I love giving students genuinely different ways to show their understanding. In Geometry this year, I have been having students record screencasts to explain paper folding phenomena. Basically, I walk them through a paper folding exercise (details below on the two I have done so far) that has a surprising or interesting result. Then we talk about as a class why it’s happening – they try to figure it out together, and I help them figure it out through a full class discussion. Then, they go home and record a video of them explaining the idea, showing me physically on the paper what is happening and why. I give them feedback and they record again! I have found it a great way to engage them in geometrical argument without the annoying technicalities of written proofs.
(Here, a student is using the physicality of the paper to show why when you fold a point onto another point, all the points on the fold are equidistant from the two points)
For video collection, I use Flipgrid which makes things SO EASY. They all go in one place and no one has to worry about saving or uploading files. I limit them to 2 or 3 minutes so that they have to be efficient and I can view them easily.
PAPER FOLDING CONJECTURE 1:
1. Fold up one corner of the paper in any direction so long as the crease goes between two adjacent sides.
2. Then, fold an adjacent corner up so that it meets the side of the fold already there.
Any conjectures? Students will come up with lots of things, but the fun ones to argue are: Why is that bottom angle a right angle? Why are the two triangles that you made from the folds similar?
PAPER FOLDING CONJECTURE 2:
(from an Illustrative Mathematics Task that I CAN’T FIND right now, halp!)
1. Draw two points on a piece of paper. Fold the paper so that
Any conjectures? We had been talking about perpendicular bisectors, so most students immediately saw that this was a perpendicular bisector. Can you argue that all the points on this line are equidistant from the original two points?
2. Now draw a third point.
3. Fold the other two combos of points onto each other (so if the first fold was from A to B, then fold B to C and A to C).
4. Locate the point that they all meet.
Wait why do they all meet at one point?
5. Now draw a circle with the center at that point, and use the radius as one of the original points.
My circle goes through all 3 points! Why did that happen?
In my precal (pre-cal, pre-calc, precalculus, Precalc, p-Rec-aLk) class, I have multi-day homework assignments that I collect infrequently, a structure that works great for older kids who can plan their own time out well. But I was struggling figuring out how to deal with homework in my geometry class, as I think freshman needed the daily *umph* to keep them going. I wanted a structure allows for:
- Accountability to work hard on it for both completion and understanding
- Feedback on their work
- A workload that I can handle
So I adapted a mode from colleagues at my last school who would roll a dice to see what happens. The system incorporates a little bit of randomness and has been kind of fun. Students, as a class, pick a number from 1-6 and behind that black box is the option for what is going to happen for that day:
If 3 is picked day 1, then that is used up, and we pick the other numbers on successive days until we get through the cycle. Then I shuffle the options behind them and we start the cycle anew. My options right now are:
- Homework Quiz (no notes)
- Homework Quiz (notes)
- Sight Check (x2)
- No Check – everyone gets full credit
I give them 10 minutes in the beginning of class to check homework, and the homework quizzes are literally just a problem directly from the homework, so the idea is, if they worked hard on the homework and fixed any small issues they had with them in the first 10 minutes, they should have no problem on the quizzes (spoiler alert: the kids who don’t do homework well have a big problem here, but at least they are realizing it?). The sight checks are just for completion, and the collected homework is graded on completion PLUS the corrections that they did in the first 10 minutes of class – I’m trying to encourage them to use that time really well and then give them feedback on how to do that…
Things that I have liked about this:
- It has been fun! The reveal every day is actually hilarious, though I have gotten in trouble with the teachers next door for noise a few times :).
- It has reduced my workload without really reducing what they get out of homework. Sometimes, I tip the scale by hiding what I want to happen behind all the boxes (i.e. if it’s a good day for me to collect).
- The homework quizzes are good for both me and them for REAL feedback opportunities (no one looks at what you write on their homework…) and as a low-stakes way for them to assess their own knowledge
- I also thought the homework quizzes would take forever – they take about 7 minutes or so, but it’s 7 minutes where they are rehashing their thinking about an important problem, so I’ve found it useful and a good tradeoff for instructional time.
- Going through the cycle as opposed to rolling evens out the workload a lot and ensures that there are quizzes at regular intervals.
One of the things that I changed most about my teaching approach at my previous school (I’m joining a new one this fall) is an expansion of my approach to summative assessment. Our curriculum there was a collaborative problem solving curriculum based on the Exeter materials. Students would work together in class discussing and debating about ideas, and exploring the problems presented to them, so by giving traditional, sit-down, silent, period-long individual tests, we found that we weren’t really assessing them in the way we were teaching. Due to a scheduling snafu one winter, we were left without a midterm during our exam period (lollll), but we were all like… oh well, who cares? Let’s do creative things! We all tried oral exams that winter in class, and then our minds were open from there…
Here are some of the different things that I have in my toolbox now:
The Oral Assessment
Good for: Areas of math with layered conceptual ideas underpinning them
How it could work: This was a staple for our math department. We would assign a set of problems (maybe 6-8 meaty problems) and students would have the couple of days beforehand to complete them (either with or without each other, I preferred with!). They could use whatever materials they wanted, but knew that they not only needed to solve a problem, but needed to understand it deeply, so there in essence was no way to cheat (just having the correct work on the paper didn’t really help!). Then students would come in one at at time for a 5-10 minute oral where we would roll a dice and randomly pick a couple of the problems to talk through. I preferred students just to show me work they had already completed to save time, but some colleagues had students re-do the problems for them on the board. Then I would ask questions like, “Hmm, how do you know that?” and “Does that work all the time?” or “Was that the only way to do that?” For each problem we discussed, I would give them a grade on the accuracy of their work, their mathematical discourse with me, and then a completion/accuracy grade for the rest of the assessment we didn’t get to talk about it.
I loved this because it’s pretty immediately clear who knows what they’re talking about and who doesn’t. How often do we try to assess this on written tests and get stuff like this that seem to go viral all the time (usually attacking Common Core, which is not why I’m sharing this):
Instead, you can ask specific questions, and just rephrase a bit until students understand what you are asking. I also loved when a student would have incorrect work, and I would ask a question, and they would figure out a mistake and correct their work – learning happens DURING the assessment!
If you’re wondering, “when do you have time to do something like this?” I would often do it DURING a normal written test, or give them problems to work on and try to power through my whole class in one period. I trust my students, and I know that is a luxury, but that works for me.
The Group Assessment
Good for: When you want to assess something that takes slow thinking, or is too complicated for a written test, or when you just want to get your students sharing knowledge and ideas with each other
How it could work: There are some things that students just need to know how to do individually (how to factor, how to take a derivative etc), but problem solving skills are amplified by others, and *real* mathematicians work together. I would have the students work on a problem together on a big whiteboard and then whenever they felt ready, they would erase their work, and then silently and individually solve the problem on their own, and this is what their grade would be based on. Some colleagues just had them do the problem together and grade that product, and my way takes longer, but I preferred not having students’ work determine each other’s grades. This was often great evidence to help people see that their way of learning and collaborating wasn’t working (if their group mates got a problem that they totally didn’t).
A fun modification is to give them the problem with the values blacked out, like this example:
They can then discuss HOW they would go about the problem, focusing on ideas, instead of specific numbers. And then, when they felt like they were done with the group, I would give them the problem with the numbers included to complete individually.
I usually would combine this with a standard test (maybe the first question) and found that it really cut the tension in the room around an assessment (it was exciting!).
Good for: Presenting challenge problems, assessing understanding with homework, doing oral tests without taking up class-time, assessing understanding with coding
How it could work: This is an idea taken from Andy Rundquist (@arundquist) and many other science teachers that do this – students would take a picture of their work, upload it and record a brief 2-3 minute explanation of their work. They talked through the WHAT of their work, but also the WHY. Similar to the oral assessment, it was always super easy to tell who really understood what was going on and who was faking it. I would watch the videos and give them feedback, sometimes even requesting another video or a written response to my feedback. I especially loved this as a way to change up homework, and as a way to assess students’ understanding of the really tricky problems that we went over in class, or solved collaboratively. It was also great for assessing code because, again, it didn’t matter if everyone had the same code, I was assessing their understanding of it.
You really need an LMS, and to insist students do it the way that makes it easy for you to grade them in order for this to work, because otherwise it’s a technological hassle. I found students figured out the easiest way to do this, but would suggest Screencast-O-Matic if they needed a suggestion for an easy way to do this.
The Toolbox Assessment
Good for: Reviewing many learning objectives, forcing students to find their own examples of things and do the art of “problem-finding”
How it could work: When I have taught Statistics, I have used coding, which makes a lot of things quicker, and generally “get through” most of the material with 3-4 weeks to go at the end of the year. I found that students struggled with APPLYING the ideas though, so came up with the toolbox assessment. I had a list of standards and skills that covered the whole year (“Running a t-Test”, “Using a Boxplot” etc.), and they needed to design small, quick statistical studies that showed proficiency on these ideas. I had 13 of them, but they could check off multiple with a single study. They would complete a study and then “present” to me and have a conversation to show me their work and understanding. I told them not to worry too much about what their product looked like (i.e. no need for tri-fold poster presentations) and would sometimes send them back to fix or re-do something. Their grade at the end was about how many of the objectives they checked out. I loved this because it put the onus on them to really figure out where something applied, instead of just regurgitating problems I made up. Though it’s not as applicable to other math classes, I could totally see this working as a review activity before a final exam.
Twas the week before Christmas Break, when all through the school, students were complaining, and trying to do new content would make me a fool… so we did something kind of interesting in my statistics class that was a cool application and reviews a lot of great stuff with inference testing. Inspired by this book called Nabakhov’s Favorite Word is Mauve by Ben Blatt, a super cool statistics-based analysis of literature, and this post on the Stats Medic, “Does Beyonce Write Her Own Lyrics?”. The basic questions are “How can we use basic statistics to examine and tell apart writing styles? What do statistics about your own writing say about your style?”
STEP 1: WHICH OF THESE AUTHORS ARE THE SAME?
To start, I gave them a page (available here, spoilers down below) from three different books (page 154 from each book, thanks Siri for the random number!). I told them two of these were written by the same author and one was written by a different author. How could we tell who wrote what? I told them the story of Hamilton, Madison and the disputed Federalist Papers to whet their appetite as a “real-world” example of this, but to be honest, they didn’t care about this, but were VERY intrigued just by the puzzle of figuring out which authors were the same.
And the statistics began, but not from a canned dataset that they ran pre-prescribed tests on – in fact, I was scrambling that week and hadn’t tried anything myself. I had no idea this was going to work! What things could we measure about the text to tell the difference between them? Some suggestions were too difficult to measure (i.e. tone), some had nothing to do with the writer (i.e. how many lines there were on the page), but others seemed easy to measure and perhaps distinctive of a writer (frequency of commas, length of words etc.). The students were skeptical that those things could distinguish authors, but we went after it anyway! We spent about a half-hour counting various things about the text, collected them on a document and then highlighted which two of the three were roughly more alike on each measure:
Of the 16 things we measured, 8 were the same between writers G and U (and 2 others were pretty much the same between all 3). Here come some interesting statistical questions… Why might one random page be off (one sample could be skewed for no reason other than randomness)? What’s the advantage and disadvantage of measuring a bunch of things (more things = more opportunity for random associations, but more opportunity to see a pattern). Which of these differences are “significant”?
We then spent about a class on that last question. Given that we know a chi-squared test and a t-test, how could you use those on these things we measured? We did this in R, and I can give some details about that for anyone interested, but the interesting part here is getting kids to imagine how you format data so that you could use a statistical test. What do you stick in about the sentence length in a t-test? How could -ly adverbs be a chi-squared test? Are either even appropriate here? (Meh, mostly… )
Wait… what? Those are three different authors. NOT SO FAST! Robert Galbraith is actually a pseudonym for… J.K. Rowling! (I wish I had played that up a bit more) So our statistics worked in a way – there were more similarities between G and U than the other combinations. So even when J.K. Rowling was writing under a pseudonym, her writing style was similar Cool!!!!!!
STEP 2: WHAT DOES YOUR WRITING STYLE LOOK LIKE?
Now, I wanted them to do something similar with their own writing. They had just written a joint paper with a partner, and I wanted them to see if their joint paper more closely resembled their own writing or their partners HAHAHAHAHAHA. They were hilariously sheepish about this idea, which told me immediately who had done what 🙂 (but it was all in good fun).
Enter a new tool, Count Wordsworth, an online tool that automatically measures a WHOLE BUNCH of statistics about any text that you paste in there (at which point they got mad at me because they had done so much by hand for the pages of the books, but they’re always mad at me for stuff like that). For example, here is just part of the output when I put in my teaching philosophy from my teaching portfolio:
I had them all put in a recent English paper and then find the THREE biggest differences between their paper and their partners. Again, a bunch of fun data questions – do the quotes in the paper mess things up? How about the number of words? What about the topic (English vs. a lab report)?
Then, once they had discovered the three biggest differences, I had them put in their joint paper and try to figure out whose writing style is more closely resembled. This class was a blast, and once they finished this, they were so curious so just kept exploring… Some kids put in their freshman year papers, some put in the headmaster’s emails etc. Lots of fun curiosity!
STEP 3: HOW DOES A PROFESSIONAL STATISTICIAN DO THE SAME SORT OF ANALYSIS?
Lastly, we read a short 10-page segment of the book I mentioned in the beginning,Ben Blatt’s Nabokhov’s Favorite Word is Mauve, specifically a chapter called “Searching for Fingerprints.” It was fun to see what a professional statistician does and we talked about how he could possibly measure some of the things that he did with the computing power we have nowadays.
Good stuff! Happy Holidays everyone!
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.