I have an adjustable screen for my window, the kind you expand horizontally to fill up the windowsill. It’s somewhat effective at keeping bugs out of the house.
When it’s not opened all the way up, the two layers of screen overlap in the middle. Depending on the angle you are looking from, you can see some cool images.
At this angle, for example, I see a contour map of a function of several variables.
I wish I understood where the curves come from!
This is a useful website on math careers from the Brigham Young University Math department:
About 40 or so careers are profiled here, like Mathematical Biophysicist, Foregin Exchange Trader, and Psychometrician. Each profile includes a salary range, educational requirements, job description, and potential employers.
Another nice potential answer to “When am I ever going to use this?”
This site has hundreds, if not thousands, of free e-books on a wide range of topics:
http://www.freebookcentre.net/
In addition to hundreds of math e-books on topics such as Linear Algebra, Geometry, Applied Mathematics, and Probability Theory, there are also selections of free books on many Computer Science topics, Chemistry, Physics, Languages, and much more!
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!
Finding 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.