grand Challenge for Mathematics Education

In the spirit of thinking big, the National Council of Teachers of Mathematics (NCTM) recently invited its members to suggest grand challenges for mathematics education. The NCTM defined a grand challenge as ambitious but feasible, positively affecting many people, and capturing the public interest.

So, here’s my grand challenge for mathematics education.

Build and maintain a free, comprehensive, modular, and adaptable repository of learning materials for all secondary mathematics content.

The foundational resources in this repository would be a variety of free, modular texts covering all secondary mathematics topics. The content could be packaged together as “textbooks” that cover typical courses, or sliced-and-diced to meet the particular curricular needs of individual states, districts, and schools. The texts could be downloaded or printed for student/teacher use and would include a wide variety of exercises and problems.

The repository would also contain supplemental curricular materials like enlightening videos made by teachers and mathematicians, rich tasks, technology-based demonstrations and explorations, assessment items, and more.

A full-time staff of professional teachers, mathematicians, designers, and editors would be charged with managing the repository. This would include creating the texts and curricular materials, curating and integrating freely available content, and advancing the project.

To be clear, my grand challenge is not build the textbook of the future. This is certainly a worthy goal, and the forward-thinking work of people like Dan Meyer is helping us to define what a digital textbook could, and should, look like.

But my grand challenge is more modest: create a high-quality, comprehensive, customizable, freely available alternative to the standard textbook model. Of the many benefits such a project would have, three immediately come to mind.

Reduce Costs, Dependencies on Publishers

It is estimated that schools in the US spend between $8 and $15 billion every year on textbooks. For a fraction of that cost, a modest team of teachers and mathematicians could build and maintain a repository to satisfy the textbook needs of the vast majority of mathematics learners and teachers nationwide.

The mathematics taught in schools doesn’t change much, yet new math textbooks are produced and purchased every year. This is mostly due to superficial changes from textbook publishers and short-lived reform movements. A high-quality, free alternative would not only reduce the costs of acquiring textbooks, but could also put pressure on the multi-billion-dollar professional development industry, which is deeply influenced by textbook publishers.

Create Opportunities for Teacher Leadership

In conversations about improving the educational environment, we often hear about elevating the profession of teaching. This project would create many teacher-leadership positions at all levels of education. Teachers would be working directly to create, curate, and maintain the content in the repository, and teachers would also work at local and state levels to select and customize materials from the repository to best serve the needs of their districts and schools.

Giving teachers an active leadership role in adopting and customizing curricular materials would have a positive impact on schools and districts, as well as on the profession as a whole.

Promote a Common Language Around Mathematics Education

One of the purported goals of the Common Core State Standards Initiative was to create a common language around mathematics teaching and learning. But the CCSS movement has stalled, due in part to serious implementation problems and inflexible mandates.

The proposed repository could help promote a common language around mathematics through the widespread use of high quality texts, problems, tasks, and other common resources. Students from all across the country could work on shared problem sets and projects, and teachers could engage in professional dialogue about common tasks, texts, and demonstrations. Because the materials in the repository would be free and editable, states and districts would have full power to choose only those materials they wanted to use, and to adapt them as they saw fit.

There are many more potential benefits to this project, and I think it satisfies the NCTM’s criteria for a grand challenge: it’s ambitious but feasible, and it would have great impact.

To see the NCTM’s original call for Grand Challenges in Mathematics Education, click here. And Robert Talbert’s excellent response at the Chronicle of Higher Education is also worth reading.

By MrHonner, 10 years ago

3D Printing Application Teaching Technology

3D Printed Torus Knot

My students and I experimented a bit with 3D printing at the end of the year (as part of my always-do-something-new strategy). This (5,3) Torus knot was one of our successes.

I see a lot of possibility for 3D printing in my classrooms next year, and I’m looking forward to thinking and learning more about it.

By MrHonner, 10 years10 years ago

Geometry Teaching

The Closest Point on a Parabola to a Point on its Axis

While playing around with distances from points to graphs, I discovered the following interesting property of parabolas.

Suppose you have a standard parabola, $y = x^2$ , and a point on its axis of symmetry, $(0,a)$ . Then the point on the graph of the parabola closest to the point $(0,a)$ either has y-coordinate equal to $a - \frac{1}{2}$ , or is the vertex of the parabola.

A calculus-based proof is fairly straightforward.

The distance between a point on the parabola, say $(x, x^2)$ and the point $(0,a)$ is given by

$d(x) = \sqrt{x^2 + (x^2-a)^2}$

To find the minimum distance, differentiate $d(x)$ to get

$d'(x) = \frac{2x + 2(x^2-a)2x}{2\sqrt{x^2 + (x^2-a)^2}}$

This derivative is undefined when the denominator is zero, which only happens when the point $(0,a)$ is the vertex. The derivative is zero when

$2x + 2(x^2-a)2x = 2x ( 1 + 2(x^2 - a)) = 2x(1 + 2x^2-2a)=0$

By the zero-product property, either $x = 0$ or $1 + 2x^2-2a = 0$ .

The critical value $x = 0$ corresponds to the vertex of the parabola. The distance from $(0,a)$ to the vertex is a minimum if $a \leq \frac{1}{2}$ , or a relative maximum when $a > \frac{1}{2}$ .

Now, if $1 + 2x^2 - 2a = 0$ , we have $x^2 = a - \frac{1}{2}$ . Since $y = x^2$ on the parabola, the point associated with this critical value has y-coordiante $a - \frac{1}{2}$ . Some simple analysis shows that the distance to this point is minimal.

As an alternative to optimization, one could approach the problem geometrically. Since the shortest path from a point to a line is the perpendicular line segment connecting them, the shortest path between a point and a curve should be a line segment perpendicular to a tangent to the curve. At a point on the parabola $(x,x^2)$ , the slope of the tangent line is $2x$ , and so the slope of the line perpendicular to the tangent at $(x,x^2)$ would be $-\frac{1}{2x}$ . This is equivalent to $-\frac{\frac{1}{2}}{x}$ , and since $x$ is the distance to the axis of symmetry, it follows that this line intersects the axis of symmetry at a point a half-unit up in the y-direction.

This fun and surprising result would make a nice exploration in a Calculus class. It easily extends to general parabolas, and the situation naturally suggests some compelling extension questions.

You can explore the problem, and create variations, in this interactive Desmos demonstration I put together.

By MrHonner, 10 years10 years ago

Application NYT Resources Teaching

Exploring Fair Division

My latest piece for the New York Times Learning Network is a math lesson exploring basic techniques of fair division.

Fair division is concerned with partitioning a set into fair shares. “Fair” can take on different meanings in different contexts, but at its most basic level, a share is fair if someone is willing to accept it.

This lesson builds on an excellent article in the NYT about a technique in rent-splitting based on Sperner’s Lemma, an important result in Topology. The author tells the story of how he and two roommates used the technique to settle on a fair division of rent for three different-sized rooms.

“The problem is that individuals evaluate a room differently. I care a lot about natural light, but not everyone does. Is it worth not having a closet? Or one might care more about the shape of the room, or its proximity to the bathroom.

A division of rent based on square feet or any fixed list of elements can’t take every individual preference into account. And negotiation without a method may lead to conflict and resentment.”

After reflecting on the article, students use the related NYT interactive feature to explore the algorithm and then research basic techniques in fair division like divider-chooser, sealed bids, and the method of markers. The full lesson is freely available here.

By MrHonner, 11 years10 years ago

Teaching

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