Making Experts - A Project Proposal

tl;dr A Project Proposal:

I'd like to see expert 'knowers' in different fields each record a 2-4 minute video (uploaded to YouTube) in which they respond to one of the following prompts:

  • Describe a situation in which a simple change to what you knew made something that was previously impossible, possible.
  • Describe a moment when you had to unlearn what was known so that you could construct new ideas.
  • What misconception in your field did you need to overcome in yourself to become successful?

I think that teachers model knowledge creation by devoting time to exploring it in their classes. I think we can show them that this process isn't just something you do until you've made it - it is a way of life, especially for the most successful people in the world. I think a peek behind the curtain would be an exciting and meaningful way for students to see how the most knowledgeable in our society got that way.

Long form:

One thing we do as teachers that makes students roll their eyes in response is this frequent follow up to a final answer: How do you know?

This is a testament to our commitment to being unsatisfied with an answer being merely right or wrong. We are intensely committed to understanding and emphasizing process as teachers because that's where we add the most value. Process knowledge is valuable. An engineering company can release detailed manufacturing plans of a product design and know they will remain profitable because their value is often stored within the process of building the product, not the design itself. This is, as I understand it, much if the power of companies dealing in open source technologies.

In a field like ours, however, students often get a warped sense of the value of process. They don't hear experts talking about their process of learning to be experts, which inevitably involves a lot of failure, learning, unlearning, and re-learning. In some of the most rapidly changing fields - medicine, technology, science for example - it is knowledge itself that is changing.

An important element of the IB program is the course in Theory of Knowledge (abbreviated TOK). In this course, students explore the nature of knowledge, how it represents truth, how truth may be relative, and other concepts crucial to understanding what it means to 'know' something to be true. From what I have heard from experienced IB educators, it can be a really satisfying course for both teachers and students. Elements of TOK are included as essential parts of all of the core courses that students take.

I can certainly find lots of specific ways to bring these concepts up in mathematics and science. Creating definitions and exploring the consequences of those definitions is fundamental to mathematics. Newton 'knew' that space was relative, but time was absolute. Einstein reasoned through a different set of rules that neither was absolute. These people, however, are characters in the world of science. Their processes of arriving at what they knew to be true don't get much airtime.

What if we could get experts in fields talking about their process of knowing what they know? What if students could see these practitioners themselves describing how they struggled with unlearning what they previously believed to be absolutely true? I see only good things coming of this.

What do you think? Any takers?

Summer Updates

One of my favorite parts of summer is reflecting on the past year and brainstorming new ideas for the next. On my mind these days:

  • Refining my standards based grading system after this past semester and year's implementation
  • Building my IB courses for math(s) and physics, which will have both HL and SL in the same class period
  • Sharing ways that programming has made my teaching life easier and richer
  • Better making the most of in-class time, as well as maximizing the benefit of time students spend on their own

There are posts brewing in my head on each of these. At the moment, I'm on a road trip headed west and plan to enjoy my time enjoying the views and life, so these will likely live only in my head for now.

Stay tuned for the roll out.20140716-220459-79499383.jpg

Curated review for finals

I really don't like reviewing for exams. I don't think I'm the only one that thinks this, by far.

If I create a the review sheet, I'm the one going through all of the content of the unit and identifying what might be important. It would be much more valuable to have students do this. I've also been filling the school server with notes and handouts of what we do each day, so they could be the ones deciding which problems are representative of the unit.

Suppose I do make a new set of review problems available to students. If students have this set of problems to work through during class, I spend my time circulating and answering questions and giving feedback, which is the best use of my time with students. Better yet, students answer each other questions, and give each other feedback. They lose the opportunity to see the scope of the entire semester themselves because, outside of the set of problems I prepare for them, they don't actually take the time to see that scope on their own. They only see my curated sample and interpret it according to their own understanding of the relationship between review problems I select and problems I select for an exam.

I've had students themselves create review sheets, but this always has its own set of issues. Is it on paper or online? If on paper, how does this sheet efficiently get shared with other students? The benefit of an online resource is the ease of sharing. The difficulty comes from (1) the difficulty of communicating mathematics on a computer and (2) compiling that resource in one place. It's a lot of work to scan student work and paste it into a document. Unless I am meticulous in making sure that all students are using the same program (which is a lot of work for a class of twenty-four students all with their own laptops) this becomes a lot of work (again) for me. I'll do it if I really believe it is worth the effort for students, but I'm always looking to be efficient in that effort. I also don't want to put this effort on the shoulders of a student to together. And before someone tells me to use Google Docs and its amazing collaborative tools, I'll bring up the governmental disruption of Google services and leave it to you to figure out why that isn't an option for me and my students.

In the end, I have to decide which is the most valuable for students relative to a review. Is it getting feedback on what a student does and does not understand? Is it going back over the entire semester's material and figuring out what is important relative to a cumulative final?

If I have to pick a theme of my online experiments this year, it has been the search for effective ways to leverage social pressure and student use of technology to improve the quality of the time we spend in the classroom together. In the past, I have been the one collecting student work and putting it in one place when I've tried doing things differently for exam review. That organization is precisely something computers do well if we design a scheme for them to use.

Here's what I have had students do this year:
Screen Shot 2014-06-10 at 4.35.37 PM

Each student has a blog where they post their own review sheet for one standard. They submit the URL of their post and their standard number through the same site through which they sign up for SBG reassessments. They see a list of the pages submitted by other students:
Screen Shot 2014-06-10 at 1.09.08 PM

This serves as a central portal through which students can access each other's pages. Each student controls their own page and URL information, which saves me the effort to collect it all.

Why am I really excited about this list?

  • I curate the list. I decide whether a page has met the requirements of the assignment, and students can see those pages with a checkmark and a WB for my initials. If a student needs to improve something, I can tell them specifically what isn't meeting the requirements and help them fix it. Everyone doesn't have to wait for everyone else to be finished for the review process to begin. I don't decide what goes into each page generally, but I do help students decide what should be there. Beyond that, I don't have to do any compilation myself.
  • Students (ideally) vote on a page if they think it meets the requirements. Students can each vote once for each page, and see a checkmark once they have voted. This gets them thinking about the quality of what they see in the work of other students. I have been largely impressed with what students have put together for this project, and students are being fairly generous with this. I'm ok with that at this point because of the next point:
  • Students have an incentive to actually visit each other's pages. I have no idea how many students actually use the review sheets we've produced together in the past. I doubt it is very many. There's some aspect of game theory involved here, but if a student sees that others are visiting his or her own pages, that student might feel more compelled to visit the pages of other students. Everyone benefits from seeing what everyone else is doing. If some review happens as a result, that's a major bonus. They love seeing the numbers adjust real time as votes come in. There is a requirement that each vote include a code that is embedded in the post they are voting for, just so someone isn't voting for them all without visiting the page.
  • Students were actually using the pages to review today. Students were answering each other's questions and getting feedback sometimes from the authors themselves.
  • I get to have valuable conversations about citing resources online.

Right now, students can vote as much as they want, but I plan to introduce one more voting option before this is entirely done which allows students to vote on their top three favorites in terms of usefulness. I am not sure how I would do this without it turning into a popularity contest, but I might try it and see how their sense of quality relates to mine. I would also love to use this next year as a Reddit style resource where students are posting problems and solutions potentially for specific standards and can vote on what is particularly helpful to them. Again, just an experiment.

I really loved how engaged students were today in either developing their pages or working on each other's review problems. It was one of the most productive review days I've had, particularly in light of the fact that I didn't have to write a single problem of my own. I did have to write the code, of course, but that was a lot more interesting to me today than thinking of interesting assessment items that I'd rather just put on an exam.

Testing probability theories with students

One of the things that has excited me after building computational tools for my students is using those tools to facilitate play. I really enjoyed, for example, doing Dan Meyer's money duck lesson with my 10th grade students as the opener for the probability unit. My experiences doing it weren't substantially different that what others have written about it, so I won't comment too much on that here.

The big thing that hampered the hook of the lesson (which motivated the need for knowing how to calculate expected value) was that about a third of the class took AP statistics this year, so they already knew how to do this. This knowledge spread quickly as the students taught the rest how to do it. It was a beautiful thing to watch.

I modified the sequel. I'll explain, but first some back story.

My students have been using a tool I created for them to sign up for reassessments. Since they are all logged in there, I can also use those unique logins to track pretty much anything else I am interested in doing with them.

After learning a bit about crypto currency a couple of months ago, I found myself on this site related to gambling Doge coins. Doge coins is a virtual currency that isn't in the news as much as Bitcoin and seems to have a more wholesome usage pattern since inception. What is interesting to me is not making money this way through speculation - that's the unfortunate downside of any attempt to develop virtual currency. What I've been amazed by is the multitude of sites dedicated to gambling this virtual currency away. People have fun getting this currency and playing with it. You can get Dogecoins for free from different online faucets that will just give them away, and then gamble them to try to get more.

Long story short, I created my own currency called WeinbergCash. I gave all of my students $100 of WeinbergCash (after making clear written and verbal disclaimers that there is no real world value to this currency). More on this later.

After the Money Duck lesson, I gave my students the following options with which to manage their new fortune in WeinbergCash:

Screen Shot 2014-05-21 at 5.17.26 PM

Then I waited.

After more than 3,000 clicks later, I had quite a bit of data to play with. I can see which wagers individual students are making. I can track the rise and fall of a user's balance over time. More importantly, I can notice the fact that just over 50% of the students are choosing the 4x option, 30% chose 2x, and the remaining 20% chose 3x. Is this related to knowledge about expected value? I haven't looked into it yet, but it's there. To foster discussion today, I threw up a sample of WeinbergCash balance graphs like this:

Screen Shot 2014-05-21 at 5.24.49 PM

Clearly most people are converging to the same result over time.

My interests in continuing this experiment are buzzing with two separate questions:

  • To what extent are students actually using expected value to play this game intelligently? If you make the calculations yourself, you might have an answer to this question. I haven't parsed the data yet to see the relationship between balances and grade level, but I will say that most students are closer to zero than they are their starting balance. How do I best use this to discuss probability, uncertainty, predictions, volatility?
  • To what extent do students assign value to this currency? I briefly posted a realtime list of WeinbergCash totals in the classroom when I first showed them this activity. Students saw this and scrambled to click their little hearts away hoping to see their ranking rise (though it usually did the opposite). Does one student see value in this number merely because it reflects their performance relative to others? Is it merely having something (even though it is value-less by definition) and wanting more of it, knowing that such a possibility is potentially a click away?

I had a few students ask this afternoon if I could give them more so they could continue to play. One proposed that I give them an allowance every week or every day. Another said there should be a way to trade reassessment credits for WeinbergCash (which I will never do, by the way). Clearly they have fun doing this. The perplexing parts of this for me is first, why, and second, how do I use this to push students toward mastery of learning objectives?

I keep the real-time list open during the day, so if students are doing it during any of their academic classes, I just deactivate them from the gambling system. For me, it was more of an experiment and a way to gather data. I'd like to use this data as a way to teach students some basic database queries for the purposes of calculating experimental probability and statistics about people's tendencies here. I think the potential for using this to generate conversation starters is pretty high, and definitely underutilized at this point. It might require a summer away from teaching duties to think about using this potential for good.

After a hiatus: circular functions

It has been a busy time in gealgerobophysiculus land. By land, I of course mean school, and by busy, I mean what results when you have multiple exciting projects going on, school functions to organize, and the normal operations of a classroom to sort through and organize.

I haven't taught the unit circle in three years. Before that, I took the approach of throwing a definition of the radian up on the board and discussing it as this strange thing that mathematicians decided would be a good idea. When I learned this in high school, we did some cool activities involving string and wrapping functions. At that time, it wasn't clear to me how the string wrapping around a circular object really related to measuring an angle around it. I was always relating the idea of the radian angle back to degrees, because the angle part never made sense.

After some thinking and coding, I put together an activity that I thought would make this concept more concrete for the students in my tenth grade class. You can check it out at

Screen Shot 2014-05-20 at 4.40.19 PM

It starts with the premise of moving around a circle at distances of integer multiples of the radius. Looking at your own work doesn't really establish how this relates to measuring angles at all. When you look at what happens when many people do the same thing to differently sized circles, the result makes clear that this could be a fairly natural way to measure out angles:
Screen Shot 2014-05-20 at 4.44.06 PM

I didn't have the networked part of this applet working when I did this with students, so I collected screenshots of students and their different circles together. I asked students what they expected would be different about the locations of these six points for circles of different sizes, and there was pretty solid agreement that they would be in roughly the same point around the circle, but this was still too abstract to establish the idea that these points measure out angles. The students weren't too surprised by the result, either, but I think the activity in this form still left me as the teacher to connect the dots.

I wish I had an extra day to configure the final screen of this activity. I wouldn't have had to work so hard.

The rest of the unit walked the line from this concrete idea of moving around the circle up the ladder of abstraction to what we ask students to typically do with these functions. We went from identifying points around the circle for a given angle measured in radians, to using our knowledge of 30-60-90 triangles to find the coordinates of some of these points, to formal definitions of sine, cosine, and tangent functions using these points. Every time I could, I related this idea back to the first activity of moving around the circle, but by the time we got to graphing these functions, I think I was demanding a high level of abstraction without also demanding the deliberate practice necessary to connect the angles and coordinates to each other. Students struggled to evaluate the trigonometric functions at different angles not because they couldn't piece it together with time, but because they always felt compelled to go all the way back to the circle. I suppose it's the trigonometric equivalent of going back to counting on your fingers.

I also was a bit disappointed to see that only a third of the class answered this question correctly on the unit exam:

Screen Shot 2014-05-20 at 4.57.41 PM

To those that recognized the similarity to our opening activity, it was quite easy. The bulk did not see it this way though.

At this stage, however, I'm not too concerned. Many students admitted immediately after the exam that they did not practice the unit circle as much as they should have. They reported that they understood much of the unit up to the graphing part, where I think I pushed them a bit more quickly to piece together the graphs than would have been ideal for them to get an intuitive sense for them. I'm confident that a second and more rigorous look at these functions next year in IB year one will help solidify some of these concepts for them.

Ratios & Proportions - Day One Antics

Yesterday was our first day into a unit on similarity with the ninth grade students.

The issue that comes up every year is that students like to cross multiply, but are incredibly mechanical in their understanding of why they do so. They don't like fractions that aren't simplified, and can usually simplify them well. They bring up the fact that multiplication of numerator and denominator by the same number is equivalent to multiplying by one. They seem to have very little understanding of how this relates to units and unit conversions as well.

I changed my approach this year to be much less review of how to solve proportions. I wanted to get at the aspects of measurement that are inherent to math problems involving similarity. I wanted to get them to ask themselves a bit more about why they took the steps they took in solving proportions in the process.

I started with a couple simple problems in the warm-up. Here was one:
Screen Shot 2014-03-08 at 2.16.05 PM

I took pictures of two students' work, put them side by side, and asked the class which one they thought was a better answer to the question:
Screen Shot 2014-03-08 at 2.17.40 PM

The resulting vote and conversation was especially spirited, particularly for a class that normally rejects whole class discussion. We talked about the ideas of approximate and exact answers, a couple of students pointing out that substitution of the approximate answers would result in a false statement in the equation.

After this, I showed them another picture and asked if the LEGO pieces in this picture would go together:
Screen Shot 2014-03-08 at 2.30.59 PM

Every hand went up.

I then showed them the bricks, which I had made on our school's new 3D printer:
Screen Shot 2014-03-08 at 2.33.47 PM

Pause for groans. Some key things were said in response to my 'playing dumb' question of why the two bricks won't fit together. One student even directly said that they looked similar to each other, but that they weren't the same size. I wanted them to have in the back of the heads that I was going to be pushing them to always think about figure with the same shape, different size.

We then made it to the second task of the warm-up activity. I asked them to estimate (and subsequently measure) the ratio of one of my heads in this image to the next:
Screen Shot 2014-03-08 at 2.45.32 PM

I developed the following points:

  • When communicating ratios to another person, begin explicit and clear about order is extremely important.
  • Despite the different units, these ratios are all communicating the same relationship from one head to the next. This relationship is even more obvious when we write the ratio as a fraction instead of using the colon notation.
  • The approximate values of this fraction are all roughly the same. We don't need to convert units either for this to happen - the units divide themselves out in the fraction.

I went on to define a proportion and reviewed the idea of cross products. They were a bit surprised when I showed them that cross products were equal for equivalent fractions. Part of this was because they saw me equate 2/5 and 4/10 and immediately said they were equal because one simplified into the other. I gave them 2/5 and 354453764/886359410 and they were a bit more willing to see that cross products can be a slicker way to check equality.

One more point that I made was that a proportion with a variable in it was really a question. If we are saying two fractions are equal to each other, and one (or more) of the fractions has a variable, what does that mean about the value of the variable? It led to a bit more conversation about the reasons for cross multiplication as a method of solving proportions, and I was satisfying then leaving students to work through some more review problems on their own.

The final piece we talked about whole group was this open ended question:
Screen Shot 2014-03-08 at 3.02.00 PM

They were able to come up with some, but struggled to make ratios that were more than simple multiples. This was surprising, as their mental calculation skills are generally quite strong. As shown in the example, I gave them one way to see how to come up with an arbitrary set of lengths that fit the requirement.

I then showed them this question:
Screen Shot 2014-03-08 at 3.05.31 PM

Some of the students realized (and explained eloquently) that they could divide the length by 7 and find the length of a single 'unit', and then multiple that unit by 3 or 4 to get the length. Explanations for why this worked didn't really materialize. I introduced the algebraic approach, and students saw it as an explanation, but seemed to be fine with just remembering it as a method rather than as a rationale.

The more that I teach proportions and similarity, the more I feel compelled to have students ground the concepts in measurements. Making measurements, especially by hand, is not something they typically do on a day-to-day basis, so there's a bit of a novelty factor there. These conversations about measurements, units, and fractions were authentic - there was a need to talk about these ideas in the context I established, and the students did a great job of feeling and then filling that need during the class. Nothing we did was a particularly real world task though. What made this real was my attempts to first frame the skills that we needed to review in the context of a need for those skills. I try to do this often, and I'd like to mark this as a success story.

Magnetic Fields, Data Collection, and You(r dog)

Assignment 1: Read the following abstract for a scientific paper.

Dogs are sensitive to small variations of the Earth’s magnetic field

Not too bad, right? Now read this more palatable explanation of what this really means:
Dogs align their bodies along a North-South axis....

Finally, here's a composite image I've put together:
Screen Shot 2014-02-16 at 8.09.41 AM

For all of these, the top of the phone was facing the same direction as my dog's nose during the act. These were all in different locations in the same yard, so it was clear to me that he wasn't just finding the same spot every time. It took copious treats and showing my dog the photos to convince him that I was not taking a picture of him while he did his business.

It's possible to get over the social peculiarity of remembering to pull out your phone, start the compass application, and take a screenshot whenever your dog pops a squat. To me, this seems like a ripe opportunity for a student project in statistics and data analysis. Furthermore, the potential for doing this now (compared to just a few years ago) is better than ever. Why?

  1. There are lots of dogs around, because dogs are awesome. They all have to unload at some point in the day. That makes for a large potential sample set of data to work with.
  2. It's all about number two, which is (of course) hilarious and engaging to all of us. I chuckled first when I read the headline of this story, and then a second time when I realized that scientists observed 1,893 defecation events and then sorted them according to magnetic field activity. I propose that this paper might include the term 'defecation events' more frequently than any other academic paper, and for that reason alone, it is special.
  3. Now more than ever, we have these great devices in our pockets that are not just capable of capturing such screenshots easily, but combine other useful information that might be important factors for students to consider. Time, date, geotagging information for location - all useful things students might choose to analyze in seeing if the results of this study are repeatable.
  4. Crowdsourcing. I took eight of these pictures, my wife took a few more. Imagine the potential of getting a photo stream of defecation event data for students to analyze from dogs around the world. Wolfram Alpha pegs the number of dogs in the US at 78.2 million.

Here's what I propose. If you're into this, take some data on the next outing with your dog. Some suggestions to maintain data integrity:

  • Stand behind the test subject and align the phone so that the top of your phone points in the direction your dog is looking while he/she concentrates on the task at hand.
  • Make sure your compass is calibrated when you take data.
  • Snap a screenshot of the compass screen. On iOS, hold the Home and Power buttons simultaneously, then release the power button. I found out from Lifehacker that in Android 4, you just hold down the power and volume-down buttons.
  • Now's the time to take a big (data) dump - upload those screenshots to Flickr, Instagram, etc. with the hashtag #DogsPoopNorth .

I don't teach statistics, but I'd love to see a class take a chunk of data and show that there is signal in the noise. The original researchers clearly showed this, but it's a great experience to have students do their own analysis work and come to their own conclusion about whether dogs have this unique ability or not.

Get to work, interwebs. I'm really interested to see what comes out.

What do you mean by 'play with the equation'?

My professional obsessions lately have focused on using technology to turn traditional processes of learning into more inquiry based ones. A really big part of this is purely in the presentation.

Here's the traditional sequence:

Screen Shot 2014-01-23 at 5.08.01 PM

  • Calculate the force between the Earth (mass 5.97 x 10^24 kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^6meters.
  • If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
  • Find at what distance from Earth's center the gravity force would be equal to 0.5 Newtons.

My observation in this type of sequence is that the weaker algebra students (or students that use the presence of mathematics as a way out of things) will turn off the moment you say calculate. The strong students will say 'this is easy', throw these in a calculator, and write down answers without units.

A modified presentation would involve showing students a spreadsheet that looks like this:
Screen Shot 2014-01-23 at 5.21.20 PM

  • Calculate the force between the Earth (mass 5.97 x 10^24 kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^6meters.
  • If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
  • Find at what distance from Earth's center the gravity force would be equal to 0.5 Newtons.

I would argue that this is, for many students, just as much of a black box as the first. In this form, however, students are compelled to tinker and experiment. Look for patterns. Figure out what to change and what to keep the same. In the last question, students will likely guess and check, which may get tedious if the question changes. This tedium might motivate another approach. A mathematically strong student might double click on the force cell and look up the function, or look up Newton's Law of Gravitation on Wikipedia and try to recreate the spreadsheet on his or her own. A weaker student might be able to play with the numbers and observe how doubling the mass doubles the force, and feel like he or she has a way to answer these questions, albeit inefficiently. Both students have a path to wrestle with the question that forms the basis of physics: how do we model what we observe in our universe?

This approach makes obvious what it means to play with a mathematical object such as an equation. Playing with an equation is something that I've admittedly said to students before in a purely algebraic context. I know that I mean to rearrange the equation and solve for the variable that a given question is asking for. Students don't typically think this way in math or science, or any equation that we give them. If they do, it's because we've artificially trained them to think that this is what experimentation in math looks like. I think that this is primarily because the user interface of math, which has been paper and pencil for thousands of years, doesn't lend itself to this sort of experimentation easily. Sure, the computer is a different interface, and has its own input language that is sometimes quite different from mathematical language itself, but I think students are better at managing this gap than we might give them credit for.

Moving in circles, broom ball, and Newton's cannonball

In physics today, we began our work in circular motion. I started by asking the class three questions:

  • When do you feel 'heaviest' on an elevator? When do you feel 'lightest'?
  • When do you feel 'heaviest' on an airplane? When do you feel 'lightest?'
  • When do you feel 'heaviest' on a swing? Lightest?

We discussed and shared ideas for a bit. I tried my hardest not to nudge anyone toward thinking they were right or wrong, as this was merely a test for intuition and experience. We then played a few rounds of circular 'book'-ball, a variation of the standard modeling curriculum activity of broom ball from the modeling curriculum in which students use a textbook to push a ball in a circular path on a table. The students could not touch the ball with anything other than a single book at one time. A couple of students quickly established themselves as the masters:

I then had students draw the ball in three configurations as well as the force and velocity vectors for the ball at those locations. Students figured the right configuration much more quickly than in previous years:
8C4384A1 - image I think some of our work emphasizing the perpendicular nature of the normal force on surfaces in previous units may have helped on this one.

We then took a look at some vertical circles and analyzed them using what we knew from the last unit on accelerated motion together with our new intuition about circular motion.

We finished the class playing with my most recent web-app, Newton's Cannonball. We haven't discussed orbits at all, but I wanted them to get an intuitive sense of the concept of how a projectile could theoretically go into orbit. This was the latest generation of my parabolas to orbits exploration concept.