Saturday, December 8, 2012

11 Recommendations to Middle-ish Grades Math Teachers

Maybe some of these are "radical" and "offensive", or maybe they're not. I'm just throwing them out there. Please don't hate me if you do some of these (I know a couple of them are quite commonly done by many math teachers). These are just my personal opinions, but I feel quite strongly about them. I figured I'd say it here, because some of these things drive me nuts, and I need an outlet.

11. Do not tell a kid to "move things across the equal sign and just change the signs". Preferably you don't ever say this phrase to a kid ever, but if a kid comes to you and says that someone else somewhere (ie. a parent or another teacher) has taught that to them, you need to drill into them the reason why this shortcut works. Grill them until they can articulate why the term necessarily shows up on the other side with the opposite sign.

Reason: If you teach shortcuts like this without the correct foundation of understanding, then down the road the kid cannot approach a complex situation intuitively, because they cannot visualize the equation as a balance.

10. Do not allow your students to do simple equations only in their heads / showing little or no work at the early stages.

Reason: Even if the kid is bright and they can do the equation correctly every time, you are helping them to develop a bad habit that is hard to break. Down the road, when the equations get more complicated and the kid starts to make procedural mistakes, they'll have such a hard time finding their mistakes because they've habitually skipped so many steps.

9. Do not allow your students to "open parentheses" without knowing why they do this and where it is the most useful.

Reason: Normally, it just irks me to see kids do 2(3 - 5 + 1) = 6 - 10 + 2 = -2 instead of following PEMDAS to do 2(3 - 5 + 1) = 2(-1) = -2, but it's really feckin' scary when I see students do
2 (4·5 + 7) = 8 + 10 + 14 = 32 in their second year of algebra. When all the values are known, they don't need to be distributing!!

8. Do not introduce integer operations without explaining the meaning of addition and subtraction of signed numbers! You can use number lines or you can use the idea of counting and neutralizing terms, but in the end, the kids need a framework of understanding.

Reason: I recently tutored a kid on a one-time basis (I was doing a favor for her parent, who's my friend, even though the kid doesn't go to our school), and the kid only knows memorized rules (she rattled them off like a poem), with no conceptual understanding whatsoever when I probed further. Unfortunately, that kind of problem is not really fixable in a day. It can take a teacher weeks to instill the correct habit of thinking. If you teach a kid to memorize a rule for adding or subtracting integers without understanding, then I can almost guarantee you that after the summer they won't remember it. And, if you first teach the rules by memorization, even if you try to explain afterwards the conceptual reasons for the rules, there ain't no one listening.

7. Do not teach "rise over run".

Reason: In my experience, a majority of the concrete-thinking kids out there do not know what the slope means. When you ask them, they just enthusiastically say "rise over run", but they cannot identify it in a table, they cannot find it in a word problem, they certainly cannot say that it's change of y over change of x, and half of them cannot even see it in a graph.

You have other alternatives. I teach slope as "what happens (with the quantity we care about), over how long it takes." For example, the speed of a car is "miles, per hour"; the rate of growth of a plant is "inches, per number of days"; the rate of decrease of savings can be "dollars, per number of weeks". You have alternatives to "rise over run". Learn to teach kids to visualize the coordinate plane as a sort of timeline, where the y-axis describes the interesting values we are observing, and x-axis describes the stages at which those values occur. Once they understand this, then they can correctly pick out the slope given any representation of the function.

6. Do not let any kid in your class get away with saying "A linear function is something that is
y = mx + b."

Reason: Again, in my experience, I can habitually meet a full class of kids who had spent a good part of a previous year learning about lines and linear equations and linear operations, and in the following year when they move to my class and I ask them to tell me simply what is a linear function, the only thing they can say is: Y = mx + b. That is really, really bad. The first thing they should be able to say is that it is a straight line, or that it is a pattern with a constant rate.

Please, please, do teach lines in context. Make sure that kids know what y, m, x, and b all mean in a simple linear situation.

5. Do not introduce sine, cosine, and tangent without explaining their relationship to similar triangles.

Reason: The concept is too abstract, too much of a jump from anything else the kids have done. Give the kids some shadow problems to allow them to see what the ancient mathematicians saw, and why they recorded trigonometric ratios in a chart. Let the 3 ratios arise as an outcome of a natural discovery, a natural need. And then, when you introduce the terms sine, cosine, and tangent, the kids will at least already understand them as sweet nicknames for comparisons between sides.

From then on, whenever you say sin(x) = 0.1283... , remind kids what that means simply by immediately saying, "So the opposite side is about 12.8% as long as the hypotenuse." If you do this consistently, kids will never grow afraid of those ratios.

4. Do not teach right-triangle trigonometry from inside the classroom!

Reason: Kids need to be able to visualize geometry relationships, especially in word problems. Some kids have trouble doing this on a flat sheet of paper. Don't disadvantage those learners. Take them outside, make them build an inclinometer, make them experience angles of elevation, angles of depression, and the idea of measuring heights through triangular ratios. This really only takes a day, or at most two. In the end, when you take them back to the classroom, you will be amazed at how well they can now visualize even complicated word problems involving multiple right triangles.

3. Do distinguish between conceptual and procedural errors. Your modes of intervention should look very different for those two types of mistakes, and what you communicate to the kids as recommended "next steps" should look fairly different as well.

2. Do incorporate writing into your classroom. Writing is an extremely valuable way of learning mathematics. Every project should have a significant writing component. When kids write, they are forced to engage with the subject on a personal level, and so they learn much more.

1. Do make learning fun. From a biological perspective (of primitive survival), our brain makes us remember things permanently when our emotions are triggered. Fun learning isn't a waste of time. It's necessary to help the students build long-term memory!


  1. :) I'm glad you don't just think I'm a grinch! I tried to offer replacement methods for all the traditional ones I'm shooting down.

  2. I think a lot of that is founded in integers, not "radical" at all. *cymbal crash*

    In all seriousness, #10 is a bit of a slippery slope. Um, so to speak. If the students who can see it in their heads are forced to write things down, they may start to see math as just a set of procedures leading to the "right" answer, rather than something intuitive. Or worse, they may start to hate math because they're being forced to write down the blindingly obvious. I think what might be better is to simply toss the "intuitive" students one with a fraction or something, so that they see that there are times when a method is necessary to validate (for them or their peers) what they may already suspect.

    As for #7 (and #6 for that matter), I think some of that is a facet of human memory. You can teach an entire unit on rate of change and scatterplots and patterning in tables, but then spend just half a period at some point in the subsequent unit on "rise"/"run" (or y = mx + b) and what's the thing that sticks with students at the end of the course? Yeah. Been there. (I think they also teach it in science.) I realize the suggestion leans towards don't do it AT ALL, but in this age of communication, that's impossible. When I first start in on equations for lines of best fit, with them having NEVER done two variable equations before, there's always at least 2 or 3 students who are all "is this a y=mx + b then?" ("NO. This is unit rates. That's later.").

    Supplement to #2, communication questions or reflections can help solidify some terms that they scramble up. Finally, for some reason, never considered talking about trig ratios as percentages (tend to go with 'it's a decimal ratio to one'. I like that, hopefully I remember to use it. Thanks!

  3. Absolutely agree in all the points! (greetings from spain!)

  4. I love your point on rise/run.
    I do research in teacher and student understandings of slope and it is so, so true, that people can know this phrase and have no idea how to solve a problem involving slope if it looks just a little different than what they are used to seeing.

  5. I really like all of these, and how you emphasize that the MEANING behind all of the concepts that have the "tricks" to remember them is the only way kids will retain the information.

    I also like you how follow up with WRITING in math class, and a fun learning environment - all in the name of retention!

    But what I think I love the most, and what I'd love to hear more about from you (and I apologize if you have written a post about this already) -- distinguishing between conceptual and procedural errors in students work. How do you organize your life to provide individualized intervention? I give out exit tickets and then correct them and sort them into similar mistakes. Usually I use these mini groups working together to figure out their mistakes as a beginning of class activity, but its often crudely designed - I am a master at writing a directive for a group on a post it note 2 minutes before class starts.

  6. I think your idea sounds great! I'll have to try that. I've written a little bit about this, but nothing substantial or earth-shattering.

  7. Hi, Mimi,

    I came across your blog via David Wees, and love that your #1 is making learning fun!!

    As a fellow mathematics educator I thought you might be able to help in spreading the word about an educational TV show for preteens about math that we're putting together. "The Number Hunter" is a cross between Bill Nye The Science Guy and The Crocodile Hunter -- bringing math to children in an innovative, adventurous way. I’d really appreciate your help in getting the word out about the project.

    I studied math education at Jacksonville University and the University of Florida. It became clear to me during my studies why we’re failing at teaching kids math. We're teaching it all wrong! Bill Nye taught kids that science is FUN. He showed them the EXPLOSIONS first and then the kids went to school to learn WHY things exploded. Kids learn about dinosaurs and amoeba and weird ocean life to make them go “wow”. But what about math? You probably remember the dreaded worksheets. Ugh.

    I’m sure you know math is much more exciting than people think. Fractal Geometry was used to create “Star Wars” backdrops, binary code was invented in Africa, The Great Pyramids and The Mona Lisa, wouldn’t exist without geometry.
    Our concept is to create an exciting, web-based TV show that’s both fun and educational.

    If you could consider posting about the project on your blog, I’d very much appreciate it. Also, if you'd be interested in link exchanging (either on The Number Hunter site, which is in development, or on which is a well-established site with 300,000 page views a month) please shoot me an email. We're also always looking for input and ideas from other math educators!

    Thanks in advance for your help,