What are your “top ten”? Sue VanHattum, a full time faculty member at Contra Costa College inspired me with her wonderful top ten list .
Below are my top ten Issues in Mathematics Education. While this is my opinion, I do highly encourage you to check our Ms. VanHattum’s post as well as her blog Math Mamma writes…
10) Math IS, by its very nature, FUN! A coworker of mine told me “I am never bored”. He did not mean to say that he was constantly entertained. The cop-out statement “Math is boring” or the equally ridiculous “Let’s MAKE math fun!” betray a general trend in society to dismiss what we don’t understand as simply being unappealing. I was tempted to write about this earlier this year when the NY Times article “Who Says Math Has to be Boring?” and response piece in Slate “Math Has to Be at Least a Little Boring“came out, but I didn’t want to bore my readership.
9) Discovering and uncovering content should take precedence over covering and recovering content. A quote from RL Moore “He who is taught the least learns the most.” Or from Paul Halmos “The best way to learn is do; the worst way to teach is talk.” Also see a great recent post from Grant Wiggins concerning when/if lecturing is an effective a teaching technique.
If you know you want to shut up more, but you have trouble (like me) not filling the dead space with your own voice sometimes, try Bob Kaplan’s advice for becoming invisible . Read about how to help students become “productively stuck” at Math For Love. Or for more information on Inquiry Based Learning, check the IBL blog. Lastly, I’ll share what I told an administrator who told me to “cover” more material while I was teaching High School. I pointed to my desk (which was COVERED in student work) and I told him that the textbook was under there already!
8) Underrepresented groups in mathematics will remain underrepresented (especially in academia) unless measures are taken to recruit and retain them. I am familiar with programs like EDGE, SK Days, Women in Number Theory, MSRI Connections for Women, etc, exist, but many women don’t know about them or aren’t actively encouraged by their departments to get involved in the mathematical community. We can bring new perspectives into our field by providing role models for those who are traditionally underrepresented in our field, making the academic workplace more family friendly, and by breaking down stereotypes. See Adriana Salerno’s recent post in PhD+Epsilon on the subtle ways in which women can be discriminated against.
7) Mathematics Educators deserve respect and more autonomy. Without the freedom to teach as they see fit, educators cannot be experimental and take risks in their approaches. Departments and school systems should reward creative teaching styles by having regular teaching observations of junior faculty by qualified individuals who can supply meaningful feedback. These observations should be formal, made regularly by a small group of individuals, and play a greater role in advancement than test scores and/or student reviews. As a postdoc, I would have loved more observations of my classroom. If some courses need to have uniformity in curriculum, the instructors should be given a concise outline (such as the Common Core) of ideas to be studied.
6) Mathematics Educators deserve opportunities to further their own content knowledge for teaching. Opportunities for ongoing professional development that truly connects research in education to implementation in the classroom are scant. Both university and K-12 teachers tend to model their teaching after what they experienced as students regardless of whether it was truly effective. Having reading groups on math ed papers is an activity done at some universities like the University of Arizona.
5) Mathematics (not the instructor) IS the Authority – this one is stolen directly from Ms. VanHattum’s list. Part of the beauty of mathematics is that `proof by intimidation’ is not a valid method of proof.
4) All students can be trusted to learn mathematics, there is no Math Gene, and math courses should NOT be mandatory. This does not mean that teachers should stop trying to inspire and excite students, but it does bear repeating that everyone can do math. Many teachers unknowingly perpetuate the Math Gene myth by saying things like “Well, you COULD subtract ‘x’ from both sides… but that wouldn’t be very smart now would it?”. Along this line of reasoning, we should trust students to take relevant courses. They are grown-ups and eventually they will figure out (perhaps with some advice) what skills they need to succeed in their area of interest. I’m sure many people will disagree with this last point.
3) Students don’t realize that Math is fiddling around, turning your drawing upside down, looking at it through the paper, being befuddled, being sure that you are a genius….being sure that you are a moron, waking up and realizing that you didn’t actually prove the Beal Conjecture in your sleep, waking up and realizing that that lemma you thought was wrong in your thesis is actually right! In other words, mathematics is joyful and unexpected. See Math With Bad Drawings and Math Ed Matters “Be Predictably Unpredicatble”. Share your own mathematical learning experiences with students.
2) Students don’t know that Math comes in many flavors. It’s hard to stay abreast of all the developments in your own field, much less others, but being curious is leading by example. If you go to a colloquium that isn’t in your area it may pay off. I’m often surprised at the number of ways there are to look at one problem like linear regression: as a machine learning problem where the data is a training set, as a geometric problem to be solved using singular value decomposition, as a parameter estimation problem involving Fisher’s matrix, as a classical minimization problem. Help students explore one toy problem from many perspectives.
1) Teachers should ask deep questions about basic ideas and be ready for questions or answers that might be better or different from what was anticipated. Sometimes when I go where students take me, it doesn’t align with my initial ideas. This is both vexing and exciting! It requires careful thought about definitions, purpose, and motivation behind concepts, not just examples and theorems. Anyway, in the blog Math for Love, Dan Finkel talks about the rewards in pursuing a student-proposed problem:“A dollar that cost a dollar”. I imagine that the ultimate goal in being an adviser is to have a PhD student who was “depending on you” become one who just totally blows you away with his/her conjectures and proofs. Of course, this has the potential for being simultaneously invigorating (“Yay! I’m an awesome mentor”) and depressing (“Was I ever that creative?”). So I put this at number one because I think that it is the item on this list that is potentially the most challenging and under-appreciated.
Wow! Writing a top ten list is hard.
I would add to #2 that there are people who like to focus on finding nice definitions or building grand theories, and others who focus on problem solving, and still others who like to apply their favorite technique to every problem they can (and make a career out of it!).
In other words, mathematics has lots of “styles.”
I just love reading this. I feel so much hope that the ways of teaching math are multiplying. Will share.
My non-professional take would be:
1. Teachers need to be engaged with the subject. A teacher can’t just go through the motions, teaching algorithms without understanding, instructing by rote, and assigning endless homework.
2. Children should be introduced to puzzles. The earlier the better. Let them work out all sorts of puzzles. Let them develop their ability to handle new and unusual brain challenges.
3. Teach alternative ways of performing calculations. (Foreign students do things differently. British and French children learn to do math in their head.) As I went between school systems, there were even differences in methods between American school systems. The point is to engage the underlying concepts not just the methods.
4. Introduce logic. Predicate logic can get very complex, but simple logical forms are very manageable.
5. From logic, introduce mathematical proofs, so that quantitative reasoning becomes more than just “plug and chug”.
6. Familiarize young people with additional mathematical symbols and their meanings. Even if advanced tools are need for performing the operations indicated by the symbols, children can be introduced to what the operations do. The summation sign is an example. It’s a simple concept that be mastered by an elementary school child.
7. Mathematics is much more than arithmetic. Introduce children to math topics beyond arithmetic at an elementary level. Topics might include probability, topology, combinatorics, or other subfields of mathematics.
8. Find applications to real world problems that involve calculating fluid flow, arrival times, distances, needed quantities of materials, processing times and other real world applications.
9. I’ll think of something else later.
This whole rant was great. My only addition is that I’d like to see students with mathematical learning disabilities identified and assisted when they’re young. I was good at Geometry, great at pre-Alg. But trying to get through Elementary Algebra courses four times, four ways, with all sorts of help as an adult, doubled my time in college and I still never passed. It really was not for lack of trying. I think if my dyscalculia had been identified when I was a kid and I’d had the right kind of help with it, maybe I could have gotten somewhere instead of swimming in algebraic circles for years while trying to obtain a music degree.
Another important point to be taken into consideration is that Math gets only one period in entire school day and English gets more such as English, Literature, etc. We need to have more time during a school day for math such as: Quantitative Math, Computational Math, Geometry, and Applied Math. Each problem discussed in the class should be discussed with different ways keeping geometry, computational etc. In other words the problem should be dissected and explained geometrically, graphically, computationally etc.
I appreciate your thoughts and agree, but I’m betting that many teachers feel as if mathematics is taking up so much of their day (in the onerous form of test prep), that they would be shocked at your comment. However, I do think that the Common Core is an effort in the direction you’d like to see followed.
I would like to see different types of math taught in high school, maybe something that was discovered less than 200 years ago. We study modern literature, why not modern math? Why is the track always arithmetic->algebra->geometry->more algebra->infinitessimal calculus? I second the suggestion on teaching combinatorics.
I really appreciate your point number 9. I struggle with the balance between teaching and letting the kids discover things for themselves and I thought the links you set up were very helpful finding this balance. I also completely agree with you about the autonomy that teachers need. This is an issue across the board not just in math and it is definitely affecting the end results and students are the ones suffering.
I go back and forth on your point for number 4 a lot. I understand and agree that math is used in everyday life and the skills learned there are helpful to everyone. However, in high school I had to take a theater class and it was unbearably miserable for me. That was after just one semester, instead of four years that students are stuck in math classes. This may seem like an odd comparison, but while math teaches you problem solving and logic skills, theater teaches you confidence and creativity which are also important everyday skills. We can see that not all students can succeed in theater and therefore do not deem it necessary for everyone yet we do not transfer those opinions to math. However, math classes do open a lot of doors for students as far as jobs and higher education opportunities so I go back and forth on my opinion on this issue.
Mine is more of seeking for advice. Mathematics has been so underated in the developing countries due to the wrong perception and the teaching methodologies. A picture has been created that rote learning is the only available way to learn math.More so, the math teacher educators who preach against rote learning sideline the practical approach. Why?
I need advice on the strategies to incorporate in math class that will make all students inclusive across all levels.