Archive for the ‘Teaching’ Category

Regents Recap — June 2012: Erroneous Questions

Here is another installment from my review of the June 2012 New York State Math Regents exams.

Mathematically erroneous questions consistently appear on these exams.  Here are two recent examples, both from the Algebra 2 / Trigonometry exam.

According to the scoring key, the correct answer is (4).  This would be the correct answer if the angle were given as -50 degrees.  Notice, however, that no degree symbol is present.  This means the angle is actually -50 radians.  In degrees, -50 radians is equivalent to roughly -2864.8 degrees, which itself is equivalent to roughly 15 degrees.  Thus, the actual correct answer is (3).

The above problem could be considered a typo (although no correction was ever issued), but the most erroneous Regents questions demonstrate a real lack of mathematical understanding on the part of the exam creators.   Consider the following question on complex numbers.

None of these answers are correct.

The exam writers believe that (3) is the correct answer.  Given a complex number a + bi,  the conjugate is indeed a – bi, provided that and b are real numbers.  But x is a variable, and there is no reason to assume that x has to be a real number.  If x = i, for example, (3) is not the complex conjugate of  -5x + 4i.  In this case, the conjugate of the original expression is i, while (3) evaluates to – 9i.

As emphasis on standardized exam performance continues to grow, a few points here or there can make a big difference in the lives of students, teachers, and schools.  The consistent appearance of erroneous mathematics on these exams calls into question their validity as a measurement of “student achievement”.

Replace Algebra with … Algebra?

August 1, 2012 15 comments

Arguments that suggest we over-emphasize mathematics in education don’t bother me.  I love math and see its utility in every aspect of my life, but I understand not everyone feels this way.  Also, when someone says we shouldn’t teach math, or we should teach less, it encourages me to reflect on my own beliefs about math and teaching.  This is usually a valuable experience.

So I read “Is Algebra Necessary?“, Andrew Hacker’s OpEd in the New York Times, knowing that I probably wouldn’t agree with much of it, but still prepared to examine my beliefs.  Hacker offers up a few of the standard reasons why students shouldn’t be forced to take “Algebra” in high school (“it’s too hard”; “it turns kids off”; “not everyone’s going to be an engineer”), but he doesn’t really bring anything new or substantial to the discussion.

One thing I did find interesting, however, was Hacker’s suggested fix:  instead of “Algebra”, we should be teaching courses like “Citizen Statistics”.

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

This is indeed a good idea.  Ironically, exploring the mathematics of the CPI is largely an algebraic activity.

Discussing which items should be included means creating a mathematical model and declaring variables for the unknown quantities we wish to investigate.  These fundamental skills are taught and emphasized in high school algebra.

Determining the weights that these items should be given essentially amounts to finding the coefficients of some function of those variables and exploring the consequences of those choices.  In their basic forms, these skills are also taught and developed in high school algebra.

Thus, it seems to me that Hacker is suggesting we replace Algebra, with, well, Algebra.

There are worthwhile discussions to be had about what we are teaching, why we are teaching, and how we are teaching.  But those discussions should be led by people who really understand what’s going on.  If Andrew Hacker thinks we should replace Algebra with Algebra, then someone else should be leading the discussion.

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Math and Weaving — Belt Weaves

Here are some examples of what I call “belt weaving” (I’m sure there’s a better term).  The basic idea is to begin with long strips of construction paper, oriented perpendicularly, and then weave and fold your way down.

Here are two examples of 2×2 belt weaves.  In both cases, the same kinds of strips are used,  but in a different initial configuration.

The 3×3 belt weaves offer more initial configurations, and show more complexity.

There is a rich and interesting structure to explore in these “belt weaves”.  For example, these two weaves look similar, but are indeed different.

My students and I had fun exploring the mathematical relationships between the various belt weaves.  I will share some of our ideas and results in my series on Weaving in Math Class.

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Teachers and Content Mastery

July 13, 2012 4 comments

On her blog, Diane Ravitch posted a reader quote about the importance of content expertise for teachers.  Here is an excerpt:

There is nothing more important, especially in the HS classroom, than a teacher who is an expert in his/her respective field. The “tricks of the trade” are second nature for those truly called to this noble profession. A teacher needs passion and patience, but more than anything else she needs to know what she’s talking about. That is what gives the teacher authority. 

And here is a slightly edited version of my response, originally posted as a comment on Diane Ravitch’s blog.

I disagree with the sentiment expressed by your reader.

First, it’s impractical to expect all teachers to be masters of their content. If we need 200,000 math teachers in the US, we aren’t going to find 200,000 math experts for those positions, unless we dramatically redefine what we mean by ‘expert’.

Second, content expertise is not the source of a teacher’s authority. Being a teacher is more about being a leader than being an authority, and leadership is earned through a combination of respect, effort, enthusiasm, caring, and expertise.

Lastly, subject-specific content delivery is one aspect of teaching that can obviously be streamlined by technology. As education evolves, we teachers need to make our case by emphasizing the variety of other tools and expertise we bring to students, not just content.

The original post can be seen here.

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Storytelling, Teaching, and Mathematics

This short list of “Pixar Story Rules” from Pixar story artist Emma Coats offers a fun look into the mind of a story-teller, as well as a surprising source of mathematical problem-solving and teaching advice!

These particular story rules sound remarkably similar to techniques of mathematical exploration.

#7   Come up with your ending before you figure out your middle.

#9   When you’re stuck, make a list of what WOULDN’T happen next.

#10   Pull apart the stories you like.

#11   Putting it on paper lets you start fixing it.

#20   Exercise:  take the building blocks of a movie you dislike.  How would you rearrange them into what you DO like?

Working backwards, proof by contradiction, taking apart things you understand and trying to put them back together, getting your hands dirty by working out the details–these are all common and useful techniques in exploring and understanding mathematical ideas.

And as a friend pointed out, writing a story is indeed a kind of problem-solving; maybe it’s not so surprising how much that process shares in common with mathematics.

And as a teacher, the following two really resonate, for obvious reasons!

#2   You gotta keep in mind what’s interesting to you as an audience, not what’s fun to do as a writer.  They can be very different.

#1   You admire a character for trying more than for their successes.

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On College Rankings

June 25, 2012 1 comment

This essay from the President of Reed College discusses what it’s like to live outside (and inside) the world of college rankings, essentially asking “Are these rankings meaningful?”

It’s a familiar story to anyone who has ever contemplated teaching to the test.  As rankings/ratings/grades become more and more important, colleges/schools/students (and teachers) tend to focus more and more on those metrics, perhaps at the expense of what’s really important (whatever that might be).

A perfect rating system, presumably, would compel the rated parties to meet and expand the standard of excellence.  But in practice, it seems difficult to come to a consensus about what comprises excellence, and even harder, then, to construct an appropriate rating system.

So how should we measure a college or university?

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Math-Intensive Majors

This is an interesting report from the MAA and David Bressoud about the current status of math intensive majors in the U.S.

Bressoud starts with the encouraging news in the STEM fields:  Science, Technology, Engineering, and Mathematics.  In the past fifteen years, colleges have seen a 33% increase in students in these majors.  However, those numbers may be dominated by particularly large increases in Biology and Psychology.

As a mathematician, Bressoud is interested in math-intensive majors, and so he looks more closely at mathematics, engineering, and physical sciences.  As total college enrollments and STEM majors have increased, these math-intensive majors attract a consistent percentage of students.  In fact, Bressoud notes that this percentage has been stable for the past 30 years, as math-intensive degrees have shown no growth as a percentage of overall college degrees.

This is curious, given the increasingly quantitative nature of modern society, industry, and academia.  Are greater percentages of students in other countries pursuing such degrees?  Or do we only need 0.5% of our college students studying math-intensive fields?

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