Other Posting!

I’ve been posting a bit on the Cambridge Coaching website recently (they’re a tutoring service where I’ve been working). My writing there has been mostly for students and parents, rather than being geared toward other educators, like this one.

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

Loner Mathematicians

This semester, for the first time in almost a decade, I’m taking a math class with people who are not involved with math education. It’s really bringing me back.

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?)

Creativity and… Gifted Students?

As I was beginning to collect resources about creativity and how it is understood in the context of mathematics education earlier this month, I was surprised to find a large number of articles focusing on the instruction of ‘gifted and talented’ students.

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 Read a Book! “What’s Math Got To Do With It?”

In June, my boss passed around Jo Boaler‘s book What’s Math Got To Do With It? to all the math teachers at my school and asked us to read it over the summer. It was originally written back in college but was updated for this year.

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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.

Photo Aug 31, 11 40 16 AMI 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!

  • STUFF TO USE WITH STUDENTS

    • Fermat’s Enigma by Simon Singh – This is a book about proving Fermat’s last theorem. Boaler writes, “Any child – or adult – wanting to be inspired by the values of determination and persistence, enthralled by the intrigue of puzzles and questions, and introduced to the sheer beauty of living mathematics should read [it].” Well, that sounds pretty good. I like that in addition to giving students the experience of creative, problem-solving mathematics, they also get a peek into how that translates in the work of professional mathematicians. (pg 22)
    • Self-Assessment – This book explains that self-assessment can be helpful for students for the simple reason that it helps students to understand *what* they are supposed to be learning. You can give a student a check-list of ideas they should understand and ask them to fill it out. I could do this after a few days on a new topic or near the end of a unit. I definitely remember being asked, when I was in high school, “what are you learning right now?” and just having no idea. I think that presenting the topics as a check-list will make that question much easier for students to answer. Also, if I collect their responses, then I have a ton of useful information that I can use to determine next steps for the class to take. (pg 96)
    • Extend the Problem – This is something I know of and am already planning to do more, but I always enjoy reading a little more about how people do it. Boaler talks about giving out problems and asking students to ask their own, new questions about them. She also says that when students asked good questions in class, they’d go up on the wall.
    • Physical Set-Up – one of Boaler’s students, Nick Fiori, apparently created all sorts of physical set-ups for students to then create and answer questions about. There were cards from SET, differently sized/colored beads and string, colored dice, snap cubes, measuring cups, pine cones, etc. I love this idea. It’s actual kinesthetic mathematical work. I’m curious to figure out how to best make this a productive experience for algebra students. (pg 169)
    • In Code: A Mathematical Journey by Sarah Flannery – This is a book written by a woman about her mathematical development. I am curious about whether there’s anything that could be useful in here to share with students, especially with an eye for presenting them with an example of a successful female mathematician. (pg 172)
    • Making Number Talks Matter by Cathy Humphreys and R. Parker – the idea of these number talks is to help students to think more flexibly about values and to develop number sense.

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  • PROBLEMS

    • Turkey – “A woman is on a diet and goes into a shop to buy some turkey slices. She is given 3 slices that together weigh 1/3 of a pound, but her diet says that she is allowed to eat only 1/4 of a pound.” (pg 44)
    • Four fours – Can you make every number from 0-20 by using four 4’s and any mathematical operation? (pg 180)
    • Race to 20 – This is a 2-player game. Start at 0. Take turns. Each turn, the player adds either 1 or 2. Whoever hits 20 wins. (pg 181)
    • Beans and Bowls – How many ways can you split up ___ beans in ___ bowls? (pg 182)
    • Partitions – “The number 3 can be broken up into whole numbers in four different ways: 1+1+1, 1+2, 2+1, 3. Or maybe you think that 1+2 and 2+1 are the same, so there are really only three ways to break up the number. Decide which you like better and investigate partitions for different numbers using your rules. (pg 182)
  • STUFF TO USE WITH PARENTS

    • Moms – It is especially important that mothers never talk about being bad at math. When this happens, it has a nearly immediate negative effect on their daughters’ grades. (pg 177, 187)
    • Enthusiasm – Boaler recommends that parents act as enthusiastic as possible when referring to math homework, even if they need to lie to do so. Maybe my students (high school) are a little too old for this to come off as anything but their parents just being lame dorks. My task as a teacher is to assign homework that makes this easier for parents to do honestly. (pg 177)
    • Explaining – She also notes that parents should keep their homework help to the guiding, question-asking side. I like what this means for parents of my students. By the end of the year, some parents say they can no longer help students with their math work. But if the math is really over their heads, then they can easily ask their students to explain it to them, which is a really excellent learning tool. (pg 177)
    • Back to School Night – I think I would like to take some of my (very short amount of) time on Back to School Night to ask parents to do some of the things that Boaler recommends. I will say that in order for you to help your student to be successful, here are the things that I will ask you to do, as backed by research by a fancy professor at a fancy school. I’ll probably make a worksheet with these ideas on it, which are selected from the things Boaler recommends:
      • Praise effort, not ‘smartness.’ (growth mindset)
      • Be positive and enthusiastic about math.
      • Describe mistakes as an opportunity for growth.
      • When helping with homework, ask them to explain and make sense of the work.
      • Advanced: Try posing puzzles with logic or numbers to your students.     (pg 186-193)
    • Math Power by Patricia C. Kenschaft – This is a book for parents of young kids. It is intended to help them to help their kids get into math. (pg 232)

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  • USEFUL RESEARCH

    • Memorization sucks – PISA – research shows that “…the students who memorize are the lowest achieving in the world. The highest-achieving students are the ones who think about the big ideas in mathematics.” I believe that this report is the proof behind what Boaler is talking about. It’s about 175 pages long so you will please excuse that I have not read the whole thing. I’ll just leave that link there for myself to get into a bit later… (pg 41)
    • Tracking sucksTIMSS analysis, Stanford study – “…we know from several international students that countries that reject ability grouping – nations as varied as Japan and Finland – are among the most successful in the world, whereas countries that employ ability grouping, such as America, are among the least successful.” My experience definitely shows this to be true – some of my most enriching math experiences have been in groups of math teachers with wildly varying backgrounds (from not majoring in math at all, to having their masters degrees, for example). But obviously, just cramming kids into the same class, with a teacher trained to deal only with narrow sections of ability at a time, is not going to be successful, either. You have to change how you teach. (I plan to write a little bit about the group work structures – borrowed from Complex Instruction – that I’m going to use this year, which are intended to take advantage of heterogeneity.) (pg 103)
    • Thoughtful Math and Tests“Is Dealing with Mathematics as a Thoughtful Subject Compatible with Maintaining Satisfactory Test Score? A Nine-Year Study” by Carolyn Maher – this article supports the idea that teaching math as a thoughtful and creative field will still result in students scoring well on dumb tests (that is, standardized tests). Duh. In my own personal experience, I’ve always done extremely well on standardized tests not because I’ve memorized a bunch of junk, but because I’m good at reasoning and figuring out new things. (pg 225)
    • List of Lots of Studies“Evidence of Student-active Learning Pedagogies” by Jeffrey E. Froyd. A bit of a survey of the research, with some important points highlighted.

Empowering Students With Clubs

Since my second year teaching, I’ve been advising clubs. Some have gone away, and some have grown to be very popular and effective. It’s one of my favorite parts of my job, because you get to see students voluntarily give time and energy to something that is important to them. I have gotten to know some of my students very well through different clubs, which sometimes even wraps around to help us to do well together in math class. I have also learned a ton about managing goal-based organizations, which has been very useful to me outside of school.

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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:

  1. Elect leaders. Make sure they have well-defined responsibilities.
  2. Check in with the leaders from time to time to remind them of their responsibilities and to share tips about managing the other club members.
  3. Ensure that your group has a common goal or interest.
  4. Set up protocols and present them to your students. In the GSA that I co-advise, the grown-ups run the first few meetings of the year. We do so with the same simple template each time, which includes a check-in and having an agenda on the board. Now our president runs his meetings in the same way.
  5. Unless you feel absolutely certain that students will do this quickly on their own (note that I said *will* and not *can*), set up a simple way for club members to communicate. My students all have google accounts through the school, so I have found google groups to work nicely.
  6. When tasks are being discussed, make sure particular students take responsibility for particular tasks with specific deadlines.
  7. If you have a group without strong student leadership, come to conclusions by leading group brainstorms. I find that the think-pair-share model works very nicely in clubs, especially if you ask students to physically write things down. As a teacher, I will often add my own ideas to a brainstorm, but make sure that they don’t carry more weight than any other ideas.
  8. Don’t run a club if no students want it to exist. (There are definitely exceptions. Even if there were no students that wanted to have a Gay-Straight Alliance, I would probably still want to force it into existence.)

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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?

Programming – design your own video game

Hey it has been a million years! I have kind of just been doing a lot of the same stuff with my algebra 1 and precalculus classes (with a few exceptions… those should appear on the blog in a bit!). I have been teaching one new class this year, but haven’t felt like I had taken enough ownership of anything to share it here.

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

  • Lecture/video/audio
  • Student activity
    • handout: analyze games they made in this class for how they incorporate elements of good games
    • handout: list of types of games. which do they prefer and why?
    • if there is time, they can start to brainstorm what they want to create

Day 2 – Intro to team programming

Day 3 – History of games

Day 4 – Planning Protocols

Portfolio – full year project – handouts, description

Awhile back, I wrote about the portfolios that I was having my students create over the course of the school year. I have now run this project for two years, and wanted to describe more details about how I set up this experience for my students, as well as talk a little bit about how I know that it is valuable for them, and changes that I plan to make for next year.

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.

  1. On the first day of school, we spend most of the period on this activity that I’ve written about before. I give them an open-ended problem and provide them with very little information. They are forced to make decisions within the problem. These are manageable decisions, such as estimating how many students go to our school. It is also important that they need to write out all these decisions and the reasoning behind them, which is what they will be expected to do for the final portfolio.
  2. We read through the mathematicians’ habits using this handout.
  3. We set up our binders. There are instructions for how to do this on the handout mentioned in #2 and I have lots of examples from previous students.
  4. I show them portfolios written in past years.

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.