When Does Teaching Cease to Be a Challenge?

An interesting conversation about teacher retention emerged recently, beginning with Shawn Cornally thinking outloud about how to keep good teachers in the system, and later moving over to Dan Meyer’s blog where he discussed the twin pressures on novice teachers.

In describing how good teachers often ultimately feel a pull out of the classroom, Dan Meyer says

The job becomes untenable at about the same time that it becomes unchallenging.

The point he’s making is that teaching becomes easier, and as it does, it ceases to be a challenge for good teachers, who are then more likely to leave the classroom in search of other challenges.

There are valid, relevant issues raised here, but the suggestion that teaching ceases to be a challenge at some point sounds crazy to me.

Teaching is always a challenge.  Experience may make certain practices more efficient, but in some ways, that efficiency only makes the deeper challenges easier to see.

Consider the endless challenges offered by the three major components of teaching: knowledge of students, knowledge of pedagogy, and knowledge of content.

A good teacher must know their students.  Every new student, and new class, presents unique challenges to a teacher, who has to forge positive relationships and create productive environments.  This may get easier with experience, but it’s always a challenge, and can always be improved upon.

Since teaching is about understanding how learning happens, the fact that we don’t fully understand how learning happens creates another set of evolving challenges.  There are always new ideas to consider, new practices to try, new approaches to instructional design, and of course, new technologies to integrate.  Trying to figure out how learning happens is a daily challenge for a teacher, and it may never fully be understood.

And when it comes to content knowledge, no teacher could feel more challenged than a math teacher.  Some of the smartest people in the world spend their lives in a perpetual state of learning mathematics.  There is always more mathematics to study, new connections to find, new perspectives on old problems, and old problems to make new again.  Understanding mathematics is a never-ending challenge.

At some point, a good teacher may decide that these challenges are no longer meaningful enough to justify the great effort and investment that teaching requires.  It’s understandable, and in that case, leaving the classroom may be a courageous and noble decision:  walking away from something you do well in order to follow a deeper passion is admirable, and it also sets a good example for students.  But this isn’t because teaching ceases to be challenging; it’s because the individual no longer feels motivated by those challenges.

The task of teaching is infinitely deep and infinitely varied.  I often feel that, in becoming a better teacher, I simply become more aware of what I need to do differently.  Like mathematics itself, teaching becomes more complex the more you know about it.  What could be more challenging than that?

Math Quiz — NYT Learning Network

Through Math for America, I am part of an ongoing collaboration with the New York Times Learning Network.  My latest contribution, a Test Yourself quiz-question, can be found here:

Test Yourself Math — February 13, 2013

This question pertains to the GI Bill, a federal program which provides money for military veterans and their families to attend college.  Approximately how much does each participant receive in benefits?

GothamSchools on MT^2

Gotham Schools recently ran a nice piece on Math for America’s inaugural Master Teachers on Teaching (MT^2) conference, which focused on mathematical modeling in the classroom.

The piece, found here, discusses the many different ways MfA teachers are bringing modeling into their classrooms, and engaging students in the mathematics of real-world, open-ended problems.

My talk, “g = 4, and Other Lies the Test Told Me” is featured in the article, as it addresses how our desire to teach and practice modeling often “is at odds with the way that the city and state assess students”.

Thanks to Gotham Schools for helping to spread the great work of MfA and its teachers!

How Many Primes Did We Miss?

largest primesThe mathematics world is abuzz with the verification of a new largest known prime number.  The number, 2^{57885161} - 1, is a Mersenne Prime, and has over 17 million digits.  The previous largest known prime was 2^{43112609} - 1, which had a mere 12.9 million or so digits.

It’s interesting to note that, while it has been known for thousands of years that there are infinitely many primes, it is a challenge even today to find large ones.

It is also interesting to note how many primes were missed in jumping from the previous largest prime to this new largest prime.

A well-known and elegant result, Bertrand’s Postulate, states that there is always a prime number between n and 2n, for n > 1.  For example, the prime 3 is between 2 and 4; the prime 5 is between 3 and 6; the prime 11 is between 10 and 20; and so on.

In particular, this says that there must be a prime between 2^{n} and 2^{n+1}, since 2 * 2^{n} = 2^{n+1}.

Thus, there must be a prime between 2^{43112609} and 2^{43112610}, and another between 2^{43112610} and 2^{43112611}, and so on!

Thus, there are at least 57,885,161 – 43,112,609 = 14,772,552 primes between 2^{57885161} and 2^{43112609}!  We can therefore safely say there are at least 14,772,551 primes between the largest and second-largest known primes!

Let’s hope it’s not another 4 years until we have a new largest prime on the block.

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