Student GuestBlogger on MindMath

One of my students from last year attended Jo Boaler’s MindMath event at Stanford last Sunday, and wrote a nice account of some of what she learned there. I asked, and she agreed to let me share it – so without further ado:

MindMath, by Yasmeen Magaña

Seven common myths of math:

  1. Being good at math is a “gift.”
  2. Being good is about learning more knowledge
  3. Math is a closed procedural subject
  4. Being good at math means being fast at math
  5. Making mistakes is bad
  6. Math is all about learning to calculate numbers
  7. The best math learning occurs individually

MindMath was a workshop led by Jo Boaler, Professor of Mathematics Education at Stanford, where she talked about the “myths” of math and the manner in which math is portrayed in society and media. She mentioned that people tend to think that someone is either a math person or they’re not. They are either “gifted” or not. Jo Boaler has conducted research to interpret whether this is true or not and her results truly fascinated me.

Neuroscience is changing our views on math. Advances in neuroscience are changing our approach to education and the way teachers teach math. There are many assumptions about math and those assumptions may even discourage some students from pursuing higher math because they believe that people are either a math person or they are not. “Some people can do math and others just can’t,” is a hurtful assumption that sadly is present in today’s world. Evidence against this is growing due to recent discoveries about human brain plasticity.

Myth #1: Being good at math is a “gift.” Jo Boaler’s research has shown that no one is born with a math brain and due to brain plasticity, which is the ability of the brain to grow, change and develop, people can acquire these skills to succeed. People may ask, “If everyone can do well in math-all the way to Calculus. Why don’t they, then?” The answer is that today, many students feel as though once they hit their wall, their math career is all over and there is just no way past it. Another myth is that people think that they can take all the math classes they’d like until they hit their personal wall and can’t go any further. Carol Dweck wrote a book in 2006 which used research to show that everyone has one of two mindsets: A fixed or growth mindset. People with a growth mindset believe that the harder you work the smarter you will get. People with a fixed mindset believe that you’re either smart or you’re not. More people have a fixed mindset about math than any other subject. Carol found that people with a growth mindset tend to achieve more in math because they are more willing to learn from mistakes.

The Programme for International Student Assessment team collected data on how children approach math. Another myth is that math is all about learning to calculate numbers. In reality math is very visual, and people who only memorize numbers and formulas tend to achieve less than those who see math as a subject of big ideas and more than just numbers. Teaching math with many visual examples is very important to teach that math is more than just memorizing formulas and numbers, but that there are a lot of concepts behind it and is very visual.

Another myth is that making mistakes is bad in math. Sure someone may not have gotten the correct answer in that moment to that one problem when they made the mistake, but MRI research shows that making math mistakes grows your brain. Whenever someone makes a math mistake when taking a test a synapse fires in their brain. There was an MRI test done and the results showed that people who made math mistakes with fixed mindsets didn’t show signs of brain growth. Those with a growth mindset had brain growth whenever they had made a mistake. People with a growth mindset may believe that mistakes are good, and will have an enhanced response to making mistakes.

Another myth is that math is associated with speed. Laurent Schwartz was a French mathematician who considered himself a slow mathematician. He felt inferior to his class because he was one of the slowest math thinkers but went on to win the fields medal in mathematics. The reality is that the greatest mathematicians are actually pretty slow. Sometimes it takes someone longer to understand a math concept but it doesn’t mean that person doesn’t understand it as equally. All of this new research has been very helpful in understanding the way students learn math and the manner in which math should be taught.

Technology Fail! No Debate Video!

Last Friday, my PreCalc/Trig classes debated two topics: polar vs. rectangular coordinates, and radians vs. degrees. I set these debates up a bit differently than normal, in that students were told the topics in advance, but were not told which position they would be taking, or even which topic they would be debating. I wanted to give them some opportunities to debate topics which I thought would be a bit easier, especially with the time to do some research in advance. Then I set up Hangout on Air events so they would be caught on video and saved to YouTube, so I could review when it came time to grade, and if there was good material on there, could save it for later use (sharing with other teachers, presenting on math debates, etc.). I did the right stuff – tested the day before, made sure video saved, and everything I thought I needed to do. But on Saturday, when I sat down to watch the videos, I opened my YouTube channel, and there was nothing there except the previous day’s demo. This was twice as unfortunate, since I didn’t bother to take notes during the debate, since, well, why would I? There was going to be a video. But, there wasn’t. Well – in some ways, that’s probably for the best, since I didn’t anticipate just how much of a train wreck this debate was going to be. Students who otherwise are very thoughtful, well prepared, and insightful during both more formal debates and informal class discussions either froze up or descended into trivial arguments that fell far astray from the main topics. Not that it wasn’t a fun experience – students really did have a good time with this debate format, and any time students enjoy their time in my classroom I consider it time well spent. But when the debate over rectangular vs. polar coordinates descended into whether there were more squares or circles in the classroom, despite some moments of hilarity. (One comment that resonated – “The Earth is a circle, and probably one of the most important ones in our lives. Because, you know, circles.”)

It seems that doing this semi-formal debate didn’t really accomplish my goals, so the next round of debates, I’ll stick to my formal ones. They take a little more time, but not too much more time, and they result in significantly better student understanding. I really thought that by this time of the year, things would have gone more smoothly without the structured preparation, but it’s clear I was mistaken. It’s easy to forget how intimidating it can be to think on one’s feet, and let’s be realistic, at this time of the year, a lot of students just weren’t going to prepare outside of class if it wasn’t a “required assignment”. There’s too much late work for them to try to catch up on to do some research on a topic that they already sort of know well.

Next debates, then, early September.  I’m not positive of topics yet, but I’m thinking about these:

  • Algebra 1 – Should we always use the letter x as our variable?
  • Geometry – Is math an art?
  • Trig/Precalc – Are linear or quadratic models more common in real life?



I had never lost my voice before, but yesterday in the late morning, my voice started to get hoarse. By the afternoon, it felt funny, and when I picked my daughters up from daycare, it was just gone. Complete silence. Nothing but whispers. For the first time, my daughter was embarrassed by me, although I’m sure it won’t be the last. And when I woke up this morning, I faced a day of classes where I wouldn’t be able to speak.

I put the following message on my SmartBoard in place of my regular warm up:


I’ve found that students are incredibly empathetic when I’m sick, and really go overboard with trying to be nice. Unfortunately, that often means that they try to be quiet. Well, that may have been great if I’d had a headache. Instead, I just had no voice, and the silence was killing me. I’d put up a paired activity, but everyone was still so quiet. I put a new message up on the board:


Now, if this had been my first few years of teaching, a day without my voice would have been a wasted day. Back then, my teaching style was talk, talk, talk, ask a few questions, talk, talk, watch students work, talk to close. Over time, I learned to focus more on getting students to talk and communicate. Earlier in my career, it was so that I would know what my students were thinking, but my understanding evolved. I wish I knew who to attribute the original idea to, but I’ve heard many times that the person doing the talking is the person doing the thinking. Today was a day that forced my students to do the talking, and although it took some time for them to get going in each class, they eventually settled in and (mostly) stayed on topic.

Today was a gift, and tomorrow looks like it will be as well. Still, I’m looking forward to returning this gift this weekend. My daughter is soooo embarrassed to have a mute daddy picking her up at school, and she’s not even five yet. In her words – “What will my friends think?” Next year, though, maybe I’ll fake this whole laryngitis thing at the beginning of the year, and prep my new students for what it means for them to talk in class.

I Admit, I Love Multiple Choice

Yes, it’s true. I’ve come to love multiple choice questions. Oh, not for assessments of any kind. They’re horrible for that (at least in my opinion). But they can’t be avoided, so I’ve learned to embrace them, and make them work for me. Now, they’ve become an integral part of many of my lessons. I use them in a few different ways now.

  1. No Distractor Style Multiple Choice

This is a technique I learned from Scott Farrand (Professor from Sacramento State) at a CMC conference at Asilomar a couple of years ago. This is a great method for homework, especially for online homework assignments. Give a problem where the correct answer is one of the options, but make the other 3 options completely wrong. The idea is that the student knows for sure if the answer is correct because she sees it or she doesn’t. The incorrect answers aren’t trying to catch a particular mistake.

If x + 3 = 11, then …

A. x = 2

B. x = 8

C. x = 10

D. x = 12

I didn’t put in any obvious or intentional distractors, so I’m not trying to “catch” a student with the wrong answer. Instead, the student gets her answer, checks my multiple choice, and gets immediate feedback so she can check her work and change her answer.

2. How Do You Get Each Answer?

My students (hopefully) learn early on that I love mistakes, because they’re great ways to learn. A multiple choice question with distractors is a great way to discover what mistakes we all can make, and how to be on the lookout for them. This is a great thing to throw into my lesson in the middle, just after we have been introduced to a concept. However, I don’t ask students which answer is right. Instead, I ask them how they can come up with the answer that they are assigned. My classes are usually in groups of 4, so I’ll assign each group one answer to investigate. They discuss the answer, attempt to solve the problem in such a way to get that answer, and figure out if the answer is correct or if they did something wrong to get there. We then come together as a class to share results, and hopefully come to a consensus on which answer is correct, and what kinds of mistakes to be on the lookout for.

3. Choose Your Own Question

This approach is based on the WODB idea. I give students the answers to a multiple choice problem, and ask them for as many questions (relevant to our unit) as possible that would make each answer true. For example, if we are covering basic trigonometry (like we are in my geometry class), students may be given a diagram with answers:


In this case, there is just enough information for every student to come up with at least one question that has a solution, and every solution has at least one valid question that can be asked. There’s great opportunity for rich differentiation here in small group discussions. Starting with a “What do you notice/wonder?” prompt with just the diagram can lead to some great questions and incredible understanding.

4. What Was The Book Thinking?

I sometimes am amazed by the mistakes I find in resource material that publishers provide to accompany their textbooks. Maybe it’s been a while since I had a 1st edition of something, but this year, my Trig/Pre-Calculus class has been using the Glencoe/McGraw Hill Precalculus as our new textbook. There are a lot of things I like about it, and a lot that I would change. I’ve been amazed at how many errors show up in their multiple choice problem answers in their PowerPoints. This past week, I came across this gem:


What a great way to discuss with my students one of the things I don’t like about multiple choice – when the correct answer is E. None of the Above, and that option just isn’t included. Not only that, but the book decided that the answer was A, and my Algebra I students would have shot that down immediately, since the plotted point was incorrect. But what did we do with this? We discussed what a better question would have been, and what better answer choices would have been. First, why is this a two part question instead of two separate questions? Second, why are the coordinates all correct, but in the wrong places? Why not just give the points, or just the coordinates, as possible answer choices?

So no, I don’t use multiple choice questions on tests. I realize that I’m pretty privileged in that my school has small class sizes. I remember the days of having 35 students per class for 4 classes, when I had no choice but multiple choice if I wanted any semblance of a life outside of school. I also remember the days when my work as a teacher was evaluated largely on the STAR test and benchmark tests that were all multiple choice, so teaching students how to take those kinds of multiple choice tests was stressed to me as an essential skill they had to learn, even more important than the math. But I have found a great place for multiple choice questions as a point of class discussion, of useful formative assessment.  Best of all, they lead to some great revelations for many students, and help to undo the stigma of making mistakes in class, because we find we all make them. Even textbook publishers who should really know better.

Completing the Square with Desmos

A few months back, I got the chance to go into Desmos headquarters to work with some great people (see When Plans Go Awry for some information on that background to this post). tl;dr: We started an ambitious activity to represent “Completing the Square” graphically, but I wasn’t sure how good it would be, educationally.

Fast forward a couple of months, and despite my best intentions, I never got the chance to really work through the activity to the level of detail that I’d hoped, and yet here we were in my Algebra 1 class, ready to cover the process of completing the square. After a crazy weekend (too sick to work on Friday and Saturday, then recovering just in time to celebrate Mother’s Day with my mother, wife and daughter, before finishing all my grading and submitting grades for the latest grading period at about 1AM Sunday night), I had just enough time to put off last minute adjustments until last night.

Those last minute adjustments ended up being pretty minor, so I crossed my fingers and this morning during class, I had my students start up Desmos and dive in to the activity. (If you want to see it yourself, check out CTS Investigation and let me know what you think.) I tried to hold students back, since some will rush through whenever they get the chance, but this was one of those days that I really wish there was a pause option for Desmos activities.

A few observations from each of the first four slides:

  1. On the first screen, I had a graph that students were supposed to look at and then estimate the zeros. In retrospect, I should have made the graph an image. I didn’t think about the fact that students would be able to click on the zeros to get their values, which really defeats the purpose of estimating. Luckily, I was able to rectify that (mostly) with some extra instruction, but good learning experience!CTS1 The responses I got from students (below) ended up with a great discussion of estimation, what counted as zeros, when negatives matter, how to write the zeros, and more. CTS1-Responses
  2. In the second screen, students had to adjust the c value of the equations to get the parabolas to “sit” on the x-axis. I was worried about this one because I wasn’t sure how clear the instructions were, but it turned out to be clear for everyone involved. I wonder if it would look nicer if the x-axis and y-axis were a little bit bolder in projection mode. It didn’t confuse anyone as far as I knew, but I think it would be a nice feature.CTS2.JPG
  3. In the third screen, students were shown a graph, and asked how to change the equation to make it sit on the x-axis. Of course, I intended them to change the c value, but some chose to change the b value instead/as well. The question did not ask the why questions, but the discussion after the next slide got into that a little bit.CTS3CTS3-Responses
  4. In the fourth screen, I got some great responses, especially in the discussion that happened afterwards. Students were given four graphs and their equations, and had to figure out which equation didn’t sit on the x-axis and why. Sometimes, the best responses come from the students you didn’t expect, and today was one of those days. The highlighted answer came from a student who often doesn’t talk, and usually isn’t a prolific writer, but was on fire today. I asked this student to clarify the response, and got an explanation that the square root of the last number should be half the middle number, and that was true for the first three equations, but not the last, which should have a 16 instead of a 9.CTS4CTS4-Responses

That’s all we had time for in class. We got some great discussion, and yet didn’t really focus too much on the main geometric tie that I was hoping could be made – the number of units you move the vertex to get to the x-axis is the same as the number that you add to the c value. It seemed that a few kids started to make that connection, but I need to tweak some of the questions that are being asked to be less open-ended if we want to make that connection. Or, maybe I need to think about the intent – what is the focus here? Because I have a bunch of students who made connections between the b and c values that are also valuable observations and should help with conceptual understanding.

Also, and I can’t believe I missed this – initially, the last equation in each “which one doesn’t belong” was the one that didn’t fit the pattern I wanted them to see, so I really need to mix that up. All told, though, I’m pretty happy with this introduction to completing the square by graphing, and students kept themselves pretty well engaged.


One Day Project

Don’t Worry About Finishing – Just Show Me Good Math!

Last Thursday, I tweeted out a preview for my Geometry students’ alternative to a regular test:

A couple of my students came in murmuring, “I think the test is about Star Wars or something!” Yes, they were as excited as I was. Hopefully they’d still be excited when the class was over. Hopefully I didn’t ruin Star Wars for them!

I started the class by saying “I don’t expect you to finish. I just want to see evidence of good math.” This was going to be a very subjective assessment, I knew. That was less important than my goal to make it meaningful. I wanted students to be more invested than they would be in a typical test on geometric solids: “Here’s a bunch of pyramids, prisms, cylinders, and spheres. Find their volumes or lateral areas or surface areas because I said someday you may need to know this.”

Instead, I told them that I was offering them a dream job – to get to work on the models for the next Star Wars movie. I gave them two pictures, one of Starkiller Base, and one of the Millennium Falcon, and told them that I wanted scale models built for each of them, but need to know how much material to buy and how much paint to buy. They had to use whatever math tools they knew (and had full access to their notes) to make these calculations. Since I didn’t give them any numbers, they had to ask for information through Google Hangouts. Some information I gave was exact, some was an estimate, and some was LMGTFY (for unit conversions, etc.).

I reminded them again that I did not expect them to complete all the work and get the right answers in the time they had. What I graded them on was:

  • Using tools that we have learned (25%)
  • Determining what information you need (20%)
  • Thinking creatively (25%)
  • Correct Calculations and Estimates (15%)
  • Explaining Errors (15%)

Unless they spent the whole time not doing the assignment (which did happen for a couple of students), they were going to get credit for a whole lot of noticing, wondering, evaluating their work, making plans, estimating, justifying their choices, and doing those things that mathematicians do.

What I found was that almost all of my students, when given the chance, can do some great math and have become more comfortable with embracing and explaining their mistakes, going in one direction and then changing their mind and going in a different direction, and starting problems by estimating. I also found that they need more help structuring their time, planning their problem solving, and solidifying their estimation skills and number sense. (More regular Estimation 180 activities are on the agenda for next year’s Geometry classes).

There were a lot of things that I really liked about this assignment, but it definitely needs some work. Most students were on board with not having a finished problem when I explained it at the beginning of class, but some students really felt a need to finish . I designed the task with the idea that there’s always more one can do to be more exact, to help reinforce the idea that a model just needs to be good enough, and that time constraints are real. There were also more than a few students who said that they would have preferred a plain old test. How much of this was because it was new, and how much was because it is less predictable? The open ended tasks that we’ve been doing definitely have helped, but maybe a few more of these projects with more possible closure would be helpful.

I did have a few students note that they weren’t big Star Wars fans, so maybe I need some less dated pop culture references in the future. On the other hand, one student is a bigger fan than I, and pointed out the different estimates on some of the measurements that I gave, and which ones were more reliable.

Overall, I really liked this attempt at something new, and hope that I can develop it some more and help students find more success in showing what they know and how they can apply it.


I introduced polar coordinates to my trig/pre-calc class yesterday. We talked about real world examples, like FPS video games with mini-maps, RADAR screens, and air traffic controller maps. Today we reinforced the concept with a new game, Marco Polar. (Credit to Susan Russo, @Dsrussosusan, who pointed out via Twitter the obvious pun that I was somehow missing.)

In case you aren’t familiar with it, Marco Polo is a game that I remember playing in pools, sort of like Blind Man’s Bluff in the water. One person is “it” and has their eyes closed, and shouts “Marco!”. Everyone else shouts “Polo!”, and the person who is “it” has to try to tag someone who isn’t it, just based on sound.

There are two twists for playing Marco Polar. First, we aren’t playing in the water. Even if the school had a pool, this probably wouldn’t be the best use of an entire class period. Second, the person who is “it” is stuck at the pole and cannot move. The player who is “it” is blindfolded, and everyone else moves to within about 10 feet of “it”. “It” shouts “Marco!”, and everyone else responds “Polar!”. Whoever is “it” uses what they heard to identify someone else in class by name and polar coordinates. (I used 1 meter as 1 unit, and students quickly started calling out coordinates to the nearest 0.5 unit).  If that person is at the correct angle, and within a half unit of those coordinates, they become “it” and switch places, and the game starts over. If that person isn’t within 3 feet, then “it” says “Marco!” again, and the game repeats.

Marco Polar

Today’s class had a few absences, and it’s already a small class, so we had very few participants. All told, though, this was 25 minutes of pure fun. We could have stopped after 10 minutes, but they were having a great time and are ahead of my other trig/pre-calc class, so I figured a bit of a break at the end of the day couldn’t hurt anyone. They’ve been working hard all year, and deserved a bit of a treat anyway.

There are a few ways to make this harder, but I haven’t tried any of them yet. Some ideas that I have, though:

  1. After an incorrect guess, allow all players to move again.
  2. Require the angle, the radius, or both be negative.
  3. Spin “it” around and tell them what direction they are facing.
  4. Move “it” around and tell them their coordinates, as well as where the pole is.
  5. Specify degrees or radians, depending on which students need to review more.

I’m sure there are others. This is a great, fast paced (hopefully) game, and I suppose it could even be paired with the same game, but using rectangular coordinates. “Marco Rectangular” just doesn’t have the same ring, though.