Page 17 – iteration, making, building, and coding in education

Standards Based Grading, Year Two (Year-In-Review)

This was my second year using standards based grading with my classes. I wrote last year about how my first iteration went, and made some adjustments this year.

What did I do?

I continued using my 1-5 standard scale and scale rubric that I developed last year. This is also described in the post above.
As I wrote about in a previous post, I created an online reassessment organization tool that made it easier to have students sign up and organize their reassessments.
The new requirement for students signing up for reassessments involved credits, which students earned through doing homework, seeing me for tutoring
I included a number of projects that were assessed as project standards using the same 1-5 scale. the rubric for this scale was given to students along with the project description. Each project, like the regular course learning standards, could be resubmitted and reassessed after getting feedback and revising.

What worked:

My rate of reassessment was substantially better in the second semester. I tweeted out this graph of my reassessments over the course of the semester: There was a huge rush at the end of the semester to reassess – that was nothing new – but the rate was significantly more consistent throughout. The volume of reassessments was substantially higher. There were also fewer students than in the first semester that did not take advantage of reassessment opportunities. Certain students did make up a large proportion of the total set of reassessments, but this was nowhere near as skewed a distribution as in the first semester.
Students took advantage of the project standards to revise and resubmit their work. I gave a living proof project that required students to make a video in which they went through a geometric proof and explained the steps. Many students responded to my feedback about mathematical correctness, quality of their video, and re-recorded their video to receive a higher grade.
Student attitude about SBG was positive at the end of the year. Students knew that they could do to improve their grade. While I did have blank questions on some unit assessments, students seemed to be more likely to try and solve questions more frequently than in the past. This is purely a qualitative observation, so take that for what it is.

What needs work:

Students hoarded their reassessment credits. This is part of the reason the reassessment rush was so severe at the end of the semester. Students didn’t want to use their credits until they were sure they were ready, which meant that a number were unused by the end of the year. Even by the end of the year, more than a quarter of credits that had been earned weren’t used for reassessments. <p\> I don’t know if this means I need to make them expire, or that I need to be more aggressive in pursuing students to use the credits that they earned. I’m wrestling a lot with this as I reflect this summer.
I need to improve the system for assessing during the class period. I had students sign up for reassessments knowing that the last 15 – 20 minutes of the class period would be available for it, but not many took advantage of this. Some preferred to do this before or after school, but some students couldn’t reassess then because of transportation issues. I don’t want to unfairly advantage those who live near the school by the system.
I need to continue to improve my workflow for selecting and assigning reassessments. There is still some inefficiency in the time between seeing what students are assessing on and selecting a set of questions. I think part of this can be improved by asking students to report their current grade for a given standard when signing up. Some students want to demonstrate basic proficiency, while others are shooting for a 4 or 5, requiring questions that are a bit higher level. I also might combine my reassessment sign up web application and the quiz application so that I’m not switching between two browser windows in the process.
Students want to be able to sign up to meet with me to review a specific standard, not just be assessed on it. If students know specifically what they want to go over, and want some one-on-one time on it since they know that works well for them, I’m all for making that happen. This is an easy change to my current system.
Students should be able to provide feedback to me on how things are going for them. I want to create a simple system that lets students rate their comprehension on a scale of 1 – 5 for each class period. This lets students assess me and my teaching on a similar scale to what I use to assess them, and might yield good information to help me know how to plan for the next class.

I’ve had some great conversations with colleagues about the ways that standards based grading has changed my teaching for the better. I’m looking forward to continuing to refine my model next year. The hard part is deciding exactly what refinements to make. That’s what summer reflection and conversations with other teachers is all about, so let’s keep that going, folks.

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.

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:

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:

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:

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:

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 http://apps.evanweinberg.org/circlemeetsradius/

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:

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:

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.

Standards Based Grading: Bridging the Gap

I ended last semester feeling pretty discouraged about my implementation of standards based grading. I still felt pretty good about what it was doing for students. For the students that were actually taking advantage of the system and initiating reassessments, it seemed to be helping. Some examples:

Students were getting frequent assessment over the different standards of the course through quizzes, and got feedback on their learning the same day since the quizzes were short.
Some students understood that they could request reassessment on their own to show progress on standards that previously were not marked as proficient.
On a day that I gave one of these standards quizzes, I was guaranteed to have a conversation on content, albeit sometimes a short one, with every single student. I can’t guarantee that I get to every student, every period, though I’d like to think that I do.
Students seemed to be less frustrated or scared by unit exams, and I received (anecdotally) fewer blank problems since students had an incentive to try anything.

By the end of the semester, I had 248 student-initiated reassessments. I don’t know how that compares to others, but that number seemed to be an indication that something was going well. Then I looked at the numbers. Of those reassessments, eighty-eight of them were in the last week of the semester. Roughly 1/3 of the total were taken by the top five most frequent assessors.

I wasn’t getting to many of the students. Despite my efforts to constantly ask students to do so, talk about opportunities to reassess in class, and target students that needed extra help with suggestions to come see me after school, some students didn’t know that they could reassess. Some students didn’t make the time. In fact, most didn’t. I was also tolerating a fairly heavy end of semester reassessment rush that seemed higher than what I’d like to see happen.

The other concept that also bugged me was that in this system, I wasn’t placing value on those behaviors that would lead to higher standards scores. No, I wasn’t about to go back to a system of checking homework, and giving extra credit – the grades reported through standards were a pretty good measure of what students knew. On an end of semester survey, a majority of students confirmed this fact. Students also understood that reassessment without preparation wouldn’t result in increased scores. I had students that said, however, that doing homework, getting help after school, teaching concepts to other students – those things that result in improved understanding – weren’t accounted for in the grade. If I valued those as part of the process, why was I not assigning any value to them in the grade?

I spent a lot of time thinking about this over the winter break. In my system, there was a fundamental gap between the behaviors I wanted students to develop, and the way I was measuring the learning happening in my classroom. Students understand the system this way:

There is a visible wall between what students know I value and what they see measured in their grade. This is the philosophical basis of standards based grading – the grade is a measure of learning content, not necessarily of completion of activities. It is fundamentally different from other grading systems. We don’t see it this way as teachers, however.

We see it like this:

I have the conversation with students all the time about the difference between these two conceptions. They do understand. It just doesn’t fit with the rest of what they have been told in school, where learning sometimes takes second fiddle to being a student.

This is a communication design problem. After a lot of thinking and designing, I came up with a new computer based system using the Meteor JS framework that I’ve been using since January.

Some parts of it have helped bridge this gap (and other problems) in some ways that have been quite productive – I’ll go through them piece by piece.

Problem 1 – Google Forms are not reliable in China.

I previously used a Google form to collect reassessment sign-up data. Students would frequently (and with good reason) say that they couldn’t log on to sign up because of this issue. While there are still internet issues with the new site, this is less of a problem. They can log in with their own email address, their own secure password, and I can control the input/output of information much more reliably since it’s a database, not a spreadsheet.

Problem 2 – I don’t want students signing up for reassessments the same day.

I really want students planning ahead. The Google form didn’t have the option of limiting which days they could sign up for based on the time of sign up, at least I didn’t see how to do this easily. It’s true that I could simply see which students signed up from the Google form results spreadsheet, but that involved scrolling, searching, and mentally filtering results. That was more work than I wanted to be doing.

With some Javascript, I make it impossible to do this. Students log in, click on ‘Reassessment Sign-up’ and see this form:

Problem 3 – Students forget which standard they signed up to reassess on.

I received an email every time students signed up for a reassessment through the Google form. I never figured out a slick way to do this through forms – I’m sure it’s possible. With this system, students can click on ‘My Reassessments’ and see what they’ve signed up to do:

If they miss the bus and have to reschedule for later the same day, they can log in and edit the reassessment themselves. Because of Meteor’s realtime system, those changes instantly show on my browser when they happen.

Problem 4 – Students sign up to reassess without any preparation.

This was a big one I wanted to solve. On my previous form, there was just a check box that said “I promise to prepare for this reassessment” along with a box where they describe what they did to prepare. I wasn’t directly able to check this, and usually didn’t have time to do so beyond noting that it was more substantial than ‘I studied’.

When students log in and sign up for a reassessment now, they see a little badge called ‘Reassessment Credits’. Each reassessment costs them one credit. What happens when they try to sign up for a reassessment without any credits left?

They can’t do so. No way around it, at least without learning some Javascript to hack my system.

This is where I find I am doing the work to educate students in what I am looking for. I value the work that they do in preparing for reassessments. I show that value through giving students credits, a token for whatever it is they choose to do to improve their understanding. These credits then directly convert into opportunities to show what they have learned through reassessment. I have the flexibility to give credits for different things that students do – tutoring other students, sitting in the classroom and doing homework with me, or simply showing me the homework they did after the previous class. This value is separate from the grade. Students have also found that through doing the work to earn a credit, I have directly increased their standards grade because I’ve seen evidence of higher understanding there.

Problem 5 – Students don’t know that their peers are reassessing.

I’m not ashamed to say that I also wanted to call upon the fact that students (and all of us) are influenced to some extent by our social community. I added one more part to the page that some students have reported has pushed them to sign up and complete reassessments.

When I first added this feature, a student said she was disappointed when she saw her proportion of the total decreasing, so she felt compelled to figure out what she might reassess over based on her progress.

Problem 6 – Staying organized with reassessments from four different classes is difficult

This system is computerized, so it’s easy to sort which reassessments I have on a given day.

Rather than skimming a spreadsheet and figuring out which quizzes to prepare, I have an easy list that allows me to prioritize and plan which quizzes I need to have available for which part of the day.

I’ve taken a lot of the obstacles out of the way that have made it stressful for me to stay organized in reassessments, and for students to opt not to do so. I am not trying to make a point about quantity vs. quality here, but I’m really pleased with how the system has worked for students.

So far this semester, I’ve had 221 reassessments through the system at this point in the semester, and we’re only in the first quarter. Thirty seven out of 55 students have assessed multiple times. Eleven students thus far have not reassessed at all, and I’ve seen that information easily through some database queries, which has prompted some conversations with those students about why they haven’t. It’s far easier than it used to be to get this information out.

Students also have great ideas for making it better. They know themselves, and know what would help them stay on top of their needs. I enjoy the challenge of learning to build this for them, and they seem to appreciate the effort.

I am not sharing the URL quite yet since there is student user information in there, and haven’t built a nice sandbox login yet to see the features. That will come when I have the time, whenever that is.

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:

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:

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:

Every hand went up.

I then showed them the bricks, which I had made on our school’s new 3D printer:

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:

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:

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:

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:

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?

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.
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.
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.
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:

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:

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.

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