Math Teachers and Twitter

Twitter is a robust and adaptable social-networking platform that makes it easy to exchange ideas and resources with others who share your interests.  Twitter has dramatically affected how I think about teaching, how I plan for teaching, and what I do when I need ideas, inspiration, or assistance.

What follows is a brief introduction to Twitter, how I use it, and why other math teachers might want to use it, too.

The philosophy that underlies Twitter is very simple.  Every user on Twitter is essentially their own broadcast channel.  You decide which users to listen to by following them.  If you follow a user, you will see every message they publish.  If you like what they say, great:  you can re-broadcast their message to your followers (a re-tweet), you can start a discussion, or you can just listen.

What if you don’t like what someone has to say?  Well, remember, you can always just listen.  Or, perhaps you want to see what other people might say in response.  In the end, if you’re not interested in what someone is broadcasting, you simply un-follow them.  You’ll no longer receive their messages.

One feature that makes Twitter unique is message length:  all messages must be less than 140 characters.  This is a hold-over from Twitter’s beginnings as a text-message-based platform; it forces users to be concise, and it keeps conversations moving quickly.

But what really makes Twitter so powerful are the many communities that use it to share and discuss ideas.  At any given time, there are millions of conversations happening between passionate and knowledgeable people on Twitter.  The key is to find people who are talking about what interests you.  When you find them, listen to them, and find out whom they listen to.  And when you are ready, start participating.

For example, there are thousands of math teachers on Twitter from all over the world, from all different backgrounds, with different perspectives on math and teaching.  Through Twitter, we share ideas, ask each other questions, brainstorm project ideas, pass great around resources, and actively discuss math and teaching in a highly positive and constructive way.

Here are just a few examples of how Twitter has affected me, as both a teacher and a professional.

  • I enjoy taking math photos and posting them here on www.MrHonner.com.  A teacher 1,000 miles away saw them, shared them with his students, and now they are taking their own math photos!
  • Alex Bogomolny, creator of Cut The Knot Math, frequently tweets about great math proofs, paradoxes, and puzzles.  He shared a question about a curious geometric limit and asked if anyone had a different solution; I quickly became obsessed and spent a Saturday morning coming up with this trigonometric approach.
  • After many Twitter-based conversations about the NY Math Regents exams, I wrote a short series on the quality of the tests.  This connected me to other teachers around the city and state who read the articles and shared their thoughts with me through Twitter, email, and blogging.

To get started on Twitter, register for an account and start following some people who might interest you.  For math teachers, here’s a short list of recommendations:  follow these folks, see whom they follow, see who follows them, and pretty soon you’ll have your own personal learning network!

  • @CutTheKnotMath — The creator of Cut-the-knot.org who frequently shares great links and great ideas.
  • @TimChartier — Math professor at Davidson college interested in teaching, technology, and mime.
  • @StevenStrogatz — A well-known mathematician and author at Cornell.
  • @JohnDCook — Applied mathematician, statistician and consultant John Cook.
  • @divbyzero — Dave Richeson, professor of mathematics at Dickinson College, and a great blogger.
  • @evelynjlamb — Evelyn Lamb, a mathematician and writer for Scientific American and other sites.
  • @RepublicofMath — A mathematician and math educator who is actively involved in many Twitter-based math communities.
  • @DavidWees — Canadian educator focusing on math and technology.  A tireless tweeter.
  • @monsoon0 — Applied mathematician at the University of Sydney
  • @MathBloggingEds — Editors picks from MathBlogging.org, a math-blog aggregator site.
  • @mathematicsprof — Math professor sharing an endless stream of interesting ideas and links
  • @maanow — Official Twitter account of the Mathematical Association of America
  • @amermathsoc — Official Twitter account of the American Mathematical Society
  • @MathforAmerica — Official Twtter acccount of Math for America, an outstanding professional organization devoted to recruiting, training, and retaining great math teachers.
  • @MrHonner — My Twitter account.  You can find everyone above, and many others, on my “Following” list.

So sign up, start following, and start listening!  Before you know it, you’ll find yourself thinking, teaching, and learning a lot differently!

Fun With Sliceforms

I was recently inspired to make my first sliceform.

With a handful of index cards, a marker, and some scissors, I was able to make this fun representation of a surface in 3D!

Sliceform Front 1

Turn it to the side, and see the surface from a different perspective.

Sliceform Side

The inspiration was timely, as my Calculus class has been discussing cross-sections, traces, and level curves of surfaces in space.  What a perfect way to demonstrate how to understand a surface by looking at representative slices!

A great, simple tool, and you can see some examples of the sliceforms my students created, like the one seen below, here.

fun with sliceforms

Are These Tests Any Good?

When it comes to educational testing, the stakes are higher than ever.  For a student, tests might determine which public schools you can attend, if and when you graduate, and which colleges are available to you.   For schools and districts, aggregate test scores and the “progress” they show might determine what kind of state and federal aid is available.

As a means of evaluating teachers, student test scores are playing an increasing role.  Indeed, state laws have been re-written to mandate the use of standardized test data as a substantial factor in rating teacher performance.

There is controversy regarding the value of standardized tests, even as measures of student achievement (in most cases, their purported purpose).  A very public debate has emerged as politicians attempt to make education more “data-driven” and hold teachers and schools more “accountable”.   But one fundamental question is rarely raised in this conversation:  are these tests any good?

If the tests we use to evaluate students, schools, and now teachers, are ill-conceived, sloppy, and erroneous, how legitimate a measure of teaching and learning could they possibly be?  The issue of test quality and relevance seems like an important one, but it gets very little attention.

In this series, I address the question “Are These Tests Any Good?” by looking at a collection of questions from the 2011 New York State Math Regents Exams.  My cursory analysis reveals many significant issues with how these tests are created:  mathematical errors; poorly-worded questions; the de-emphasis of knowledge; and misalignment with course curricula.

If we can’t create legitimate, relevant, appropriate tests, should we really be using them to evaluate teachers?  Or students?

Are These Tests Any Good?

Part I:  Mathematically Erroneous Questions

Part II:  Ill-Conceived Questions

Part III:  Underrepresented Topics

Part IV:  The Worst Math Regents Question of All Time

Part V:  9th Grade Questions on 10th and 11th Grade Exams

Here are some other resources on this topic.

This blog by JD2718 offers a similar critique of Regents exams from 2009.

Here’s some fun I had with one of this year’s Regents questions involving the famous 13-14-15 triangle.

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Are These Tests Any Good? Part 5

This is the fifth entry in a series examining the 2011 NY State Math Regents exams.  The basic premise of the series is this:  if the tests that students take are ill-conceived, poorly constructed, and erroneous, how can they be used to evaluate teacher and student performance?

In this series, I’ve looked at mathematically erroneous questions, ill-conceived questions, under-represented topics, and what is perhaps the worst question in Regents history.  In this entry, I’ll use questions from two exams to discuss duplication, lowered-expectations, and poor test construction.

Number 37 from the 2011 Geometry Regents exam is a 4-point question which asks students to solve the following system of equations graphically:

2x^2 -4x = y + 1

x + y = 1

Number 39 from the 2011 Algebra 2 / Trigonometry Regents exam is a 6-point question which asks students to solve the following system of equations algebraically:

5 = y-x

4x^2 = -17x + y + 4

These two systems of equations are roughly equivalent in terms of difficulty.  Why is a question suitable for the Geometry exam appearing on a the Alg 2/Trig exam, and as the highest-valued question (6 points) to boot?  In New York state, the Alg 2/Trig course follows Geometry in the standard sequence, so it is strange to see the same kind of problem on two state exams that are designed to be taken at least a year apart.

It’s true that the Alg 2/Trig test question asks for an algebraic solution, as opposed to a geometric solution, but that is essentially the only difference between the two.  This being the case, this speaks to a serious problem in how these tests are conceived and designed.

Looking at these two tests, one might conclude that learning to solve this kind of system algebraically is an important part of the Alg 2/Trig course:  why else would the official exit exam require the use of this technique in solving a problem that could have been solved last year?

Solving systems algebraically is definitely is a fundamental skill; so fundamental, in fact, that it is part of the Integrated Algebra curriculum (see the Integrated Algebra Pacing guide on the official schools.nyc.gov website).  Integrated Algebra is the course students take before they take Geometry!  Since many students take IA in 9th grade and take Alg 2/Trig in 11th or 12th grade, this means that a 6-point question on the Alg 2/Trig exam is testing the student’s ability to solve a problem they should have been able to solve two math courses ago.

Students should be able to solve this kind of problem at all mathematical levels, but why is material from two courses ago playing such a prominent role on an advanced exit exam?  What Alg 2/Trig course material is being shortchanged in order to re-test more elementary skills?  And to the point, how can this be considered a legitimate assessment of what a student learned in an Alg 2/Trig course?

Furthermore, in each case the scoring guide allows for half credit if the problem is solved using a method different than the one specified.  This is a reasonable policy, but what then is the purpose of a question specifically designed to test knowledge of a technique?  On the Alg 2/Trig test, a student can earn half credit for solving the system graphically; that means a student can get 3 of the 6 points by simply doing exactly what they did on the same problem on last year’s Geometry exam.

This example highlights how some questions on these exams aren’t directly connected to the content of their respective courses.  If a test isn’t legitimately designed around the curricula and content of the course, how can teachers and students effectively prepare?  How could such tests be valid assessments of what a student learns in that class?  Or how effectively a teacher teaches?  These are all questions that aren’t asked enough in the debate about standardized tests, student performance, and teacher accountability.

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