I played a quick, but fun and mathematically rich game in precalculus the other day that I thought I’d share. Let’s call it Rock, Paper, Triggers for now, (it’s kinda like Rock, Paper, Scissors but with Trig functions) but if you have a better name, let me know.
Each person secretly picks a trig function (SINE, COSINE or TANGENT) for themselves, and an angle to send to the other person. Then, once ready, both reveal and each person thinks about…
Whoever’s value is higher wins. No need for exact values, just figure out which one is bigger (and DNE automatically loses). So for example:
Person 1 has sin(190°) and person 2 has cos(269°). Well, both are negative, but 269° is so close to 270° that cos(269°) is a little less negative. So person 2 wins!
This was really good for number sense (no calculators), for thinking about what values of the different functions are possible, and where those values are on the unit circle.
Our school has a bi-weekly community newsletter that goes out to the school, alumni, parents and whoever else wants it. Often, a teacher writes a little introductory letter about their philosophy of teaching or their journey to the profession. I wrote for this week’s newsletter, and got a great reaction from a lot of lay people (i.e. non mathletes) so I thought I would just share it here too. The ideas in it should be familiar to the MTBoS, so get your head nod ready…
Dear St. Andrew’s Family,
When I meet new people out in the wild, I can usually predict their reaction when they hear that I’m a high school math teacher. Often, they immediately express to me how much they hate math. I have to admit I think it’s rather odd to tell someone you just met how you loathe the very thing to which he has dedicated his life’s work. (“You work for the Red Cross? Yeah, I absolutely detest charities.”) Another, even more common reaction is to tell me just how awful they are at math, taking pride in how colorfully they can describe the extent to which they struggled with the subject in school. Again, I find this a bit odd. Would we boast of our inability to read or write to an English teacher? Why is it not only okay but apparently a point of pride to be “bad” at math?
I love math. To me, it is a beautiful, complex web of ideas that can delight us with a puzzle, or shed light on the world around us. How could the math I love be a groan/panic/boredom inducing menace for so many people? The only resolution to this paradox that I can see is that the math I love and the math they hate are really two totally different entities. Without a focus on beautiful ideas, math’s procedures and operations lose their larger meaning and purpose, and math becomes a boring, repetitive, unconnected series of challenges that demand rote memorization without real understanding. This lack of connection to the deep conceptual backdrop of mathematics is not only the reason math haters don’t enjoy the subject—it’s also the reason they struggle mightily to learn it well.
As a math teacher, the painful part of this disconnect is that I believe it’s all our fault. The way math is taught often creates an oppressive and obfuscating imposter subject.
I aspire every day to fight against this imposter math, and to connect my students to the idea-rich math that I know and love. I try to make every problem we tackle in class or in homework one that a student cares about solving, whether by framing the class with a running conceptual thread that makes learning feel like unearthing the next piece of a mathematical mystery, or by investigating an application of real import, or by just engaging with a curious puzzle. I try to never tell a student something that they can figure out for themselves, because math is about discovery and exploration. Newspapers don’t print already-filled-in crossword puzzles; it’s not the answers but getting to the answers that’s the point. And I try to help students become vulnerable enough to take risks productively and make mistakes confidently, so that the more difficult, but more satisfying, work of idea-making (as opposed to procedure-regurgitating) is accessible to them.
As I write out these aspirational teaching goals, I am struck by how often I fail to meet them, and, how when I don’t, I am contributing to the creeping oppressiveness of “imposter math” by default. But it’s this awareness of my sworn enemy that keeps me engaged and excited about my profession every day.
Even if I can’t lead every student I teach to fall in love with math the way I have, I hope that at the very least I am connecting them with math’s big ideas in some real way. I like to think I am helping to rear a generation of students who won’t, twenty years down the road, regale every stray math teacher they meet with stories of how much they hated nasty old mathematics.
All the best,
Mathematics Faculty; Cross-Country & Swimming Coach
What is authentic assessment in the math classroom? It’s probably not a math test. Tough to admit, as I give lots of math tests, but a test is so limited, so contrived, so singular. The most authentic assessment I have been part of in the math classroom was our culminating project for Data Driven this summer – a business case study presented to people in the business world.
A friend of mine from college who now works for a predictive business analytics company ran a case study on my students. The case was for a bagel store that wanted to expand – they had data on the profit of their current stores over time, and data on features of the current stores. In teams of four, students had to advise the bagel company on where the company should build 10 new locations, and what the layout should be. My friend served as the lead of the company’s expansion team – the students had a halfway call with him and could email him at any point during the week with questions or requests for data. At the end of the week, students presented (via Skype) their recommendations and defended them with questions.
HOW WAS THIS AUTHENTIC?
- We learned a billion things and amassed a ton of data analysis tools this summer – instead of being directed what to use where, students had to sift through their knowledge to figure out what was appropriate. Though they received an initial prepackaged dataset, the problem was wide open and had very little hand holding. If they wanted to use census data about median incomes in zip codes, they had to go find that data, clean it up and attach it to the given dataset before they could use it.
- All the math that they were doing was supporting a genuine and interesting, multifaceted problem, instead of being motivated by just being a question on the test. If they needed to do a multiple linear regression, it was because they wanted to figure out something about the data, not because a question asked them to do a multiple linear regression.
In addition, it was a problem that forced them to translate their mathematical knowledge into human decisions. They had to tell the story that the data was presenting, had to make choices that didn’t have a “correct” answer, and had to defend everything they were doing in a way that a naive non-math outsider could understand.
- Presenting to an outside audience forced them to be as prepared as possible, and also taught them a lot of lessons about communication! I wish I had taken a picture of one group when they were on a conference call with my friend. They were pacing around the room, hands on heads, brows furrowed, goofy smiles from feeling awkward – so much more learning was happening than if they were presenting to me! I also just had to sit and watch them struggle through things, like explaining what a t-test was, during their final presentation, which gave me deep insight into the results my teaching.
- There were many points of entry and many different depths that students could take it. There were immediate things that anyone could do, and things that only a professional data scientist could have done, which made the problem perfect to test everyone, but give the students needing a bigger challenge a place to go.
HOW WAS IT STILL INAUTHENTIC?
- The data was fake, the business fake, the audience fake. The advantage to this was that I could ensure that the math involved was the right level, and that the problem was doable, but perhaps this took something away from the motivation for the students.
- There was no followup from the final result. Wouldn’t this have been even more awesome if they were presenting to a real company, or community organization, that was trying to make a real decision? And then they could see what the company actually decided and see what the results were.
- There were students in each group that didn’t contribute. I don’t think anyone didn’t want to contribute, but it’s really hard to work in teams. I think that this exercise tested their collaboration skills, but perhaps didn’t assess every single student’s math skills.
Though this course was unique in its format (long 4 hour classes, only 12 students, no curricular pressure) and did not come with grades, there is so much from this to take to my school-year classroom. How can I include more authentic assessments in my day to day classroom life? Assessments with multifaceted, human problems that motivate great math along the way; ones with many points of entry and many places to go; and ones where they have to defend their decisions to audiences other than me.
It’s important to remember that “authentic” is not a binary designation, so my goal is to add pieces of the above to my normal classroom assessments one step at a time.
As I start reflecting on the course I taught this summer, I thought I’d start by sharing my Syllabus for anyone curious. The course was a functional data course – the focus was more on being able to DO things rather than on abstract statistical work. We used data visualization software geared at businesses (Tableau), coded in R, conducted election polling, performed original research projects, wrestled over issues of data privacy, cracked codes, and put together advice for a business on how they should expand (amongst many, many other things). It was exhausting and awesome. More reflections to come!
(if that is too small below, here is a google drive link)
This summer, I’m teaching a 5 week intensive course called Data Driven (course description) at this amazing summer program at St. Paul’s School in NH called the Advanced Studies Program. It’s an enrichment program for rising high school seniors. We are doing class 3-4 hours a day, 6 days a week for 5 weeks, with tons of time for independent work at night. The class is about creating functional data mavens – think statistics, plus data science, plus research, plus data ethics/privacy, plus cryptography, with a whole lot of reading, coding, writing, computing and interacting with the community along the way.
DATA SPEED DATING
After a quick math-themed icebreaker, we started our data class this summer with a few data themed get-to-know-you activities, the first being data speed dating. Each student picked a categorical variable and a quantitative variable that they wanted to collect from every student in the class. They then sat across from each other and “speed dated” to collect the info from each person in the class.
It was nice to knock out the kind of dumb and easy idea of variable types in an icebreaking activity, and it was great that every single student had a conversation with every other student in the class (only 12 students).
Then, I paired them up and each pair had to pick one of the sets of data to present visually to the class. I wanted to get them started on culling the most interesting data from a data set, picking appropriate visualizations, and translating data for others. One group did this kind of funny infographic describing how many pairs of pants were owned by people who preferred certain movie types. Problems with the visualizations, of course, but interesting nonetheless (and hey, it was the first half hour of class). In retrospect, I wish I had explicitly said “Combine TWO of your pieces of data in a visualization” because I think that would have been a much more interesting intellectual challenge (and would have led to a bunch of silly things!).
Then, I introduced our homework for the night, which fell on similar lines. It was based on the project Dear Data by two data scientists Giorgia Lupi and Stephanie Posavec. They picked a broad topic (like “laughter”, “books”, “thank yous”) at the beginning of a week, and each chose what data they were going to collect about that topic. At the end of the week, each turned their data into a beautiful visualization on a postcard, with the key on the back, and sent the postcards to each other (one was in London, one in NYC).
For my students, we picked the topic “New Encounters,” as they are all starting this program with a bunch of people they don’t know. They each brainstormed the data they were going to collect, and I gave them mini-reporter notebooks to carry around. From what I saw when they were working on them earlier tonight, some of the visualizations that the students did were just as beautiful as these professional data scientists (and some managed to collect 70-80 points of multidimensional data in a day and a half). Will post once I see them tomorrow!
I have had MANY requests for the actual files for my AP Calculus Skill Drills – a 5-10 minute start to class every day for a couple of weeks leading up to the AP Calculus AB exam. Below is the file. Know that it is fairly specific to my class – they are categorized based on my standards and we voted which standards to keep reviewing that day – but still should be a decent review for anyone if you want to modify. There are 10 days of review goodness, which according to the file, I started on April 9th a few years ago. Forgive any errors of course.
- Skill Drill PDF
- Skill Drill Word Doc (will show up as a file on Google Drive missing tons of stuff, but hit download)
Best of luck prepping kiddos for the exam soon.
This year, I have tried to engage my students in a more thoughtful homework process. I don’t think any math teacher, ever, has been satisfied with the way homework works in their class, and I would certainly put myself in that boat. My frustrations in the past have been that students sometimes would do something wrong and then continue to cement that wrong thing by repetition, I would get 30 homework assignments that look basically the same and spend tons of time giving useless feedback that they didn’t really even look at, and students were focused on completion over learning. I attribute this to the structure of the homework over students being their nutter butter selves. Here are the changes I made this year:
1. Every homework assignment comes with a full solution (not just answer) guide. It’s more work for me, but also makes me assign a reasonable amount of homework.
2. Students go through the assignment and do whatever they can without the solution guide.
3. Then they check the solution guide to check what they did and finish what they couldn’t. Anything they write after this point (or using the solution guide) is in a different color – which is a crucial point. They check their answers, fill in the rest of incomplete solutions and give themselves feedback on what they did well and what they did poorly.
It takes a little longer for the students, so I try to assign a little less. And some students haven’t bought totally into it yet (slash never will). But as a teacher grading it, I can see so much more. Like…
- Where students struggled and what they still don’t understand well, which is so obvious with the colored pen. What they did in pencil is their work and what they did in pen is their work with the solution guide.
- Evidence of learning – instead of doing something wrong over and over, they correct it and do it better the second time around, or at least know that what they did is wrong and need to get help from me.
- Where to give them feedback on the specific things that they are struggling on.
- Who is engaging with the homework and trying to learn from it, vs. who is just tryna get-r-done.
I also spend less time grading homework while still giving better quality feedback. I think they spend about the same amount of time doing homework but get more out of it.
The training process for this has been an investment, but worth it. I share with the class examples of things they can do to do this better, like this:
Feedback from students has been that they almost either really like it, or are fine doing it. They almost all indicate that it’s better for learning, which is what I care about.
How do you feel about the method of doing homework where you check your own answers?
It is very helpful XXXXXXXXXX
- It allows you to learn the right way of doing it while it’s still fresh in your mind.
- I like understanding what I did wrong right after I did it so that I can grasp what I did wrong.
- Being able to look at the answer and find what I did wrong at my own pace helps me understand the problem and how I should do it next time.
- Writing my own feedback is more helpful than skimming any you would give on homework.
- Self check is a way to see what you did wrong right after you did the work instead of a couple of days later,
It’s fine XXXXXXX
- I feel as though that making corrections and not totally understanding my mistakes is perhaps the biggest downfall.
- maybe if i came back after a longer period of time it would be more helpful to me in particular.
- I understand that it’s good to correct ourselves but I think I get more out of simply going up to you to clarify he things I’m struggling with.
- I only feel like feedback is necessary for some problems if I really don’t get it
- Well it is helpful some of the time but it does take a really long time to do this.
- I think that it’s helpful like 85% of the time, and then other times it confuses me
Meh, I don’t really do it. XX
Still experimenting! Would love some thoughts.
I have been quieter here than I normally would be during summer planning because I have been blogging at Klingspace, a blog run by the graduate school program from which I graduated in May. Below are the posts I have published there with a brief summary, if you are interested in reading about topics that may not be as math education focused as I usually am.
I am excited to rejoin a math classroom in the fall and hope to re-engage in the math education discussion on this blog that I am used to!
- 5/22 – Structure Is Not the Opposite of Autonomy – We shy away from procedures, structures and limitations in the name of creativity, but that structure can actually promote creativity.
- 5/28 – Keeping the Change: How > What – Success naturally breeds resistance to change, which means we must be sensitive to the fact that our change-filled futures are challenges to our success-filled pasts. Give people time to process change.
- 6/4 – Teacher Observation: Informing Practice, Not Judgment – The way most schools structure observations and evaluations make us see them as moments of judgment instead of opportunities to improve our practice.
- 6/11 – Have a GSA? Great! But It’s Probably Not Enough – There are queer students at our schools who aren’t served by simply having a GSA. More generally, we should not assume that because we have programming for X type of students that it serves every student who identifies as X.
- 6/28 – Using “Creative Tension” To Communicate Change – If leaders effectively show faculty the gap between their vision and the current program, faculty will be more likely to feel the need to move toward the vision
- 7/11 – Cultivating a Growth Soulset – Just as we can always learn more with a growth mindset, we need to tend to our emotional intelligence with the attitude that we can always become more emotionally adept.
- 7/21 – The Case Against a Linearly Sequenced Curriculum – Research about distributed practice suggests that studying something with space between is always more effective than studying it for the same amount of time uninterrupted. How can we incorporate this finding into our curriculum design?
- 7/29 – TBA
- 8/15 – TBA
(here are some excerpts from a paper I wrote for grad school about structuring math homework for better learning – the full paper is below)
The traditional structure of math homework (e.g. 1-67 odd) forces students to work hard, but not effectively, as all students blindly do the same assignment consisting of a similar number of each type of problem regardless of each student’s personal weaknesses. In “Practice Perfect,” Doug Lemov’s book on how to practice more effectively, the authors compare this type of practice to shampooing your hair, something we repeat daily but probably never improve on.
Math students ought to practice math the way that experts in other fields practice. When a musician learns a piece of music, they do not just play whole piece over and over. Instead, they workshop specific parts that they need to work on. When I was learning to play piano, my teacher would have me play difficult parts repeatedly – first each hand separately, then both hands together slowly, and finally at full speed. Students do the exact opposite on math homework – they do the problems with which they are comfortable and then leave blank those they do not know how to do. Thus, they are only practicing the very material that they do not need to.
If teachers would like them to engage with it differently, they need to make intentional changes in the structure. To make math homework more like expert practice, teachers should:
Force students to differentiate their homework experience.
Though the other suggestions below would be a helpful addition to traditional homework assignments, this first one would require a more radical shift. Instead of a linear assignment that encourages students to spend an equal amount of time on each part of the course, a math assignment should consist of minimal core problems for each learning objective that each student must complete, and a bank of other problems that the student could use to remedy any misconceptions. The set of core problems should be small enough that students can complete all and still have time to tackle their weak areas. Students should be instructed that a wrong answer should be a sign to reflect for a moment about what went wrong, perhaps even formally, and then immediately try more problems until they understand. To give space for this thoughtful type of work, a teacher might have to assign less work, but the quality of the work that is completed has the potential to be much higher.
(some thoughts on grading and accountability in the full paper)
Make homework objectives transparent.
With a differentiated homework assignment that required metacognition about weaknesses, students would need the tools to pick out their own weaknesses. When picking out problems from a math textbook to assign, I have an objective in mind for each group of problems. In retrospect, it seems obvious that I should simply share these objectives with students, paralleling Standards Based Grading for the wider structure of the class. Simply grouping math problems into learning objectives would help students focus their effort more effectively by allowing them to isolate skills and measure their success.
Ensure students can get immediate and actionable feedback
But even with clear learning objectives, students can’t make progress without feedback. Too often, students will power through their entire homework doing something wrong the entire time, encoding something in their brains the wrong way. Worried that students will copy answers out of the back of the book, teachers will assign problems that do not have attached solutions. We have to get over this fear – if students cannot check their work, or are not in the habit of doing so even when they are confident they got a problem correct, they risk not knowing that they are doing something wrong. Doing one problem wrong, fixing a misunderstanding and then doing a few more correctly will lead to far better results than doing five wrong and having to unlearn something incorrect a week later. For complex problems that only have a simple answer in the back of the book, teachers could post a solution guide that details not only the answer but the process that it takes.
No idea if this will work! I’m interested to try it out when I get back to the classroom. Here is the full paper: