Four Mathematicians You Should Know – quick bios of famous mathematicians and how they tell a different story about what math is than what kids sometimes learn in school
How to Help Your Child with Math Homework: 5 Easy Questions You Can Ask – geared toward parents who don’t feel comfortable with math; I explain how important they are to supporting their kids and how valuable they can be
Working in my classroom, I found that having students work in groups was one of the most powerful tools of instruction that I had. This view is supported by research and by experiences I have had, since college, as a learner. There’s nothing that can replace students working through new ideas together — a teacher who already knows the content runs the risk of giving too much away, whereas students working together will (usually) be able to give each other new ideas or questions that provide each other with just the right amount of new information. Even when this doesn’t work perfectly, students get to watch mathematics in action as somebody similar to them in content knowledge works through new ideas.
So, I’ve been living in that bubble. Pretty much everyone I talk to about teaching agrees that group work is an important aspect of mathematics pedagogy, and that math is a collaborative pursuit. Then I go to class one day and hear the graduate student filling in for my professor say, “If you don’t understand this, work it out alone later. Mathematics is something you do by yourself.” (I’m paraphrasing.) He repeated a similar sentiment in the following lecture.
A few days later, I was chatting with a classmate before the professor began, and he spoke about how he liked mathematics so much because it was a solitary pursuit. He said he probably would not be interested in the subject otherwise.
Oh right. I remember now why I hated math class so so so much.
We’re missing out on a lot of great mathematicians. Folks see many people working in math who enjoy working on their own, so they assume that there is a connection between liking math and liking working alone. But! Maybe we see people like that in mathematics because math is taught as an independent pursuit, so most of the people who persevere are those who have an innate preference toward working that way.
Sure, sometimes you need to sit and ponder something on your own, but there’s no reason that math should be so incredibly solitary! If math class were to incorporate more collaborative pedagogy, students who enjoy working with others may be more likely to find success in the field and therefore persevere — if only for the simple fact that they enjoy it more. We may discover strong math ability in people who would traditionally be turned off by the way they see mathematics traditionally being done. Then, beautifully, we have a more diverse group of role models for the next generation.
I won’t even get into the gender stuff this brings up. (Well, maybe just a teaser: young girls are taught how important it is that they orient themselves toward other people and be nice and helpful; how can we expect very many women to break down that expectation or the popular view of math as a solitary endeavor?)
Since I began teaching, I have had no real interest in students identified in this way. They’re somewhat less interesting to teach, quite frankly, because they’re most likely going to learn the content no matter what you teach. Most of students have a great deal of support at home, which has allowed their intellect and curiosity to be displayed to their full extent. Many of these students also are identified in this way also just because they happen to learn in a way that schools typically expects students to learn. As a teacher, you don’t have to do much in the way of reflecting upon and adjusting your practice in order to get these students to reach course curriculum goals.
I learned a lot more about teaching from my ‘level one’ classes than my ‘honors’ or AP groups. These students forced me to think about how to present the essence of mathematics that makes it so captivating.
Seeing the ‘gifted’ students being called out in particular as being creative made me think more about the difference between the students who are viewed as ‘good at math’ and those who are not.
Is creativity the thing that makes people think certain students are gifted? If so, doesn’t that imply that creativity is the sign of a good mathematician, which should make it the goal of math instruction? And yet, we focus on teaching mostly skills to the students who are already succeeding, rather than helping them to become creative in mathematics. This means we are giving up on these students ever being truly proficient in mathematics.
Or, maybe the implication is slightly different — creativity isn’t the thing that causes folks to identify a student as gifted; ability to answer math questions of some sort is. But, the amount of studies on gifted students and creativity still demonstrates that we see more creativity in the gifted students. This tells me that creativity is a tool that helps these students to succeed in their math classes. Again, why are we not finding a way to give this tool to all students?
Overall, it seems to me that if researchers and educators note overlap in the students they consider to be ‘creative’ and the students they consider to be ‘gifted,’ then not making creativity an important part of math education for all students is hugely problematic. The issue further complicated by the economic, gendered, and racial differences between students who are identified as ‘gifted’ and those who are not.
I must admit that what with traveling off to Utah for PCMI and then a million other places for the rest of the summer, and also plowing my way through a bunch of Tom Robbins books, I kiiiind of forgot that I was supposed to be reading this until two weeks before the start of school when I looked into the short stack of books I keep by my bed after finishing Fierce Invalids Home from Hot Climates (which I of course heartily recommend). Luckily, Boaler’s book is a super fast read. I ate it up over about three days.
This marks the first time in my life that I have ever felt the need to put those skinny little post-it notes into a book to keep track of what I came across. The content is perhaps more directed toward those who haven’t already drunk the creative-math-teaching kool-aid, but there were a number of specific ideas that I’d like to use and references to research backing up ideas that I’m always trying to convince people (parents, students, other math teachers) are true.
I was planning to write out my notes and that I took and review the bits that I earmarked so they would be easily at hand rather than stuffed into a book that I accidentally spilled green tea onto when I fell asleep on the airplane (sorry, library…). I figured I’d just do it here in case any of it was useful to anybody else!
Obviously, it is best if the students accomplish as much as possible on their own. When the students are the ones making change, they are doing things that speak most truly to their and to other students’ needs. They also get the opportunity to learn a lot of very useful skills as they plan and execute fundraisers, events, or informational campaigns.
On the other hand, kids are kids. They haven’t lived for very long yet, so sometimes they’re not very good at things. I know that in some cases (but not all!), if I just do something myself, it will get done so much more efficiently. Sometimes this is important, if the goal is something that the wider school community could benefit from.
My goal is always that my students accomplish as much as possible, which sometimes requires me to lend my experience. A certain amount of failure and floundering is important for learning, but I don’t expect my students to always come up with effective tools from scratch. My M.O. is to share my knowledge, but to have the students be the ones to actually do as many things as possible, including planning, goal-setting, and making decisions.
Here are some guidelines that I have developed over the past few years:
I’ll probably go into detail about a few particular things that the students in the clubs that I work with have accomplished in future posts, because they do oftentimes wow me. The GSA just raised $200 for a homeless shelter for youth, the Feminist Coalition is putting together a zine, and the student director for the talent show was the only reason that we grown-ups were able to relax and enjoy the show.
Does anybody have any other tips about empowering students to take charge and make change in school-based clubs?
The class an introductory Computer Science is run by TEALS which means the projects and plans are theirs. It’s a cool class. We program in Snap!, a visual block-based language. I have a few outside programming professionals who come in to help, which is good. The material is all pretty darn simple, but I don’t have the kind of scope that tells me what people will really use when they work as programmers, in the way that I can do with math.
Anyway, for our last unit, we’re having our students design their own game. Rather than teaching anything like loops or lists or informative things like that, this unit is about more general concepts like good game design, project management, and working in a team. Since the unit is right after break, I had some time to really get into it and I’m pretty excited about what I’ve come up with. The first few days talk about some background information and allow for a bit of brainstorming before diving headlong into making a project.
After these things, I’ll give them a few days to code. Then we’ll spend a day each on debugging, testing prototypes, and the life of a video game developer. I’ll post those lessons later (aka, after I make them, as they do not yet exist).
Here’s the stuff! Use it if it is useful to you!
Day 1 – Intro to game design
Day 2 – Intro to team programming
Day 3 – History of games
Day 4 – Planning Protocols
The portfolio that my students create is a little booklet. I leave the presentation entirely up to them, besides requiring that they submit it in person rather than electronically (just because I’m sick of kids claiming to have had email or google drive issues). Each portfolio consists of a cover page, a one to two page reflection on problem-solving, and written explanations of the four problems that the student solved that year.
I’d also like to note that a lot of the ideas described here come from Bryan Meyer’s blog Doing Mathematics and Jim Mahoney’s book, Power and Portfolios. Obviously I’ve tweaked everything so that it makes more suits my practice best, but I’ve definitely borrowed very heavily from them.
BEGINNING OF THE YEAR
There are a few important things that I do at the beginning of the year to set up the expectation that my students will be creating there own portfolios.
Here are the habits I use:
Collaborate, Be confident/patient/persistent, Create data, Generalize, Look for patterns, Multiple representations, Seek why and prove, Solve a simpler problem
BI-WEEKLY
Students select a problem they solved in the last two weeks that is an example of them using one of the habits. They fill out a little cover sheet (I keep them in labeled pockets on my back wall) and put both the problem and the cover sheet in their binders. It’s super important that they actually submit the problem, so they can potentially write about them later.
Here‘s an example of one of the cover sheets. The others are nearly identical, with the name of the habit changed.
QUARTERLY
Each quarter, students select the problem of which they are the most proud and write a detailed account of how they figured it out. We call these ‘portfolio write-ups.’ At this time, I do not include any discussion of the mathematician’s habits in this assignment (see below for future changes I plan to make).
One issue is that sometimes the write-ups look more like how-to’s than I would like. It helps to encourage students to include things that they did that were incorrect, or times that they felt confused or overwhelmed. This also reminds students that you can still be a good mathematician even if you don’t understand something immediately.
I give them this handout, which contains instructions and a grading rubric.
MID-YEAR
In addition to their second-quarter portfolio write-ups, I also ask students to answer questions about the collection of problems in their binders and about how they are doing with the habits. I use this handout.
END OF YEAR
By the end of the year, my students have written 3 write-ups. Their final portfolio must have four, so they have one more to write. Of course, many students have more than one more to write, if they missed assignments earlier in the year or aren’t super well-organized.
Every quarter, I give fairly detailed feedback to the portfolio write-ups, so I do ask students to rewrite their work, unless they got a 100%. I don’t require this, though maybe I should.
This is the handout that I give to students.
This is the grading sheet that I use for each portfolio.
My only requirements for the presentation is that it has a cover sheet and that students hand in a physical copy. I give full points for presentation as long as they are neat and legible – I don’t care whether they simply typed “Meghan Riling’s Portfolio” in black ink, or created an intricate design in colored pencil. I recommend students type the write-ups, but have given full credit to immaculately handwritten ones.
HOW DO I KNOW IT’S WORKING?
First of all, what do I mean by ‘working’? Ideally it will do a few things for students: develop their problem-solving skills, make them believe in themselves as problem-solvers, make them believe that being good at math isn’t the same thing as being fast at math, and encourage them to try new things instead of give up when they are confused.
For one thing, my students take this project pretty seriously. For the most part, their write-ups go into a lot of depth. (I’m considering a length cap next year, as some honors kids handed in 15+ page portfolios.)
I also gather from what they write in their reflection letters that they do feel that their problem-solving skills are improving. It’s possible that they’re just sucking up to me. But I don’t think so — some kids were talking to me about how they thought they had pulled one over on me. In the reflection questions, I asked how they would use what they had learned in the future, and these dudes thought they were just so slick for saying they would use it in real life, rather than in math class. So, I guess they don’t even know that they’re saying things that would count as sucking up to me.
Here are some quotes I grabbed from their write-ups that give me hope:
Senior – precalculus
This year I learned I am actually very capable of figuring out Math. I used to be hard on myself when I couldn’t get the right answer and give up right then. That isn’t the case this year; I have learned there is possibility.
Junior – precalculus
I have learned not to shy away from things because failure is normal. I will fail numerous times in life but I need to bounce back and shake it off. If you set your mind to getting the problem right, you’ll get it right. … If I don’t get it right the first time, why can’t I get it right the second time?
Sophomore – honors algebra 2
What I learned about solving problems is that there is no correct way to solve a problem and there are almost always multiple ways to look at it.
Junior – precalculus
I have become much more creative as well as talkative and collaborative about how to solve the problems in front of me. I believe that the most useful skill I have learned was to really experiment with all different sorts of ways.
Freshman – algebra 1
See now one of my biggest problems when it comes to math is giving up easily. I used to get overwhelmed and anxious when I saw a difficult equation, which would result in me not completing a lot of problems. So by picking up the habit of being confident, patient, and persistent I’ve been able to be open minded to lots of new possibilities and outcomes.
I know that’s a bunch of them. The things that pop out a lot are that they now see there are multiple ways to solve a problem and that they are finding that they are solving more problems just by not giving up. I think the other side to not giving up is that they must be trying new approaches. And frankly, even if they are sucking up to me, I’m not that worried about it — at least they know that talking about perseverance and creative problem-solving are ways to suck up to me!
Anyway, I think that just about explains everything. I’m more than happy to explain anything at all in more detail, if anybody is interested.
The set-up that I used for the first two lessons was:
-a written hand-out with both questions requiring written responses and instructions for coding (lesson 1 and lesson 2)
-an entry on my programming club blog with code to copy (lesson 1 and lesson 2)
-kids coded in http://www.coffeescript.org
Then for the third, which only a handful got to, I used:
-a written hand-out with explanations only
-coding in sublime and running programs in firefox
The first two days went really well. Then on the third day, kids who hadn’t gone very quickly and were still on earlier lessons started to flag. I wasn’t really able to help all 22 kids in my big class as much as they needed, what with having to constantly add dropbox to various computers that it had mysteriously disappeared from, and the momentum had run out, so on the third day I saw much less enthusiasm. I really should have just moved them all onto the day 3 activity no matter what, because it was visual, which always helps, and since it was just a 3 day thing, it wasn’t crucial that they all did each lesson straight through.
A consistent problem was getting students to understand WHERE in my prepared code they were supposed to add their own things. I did absolutely no direct instruction, which may have helped with this. Then again, since we don’t have programming classes beyond the club that I run with Adam (aka an actual pro programmer), I thought it was nice to show them that they could just sit down at a computer without anybody telling them what to do, since that would most likely most clearly mirror their experience for the next few years if they chose to continue.
One very nice thing about this all is that one of my students who has recently had very low math self esteem, to the point of handing in blank tests and quizzes, was a total rock star and was even helping other kids get started with the third lesson.
So this year, first we learned all sorts of things about sine, cosine, and tangent, including how to graph them and variations on them. Then after a test, I started on secant, cosecant, and cotangent.
We began with a do now that wasn’t about trig at all. I had taken pictures of the stuff on the board, but apparently I don’t properly know how to manage storage on a smart phone, so they’re gone now. Anyway, I’m going to do my best to recreate them.
Here were the instructions on the board when my students came in:
So anyway they did that, and got this graph:
They weren’t sure what was going on with the 1/x^2 graph, so I added some x-vales between -1 and 1, and then they got the idea:
After that I asked them to help me fill in a table comparing the two graphs. They came up with something like this:
All true things. So I gave them a sheet with 6 graphs on it: sine, cosine, tangent, cosecant, secant, and cotangent. Then I asked them to cut them out and match them into pairs — pairs in which each graph was the reciprocal of the other. Obviously they also had to explain how they made their matches.
I also made up an instruction sheet, but I didn’t end up using it, except for with a girl who did the activity the following day.
I thought it was interesting that many of the students’ first instincts told them to just pair up the graphs that looked alike — sine and cosine, tangent and cotangent, secant and cosecant. (I didn’t label the new graphs with their full names — just the shortcuts, sec and csc, so I don’t think that was part of it.) When I reminded them of the table we had created on the board, that seemed to help, and students sorted it out quickly enough.
Afterwards, I graphed the reciprocals together on the board (using desmos of course) so that we could double check that they followed the table’s guidelines that we had come up with earlier.
This is the first time that I’ve gotten the sense that my students understand why the secant and cosecant graphs look the way that they do. I will definitely be doing this again next time.
My very first student interaction of the day was with a freshman girl — let’s call her Minnie — who usually comes to school early and chats with me a little bit. She is extremely sweet and hard-working, and I know she cares a lot about school and her family. It turns out that her father is a police officer who was part of the gunfight in which the older bombing suspect was killed. Again, this is the very first student conversation I had. Usually, I must admit, I shuffle around the room getting things ready, but of course I put my white-board marker down to listen to her. She told me about the insane hours that her father had spent not that far from home, and excitedly explained how the Bruins had signed a hat for him and how many people had been treating him like a celebrity. She was rattled but clearly saw her father as a hero.
In each class yesterday (aka the first day back), I started off by asking each student to share both a ‘brag’ and a ‘nag,’ which I have blatantly ripped off from some wonderful dancing friends of mine. I gave basic examples that didn’t reference the weeks event, because I wanted to wait to see whether they would, and if so, what they would say. I allowed students to say only one or the other, but nobody could totally pass, even if they just said ‘I got a lot of sleep’ or ‘my dad baked brownies.’ The second you let somebody pass, suddenly nobody wants to share anything.
In my very first class, a rather precocious freshman said that the thing ‘nagging’ him was that all these terrible things had lumped up in the same week — the bombing, gunfights in town, Texas explosions, some more miners were apparently trapped, Denver gunshots, et cetera. A few students in addition to Minnie shared that their parents were police officers who were called into action in Watertown. Many said that the thing that ‘nagged’ them was that the lock-down kept them from enjoying the nicest day on break or going to New Hampshire for the weekend. I was surprised by the number of students whose families had planned weekend trips to New Hampshire and Maine.
After the initial check-in, I opened the class up to a general discussion of what had happened. Again, I wanted the kids to talk to what they wanted/needed to talk about. I made a point of saying that, and also telling them that we didn’t need to talk about anything if they felt that they didn’t need to, or if it was too difficult.
My students were great. They expressed sympathy for the people who were hurt. Generally, they just swapped stories and asked each other questions about what they had seen. It seemed like they had been scared. Which makes a whole lot of sense. Many of them discussed hearing gunshots and explosions. One boy had heard the bombs explode at the marathons. One boy said that there was still a bullet in his house. Many live very close to where the second brother was found hiding in the boat, and some saw what was happening. Some had connections to people killed at the marathon and to the younger brother. Besides the most violent events, many clearly were very affected by the general situation. One girl told me she cried because she was worried about her grandmother who lived in the most closed off part of town. Another told me she had shut and locked her windows at night in worry.
In leading the conversation, I tried not to do too much besides add factual information and get students to answer each others’ questions. I said “thank you for sharing” about a thousand times.
One thing that is interesting about the whole situation is that people in the world are talking about how these terrorists happened to be Muslim. Some ignorant people have been doing terrible things with these facts. The really cool thing about Watertown is that it is incredibly diverse. We have a large number of students who have come from Armenia (lots of kids from Armenia), Lebanon, Russia, Jordan, Iran, and other students from Brazil, Guatemala, Honduras, and so forth. Many girls, some recent immigrants and some who in every other way fit the American teenage girl norm, cover their hair or wear more full traditional dress.
The religion of the bombers came up only three times. The first time, a student said that there were non-Watertown affiliated strangers who had said angry, cruel things to some Muslim Watertown students on their twitter pages. My kids didn’t understand this. In another class, I mentioned that I was proud of the fact that students at my school had been so cool about the issue, and one of my students was so confused that people wouldn’t be. He said “these are just two horrible people — why would anybody think that’s what all Muslims or all Russians are like?” What a freaking awesome kid.
The last Muslim-related comment was from a whip-smart Muslim girl, who covers her hair (but once let me see it!). I noted that she had been very quiet in class, which was unlike her, especially when political sorts of things come up. She told me that everyone at school was fine, but that the event had been difficult on the Muslim community at large. I hate that. Then again, she was giggling with her pals about some silly youtube video they had seen for quite some time. I think she’ll be okay, though I’m definitely going to check in on her.
Oh! I also gave my students the opportunity to write letters to the local police officers. I stressed that they didn’t need to, and that I wouldn’t think any less of them if they didn’t want to. I think about 2/3 of my students wrote letters. Minnie offered to give them to her dad the next day to bring into the station. I took her up on the offer — I think she was proud to be able to do that.
Today, Tuesday, kids seemed better. They still talked about it. In the beginning of my classes, I made sure to remind them that they could still go see a guidance counselor if they needed to at any time, and if they felt that anything we were doing in class was beyond them right now, to let me know. When I was one on one with various students, I made sure to ask them how they were feeling. Again — they’re still skittish, but I think they’re nicely on their way to feeling settled again, even if this ‘settled’ feeling is different than before.