Response: Several Ways to Become Better at Teaching Math -- Part Two
Last week's question was:
What is the best advice you would give to help an educator become better at teaching math?
As I mentioned in Part One of this series, I have very limited teaching experience with math. Therefore, I'll defer to guest responses -- and reader comments -- from math educators. I do want to say, though, that people might be interested in the math resources I offer to my English Language Learner students, as well as the math materials I use in the International Bacculaureate Theory of Knowledge course I teach.
Part One included responses from educators José Vilson, Shawn Cornally and Dan Meyer. Today, Bob Peterson and Eric Gutstein offer an excerpt from their book, Rethinking Mathematics; Gary Rubinstein contributes an excerpt from his book, Beyond Survival; and I'm sharing several comments from readers.
Response From Bob Peterson and Eric Gutstein
Bob Peterson is the President of the Milwaukee Teachers' Education Association. Prior to his election, he taught fifth grade at the La Escuela Fratney in Milwaukee, and is a founding editor of Rethinking Schools.
Eric (Rico) Gutstein teaches mathematics education at the University of Illinois-Chicago, and taught middle school mathematics for several years in a Chicago public school. He is co-founder of Teachers for Social Justice in Chicago and a frequent contributor to Rethinking Schools magazine.
This is an excerpt from the introduction to the recent book they edited, Rethinking Mathematics:
I thought math was just a subject they implanted on us just because they felt like it, but now I realize that you could use math to defend your rights and realize the injustices around you.... [N]ow I think math is truly necessary and, I have to admit it, kinda cool. It's sort of like a pass you could use to try to make the world a better place.
- Freida, ninth grade, Chicago Public Schools
We agree with Freida. Math is often taught in ways divorced from the real world. The alternative we propose is to teach math in a way that helps students more clearly understand their lives in relation to their surroundings, and to see math as a tool to help make the world more equal and just.
To have more than a surface understanding of important social and political issues, mathematics is essential. Without mathematics, it is impossible to fully understand a government budget, the impact of a war, the meaning of a national debt, or the long-term effects of a proposal such as the privatization of Social Security. The same is true with other social, ecological, and cultural issues: You need mathematics to have a deep grasp of the influence of advertising on children; the level of pollutants in the water, air, and soil; and the dangers of the chemicals in the food we eat. Math helps students understand these issues, to see them in ways that are impossible without math; for example, by visually displaying data in graphs that otherwise might be incomprehensible or seemingly meaningless.
As an example, consider racial profiling. This issue only becomes meaningful when viewed through a mathematical lens, whether or not the "viewer" appreciates that she or he is using mathematics. That is, it is difficult to declare that racial profiling occurs unless there is a sufficiently large data set and a way to examine that data. If, for example, 30 percent of drivers in a given area are African Americans, and the police stop six African-American drivers and four white drivers, there is weak evidence that racial profiling exists. But if police stop 612 African-American drivers and 423 whites, then there is a much stronger case.
The explanation lies in mathematics: In an area where only 30 percent of the drivers are black, it is virtually impossible for almost 60 percent of more than 1,000 people stopped randomly by the police to be black.
The underlying mathematical ideas -- (dis)proportionality, probability, randomness, sample size, and the law of large numbers (that over a sufficiently large data set, the results of a probability simulation or of real-world experiences should approximate the theoretical probabilities) -- all become part of the context that students must understand to really see, and in turn demonstrate, that something is amiss. Thus with a large data set, one can assert that a real problem exists and further investigate racial profiling. For youth, racial profiling may mean being "picked on," but the subtleties and implications are only comprehensible when the mathematical ideas are there.
When teachers weave social justice into the math curriculum and promote social justice math "across the curriculum," students' understanding of important social matters deepens. When teachers use data on sweatshop wages to teach accounting to high school students or multi-digit multiplication to upper-elementary students, students can learn math, but they can also learn something about the lives of people in various parts of the world and the relationship between the things we consume and their living conditions.
Engaging students in mathematics within social justice contexts increases students' interest in math and also helps them learn important mathematics. Once they are engaged in a project, like finding the concentration of liquor stores in their neighborhood and comparing it to the concentration of liquor stores in a different community, they recognize the necessity and value of understanding concepts of area, density, and ratio. These topics are often approached abstractly or, at best, in relation to trivial subjects. Social justice math implicitly tells students: These skills help you understand your own lives -- and the broader world -- more clearly.
We believe it's time to start counting that which counts. To paraphrase Freida, the ninth grader quoted above, we need to encourage students to defend their rights and to recognize the injustices around them. By counting, analyzing, and acting, we will help students and ourselves better read the world and remake it into a more just place.
Response From Gary Rubinstein
Beware of the Exercises in the Textbook
You generally have to make up your own exercises that progress from a really easy question to a very difficult one. In many texts, though, there will be an easy question followed by an extremely difficult one and then another easy one. Students often give up when they encounter the first difficult question. Maybe the rationale is that on a standardized test the questions won't always progress so smoothly from easy to difficult. But this isn't the standardized test yet. This is the part where they practice and assimilate their skills, and if they give up on that part, they won't do so well on the standardized test anyway.
Should I Let the Students Use Calculators?
The philosophy is that with the rise in technology, we should not waste time with things like multiplication drills. The only reason that math class included such a large amount of drill in the past was that there were no calculators yet. Now that calculators are here, it would be anachronistic to continue using pencil and paper for skill drills. In place of the monotonous drudgery that used to dominate math classes of yesteryear, students can be engaged in problem solving where the calculator is merely a tool like a protractor or a pencil. The real work can only be accomplished in the mind. As a result of this philosophy, there are many students who cannot add 9 and 3 without reaching for their calculators.
I'm currently teaching at a specialized high school where some students are so accustomed to using their calculators that they use them with two hands for speed. Pencil and paper calculations give students a feel for how things work in more complicated math. For instance, when students are learning to multiply polynomials, they can relate it to their skills in multiplying numbers that have multiple digits in them. What good is being able to reason out a simple process if you need a calculator to do the easy part for you?
Another thing about pencil and paper calculations is that they are a lot simpler and clearer than the more involved problems that are supposed to replace the "easy" monotony. But for some students, that easy monotony was challenging. The opportunity to complete some calculations was, for some students, the only time that they have ever felt any real success in a math class. To take that away from them and replace it with a confusing multi-step problem to solve with the calculator deprives those students of the opportunity to build confidence. I have gone to an extreme to include pencil and paper math, even some topics that have been discarded from the curriculum. I've had my students study the square root algorithm and the log tables as two mechanical procedures that build mathematical experience.
Out-of-Context Percent Problems
I think the worst math questions that often appear in textbooks are the old-fashioned percent questions. With no context, they look like, "What is 90% of 250?" "280 is what percent of 300?" and "21 is 70% of what?" There are two common, yet terrible, ways to teach these. One is the word-byword translation: " is means equals," " of means multiply," and " what means x. " Even if you can get your students to answer the questions this way, they are not really learning anything meaningful.
The other way is the "is over or equals percent over 100" method. Any word-byword process is a waste of time. First, I'd make the questions more interesting by making them about sports. I can rephrase the first two questions above as "If LeBron makes 90 percent of his foul shots and he attempts 250 shots, how many will he make?" and "If Kobe makes 280 out of 300 shots, what was his percent?" These are questions for which the students can first make mental estimates. Ninety percent is "most" so most of 250 is around 240; 280 is "most" of 300 so it's a good percentage, certainly over 90. Then I use proportions, which is one of the most important concepts in middle school math.
Responses From Readers
I think the biggest thing is to understand the "why" behind all the procedures and processes. It shouldn't be a recipe for solving problems....it's really much more about the journey and the wondering. If you can hook kids into that, their willingness to attend to arithmetic precision will increase.
I also think being playful is hugely important. Playing with math builds number sense and opens a student's mind into becoming involved. Kim Sutton does a great job teaching teachers how to do this with the younger kids....and this playfulness should continue thru middle and high school. Even if "what happens if I do ______" kind of thinking is playful...and gets kids to wonder and experiment with numbers and processes.
Readers might also be interested in Marsha's article in Education Week Teacher, The Talking Cure: Teaching Mathematical Discourse.
Perhaps if we focused a bit more on how to use math instead of just teaching math? I know math really came alive for me when I started taking drafting and electronics courses in high school.
1) Join Twitter and connect to the huge number of terrific math educators posting there regularly. I find most of my new ideas from my own personal learning network of people I follow.
2) Read as much as you can. Read about the history of mathematics, so you can talk intelligently with your students about what it is they're doing. Learn about the history of math education and education in general - learn about how schools got to where they are today.
3) Get into the nitty-gritty of policy. Keep up with what is going on in the corridors of power, so you know what's coming and what your response will be when it arrives.
4) Most importantly, have a philosophy. Know why it is you're doing what you've chosen to do.
I have several pieces of advice for those wishing to become better math teachers:
#1 - Remember why you became a teacher. If you are a math teacher, it's probably because you really love math. If you're not a math specialist, it probably has something to do with the fact that you love learning. Remember that love, and try to foster it in your students. Drill and kill worksheet do not foster a love of anything!
#2 - Read! There are now a lot of great books being published about teaching math, especially at the elementary level. NCTM journals are also a great resource - if you're not a member, you should be.
#3 - Expand your personal learning network (PLN). Find a few math bloggers that you like (Dan Meyer is one of my favorites). Join Twitter and check out #mathchat.
#4 - Know how to answer that age old question "When are we ever going to use this?" with something beside "On the test next week." If you don't know, find out. Start your lesson with something related to that answer.
#5 - Study the CCSS Standards for Mathematical Practice and work diligently to incorporate them into every lesson. These eight practices really embody what it means to know and "do" mathematics. Lessons that require students to use these mathematical habits of mind will help build a better understanding of and greater appreciation for mathematics, and help kids see a purpose in their learning.
Advice I would give to educators who want to become better math teachers:
Invigorate! - Math instruction should be invigorating with purposeful movement and use of concrete manipulatives, not sitting in one place doing paper and pencil tasks all day.
Investigate - Don't TELL students what they can learn for themselves. With math concepts, it is IMPERATIVE that students master math concepts in the concrete before symbolic is used. Students construct meaning when they investigate and reach their own understanding of addition, subtraction, equality, fractions, etc. When it comes to procedural things, direct instruction and skill drill is fine, but concepts must be learned in the concrete through investigation.
Integrate - Math does not exist on worksheet pages or in the pages of a textbook, but is a beautiful symbolic representation of real world phenomena. NEVER EVER EVER divorce math class from real world application.
Communicate - Students must ALWAYS be practicing math language and explaining math activities with math terms. ALWAYS have students collaborating and communicating about their problem solving.
...let's make math teaching truly professional by taking collective responsibility for math teaching at all levels, by demanding that we decide what qualifies as appropriate professional education and development, and by demanding high standards for entry into our community. Let's develop repositories of shared, vetted knowledge about math and its teaching. Let's compete for each other's admiration through the sharing of this knowledge, and let's use such a system to evaluate our work from within our community. Let's evaluate our work using qualitative, descriptive means, let's resist external accountability measures, and let's not allow the pursuit of high scores on standardized assessments to distort teaching and learning. Instead, let's be driven by our desire to help our students think and learn.
...this is more centered towards boys. Everyday, before the boys entered class, I would write on the board a simple math puzzle. The boys had an additional five minutes to figure out the problem pass the last bell. Interesting enough this cut down tardies to class too. The problem on the board was not graded and sometimes there was even a prize for solving it. This method immediately got the boys thinking math before class started. The key to my lesson was for the boys to discover a pattern. I used storytelling and history that would make connections on how mathematics was used to solve a problem. Then I would have the boys discover how that past solution changed our way of life today. I was making mathematics relevant and boys were discovering the connections. Whenever I could, I added hands-on projects (like creating a hologram without a laser called a Scratch Hologram.) Creating the Scratch Hologram brought in topics of geometry, optics, science, engineering, history and art.
Thanks to Bob, Eric, Gary and many readers for contributing their responses.
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