Presentation at AISNE Math Teacher Workshop on October 4, 2012, “Thoughts on teaching math.”
Let’s start by considering a scene from a neighborhood bar on a Thursday night…
“Oh, what do you do?” says polished and sophisticated looking patron.
“I am a math teacher,” you say.
“I hated math,” s/he says.
You think to yourself, “what do I say to that?”
Sometime ago in our culture, it became okay to say that “I suck at math.” But no one who is educated would ever admit to being incapable of reading or writing – you would be labeled illiterate and people would feel compassion for you. Yet, there is a cult around math-phobia or should I say around people who are math-centric. It’s a love/hate relationship and your job is to be somewhere in the middle.
That’s right, I don’t want you to get your students involved in cults. That’s a good thing… but seriously, don’t mislead your students – very few of them will directly use what you are teaching them. But most of them won’t go on to perform MacBeth or shape foreign policy (so out with the Monroe Doctrine, because history never repeats itself).
We have to give students a compelling reason for why they need to learn math. That’s your first goal. Hopefully, you will only have a few students who really question why they need to learn math.
I see the syllabus as your opportunity to convey to your students who you are and who they can be. Here is my teaching philosophy, which I discuss with my students on the first day of classes (I hand this to them, briefly go over it, and then suggest they read it along with the syllabus and assignment sheet for homework – I am certain that most do not read it).
Dear Students, I want to share with you how I approach your learning and my teaching, and what follows is my teaching philosophy – it lays out what you should expect this year.
In this mathematics class, you will learn to think, reason, and write– because these are necessary skills for life! This means that the exercises and problems that I select for homework and for quiz/test review may be vastly different from the actual questions on my assessments. I know I’m asking much of you, but you are some of the best and brightest; so if you put in the effort at the beginning of your Andover career you will be on the path toward success.
I believe that students must be trained to take calculated intellectual leaps, intent to listen for valid opinions, reflect on their contribution to the larger community, and realize their potential to becoming model citizens. In the math classroom, I foster this by encouraging my students to see math in the real world − the logic in arguments, the structure of languages, the systematic way of annotating a page of math literature, and using examples to generalize arguments to reasonable conjectures, and writing proofs of theorems.
I want my students to learn to grapple with questions. I fell in love with mathematics because I enjoyed the search for the answer. Sometimes the search involves asking someone for help, looking up a reference, or sitting down with paper and pencil and patiently waiting for the answer to “come to you.”
Lastly, if I want my students to become life-long learners, tinkerers, experimenters, hypothesizers, and doers, it has to happen in the context of a structured learning environment. Homework, tests, and the final exam have to be consistent in the breadth and the scope of what is asked and set reasonable expectations for literacy. Classes should have a definite rhythm while allowing for the occasional improvisation, sharing a newspaper article, an alternative calculation/proof, or giving students the opportunity to go to the board to show their work. Students learn best when they are sitting next to each other, seeing each other’s work, and encouraged to talk about how they got from Step 4 to Step 5.
Notice that in my teaching philosophy, there is very little about solving equations. I don’t mean to be intentionally misleading; rather, I want students to see that in learning theoretical mathematics, they are learning to think deep thoughts.
So how do you set the bar high? Tests, quizzes, group projects, and alternative assessments. Try to mix it up…
Another way to sum things up is by what our veteran math teacher, Doug Kuhlmann, calls “The Rule of Four”: every topic/idea in mathematics should be taught (and thereby assessed) symbolically, graphically, numerically, and verbally.
Symbolically – students need to be comfortable with the syntax of mathematics – these are the words and phrases that make up math. Proof – why do we teach it? It needs to be a necessary part of the each math curriculum. In my PreCalculus class, I am currently struggling to find time to prove ideas. I have taught my students a ton of complex numbers, but they have yet to really see a proof. It’s a sad reality. There are many easy proofs one could do in a course (Euclid’s Lemma, , proof of the Pythagorean Theorem).
Graphically – a picture is worth a thousand words. If a student can not graph the function, find the circumcenter, or see the big idea, what’s the point?
Numerically – some students are motivated by theory. Others see things once they can get an opportunity to put values into the machine and see what comes out. Statistics or let’s put it more simply, data analysis is a valuable tool.
Verbally – fundamental theorem of arithmetic, describing how to get the quadratic formula through completing the square, and to explaining to their aunt or uncle (who does not know Calculus) the concept of the derivative. Each of these ideas could easily take up page.
Technology Ideas – can help students to reinforce or develop ideas, new ways of thinking, and make learning enjoyable and fun.
WebWork Geogebra Geometer’s SketchPad Wolfram Alpha TI-84 or TI-Nspire
iOS apps FluidMath Coursesmart.com
Collect homework at least a few times a term – even if you decide not to grade it, make sure you have an idea of how students are doing in the class and the attempts they are making on homework.
Group Review for Extra Credit – math relays, similar to GUTS rounds at some math team contests
3 week minicourses on applied topics – see financial planning
Writing math tests –
- Cull from different sources. Have a number of tests from other teachers. Don’t be afraid to use a test bank, but don’t limit yourself to those problems.
- Work together on writing exams/tests. Don’t let the Course Coordinator write the exam – ask if you can contribute problems so it can be a collobarative effort.
- Those in high school – try to grade the AP exam somewhere in the next 5-10 years.
- The more problems you do, the better the problem writer you will become.
- Aim for 70% routine homework problems, 20% somewhat challenging, and 10% left field problems (that you haven’t prepared them for, but that are not Greek to them either)
Managing class time (say a 50 minute class)
- 2 minutes get to know you
- 13 minutes homework review
- 25-30 minute lesson (includes demonstrations, proofs, group work)
- 5-10 minute wrap up, extension, or challenge
Problem based learning – see the Exeter Math Problems – http://www.exeter.edu à Academics à Departments à Mathematics à (Mathematics) Teaching Materals
Diversity in the Classroom
Be mindful of students who have learning accommodations and study plans, and women and students of color for whom math/science may require you to think about stereotype threats. See this blog post by PA’s Head of School, John Palfrey, on why “Steele’s book should be required reading for anyone who works in a school.”
- Masters Degree in Mathematics or Mathematics Education
- The Anja S. Greer Conference on Mathematics, Science and Technology, June 23 – June 28, 2013, http://www.exeter.edu/summer_programs/7325.aspx
- Independent School Focused
- Klingenstein Summer Institute: http://www.klingenstein.org/content/summer-institute
- Harvard GSE Summer Institute: http://www.gse.harvard.edu/ppe/programs/audiences/prek-12/teachers.html
- UPenn’s Grad Program in Ind School Leadership: http://www.gse.upenn.edu/degrees_programs/app
- Critical Friends Groups – using the Tuning Protocol as developed by NSRF (National School Reform Faculty)