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Equity Eyes

I’m really enjoying my Twitter Math Camp morning session. The title is, “What is the relationship between the Standards for Mathematical Practice and equity?”, presented by Grace Chen, Brette Garner, and Sammie Marshall.

One scenario we looked at was a classroom situation drawn from this journal article. Students are analyzing data for where certain Netflix moves are popular, and are looking at a specific movie with Black characters and a Black director that was only popular in a few neighborhoods in Los Angeles. William and Jessica have hypothesized that it is “more popular in predominantly Black neighborhoods because of ‘support’.”

The class grows increasingly rowdy. Jessica and William [two of five Black students in a class of 30 largely Latino students] defend the movie and point out that it was also very popular in Atlanta (Marker D). Other students make coded comments about “low budget marketing,” Tyler Perry movies, and “whack-ass movies over there.” Ramon calls William racist; other students chime in and say “that’s racist” or “it’s not being racist.” Mr. Romero asks students to calm down, to little effect. Eventually, William sits back in his chair, faces the board, and says “Okay, next slide.” Mr. Romero takes this opportunity to move on to another example of big data. William and Jessica are silent for the rest of the class period.

The last line is what got me. I’ve taught classes where similar accusations of racism have been tossed around. I’ve mostly handled those moments poorly. But I don’t think I would have noticed that two students were silent for the rest of the class. I don’t think my attention would have zoomed in on that piece of information or allowed me to address it in a way that could at least reduce the damage done to my class culture and those students’ sense of safety in my room.

On the first day of TMC, Grace, Brette and Sammie asked us to consider creating a checklist of “equity eyes”. These are ideas that we want to be thinking about, relative to equity, that are not yet automatic for us. Then, they asked us to take those equity eyes to our experiences at TMC, and be ready to share how this impacted us.

My goal for equity eyes was to look at ideas through the lens of whether previously low-performing students were learning as much as or more than previously high-performing students. I then promptly forgot about that goal and went about my learning for the rest of the day. I was reminded of equity eyes again at the start of our second session on day two.

Reflecting on that experience, it was an important reminder of how hard it is to change. It was also an important reminder of how necessary it is to do the work and get a little better every day. Jay Smooth has a great TED talk, which I found via Ben Blum-Smith. Jay shares the “dental hygiene model” of talking about race. Working against racism isn’t something you turn on and off. Instead, it’s like brushing your teeth. It’s something you need to do regularly. No one is perfect, everyone should be working on it, and no matter how great your dental hygiene is if you don’t brush your teeth for too long they’re going to turn yellow and fall out. In the same way, if we hide from hard questions we will lose that perspective and the ability to grapple successfully with hard challenges as they come.

As Grace said in our session:

The more we think about it, the more automatic it gets and the less we have to think about it.

One thing I find incredibly important is balancing an explicit focus on equity in particular spaces to build capacity with embedding equity in everything I do. It’s hard to look at everday decisions, teacher moves, and conversations through a lens of equity. That lens isn’t built overnight; it’s built painstakingly and slowly and deliberately. It’s built by taking time to dive deep into equity work, and then stepping back and figuring out how it can apply to everyday classroom teaching.

Sadie shared with me a beautiful Hawaiian word, `ukana. It means the stuff that we carry with us. I see an equity lens as something I want to add to the stuff I carry with me. It’s not easy, but it’s something that becomes more and more natural as I do it more regularly.

Today I improved my equity eyes a little bit. I engaged with some hard questions with some thoughtful people. I watched myself fail. I’m still reflecting on it and organizing my thoughts here. I have a long way to go. Looking forward to continuing the journey.

Oversimplifying Education Technology

Tucked away in a letter from the Bill and Melinda Gates Foundation last week, along with proud notes about the foundation’s efforts to fight smoking and tropical diseases and its other accomplishments, was a section on education. Its tone was unmistakably chastened.

“We’re facing the fact that it is a real struggle to make systemwide change,” wrote the foundation’s CEO, Sue Desmond-Hellman. And a few lines later: “It is really tough to create more great public schools.”

-LA Times, Gates Foundation failures show philanthropists shouldn’t be setting America’s public school agenda

I worry that the future of education will be determined by wealthy philanthropists who do not understand the realities of classrooms and jump to spend money at flashy ideas without the substance to back them up. The article references above explores in depth the Gates Foundation’s challenges in trying to do so. An article in today’s Economist attempts to take a more balanced perspective, looking at both the potential and the liabilities of education technology. While doing so, it puts a veneer of research-based authenticity on ideas that do not deserve it, and falls victim to the same faulty logic that has proliferated bad ideas in too many schools.

There are some useful ideas:

Backed by billionaire techies such as Mark Zuckerberg and Bill Gates, schools around the world are using new software to “personalise” learning. This could help hundreds of millions of children stuck in dismal classes—but only if edtech boosters can resist the temptation to revive harmful ideas about how children learn. To succeed, edtech must be at the service of teaching, not the other way around.

A solid start. But when it gets into specifics, things get rough:

In India, where about half of children leave primary school unable to read a simple text, the curriculum goes over many pupils’ heads. “Adaptive” software such as Mindspark can work out what a child knows and pose questions accordingly. A recent paper found that Indian children using Mindspark after school made some of the largest gains in maths and reading of any education study in poor countries.

I found what seems to be the study they are referencing. Students do make gains, but effect sizes of 0.36 in math and 0.22 in Hindi are smaller than numerous other, more well-understood interventions. At the same time, this program was not compared with any other intervention. Students attending an after-school session up to 80 times in half a year learned more than students who didn’t. That’s unsurprising to me; one would hope all of that time would not be wasted. And the authors of the paper acknowledge one of the challenging realities of edtech: they were able to implement their program with a great deal of fidelity because they were running a small study with great resources. They acknowledge that it’s an open question whether something like their intervention could operate at a large scale. And taking something that seems to work in a small setting and extrapolating it to millions of students is too often a hallmark of edtech philanthropists, eager to seize on thin evidence without reading the fine print.

Another example from the article:

The other way edtech can aid learning is by making schools more productive. In California schools are using software to overhaul the conventional model. Instead of textbooks, pupils have “playlists”, which they use to access online lessons and take tests. The software assesses children’s progress, lightening teachers’ marking load and giving them insight on their pupils. Saved teachers’ time is allocated to other tasks, such as fostering pupils’ social skills or one-on-one tuition. A study in 2015 suggested that children in early adopters of this model score better in tests than their peers at other schools.


This is even more vague about the research referenced, but if the author is referencing the Gates/RAND study, the results have been criticized elsewhere, and in detail. In short, conflating a wide range of pedagogic and technological strategies as personalized learning does little to help us understand what works and what doesn’t. Grouping students based on performance data, discussing data and learning goals with students, and providing spaces for students to work at their own pace are very different strategies, and each can be implemented in lots of different ways. This study groups them together under “personalized learning” and says they “work”, a level of ambiguity that helps no one. Additionally, the schools in question were largely charters, implemented their own interpretation of personalized learning, and received additional grant funding; it’s hard to know the difference between the effects of the various personalized learning interventions and the increased resources that each school had access to.

The original article goes on:

A less consequential falsehood is that technology means children do not need to learn facts or learn from a teacher—instead they can just use Google. Some educationalists go further, arguing that facts get in the way of skills such as creativity and critical thinking. The opposite is true. A memory crammed with knowledge enables these talents. William Shakespeare was drilled in Latin phrases and grammatical rules and yet he penned a few decent plays. In 2015 a vast study of 1,200 education meta-analyses found that, of the 20 most effective ways of boosting learning, nearly all relied on the craft of a teacher.

I can barely keep up with the non-sequiturs. There isn’t even a source cited for the bold claims here, and while plenty of educators would agree that knowledge enables creativity and critical thinking, others would disagree, and most of both groups would tell you that Shakespeare’s education is probably insufficient evidence for best practice in today’s schools. And conflating a knowledge-based education with the influence of teachers is at odds with how edtech often plays out in classrooms; too often technology is dehumanizing, and the “playlists” that students are learning through are associated with shallow knowledge and memorization.


I’m often reluctant to wade into debates on edtech. Emotions can run high, and I’m hesitant to come down on one side or the other; in my experience, the philosophical or technological decisions matter far less than how well they are implemented. Any tool can be used well or poorly, it’s the teachers that determine whether students learn.

At the same time, I am frustrated at popular media fawning over technology, playing fast and loose with research, and making broad assumptions divorced from classroom realities. Edtech is a hard field to understand. It’s broad and complicated, with lots of players and lots of motives. No one is helped by reading shallow takes with a veneer of authenticity. The Economist tries to take multiple perspectives, but only scratches the surface of half a dozen different ideas and pretends that reading the abstracts of a few studies is a substitute for understanding the complexities of education. Instead of trying to survey all of education technology in a thousand words, I would much rather read a well-reported exploration of a single example or a few closely connected examples. Generalizing about big, messy problems in education and oversimplifying the challenges involved gives the impression that there are easy solutions, if only the stubborn teachers and unions would get out of the way and move into the 21st century.

I’ll finish by returning to the LA Times’ article on the Gates Foundation:

Philanthropists are not generally education experts, and even if they hire scholars and experts, public officials shouldn’t be allowing them to set the policy agenda for the nation’s public schools. The Gates experience teaches once again that educational silver bullets are in short supply and that some educational trends live only a little longer than mayflies.

Task Propensity

This task propensity entices teachers and textbook authors to capitalize on procedures that can quickly generate correct answers, instead of investing in the underlying mathematics while accepting that fluency may come later.


The article linked above is a thought-provoking perspective on why some conceptually-focused math reforms have been unsuccessful. The authors explore the idea of task propensity, or the tendency of teachers and curriculum writers to focus on features of specific tasks rather than  the underlying mathematics that may be used in new tasks in the future. Teachers may have great, conceptually oriented tasks that can elicit mathematical thinking, yet if they only focus on teaching students how to solve those specific tasks that thinking is unlikely to transfer to new problems down the road.

I’m hanging out with some great folks at the Desmos fellows weekend, and I’d like to share two contrasting cases:

Case 1 
We spent some time yesterday mingling and doing math together. I spent much if working on this problem from Play With Your Math with a great group of teachers.
Screenshot 2017-07-15 at 7.02.24 AM.png
I won’t spoil it; this is absolutely worth exploring, and after what was probably an hour of work I have plenty more to learn. The most important feature of my learning was that, in a relatively short period of time, the group I was working with established the answer to the question as it was posed. We then went further, and explored different conjectures and directions to extend the problem. The vast majority of our learning came after we had solved the problem, and depended on our interest in creating new problems to further our thinking. In other words, we avoided the temptation of task propensity to fixate on the problem at the expense of additional learning.

Case 2:
I have really enjoyed both playing and watching students play Marbleslides lessons like this one. Students have to transform various functions in order for the marbles to get every star when they are launched.
Screenshot 2017-07-15 at 7.21.01 AM.png
This is one of my students’ favorite things to do in class, and is far more engaging for them than any other lesson I have on rational functions. At the same time, I find that students often learn less than I would like from the activity. They spend most of their time focused on the task at hand — getting all of the stars — and less on what I want them to learn — general rules for transforming rational functions. This is not to say that no learning happens, just that students can fall victim to task propensity and lose the forest for the trees.

I am looking forward to my Desmos fellowship and what I will learn from a great group of teachers and the stellar folks at Desmos. One of the important questions I have is around when Desmos is the appropriate tool to use, and when other tools will work just as well or better. One challenge I have with many activities is task propensity; that, while Desmos is a powerful tool for generalizing thinking, that generalization does not happen if students are too focused on the specific features of a task to make connections to broader mathematical ideas. I hope to do some writing over the next few months to explore this idea and try to better understand when Desmos is the right tool, and how to use it effectively.

Understanding Abstractions

This doesn’t feel true about mathematics. Much of the math I teach I would enjoy going down a similar rabbit hole with students, though it hopefully wouldn’t take as long.

But this comic also made me think about calculus. There are plenty of gaps in my calculus understanding — I’m not sure I could prove the product rule without some significant help, for instance. I’ve worked through proofs of Lagrange error before but I’m a long way from really understanding how that whole thing works. Not to mention the Fundamental Theorem of Calculus, which I can use pretty fluently yet don’t particularly understand why it’s true.

Maybe this is a reminder to deepen my own content knowledge. At the same time, my instinct is that there are times when it’s appropriate for a tool to remain an abstraction. I would like to verify that abstractions work — for instance, use Desmos to verify that a few product rule applications do, in fact, produce appropriate derivatives. I wonder if I could come up with criteria for when an abstraction is far more useful than understanding why that abstraction is mathematically correct.

On Teaching Collaboration

I’ve seen the phrase “The four Cs” thrown around more and more recently — critical thinking, communication, collaboration and creativity. 21st century skills, etc etc. I’m going to zoom in here on collaboration.

I want my students to be more thoughtful and effective collaborators when they leave my class. What I know less about is how to structure experiences that will teach students to do so.

Students should collaborate, sure. I find purposeful partner and group work to help students better learn math. But does it also help teach them to collaborate?

I find it interesting that teachers, who are seemingly charged with teaching collaboration, work in a profession where there is little sustained collaboration between colleagues in many schools. In reflecting on my experiences working in groups it seems I have learned much less than I would like to think about collaboration, the general, all-purpose skill, and more about collaborating with those specific people in that specific context.

This person has great ideas but when I ask him to write something up it will take two weeks and three reminders, I should just do it myself. This person is great at taking the student perspective and thinking through how it will impact their experience; make space for her to share before we make any decisions. This person gives excellent, honest feedback; even when it stings, I know that it comes from the right place and is on the mark. Those are the types of lessons I think I’ve learned from my experiences working collaboratively.

I’m sure I’ve learned broader skills of collaboration along the way. My point is just that my practice of putting students in groups “because they need to learn how to collaborate” is probably insufficient to meet the goal. Seems likely that humans learn collaboration like any other skill: practice and reflection. Plenty of practice, spaced over time, and reflection that is mindful of how lessons learned may apply in new contexts in the future.

Now to figure out how to do that.

Doing Math This Summer

Summer is here (apologies to those final folks who are still in school). Every summer I try to spend some time doing math, focused on challenging myself and learning in ways that will support my teaching. I’ve got four different ways I’m doing that this summer. No big commitments for me, but instead a few different avenues to explore and learn when I have the time and inclination.

Exeter Problem Sets
The Exeter curriculum is online and free. It starts with the upper-middle school math that leads into Algebra I, and continues through calculus while exploring some lovely math along the way. Everything is problem-based, and the curriculum is really just problem sets that build high school mathematics piece by piece. Every time I have dug into the Exeter sets I have learned new math, made new connections, filed away different problems for future use, or had some inspiration about how to better sequence and teach the ideas in my curriculum.

Park City Math Institute Problem Sets 
The PCMI problem sets are similar, but are designed for math educators. They require little prior knowledge to dive into and explore some fascinating math topics, while also providing a great opportunity to play with ideas, make connections, and discover. This summer’s sets are being posted one day at a time on this website, and prior years can be found on the same site.

Brilliant 100 Day Challenge 
I had never heard of the Brilliant website before, but they are posting one challenging problem a day over the summer for 100 days. The problems are excellent, and I’ve had fun exploring some other ideas on their site as well.

GDay Math 
James Tanton’s GDay Math site  has a few different courses to work through.  I have previously explore Exploding Dots and did quadratics; in both cases I learned for more than I expected about topics I thought I already new. He also offers courses on fractions, combinations and permutations, and area.

Happy mathing!

Reflecting On Writing

The school year has ended and I’m on to summer vacation. I’m working on a few projects, some related to teaching and some not. I’m also thinking about next year — my priorities, my goals, and my commitments.

In that reflection, I realized that one commitment I haven’t even considered ending was this blog. Writing has become a central part of my identity as a teacher. I think things through in writing. I encounter a challenge in the classroom and start thinking about how I can write about it. I set goals for what I want to learn, and writing about that process holds me accountable and helps me cut through to the essential takeaways.

At the same time, I’ve built a written record of how my ideas have evolved over time. I’ve become someone who can sit down and write when I need to — I no longer put off writing tasks as long as possible. Writing has opened doors and created relationships in my professional life that I never thought would be possible.

This is all to say thanks for reading. And for anyone out there who has considered starting a blog — I don’t know if it’s for you, but it’s absolutely worth a try. Most of all, my advice is to write for you. Don’t write worrying about what others will think or how it looks to someone you don’t know. Write because it will help you learn, help you reflect, and help you grow. You can learn a lot by sitting down to write about your teaching, once a day, once a week, once a month, or once a year.