Math Flowcharts

Flowchart in progress. Showing topics being covered in basic statistical graphing.
Flowchart in progress. Showing topics being covered in basic statistical graphing. The topics they are working on at the moment are highlighted in yellow. The worksheets attached to each topic are linked on the left side in the highlighted area.

To help students track their progress in math we’ve started requiring them to map what they’re doing on flowcharts. Right now, they’re doing it on paper, but we’re working on getting it to be all electronic.

Tracking with the charts helps them see how what they’ve done fits into the bigger picture. It allows them to be able to go back up the chart to previous topics if they need to review, and look forward to what comes next (and to work ahead if they would like).

The image above shows a sample what a student’s flow chart would look like while they are working on a subject (statistics in this case). The topics they are working on at the moment are highlighted in yellow. The worksheets attached to each topic are linked on the left side in the highlighted area. Links to references (Khan Academy for example) are linked on the right–there are only a few on right now (see the Mean topic on the upper right of the flowchart).

The topics on the flowchart can be expanded (using the green button on the top right of each topic) to show more detail.

Expanded window for the topic "Mean".
Expanded window for the topic “Mean”. Detail on the topics is a little sparse at the moment as we’re focusing on setting up all of the flowcharts first.

At the moment, I’m uploading the flowcharts that we’re currently using up on the website myself. Students can use the website to get worksheets and links to references, but if they mark what they’re doing on the webpage it’s not saved. We’re currently working on making it possible to create and edit flowcharts on the website itself. After that, we’ll be setting it up so students can log in with their school accounts and track their progress electronically. One ultimate goal is to map the entire upper school curriculum with these flow charts so a student might be able to track their work all the way from 7th to 12th grade.

Preparing Students for a Technological Future

I’m currently preparing a proposal to create a laboratory of digital fabrication machines–a CNC, a laser, and a vinyl cutter–and one of the questions I’m answering is about how the proposed project would prepare students for a technology-rich future. What you see below is my first response to this prompt. It’s a bit longer than I have space for in the proposal, and probably a bit too philosophical, but before I cut it down I wanted to post this draft because it does a reasonable job of encapsulating my philosophy when it comes to teaching technology:

Preparation for a technology rich future is less about preparing for specific technologies and more about getting students to have a growth mindset with respect to technology. We are living in a truly wonderful moment in history. Technological tools are rapidly expanding what we as individuals can accomplish. They are allowing us to see farther (think about remote sensing like lidar and tomography), collate more information (especially with more and more data becoming publicly available), and create things that push the limits of our imaginations. Indeed, to paraphrase a former student, we are already living in the future.

To prepare students to live and thrive in this ever-evolving present we need to demystify technology and give students the intellectual tools to deal with the rapid change. We can start by letting them peek into the black boxes that our technological devices are rapidly becoming.

We request electronics stations and tool kits not just to build things, but to be able to take them apart and look inside. Students greatly enjoy dissassembling and reassambling computers, for example, which provides younger students a good conceptual understanding of how most modern devices work. This foundation helps when they start building circuits of their own and realize what they really want to do is to control them–making lights blink and turning motors for example–and this is when they will start working with Raspberry Pi computers, Arduino microcontrollers and programming.

As students start to build (and even before really), they naturally start thinking about design. We all have an affinity for the aesthetic. If you’ve ever had the opportunity to see a laser in action, you’ll remember your sense of fascination the first time you saw someone’s design emerging from the raw material right before your eyes. Thus we get into graphic design, computer aided design (CAD) and computer aided manufacturing (CAM) and the digital fabrication machines we propose.

By the time they’re done with this curriculum, we intend that students will have developed an intimate familiarity with the technological world–including the ability to create and design their own, which prepares them for the technological future.

Cell Phone Shelf

Cell phone rack in use.
Cell phone rack in use.

Managing cell phone usage at school is a tricky topic. We have some teachers who’d like to ban them outright, but we also have a growing number of parents who are expecting to be able to communicate with their kids–to organize pickups and carpooling during the day for example. The phones can be great for data-collection and documentation in classes, and a lot of my upper level math students prefer the Desmos app to using their graphical calculators.

Our current compromise is that middle schoolers have to leave their phones in the front office, where they can check them at lunch time or check them out if a teacher wants them to use them.

The high schoolers are allowed to keep their phones with them, but have to put them in a basket at the front of the classroom. Since they don’t like piling them into the basket, I experimented with the CNC machine to cut some plywood into a cell-phone shelf.

The shelf can hold about 30 phones, and I can easily see how many phones are on there from across the room, so I’d say this one worked out pretty well.

Grit versus Passion

David Brooks argues that grades discourage students from following their passions (they have to spread their attention to keep up their GPA). And grit is easier to have if you’re following your passion.

Suppose you were designing a school to help students find their own clear end — as clear as that one. Say you were designing a school to elevate and intensify longings. Wouldn’t you want to provide examples of people who have intense longings? Wouldn’t you want to encourage students to be obsessive about worthy things? Wouldn’t you discuss which loves are higher than others and practices that habituate them toward those desires? Wouldn’t you be all about providing students with new subjects to love?

In such a school you might even de-emphasize the G.P.A. mentality, which puts a tether on passionate interests and substitutes other people’s longings for the student’s own.

— Brooks (2016): Putting Grit in its Place

Human Jobs in the Robotic Future

After all the time I spent working with Raspberry Pi microcomputers and Arduino microcontrollers this summer, it was interesting to see Claire Cain Miller summary of a PEW report on “AI, Robotics, and the Future of Jobs“.

Miller provides some interesting quotes from the experts surveyed for the report. One quote stood out in terms of its perspective on education and pedagogy:

“Only the best-educated humans will compete with machines. And education systems in the U.S. and much of the rest of the world are still sitting students in rows and columns, teaching them to keep quiet and memorize what is told to them, preparing them for life in a 20th century factory.”

— Howard Rheingold, tech writer and analyst .

The Key Findings from the PEW report provides a good summary of their results:

Half of these experts (48%) envision a future in which robots and digital agents have displaced significant numbers of both blue- and white-collar workers—with many expressing concern that this will lead to vast increases in income inequality, masses of people who are effectively unemployable, and breakdowns in the social order.

The other half of the experts who responded to this survey (52%) expect that technology will not displace more jobs than it creates by 2025. To be sure, this group anticipates that many jobs currently performed by humans will be substantially taken over by robots or digital agents by 2025. But they have faith that human ingenuity will create new jobs, industries, and ways to make a living, just as it has been doing since the dawn of the Industrial Revolution.

These two groups also share certain hopes and concerns about the impact of technology on employment. For instance, many are concerned that our existing social structures—and especially our educational institutions—are not adequately preparing people for the skills that will be needed in the job market of the future. Conversely, others have hope that the coming changes will be an opportunity to reassess our society’s relationship to employment itself—by returning to a focus on small-scale or artisanal modes of production, or by giving people more time to spend on leisure, self-improvement, or time with loved ones.

— Smith and Anderson, 2014. AI, Robotics, and the Future of Jobs.

The full report is worth a read.

An Equation for Happiness

An interesting article by some researchers from the University College in London describes the equation they constructed and tested that predicts happiness.

A key part of the equation is that it relates happiness to the difference between people’s expectations of rewards and the actual rewards.

… we show that emotional reactivity in the form of momentary happiness in response to outcomes of a probabilistic reward task is explained not by current task earnings, but by the combined influence of recent reward expectations and prediction errors arising from those expectations.

— Rutledge et al., 2014. A computational and neural model of momentary
subjective well-being
, in PNAS Early Edition.