Teaching

Google and Conditional Probability

Conditional probability is one of my favorite topics to teach.  Whereas normal probability calculations simply compare favorable outcomes to total outcomes, conditional probability allows us to consider the impact of certain knowledge on the likelihood of those outcomes.

For example, the probability of rolling a 6 on a six-sided die is 1/6, but if it is known that the number showing is greater than 3, then the conditional probability that a 6 is rolled is 1/3.

There are many applications of conditional probability, but a recent “Math Encounter” from the Museum of Math made me aware of an application of conditional probability that all of us see on a regular basis:  Google search autocomplete.

Suppose I type in the search term “under”:

Here, Google is trying to autocomplete my search query.  In essence, Google is trying to guess the next word I’m going to type.  How does it make its guess?  It computes a conditional probability!

Google has a lot of data on when words follow other words.  When I enter “under” into the search bar, Google looks for the word/phrase with the highest conditional probability of being next.  Here it turns out to be “armour”; the word with the second highest conditional probability is “world”, and so on.

Naturally, as more information is provided, the conditional probabilities change.

 A fascinating, and perhaps surprising, application of a powerful mathematical idea!

By patrick honner, ago

The Write Angle for Teaching Math: How to Get Students Writing in Math Class

Math WritingFinding ways to get students to write about mathematics has played a pivotal role in my development and growth as a math teacher.  Mathematical writing challenges students to express their ideas clearly and efficiently; it forces students to stop thinking of mathematics as merely equations and answers; and it opens up a new and unexpected dialogue between math teacher and student.

I have always found great value and pleasure in writing.  It is a valuable skill, a necessary tool of scholarship, and a powerful creative outlet.  And now I see its value as a math teacher.  The more my students write, the more useful and interesting we all find it.

In this post, I’ll discuss some strategies for getting students writing in math class.

How to Get Students Writing in Math Class

1)  One-page solutions

Have students choose a good math problem they like and write up a solution in one page.  Have them narrate their process and explain the choices they make.  They can also write about mistakes someone else might make, or offer an alternate solution if possible, or suggest a new problem that’s related but slightly harder.  Problems from math competitions (like the New York Math League or the American Mathematics Competition) are usually great places to start.

2)  Create New Questions

Have students choose a good math question and write up three new questions based on the original.  If the original problem asks to find the sum of consecutive integers from 1 to 10, some new questions might be “What is the sum of the consecutive integers from 1 to 100?”, “What is the sum of consecutive integers from 1 to n?”, and “What is the sum of the squares of the integers from 1 to 100?”  Make sure that the students understand that the assignment is to create the question, not answer it; not being required to answer the question takes the pressure off, and it frees them to be more creative.  Again, math contest questions are great places to start.

3)  Mathematical Poetry

Have students write short mathematical poems, like haikus, quatrains, limericks, or even sonnets.  Ask them to explain a math concept, pose a math problem, or describe a geometric object in verse.  And keep it short!  Insisting that the students only use 17 syllables or four lines to describe their mathematical topic not only forces them to refine their thoughts, but it also makes the assignment so easy that it’s nearly impossible not to do it!

4)   Writing Prompts

Have students respond to prompts like:  “What is your favorite number?”; “What’s the best shape?”; “Describe a mathematical epiphany you’ve had”; “Write an original math joke”; or “Write a mathematical aphorism”.  Create your own prompts, or better yet, have students suggest them!

For more resources, see my Writing in Math Class page.

By patrick honner, ago

Testing Maximum Performance

This is an interesting article by Jonah Lehrer of the Wall Street Journal about the limits of standardized testing.

https://www.wsj.com/articles/SB10001424052748704471904576230931647955902

Lehrer discusses the results of a study from the 1980s in which psychologist Paul Sackett attempted to measure the speed of supermarket cashiers.  A short “check-out test” was developed which involved scanning a small number of items.  The test was administered, and resulted in a list of the fastest cashiers.

What is interesting is that when Sackett compared the results of the test with long-term data collected by the electronic scanning systems, there was a surprisingly weak correlation between the results of the speed test with the data from regular usage.  That is to say, there was no real connection between being fast on the test and being a fast on a day-to-day basis.

Sackett’s misconception, and perhaps one held by many, is that there is a natural correlation between maximum performance (that on a short test) and typical performance (that is, under normal, day-to-day circumstances).   Tests like the SAT, the GRE, and other high stakes tests, are tests of maximum performance.  Our educational system relies on these  more and more, but are we sure they measure what we assume they measure?

Lehrer points out that individual success is determined more by character traits like perseverance and self-control,  but of course, it’s hard to capture that in a timed, multiple choice exam.

By patrick honner, ago

The Write Angle for Teaching Math: Why Write in Math Class?

Math WritingFinding ways to get students to write about mathematics has played a pivotal role in my development and growth as a math teacher.  Mathematical writing challenges students to express their ideas clearly and efficiently; it forces students to stop thinking of mathematics as merely equations and answers; and it opens up a new and unexpected dialogue between math teacher and student.

I have always found great value and pleasure in writing.  It is a valuable skill, a necessary tool of scholarship, and a powerful creative outlet.  And now I see its value as a math teacher.  The more my students write, the more useful and interesting we all find it.

In this post, I’ll first address the question “Why Write in Math Class?”.

Why Write in Math Class?

There are infinitely many good reasons to write.  I’ll offer three that have been on my mind lately.

1)  Writing is a fundamental mathematical skill

Many people might not be aware of it, and many might not admit it, but good writing is a fundamental mathematical skill.  A proof isn’t a proof unless others understand it, and that can’t happen if it isn’t written clearly and concisely.  Also, it’s great when we find the right answer in a math problem, but as most teachers know, it’s usually more important to understand the problem-solving process than to get the right answer.  Good writing skills help narrate and record that process, and make that process available for teachers and peers to understand.

2)  Writing is an indispensible professional skill

I’ve had many different jobs in my life, and being a good writer made me more effective at all of them.  Whether designing technology systems, meeting with clients or consultants, talking through project specifications, or working on a team, being able to document and communicate effectively about the process gave me an edge.   Writing about mathematical ideas and procedures can be hard, but it’s great training for thinking and communicating about the kinds of open-ended problems students will face in the real world.

3)  Writing helps me understand my students better

By regularly interacting with my students through writing, I get to know them in a significantly different way than through their work on exams and homework.  Through various writing activities, I can develop a better sense of what kinds of math problems they like, what kinds of problem-solving techniques they are most comfortable with, and of course, what kinds of ideas are difficult for them to consume.  Getting a different look at how my students think mathematically is incredibly valuable as a teacher, and it can be extremely fun, too!  Giving students the chance to think and write creatively about math almost always produces something unexpectedly wonderful!

For more resources, see my Writing in Math Class page.

By patrick honner, ago

The Terrible Trapezoid

Schoolbook ran a piece on yet another terrible test question, this one appearing on the New York State fifth grade math exam.  The most disturbing part of the situation is that no one really seems to understand just how bad this question is.

The New York Times framed the issue as requiring the student to use a technique outside the normal curriculum; the problem is worse than that.  The NYS education commissioner dismissed the error as a “typo”‘; the error can not be considered a typo.  The chancellor of the NY Board of Regents decreed that anyone who claims the tests are invalid is just pushing back against teacher evaluations; no one who understands mathematics can claim that this question is valid.

The problem starts with a trapezoid of sides 5, 16, 13, and 28.  After asking the student to find the perimeter of the trapezoid, the problem then states

A new trapezoid is formed by doubling the lengths of sides AB and CD.  Find the perimeter of the new trapezoid.

 And here’s where the trouble begins.

1)  Is this a right trapezoid?

The Schoolbook piece assumes that the trapezoid is a right trapezoid, i.e., that angle ADC is a right angle.  Nowhere in the problem is it stated or indicated that the trapezoid is right.  And even if we know that it is right, 5th graders are not expected to know the Pythagorean Theorem.

2)  Why does BC change while AD remains constant?

The Schoolbook piece also assumes that as AB and CD are doubled, the length of AD remains constant while the length of BC changes.  Thus, in order to find the new perimeter, the student must find the new length of BC (using the Pythagorean Theorem).

This is the critical error in the construction of this problem:  the test authors don’t seem to understand the subtleties of scaling figures.

Doubling AB and CD doesn’t specify a unique new trapezoid.  BC could change while AD remains constant; AD could change while BC remains constant; AD and BC could both change.  (It is interesting to note that it is impossible to double AB and CD while keeping both AD and BC constant).

Was the original intent to tell the students, or have them assume, that the angles stayed the same?  If so, the resulting figure could not exist.

Was the original intent to tell the students, or have them assume, that AD was also supposed to be doubled?  If so, this still doesn’t specify a unique new trapezoid (unless the angles also remain constant).

The concept of this problem is fundamentally flawed, and it demonstrates a real lack of mathematical understanding on the part of those who created, edited, and screened it.

What’s worse, education officials pretend that this is just a ‘typo’, and that this is no reason to question that validity of these tests.

If the consistent appearance of erroneous math questions on state exams year after year doesn’t constitute legitimate criticism of the validity of these exams, then what possibly could?

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By patrick honner, ago
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