I wrote nearly a year ago about my adjustment to what I had done previously to develop the topic. The idea was based on what my own pre-Calculus teacher did in high school, a series of activities related to a ‘wrapping function’ moving around the unit circle. This lesson is for a group of Algebra 2 level students that will likely move into the IB program for next year. Mastery of trigonometric functions isn’t necessary, but I do want students to feel comfortable converting between radians and degrees, locating angles on the unit circle, and evaluating trigonometric functions.
In the last class, we talked about 30-60-90 and 45-45-90 triangles and the fact that we can evaluate trigonometric functions exactly using our knowledge of ratios and the Pythagorean theorem. We also did a series of exercises having students locate angles on the unit circle during the last class.
Today’s warm-up was a continuation of these ideas through these sets of questions:
Normally at this stage, I show a development using similar triangles of finding what these coordinates are. Though I bring up this goal in a number of different ways, whether students are doing this at their seats, or I’m doing it for them, I can never the sense of understanding that I want. This development is also not what I want them to do when they are evaluating trigonometric functions either – I want them to figure out where they are on the unit circle, and then evaluate based on the x and y-coordinates of the point.
Today I made a subtle change to my sequence. I directly told students that the coordinates of these points were some combination of a set of five lengths. Two of these lengths we found in a previous lesson, but I never made a connection to it here. I asked them to put the numbers in order from least to greatest
Then I asked them to complete the coordinates in this blank unit circle. Here’s a student’s work, corrected by a classmate when it was shared:
All of the conversations about sign and value that I had to force previously happened naturally this time. The handout was folded so that as students finished, I could then nudge them into the next step of finding angles that match to particular coordinates, an exercise on the other side.
For most of the students, this wasn’t a problem. Some even looked like they were enjoying it.
It was only in the last few minutes of the class that I introduced the sine, cosine, and tangent as a shorthand way of asking the question of finding the x-coordinate, y-coordinate, or the ratio of the two. My students are pretty trusting, but they have also become used to asking why [Statement A] is true once they have the basic idea of what [Statement A] means. This lesson was just a continuation of this process. Almost every student was able to evaluate a cosine function of a different angle during the exit activity.
I felt a little bad about giving the coordinates and putting off the understanding to later. This short bit of mathematical fact, however, was followed immediately by a task that required them to reason about what they mean. It builds the need to show why those coordinates are what they are, and this process of looking at 45-45-90 and 30-60-90 triangles on the unit circle will make much more sense in the context of the student experiences here.
One student summed up my motivation for doing this beautifully as she was packing up – I love that I’m not making this quote up:
It’s good that you don’t have to memorize it because you can just see the picture in your head and know what the answer is.
I decided to apply for the Apple Distinguished Educator program this year. The primary reason is that the various ways I work toward my classroom goals tend to involve my use of their products. Their design aesthetic has had a strong influence on my own design tendencies as I create materials for the classroom, digital or not.
Update:
I was not selected for this year’s group. In hindsight, it’s possible that my use of technology is platform independent enough that I don’t really need Apple to do what I do. Oh well, maybe next time!
The process of reflection is always valuable. If nothing else, my application stands as a pretty straightforward summary of my ed-tech philosophy these days.
Here is my application video, and my answers to the questions:
How have you as an educator transformed your learning environment?
My major realization about technology in the classroom is that single-purpose devices are quickly losing their value. An iPhone in my pocket is simultaneously a document camera, graphing calculator, and assessment tool. My MacBook is a content recording studio, interactive whiteboard, and software development center. Student MacBooks combine authoring tools, answer manuals, problem generators, and nodes of an instant communication network in my classroom. All of us have access to the same tools; there is no way that I as a teacher am doing any sleight of hand. My students can learn to do what I do, make what I make, and then make completely new things on their own.
In contrast, when I first started teaching, I had a number of useful (but single purpose) technological tools at my disposal: an interactive whiteboard, graphing calculators that networked together, and document cameras. My approach to integrating these tools into my lessons was to ask myself how I could use them to enhance my presentation of content to students.
When my wife and I decided to move overseas to teach, it was to my current school which had a 1:1 MacBook program for the students I would be teaching. It felt awkward standing at the front of a classroom in front of desks of students behind screens. I was asking students in a whole class setting what they observed while I clicked through a program on an interactive whiteboard. The students had their own laptops in front of them – they should be the ones to be clicking, tapping, and sliding mathematical objects on screen. They could be making observations, drawing conclusions, and building intuition for what we were learning based on their experiences. No matter how good my direct instruction might be, students would be better served by spending more time actively working together.
This has since become the new ideal for my classroom. I do not start with the technology, and then decide what I could do with it to make my teaching better. I start by asking myself what I want my classroom environment to be, how I want students to interact, and what students should do there in order to learn. Technology then serves to help me build that classroom. My planning time consists of making or searching for tools that let students construct knowledge themselves. When direct instruction seems necessary to help students learn, I work to reduce it to its essential elements. I have recorded videos of content that students watch during class. This frees me to circulate amongst the students and listen to the conversations students have with each other.
Technology helps maximize the quality of social interaction between students and me in the classroom. It helps minimize the time spent collecting student answers and responses in one place, which then maximizes the time we can all spend discussing and analyzing that work. It provides structure to keep me and my students organized, which maximizes the brain space available to manage abstract thinking in mathematics and physics. It reduces the clerical work associated with selecting questions for a quiz or making copies, and instead moves students and me quickly to the point where we can have crucial conversations about learning.
Illustrate how Apple technologies have helped in this transformation.
The simplest shift came from unplugging my MacBook from the projector screen. I can sit anywhere in the classroom and project notes, problems, and student ideas wirelessly through an AppleTV using AirPlay. I use a USB tablet and stylus to make handwritten notes during class. I use the same set up to record short instructional videos and share them with students for use during class, or when they are on their own.
There are many applications and online tools that exist to make it easy to collate responses in a classroom, make collaborative documents, and share images. The reality of accessing these tools through Chinese internet filters makes use of these applications is unreliable and difficult, if not impossible. The features of these tools, however, would be valuable for helping create the learning environment I want for my students. I have learned to use Python and JavaScript to build tools with some of these features for my classroom. I host these applications on my MacBook and students access them through Safari over the school network.
I created a web based application that allows me to take a picture of student work with my iPhone, and then upload the file directly to a folder on my computer. We can then flip through different responses using Preview and discuss the content as a class. Students can also share images of their work using their phones or computers, anonymously or not.
I have implemented standards based grading for almost all of my courses so that students have multiple opportunities to demonstrate mastery of what they have learned. I wrote another application that sends individualized quizzes on specific learning standards to students through a web page, also hosted on my laptop. Students can access their individual quiz site through whichever device they have availiable. I experimented with the Meteor JavaScript framework and built a site that lets students sign up for these quizzes at any time, from anywhere.
I let the technology handle the collecting, organizing, displaying, and calculating, as these are what computers do best. As a result, the valuable but limited time that I have with my students can be spent learning to do the thinking and develop the skills that are uniquely human, and that will be necessary long after students leave my classroom. The versatility of the tools that Apple provides makes that process possible.
What successes have you seen with your learners?
I survey my students frequently on what is or is not working well in the classroom. Listening to me talk and go through problems, though it is easiest for me in terms of planning, is consistently at the bottom of student preferences. The more student-centered methods are, by far, the most effective and preferred methods for students to learn in my classes. My presence in the classroom is most valuable when spent moving from student to student, listening to conversations, and asking questions based on my assessment of their comprehension level. In the lessons that involve my recorded videos, the ELL students appreciate being able to pause the videos and switch their focus between the concepts being taught and the language. The more advanced students often start with the assigned problems, and then work backwards with the video content when they need to get unstuck in solving a problem. I can monitor how students are engaging with these videos through written notes and solving problems, and can provide assistance on an individual basis.
Many of the students in my classes are used to rote instruction, as this is what they experience in schools in their home countries. My use of technology as a tool for investigation, and emphasis on sharing student ideas to develop understanding, helps reduce the belief that memorization and obtaining answers are the primary goals in mathematics and science. My students understand that there are many tools available to help them arrive at an answer. They use one tool to verify the results of another.
I have had excellent results with students in my AP Calculus and AP Physics courses over the past five years. I attribute much of this success to the positive learning habits that students have developed through my classes. Students know how to get unstuck. They know how to use each other’s presence in the classroom to build on their understanding.
The best feedback on my teaching often comes from students that are no longer in my classroom. One student from last year’s physics class was often frustrated that I would not generally not lecture on how to solve every type of problem. Here is an excerpt from an email I received from this student earlier this year:
“…I am very happy that you made me struggle with physics last year because now when I don’t see how to solve a problem immediately, I know how to use the tools available to me to experiment to find the right answer. ”
I often wonder if I am doing what is best for my students. Comments like this one lead me to believe that I am moving in the right direction.
How do you share these successes to influence the broader education community?
When I first moved abroad, I left a large department of teachers to be a member of a one person team at my current school. While this team has since grown to include amazing collaborators, I get a lot of my best ideas and encouragement from teachers that I have never met in person. They push back when I think I have everything figured out, and never let me stop tweaking a lesson to be its best. I am in communication with this network of teachers from around the world regularly through Twitter, blogs, and email. Many of these teachers are already in the ADE community, and their feedback was important in deciding to apply to the program myself.
Any time I have an experience in the classroom, successful or not, I turn to my online community. It has been important to share the good ideas, but it is increasingly more beneficial to also share uncertainty. I blog whenever possible at my website about my experiences with students. When an activity has materials that can be shared in their raw form, I make these materials available on my website. Otherwise, I include enough details that teachers that want to imitate what I have done can do so with minimal effort. When computer code is involved, I share it through Github or other online repositories.
I have presented at conferences in my region about my use of technology for teaching. This includes the EARCOS Teachers Conference in Bangkok, the 21st Century Learning conference in Hong Kong, and Learning 2.0. On my personal website, I post videos of these workshops and presentations so that anyone can benefit from what I have to share. I also have presented to my colleagues about mathematics, technology, and assessment.
These experiences have led to invitations to join online communities for teacher education. I have collaborated with leaders in mathematics education to build online learning experiences for students around the world. I have spoken to online groups such as the Global Math Department, Global Physics Department, and a Google Hangout on computational thinking.
In short, I am eager to share my ideas and learning with others. Doing so helps me develop as a teacher and stay active as a learner, which also lets me model life long learning for my students.
After learning from Jessica Murk before our spring break about the idea of revising mathematical writing in class, I decided to try it as part of an introduction to the fourth topic in the IB Mathematics curriculum: vectors. The goal was to build a need for the information given by vectors and how they provide mathematical structure in a productive way.
I asked all students to pick one dot, and then asked a student to give the class instructions on which one they picked. They did a pretty good job with it, but there was quite a bit of ambiguity in their verbal descriptions, as I wanted. This is when I sprung Dan’s helpful second slide that made this process much easier:
Key Point #1: A common language or vocabulary makes it easy for us to communicate our ideas.
I then moved on to the next task. Students individually had to write directions for moving from the red dot to the blue dot. I gave them this one to start as a verbal task, but nobody was willing to take the bait after the last activity:
Fair enough.
I then gave one of the following images to each pairs of students, with nothing more than the same instruction to write directions from the red to the blue dot.
Here is a sampling:
Move across 5 dots on the outermost layer counter-clockwise, with the blue dot at the bottom of paper (closest to you)
Move 7 units to right, and move about (little less) 3 units up so that the blue dot is right on the vertical line
Fin the dot that is directly opposite to the red dot that is across the diagram. Once there, move down one dot along the outermost layer of dots.
Stay on the circle and move right for five units
Move from coordinate $latex \frac{7 \pi}{6}$ to the coordinate of $latex 2 \pi$ on the unit circle.
Then, without any input from me, I had students sit down and each write a new description. Just as Jessica promised, the descriptions were improved after students saw the work of others and focused on what it means to give specific and unambiguous directions.
This is where I hijacked the results for my own purposes. I asked how the background information I gave helped in this task? They responded with:
Grid/coordinate system in background of the dots
Circle connecting dots – use directions and circles to explain how to move
Connected all dots – move certain number of ‘units’
One student also provided a useful statement that the best description was one that could not be misinterpreted. I identified the blue dot as (3,0), and asked if anyone could give coordinates for the red dot. Nobody could. One student asked where (0,0) was. I pointed to some other points as examples, and eventually a student identified the red dot as (3,8). Another said it could also be (3,-5). I pointed out that if I had asked students to plot (3,-5) at the beginning of the class, the answer would have been totally different.
This all got us to think about what information is important about coordinates, what they tell us, and that if we agree on common units and a starting point, the rest can be interpreted from there. This was a perfect place to introduce the concept of unit vectors.
We certainly spent some time wandering in the weeds, but this ended up being a really fun way to approach the new unit.
If you are interested, here is the PDF containing all of the slides: Point Circle
During the IB Exams, students get a set of equations and constants to use. Part of the motivation behind them is to reduce the amount of memorization required. There’s no sense in students memorizing Planck’s constant or the Law of Cosines in a context that emphasizes application of these ideas.
That said, I’ve heard variations on the following from different students just in the past three days:
I thought I was right, then I looked at the formula sheet, and realized I was wrong. (She was right the first time.)
I didn’t study it because I knew it was on the formula sheet.
I don’t know what formula to use.
If you read my blog, you know that I don’t test formula memorization for all sorts of reasons. You get it. I get it. It has a place, but that place isn’t one I want to be spending my time.
You might also know that I’ve experimented with different versions of resources available to students during a test. I’ve done open note-card, open A4 sheet, open A5 sheet, open computer/closed network, open computer/open network, open notebook, and open people (i.e. a group test) formats.
I believe that the act of students creating their own formula sheets is more effective than handing one to them. The process of seeing how a formula is applied in different contexts and deciding what needs to be remembered is valuable on its own. Identifying that one problem is similar to another for reasons of physics shows understanding. I want to make opportunities for that to happen. Reducing the size of the resource requires students to prioritize. These are all high level skills.
The difficulty is that students see formulas directly as a pathway from problem to solution. Most problems worth solving don’t fit with that level of simplicity. Formula sheets give you the factual information, and rely on the user to know how to connect that information to a problem. The student thinks that the answer is staring at them in the face, and they just have to pick the right one. As teachers, we want students to identify information they need, then look at the reference to get it.
This is part of the reason I like standards based grading, as it justifies assessing students through conversation. A student asks me for a specific piece of information. If it’s how to calculate something, I’ll tell them if the related learning standard is about applying a concept, not calculating a quantity. If their request directly asks for the answer to the question, I don’t tell them. If they ask for a hint, I give them enough to get them moving, and adjust their proficiency level for the related standard according to the amount of help I give them.
In the long run, however, students need to know how to use the resources available to them. This is one of those big picture skills everyone talks about. Students need to know how to use Google to effectively find what they are looking for. They need to know that typing the text of a question into Yahoo Answers is not going to get them the answer they are looking for. I do know that if a student directly says “I can’t remember a formula for [ ]”, and I give them an equation sheet, they can usually find it. If they use the formula sheet as step one, they are not likely to complete the problem on their own. Having the sheet there in front of them makes it far too easy to start a problem that way. Would having students tally the number of times they looked at their sheet be enough of a feedback mechanism to keep this in check?
I don’t know what the answer is right now.
How do you help students treat a formula sheet more like a tool box, and less like a restaurant take-out menu?
In my Math 10 class, did my lesson today involving solving exponential equations that cannot be solved using knowledge of integral powers. My start was the same as it has been for that lesson over many years:
I have students start with an iterative guess-and-check method since it’s something that will pretty much always work. This was no big deal to the students. When one student said her TI calculator gave the exact answer, I asked if she really thought that was the exact answer. She said no, but I used Python to rub it in a bit.
This was another opportunity to show the difference between exact and approximate answers – always something I try to teach implicitly whenever it comes up. As with many of the Common Core Standards for Mathematical Practice, I think this (MP6 – Attend to Precision) is always an idea that comes with context.
The big shift in this lesson came when we started solving the equation algebraically. I always do a bit of hand-waving at this point saying ‘isn’t it great that these logarithm properties let us do this?’, while getting a class full of students giving me just enough of a sarcastic head nod to make me feel bad about it.
Instead, I made reference to the process of switching back and forth from logarithmic and exponential form.
The students are pretty skilled at doing this. I wrote it up in the notes myself because most students wrote it faster than I could get anyone to explain the process.
The key here was that when I asked students to calculate these values on the calculator, nobody could do it. One found the LOGBASE command on their TI, but for the most part, this stayed as an abstract number. It made sense to them that they ended up with ‘x =’ in the end, but that didn’t make a big difference in terms of being able to talk about what that meant. They did a couple of these on their own.
Only then did I show them the logarithm property trick that lets us get the answer in a different form:
I admittedly connected some dots here, but I didn’t do so in a formal way of introducing change of base. A couple of them figured out that this was a form that they could calculate using the common logarithm button on their calculators.
I’m not emphasizing log properties this year outside of what they allow us to do in solving equations. This is something that we will devote more time to next year in IB Mathematics year 1 class. I will mention this change of base property as a nice tool to use for confirming graphical and iterative solutions, but probably won’t assess them knowing how to apply change of base directly.
Any time I can get rid of hand-waving and showing mathematics as a list of tricks to be memorized, it’s a win.
The relatively recent emergence of the Internet, and the ever-increasing ease of access to web, has unmistakably usurped the teacher from the former role as dictator of subject content. These days, teachers are expected to concentrate on the “facilitation” of factual knowledge that is suddenly widely accessible.
This line of reasoning inevitably comes up in my conversations with those that don’t teach, including those that have children currently in the system. What is the role of the teacher in today’s classroom?
My response usually pays lip-service to the idea that the role of teachers is certainly changing in response to the presence of technology. I think it’s obvious that is the case. I don’t believe that most of us are turning our classrooms into rows of students doing computerized lessons because of their effectiveness – that certainly isn’t he case either. My arguments for there being a place for teachers in the classroom surround the social situation that exists in having learners together in one place. In the best classrooms, historically, it has never really been about transferring knowledge from the front to the back. It has instead always been about the community.
Here are my main ideas on this concept:
Making the social network of the classroom into a learning resource requires careful planning and experience in managing the process.
Students need to learn that it is normal to make mistakes along the road to understanding. This isn’t easy when done in isolation.
Making big picture connections is done best in conversation with others having a diversity of experiences and understandings.
Some skills are learned best in context with someone knowledgable in their use.
Asking a question of a source you know and trust is easier than taking a shot in the dark on an online forum or through a chat window.
I’m not saying these processes can’t be completed online. Our students certainly have experience communicating through online channels. They need our guidance as teachers in using these networks for learning, however, and the classroom is a great place to give them that guidance. In light of the social, emotional, and finally academic needs of teenagers, I think we will be needed for a while yet before computers can fully take over the classroom for good.
The seniors completed their final presentations this week. This was a series of TED style talks on subjects ranging from 3D printing and product placement to the connections between meat and cancer and the lack of women in foreign policy. I’ve been really pleased with how this group has developed their skills in communicating ideas, both through writing last semester, and in visual communication more recently.
We still have a couple of months left in the year, so when the seniors and I got back together for one class before spring break, they wanted to know what we were going to do with the time left. This point of the year for seniors, more so than other times, has a consistent theme of time ticking down in all sorts of ways. They keep an accurate count of the number of days left in school on a small chalkboard in the lounge. They keep track of college acceptances on a big map there as well. Keeping them in the present is much more easily said than done, so I tend to push seniors to think through big picture stuff at this stage.
So when we sat down in class this past week, I had rearranged the tables into a big family style U-shape to make. Lear that something would be ‘different’ from that point forward. I talked to them about my history in education. I described different schools I went to, how they nudged my personal path one way or another. I then showed them two talks, one from Ken Robinson about the learning revolution, and the other from Shawn Cornally describing the Iowa BIG school.
My questions after both of these were simple:
In what ways are you who you are because of your school experience?
In what ways are you who you are in spite of your school experience?
We had a brief conversation about this, and students had really insightful and revealing comments about it. I didn’t want to give a big assignment or written reflection for the break though. This is family vacation time, and I didn’t feel the need – plus my plans for the next steps are still in the formative stages. I did want to get seniors at least thinking big picture about the role of school as part of their identity. One senior said on the way out: “pretty deep for the day before spring break, Weinberg.” For someone who thinks about education as much as I (and most teachers I know) do, this type of question is the norm.
I spent the day in a room full of my colleagues as part of our school’s official transition to using the Common Core standards for mathematics. While I’ve kept up to date with the development of CCSS and the roll-out from here in China, it was helpful to have some in-person explanation of the details from some experts who have been part of it in the US. Our guests were Dr. Patrick Callahan from the Illustrative Mathematics group and Jessica Balli, who is currently teaching and consulting in Northern California.
The presentation focused on three key areas. The first focused on modeling and Fermi problems. I’ve written previously about my experiences with the modeling cycle as part of the mathematical practice standards, so this element of the presentation was mainly review. Needless to say, however, SMP4 (Model with mathematics) is my favorite, so I love anything that generates conversation about it.
That said, one element of Jessica’s modeling practice struck me by surprise, particularly given my enthusiasm for Dan Meyer’s three-act framework. She writes about the details on her blog here, so go there for the long form. When she begins her school year with modeling activities, she leaves out Act 3.. Why?
Here’s Jessica talking about the end of the modeling task:
Before excusing them for the day, I had a student raise their hand and ask, “So, what’s the answer?” With all eyes on me, a quick shrug of my shoulders communicated to them that that was not my priority, and I was sticking to it (and, oh, by the way, I have no idea what time it will be fully charged). Some students left irritated, but overall, I think the students understood that this was not going to be a typical math class.
Mission accomplished.
Her whole goal is to break students of the ‘answer-getting’ mentality and focus on process. This is something we all try to do, but perhaps pay it more lip-service than we think by filling that need for Act 3. Something to consider for the future.
The other two elements, also mostly based in Jessica’s teaching, went even further in developing other student skills.
I had never head of Bongard problems before Jessica introduced us to them. This involves looking at well defined sets of six examples and non-examples, and then writing a rule that describes each one.
Here’s an example: Bongard Problem, #1:
You can find the rest of Bongard’s original problems here.
In Jessica’s class, students share their written rules with classmates, get feedback, and then revise their rules based only on that feedback. Before today’s session, if I were to do this, I would eventually get the class together and write an example rule with the whole class as an example. I’m probably doing my students the disservice by taking that short-cut, however, because Jessica doesn’t do this. She relies on students to do the work of piecing together a solid rule that works in the end. She has a nicely scaffolded template to help students with this process, and spends a solid amount of time helping students understand what good feedback looks like. Though she helps them with vocabulary from time to time, she leaves it to the students to help each other.
Dr. Callahan also pointed out the importance of explicitly requiring students to write down their rules, not just talk about them. In his words, this forces students to focus on clarity to communicate that understanding.
The final piece took the idea of peer feedback to the next level with another template for helping students workshop their explanations of process. This should not be a series of sentences about procedure, but instead mathematical reasoning. The full post deserves a read to find out the details, because it sounds engaging and effective:
I want to focus on one highlight of the post that notes the student centered nature of this process:
I returned the papers to their original authors to read through the feedback and revise their arguments. Because I only had one paper per pair receive feedback, I had students work as pairs to brainstorm the best way to revise the original argument. Then, as individuals, students filled in the last part of the template on their own paper. Even if their argument did not receive any feedback, I thought that students had seen enough examples that would help them revise what they had originally written.
I’ve written about this fact before, but I have trouble staying out of student conversations. Making this written might be an effective way for me to provide verbal mathematical details (as Jessica said she needs to do periodically) but otherwise keep the focus on students going through the revision process themselves.
Overall, it was a great set of activities to get us thinking about SMP3 (Construct viable arguments and critique the reasoning of others) and attending to precision of ideas through use of mathematics. I’m glad to have a few days of rest ahead to let this all sink in before planning the last couple of months of the school year.
I created an interactive lesson called Thinking Machine for use with a talk I gave to the IB theory of knowledge class, which is currently on a unit studying mathematics.
The lesson made good use of the Meteor Blaze library as well as the Desmos Graphing Calculator API. Big thanks to Eli and Jason from Desmos for helping me with putting it together.
I was asked by a colleague if I was interested in speaking to the IB theory of knowledge class during the mathematics unit. I barely let him finish his request before I started talking about what I was interested in sharing with them.
If you read this blog, you know that I’m fascinated by the intersection of computers and mathematical thinking. If you don’t, now you do. More specifically, I spend a great deal of time contemplating the connections between mathematics and programming. I believe that computers can serve as a stepping stone between students understanding of arithmetic and the abstract idea of a variable.
The fact that computers do precisely what their programmers make them do is a good thing. We can forget this easily, however, in our world that has computers doing fairly sophisticated things behind the scenes. The fact that Siri can understand what we say, and then do what we ask, is impressive. The extent to which the computer knows what it is doing is up for debate. It’s pretty hard to argue though that computers aren’t doing similar types of reasoning processes that humans do in going about their day.
Here’s what I did with the class:
I began by talking about myself as a mathematical thinker. Contrary to what many of them might think, I don’t spend my time going around the world looking for equations to solve. I don’t seek out calculations for fun. In fact, I actively dislike making calculations. What I really enjoy is finding interesting problems to solve. I get a great deal of satisfaction and a greater understanding of the world through doing so.
What does this process involve? I make observations of the world. I look for situations, ideas, and images that interest me. I ask questions about what I see, and then use my understanding of the world, including knowledge in the realm of mathematics, to construct possible answers. As a mathematical and scientific thinker, this process of gathering evidence, making predictions using a model, testing them, and then adjusting those models is in my blood.
I then set the students loose to do an activity I created called Thinking Machine. I styled it after the amazing lessons that the Desmos team puts together, and used their tools to create it. More on that later. Check it out, and come back when you’re done.
The activity begins with a first step asks students make a prediction of a mathematical rule created by the computer. The rule is never complicated – always a linear function. When the student enters the correct rule, the computer says to move on.
The next step is to turn the tables on the student – the computer will guess a rule (limited to linear, quadratic, cubic, or exponential functions) based on three sets of inputs and outputs that the student provides. Beyond those three inputs, the student should only answer ‘yes’ or ‘no’ to the guesses that the computer provides.
The computer learns by adjusting its model based on the responses. Once the certainty is above a certain level, the computer gives its guess of the rule, and shows the process it went through of using the student’s feedback to make its decision. When I did this with the class, more than half of the class had their guesses correctly determined. I’ve since tweaked this to make it more reliable.
After this, we had a discussion about whether or not the computer was thinking. We talked about what it means for a computer to have knowledge of a problem at hand. Where did that knowledge come from? How does it know what is true, and what is not? How does this relate to learning mathematics? What elements of thinking are distinctly human? Creativity came up a couple times as being one of these elements.
This was a perfect segue to this video about the IBM computer Watson learning to be a chef:
Few were able to really explain this away as being uncreative, but they weren’t willing to claim that Watson was thinking here.
Another example was this video from the Google Deep Thinking lab:
I finished by leading a conversation about data collection and what it signifies. We talked about some basic concepts of machine learning, learning sets, and some basic ideas about how this compared to humans learning and thinking. One of my closing points was that one’s experience is a data set that the brain uses to make decisions. If computers are able to use data in a similar way, it’s hard to argue that they aren’t thinking in some way.
Students had some great comments questions along the way. One asked if I thought we were approaching the singularity. It was a lot of fun to get the students thinking this way, especially in a different context than in my IB Math and Physics classes. Building this also has me thinking about other projects for the future. There is no need to invent a graphing library on your own, especially for use in an activity used with students – Desmos definitely has it all covered.
Technical Details
I built Thinking Machine using Bootstrap, the Meteor Blaze template engine, jQuery, and the Desmos API. I’m especially thankful to Eli Luberoff and Jason Merrill from Desmos who helped me with using the features. I used the APIto do two things:
Parse the user’s rule and check it against the computer’s rule using some test values
Graph the user’s input and output data, perform regressions, and give the regression parameters
The whole process of using Desmos here was pretty smooth, and is just one more reason why they rock.
The learning algorithm is fairly simple. As described (though much more briefly) in the activity, the algorithm first assumes that the four regressions of the data are equally likely in an array called isThisRight. When the user clicks ‘yes’ for a given input and output, the weighting factor in the associated element of the array is doubled, and then the array is normalized so that the probabilities add to 1.
The selected input/output is replaced by a prediction from a model that is selected according to the weights of the four models – higher weights mean a model is more likely to be selected. For example, if the quadratic model is higher than the other three, a prediction from the quadratic model is more likely to be added to the list of four. This is why the guesses for a given model appear more frequently when it has been given a ‘yes’ response.
Initially I felt that asking the user for three inputs was a bit cheap. It only takes two points to define a line or an exponential regression, and three for a quadratic regression. I could have written a big switch statement to check if data was linear or exponential, and then quadratic, and then say it had to then be cubic. I wanted to actually give a learning algorithm a try and see if it could figure out the regression without my programming in that logic directly. In the end, the algorithm works reasonable well, including in cases where you make a mistake, or you give two repeated inputs. With only two distinct points, the program is able to eventually figure out the exponential and quadratic, though cubic rules give it trouble. In the end, the prediction of the rule is probability based, which is what I was looking for.
The progress bar is obviously fake, but I wanted something in there to make it look like the computer was thinking. I can’t find the article now, but I recall reading somewhere that if a computer is able to respond too quickly to a person’s query, there’s a perception that the results aren’t legitimate. Someone help me with this citation, please.
Last fall, when I was teaching my web design students about jQuery events, I included an example page that counted the number of times a button was clicked and displayed the total. As a clear indicator of their strong engagement in what I asked them to do next, my students competed with each other to see who could click the most number of clicks in a given time period. With the popularity of pointless games like Cookie Clicker , I knew there had to be something there to use toward an end that served my teaching.
Shortly afterwards, I made a three-act video activity that used this concept – you can get it yourself here.
This was how I started a new unit on exponential functions with my Math 10 class this week. The previous unit was about polynomials, and had polynomial regression for modeling through Geogebra as a major component. One group went straight to Geogebra to solve this problem to figure out how many clicks. For the rest, the solutions were analog. Here’s a sample:
When we watched the answer video, there was a lot of discouragement about how nobody had the correct answer. I used this as an opportunity to revisit the idea of mathematics as a set of different models. Polynomial models, no matter what we do to them, just don’t account for everything out there in the universe. There was a really neat interchange between two students sitting next to each other, one who added 20 each time, and another who multiplied by 20 each time. Without having to push too much, these students reasoned that the multiplication case resulted in a very different looking set of data.
This activity was a perfect segue into exponential functions, the most uncontrived I think I’ve set up in years. It was based, however, on a useless game with no real world connections or applications aside from other also useless games. No multiplying bacteria or rabbits, no schemes of getting double the number of pennies for a month.
I put this down as another example of how relevance and real world don’t necessarily go hand in hand when it comes to classroom engagement.
