Category: Algebra 1
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.
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 postit 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 creativemathteaching koolaid, 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!

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, problemsolving mathematics, they also get a peek into how that translates in the work of professional mathematicians. (pg 22)
 SelfAssessment – This book explains that selfassessment 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 checklist 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 checklist 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 SetUp – one of Boaler’s students, Nick Fiori, apparently created all sorts of physical setups 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.

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 020 by using four 4’s and any mathematical operation? (pg 180)
 Race to 20 – This is a 2player 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, questionasking 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 186193)
 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)

USEFUL RESEARCH
 Memorization sucks – PISA – research shows that “…the students who memorize are the lowest achieving in the world. The highestachieving 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 sucks – TIMSS 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 NineYear 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 Studentactive Learning Pedagogies” by Jeffrey E. Froyd. A bit of a survey of the research, with some important points highlighted.
Trying to Show Off — Observation in Algebra 1
A few weeks ago, I was observed by the Assistant Superintendent of my district, as it’s my third year and it’s time to decide whether or not I receive professional status (like tenure).
I ended up having her come in on a rather terrible day. It was first period after midterms, so my students hadn’t been in class in a week. We also had a student who was brand new to the school. It would have been a good day for a slower lesson, but I wanted to wow, of course.
We met a few days before the lesson and I explained what I planned to do. I made up this lesson plan sheet to show to her. It’s loosely based on what they taught me to do in graduate school. I remember seeing these and wondering “but what the heck are you actually DOING in class?” and “do teachers actually do this?” My personal answer to the second question is no, based on the fact that these really don’t explain too much about what actually gets done. Even the assistant superintendent didn’t ask about the CCSS topics that I was addressing.
The day was okay. The kids were kind of crazy and we barely began the main course activity that I was most excited about. But I can say that even if that didn’t go exceptionally well while I was being observed, it did seem to go nicely and teach some things to my kids the next day (and for some, the day after that) when they finished it up.
We started with a Do Now that went okay. Then I tried a handout that asked them to convert a pointslope form equation into slopeintercept form. My goal was that before seeing a bunch of different types of linear equations, the students would see that they were all mathematically equivalent. But it turns out that they had no idea what I was asking them to do and that they already believed that linear equations could be pushed around into different forms. So I dropped this part with the next class. I only wish I had been able to drop it for the class that was being observed, seeing as it was a huge waste of time and the kids became rather flustered until I did it FOR them on the board, which is lame.
Then we got to the good stuff.
But we only had about 20 minutes left. This activity needs a full class period.
I gave each group of 2 or 3 students a bunch of little cut out equation and graph cards. Here are the pages to cut up. Each group also got a packet. The first two pages consist only of a big table where students would glue/tape/staple each equation with its matching graph and write in an explanation of how they made their choices.
Here’s the packet:
Very few students made any mistakes. The explanations weren’t particularly interesting. Most students used the same techniques the whole way through. Some found points that worked in the equation and then figured out which graphs worked the best. Some took more advantage of yintercepts. Interestingly, even though the kids were talking about slopes when I sat and asked them about what they were doing, they didn’t mention slope that much in their explanations.
The rest of the packet was intended to help students generalize what they had seen. I found myself explaining what they were supposed to be doing in person with most students. Granted, I mostly just read what the instructions said, approximately word for word. There must be some way of reformatting what I write so that kids will read it — this has happened a fair amount in my classes lately. I guess I should just stop giving in by reading it, and instead say “hey read the directions.”
Anyway, it seemed to go okay. They were good at explaining what the variables in both pointslope and slopeintercept form represent, without me prompting them, and that was the goal of the activity.
One other thing that I added to the end of the worksheet is something that seemed to go well and I plan to do more in the future, which was to ask which two of our mathematician’s habits they used the most in the activity and how it helped them.
All in all, this went well, though the amount of transfer to looking at a new equation and seeing shortcuts for how to graph it varies among my students.
Very basic stats, day 2 and a half or so
So on my first day of doing simple stats things with my freshmen, we focused on them making up their own ways to represent data. The next day, I wanted to continue looking at different representations critically, but also introduce other forms that people already use. Specifically, I had to introduce the ones that the state uses on the MCAS, of course. I had already talked about boxandwhisker plots, and the three I needed to introduce were circle graphs (aka pie charts), histograms (similar to bar graphs), and stemandleaf plots. The last representation remaining was scatter plots, which I introduced via a simple project for the break.
Rather than give instructions about how to make each type, I decided to take one collection of data and show three different representations of it. Each thing is fairly straight forward, so I thought I would give students the satisfaction of figuring it out themselves. To make it be data relevant to them, I used the yeartodate grades of students in the class. Once again though, I didn’t do a good job of having kids use the representations to make comments on what the data means.
Here’s the worksheet I made:
*Originally I was going to do this with three different types of data, so that we could see that some representations are better for some data, but this was taking foreverrrr so I gave up on that. I ought to plan ahead more than 1 or 2 days so I have time for this stuff.
*Another note: it didn’t look so sloppy on my work computer. The lines don’t seem to match up in competing versions of word.
I was a little surprised by how much students liked the stemandleaf plot, even before they had to choose one of the representations to make on their own. They talked a lot about how they could find the actual mean/median/mode that way, while the other types obscured the data. Very few kids liked the circle graph, since you can’t tell as easily where the high/low scores were. I wish I had another data set to show them an example of how stemandleaf plots can fail — such as when you have thousands of numbers.
Here’s how a pair of girls answered the questions:
These are a couple of girls who work super hard all the time, but take longer to build their own big picture. You can see them doing that here, as they point out detail after detail and finally at the end of each section give a summarizing thought.
Students also did a nice job creating their own stemandleaf plots, and a decent job with the histograms. They didn’t quite catch on the the historgram xaxis has to be evenly spaced, but that can be built up over time. I’m also confident based even on their imperfect work that they would be able to correctly interpret a histrogram made by somebody else.
Also in case you’re dying to see it, here’s the ‘project’ I assigned for break. I wanted to make something very doable, since I usually give a decent amount of classtime, but did not this time.
I got real hip and used data about record sales in the US. They’re not fitting any lines or curves to it, just generally talking about the trend. Maybe that’s something I can do later in the year, or with another class.
Grades to Grades
I teach a couple classes of freshmen. Next year they’ll take the 10^{th} grade math MCAS (our state’s official test suite) which has an awful lot of questions about data and very basic stats – bar graphs, pie charts, etc. They won’t cover that in geometry next year, so we just did a little unit in the week before winter break.
I decided to use two main pieces of information: their grades on the last test, and their yeartodate grades. At first I was worried that this would mean we wouldn’t be able to speak as candidly about how the data was distributed. But I did appreciate how it made students realize they did have to be nice about all these numbers, and we couldn’t just have a scapegoat data point to make fun of. They were very good about it.
First I used the test scores. There were 62 points on the test, which obscured the actual grades a little bit. I printed and cut out little cards to give to each group so that they could easily manipulate the numbers.
The first thing I did was to very simply ask them to find the mean, median, and mode. They definitely took advantage of the fact that they could move the numbers around. It also forced them to work together in useful ways. This was especially awesome for my class that only has 6 students in it who typically don’t like to acknowledge one another’s presence.
It might be interesting, or it might not be, to consider why some students spread all the cards out into one line, while some snaked them along on one desk. Both arrangements seem to have merit.
The next thing that I did was to explain how to make boxandwhisker plots. They’re all over the MCAS. I think having little slips of paper helped with this, since they had to find not only the median, but also the first and third quartiles. I will never understand why this is so tricky for so many students.
The last thing that I did with the slips with test scores on them was something that I stole from a blog that Dan Meyer linked to, run by Nico Rowinsky. This teacher had a small group of students arrange index cards in different ways, periodically checking in and making different recommendations. His goal seemed to be for them to make something like a histogram, and specifically for them to understand why it would be useful to arrange data in that way. I decided to do the same thing with my students more or less, but with everyone in the class at the same time. I made comments requesting students to make their arrangements easier to read quickly. I think if I do this again, I need to ask my questions differently, as I felt like I was being a little bit leading. I really should have said, “why did you do such and such.”
Something they different on in a big way was how big to make the groups. Some students went by the number in the tens column, some found the percentage score and grouped by the letter grade, and some did things that I still don’t quite follow, but are probably purposeful, but maybe not filled out yet.
The next day, I wrapped up that activity by making a worksheet using pictures I took at the end of class of the ways that a few groups students had arranged their data. Then I had a couple of questions.
I was being observed by my department head that day, so I had extra reason to be fancy:
I wanted to focus on understanding that we arrange data to let us get a sense of data without working too much, which is why I asked about what was easy to figure out, and what took work.
Firstly, they did not get into the naming thing. This might have gone better after looking at ‘circle graphs’ and other things with cute names, I think.
Secondly, they did take these questions seriously, but I was a little disappointed by how little detail they seemed to pick out. Each kid pointed out an interesting thing or two, but not much more. This was a pretty typical paper:
a) Easy to get:
more people got 50’s
b) Work to figure out:
How many 49’s there are
c) What would you change?
I would spread out the 49’s
I think it would have been more useful if I had first asked them to explain how the displays had been created, or if they could take another group of data and arrange them according to the same structure. For example, very few students noticed that in the first display, the students didn’t put everything in numerical order, which seemed obvious to me.
If I had unlimited time, it might have been nice to wrap everything up by asking students to take a final stab at their ideal way of arranging the information, and then explain why that tells people the most about the data. I definitely should have added more in about giving a final summary about the distribution of grades in the class.
First Project
At the end of every unit, I have my students do a project. This year, I had all my kids (algebra 1, algebra 2, and precalculus) do the same thing, which was to design an app or make a video that would help students like them learn one of the topics that we had worked on in the beginning of the year.
Here’s the assignment:
What I like about the project is two things in particular:
 It’s easy to make it creative and cute. Projects are a big deal in my class and I think it’s important for students to take pride in them, which bleeds from content into presentation. It’s easy to demonstrate that in this project.
 The students always do a nice job explaining their topics, some of which are simple and some of which are more advanced. (They get to choose, and usually choose something that’s a good level for themselves.)
The students tend to give nice explanations and provide example problems. But I wish that they would make some apps that weren’t just lecture and practice. Students get excited about dressing them up with glitter and weird internet memes, but they’re mostly just following a model that they know.
So my question is how I can get more stuff from them where they’re creating new ways of looking at the material, rather than regurgitating information in a format that has been made by somebody else. I guess that’s a lot to ask of every student, especially in the beginning of the year, but I feel like I could be doing a better job. One easy answer that occurs to me is that I ought to provide them with some examples, ideally of topics that they would not be covering in their own projects.
Diagnosis
It seems traditional to start the math year off with a BANG and having a nice, quiet, individual test. That’s how to get kids PUMPED about math, right? tests tests hurray boo rah! But I kind of did want to know what they knew, so I made up a bunch of questions about stuff my students supposedly learned last year anyway. I decided to let them do it in partners. Each pair got one packet and I said “hey I want to see both your handwritings kay?” but gave them an individual reflection sheet in order to see if JimBob actually had no idea what was happening but got lucky with his partner.
Here’s the one for my precalc class:
And honors algebra 2:
http://www.scribd.com/doc/106206335/Alg2HDiagnostic#page=1
And algebra 1:
http://www.scribd.com/doc/106206163/Alg1DiagnosticQuestions#page=1
It went pretty well. They seemed to remember a bunch and weren’t intimidated and worked nicely with each other. I tried to focus on still giving them more advanced, problemsolving questions rather than simple recall stuff. (My questions weren’t brilliant or anything, though.) Even though I was a little worried that would be too much, since it had been awhile, I had high hopes since they had been really open to answering the weird first day question, and they definitely kept up the cando attitude and barely even whined!
The reflection sheets did totally bomb, though. As in, they did them, but
a) they didn’t care about them
b) some of them didn’t fill them out, or did quickly at the end
c) some of the partners just wrote the exact same things as each other
d) I didn’t really get much information out of them.
Here’s the reflection sheet for the precalculus class:
The last questions were more general questions about how they are as students in general, which were maybe more helpful. Or maybe not? I think to make any of this more helpful, I need to really teach them how to selfreflect. I need to give them feedback and some really reason to get better at it.
But I did definitely get to know how they did individually just because of going around and helping out/responding to questions as they worked on the packet. So in that sense it was okay that the reflection sheet didn’t work, though I would like to learn how to do that well anyway, for the future and my growth as an educator etc.
Separate from the question of figuring out what my students know, the packets were a nice way to start introducing the topics from the first chapters. Even though it seemed like we spent about 2.5 classes on them, we actually got a nice intro on most of the stuff in the first units in precalc and algebra 2. I didn’t even realize that in advance, but it was a nice surprise when I was looking through my tests from last year and realized that my students had already been working with most of the concepts already.
Other things that have been happening:
we’re going to do a portfolio! based on building nice mathy habits
we’ve been starting class with fun patterns and puzzles and they’ve been getting nicely into it all.
Alright everybody let’s do math! Yeah!