The museum in your classroom – exploration, discovery, play, and authentic learning

I was visiting the Great Lakes Science center with family during high school. I was busy reading information printed on a horizontal rotating triangular prism – the three sides had facts about the phenomenon being demonstrated at the exhibit, though I don’t remember what the exhibit was about. While I was reading, a younger student came along and stood in front of me to view the exhibit. He barely paid any attention to me; his attention was piqued when he placed his hand flat on the prism and it rotated slightly under the weight. He then proceeded to flick the prism as hard as he could to see how fast he could make it turn. That was how he spent a couple of minutes while I looked on, flabbergasted by both his lack of interest in the flashing exhibit in front of him and the fact that all the exhibit inspired him to do was to challenge himself to a spinning competition. Once he was satisfied that he had peaked in the spinning task, he went off to a new exhibit.

I was pretty annoyed as a sixteen year old science enthusiast. How dare he not take the time to read what was printed at the exhibit? Not even try? There was so much good knowledge there to be learned – why bother coming to the museum if you weren’t going to try to learn something new? The best part for me was coming and playing with the exhibits and then seeing what science principle was being demonstrated. This was (as I understood it at the time) what science was all about.

For me, this was not something that was limited to museum visits. I had a pretty good arrangement for doing investigation at home as well. I was always able and encouraged to go outside and explore in the woods, burn stuff with magnifying glasses (not of course while doing the previous activity), and do experiments mixing things in the kitchen. I am incredibly grateful that my mom allowed me to do these experiments in spite of my frequent habit of rushing away afterwards without cleaning up. I didn’t realize at the time how unique it was that she let me do some of the things I did, and probably would have cleaned up myself more often  if I had. (I did do so marginally more often after a particularly stern chat about the difficulties of removing hardened candle wax from the good silverware. She clearly explained that the experiments would stop if I didn’t do this sort of cleaning myself.) The other major time I realized I should be grateful was when I accidentally removed the gold coating on a fork during an electroplating experiment. (Sorry mom.)

One that sticks in my mind was after I first learned about objects from space reentering the atmosphere and burning up.  I didn’t understand how metal objects could burn – I had seen metal melt before on TV, but could it really burn? I took a penny and some cooking tongs to the gas stove and held the penny in the flame for a long time. I was able to see the penny get hot and ultimately glow. I had a bowl of cold water there to drop the penny into afterwards. The colored patterns on its surface reminded me of a picture in a magazine that showed the oxidation patterns on a sample of material that had survived reentry. I also tried wires and aluminum foil in the flame, and the way both materials twisted around themselves and changed both in appearance and material properties gave me some insight into what it meant for metal to burn.

Was there a goal? Not really. I didn’t write up a lab report or keep a notebook recording my observations. These were just experiences in which I explored what I could do with the stuff in front of me. I did get the sense that this sort of thing was distinctly different from what I was doing in school because there was no assessment. I don’t know if she ever talked to others about her son “playing on the stove” as she called it. At the time I objected to her calling it that because I thought it made it sound like I was being reckless. I had a purpose to my experiments. I was creating meaning on my own as I had done throughout my Montessori elementary education. And I was careful when carrying out these investigations.

Years later, I have a different understanding the role of play in learning. I really like this TED talk by Stuart Brown that talks about some of the reasons why play is important. Much of what I have learned about building with LEGO is in the context of playing in an unguided way. Another major influence on my philosophy on play was K.C. Cole’s book Something Incredibly Wonderful Happens, which I heard about during the summer before teaching biology for the first time to ninth graders. The book describes physicist Frank Oppenheimer (brother of Robert Oppenheimer) and the full story of his life as a father, rancher, teacher, and ultimately creator of the world class Exploratorium.

I have visited the Exploratorium twice. There is no other museum in the world that has influenced me in such a visceral way as that museum. as I can still picture numerous things at that museum and what I learned from doing the exhibits. Reading about Frank and his process of seeing the museum as vital was really important to defining something that I think I hadn’t officially acknowledged in the preceding six years I had been teaching.

First, some Frank Oppenheimer quotes from a speech he gave upon receiving an award from the American Association of Museums, along with my thoughts:

Many people who talk about the discovery method of teaching are really talking about arranging a lesson or an experiment so that students discover what they are supposed to discover. That is not an exploration. The whole tradition of exploration is being lost for entire generations.

There is a role for discovery in our classrooms. This is not, as is often thought, the expectation that students will spontaneously figure out Newton’s laws or the quadratic formula. These are instead carefully designed activities through which students arrive at an idea. Our world needs more interactivity. People, not just students, are spending less time constructing their own understanding, and more time (since we are all inherently busy) hoping that others can explain things to us since it will invariably be faster this way.

If one of the things, however, we want to teach students is how to construct their own understanding, this is not going to come from giving them information and then telling them how to use it. Any way we can engage students to interact with the material actively instead of merely receiving content moves us closer to that goal.

It is, therefore, more important than ever that museums assume the responsibility for providing the opportunities for exploration that are lacking for both city and suburban dwellers. It would be fine, indeed, if they would, but it will take a bit of doing to do so properly. If museums are too unstructured, too unmanageable, people get lost and simply want to get back to home base. On the other hand, if they are too rigid, too structured or too channeled, there are no possibilities for individual choice or discovery.

It may be useful to note that these quotes are from 1982. Certainly these issues are no different nearly thirty years later. If lesson activities for students are too unstructured, they may have no idea of the learning goal, what they are supposed to figure out, or how to get from one point to another. They get lost. They get cranky. They would rather just be told information. This, however, is the opposite extreme. While some students demand the structure, there are serious limitations to the quality of learning experiences under a classroom model that is too rigid.

Exploring, like doing basic research, is often fruitless. Nothing comes of it. But also like basic research, as distinct from applied or directed research, exploring enables one to divert attention from preconceived paths to pursue some intriguing lead: a fragrance, a sight or smell, an interesting street or cave, an open meadow encountered suddenly in the woods or a patch of flowers that leads one off the trail, or even a hole in the ground! Often it is precisely as a result of aimless exploration that one does become intensely directed and preoccupied.

A museum that allows exploration does not have to be disorganized either physically or conceptually. It does, however, mean that the museum must contain a lot of which people can readily miss, so that discovery becomes something of a surprise, a triumph, not so much of personal achievement as of personal satisfaction. It is the kind of satisfaction that invariably leads me to tell someone about the experience.

When people in a museum find something that engages them, that moment of engagement is what justifies the museum’s existence. That may be what the visitor remembers about their museum experience. It may also be the sort of experience that causes the visitor to come back, and ideally, bring a friend or child. A well designed exhibit involves its visitors in its operation, tries to engage them, and along the way provides interesting information in the off chance a visitor is interested. A good museum has many of these experiences.

Here is the key idea that changed the way I decorate my classroom, organize my lessons, and structure my time with students:

You cannot entirely control what your students will get out of their time with you.

I have spent lots of time designing what I thought were perfect lessons only to have students remember the fact that I used colors in my handout, even months later, because that was what stuck with them.

You can tell them what you want them to get out of an activity. You can assess that they got out of the activity what you wanted them to get out of it. You can also try to tell them why something should be interesting to them. (Not recommended) None of these work well, at least authentically well when it comes to evaluating our use of an activity to reach specific learning goals.

What you can do is provide a range of activities, approaches, and experiences for your students. Providing students a chance to play in your classroom is one of the most powerful tools in our tool chest. You can’t play incorrectly. You can’t get playing wrong. Play is one of the few times when the only judgments being made belong to the individual that is playing. In the world of math education where students still see math as a class where there is always a right answer, and that right answers are inherently worth more than wrong answers, we need play more than ever.

What does play look like? Not like the majority of lessons I do, admittedly, but I’m working on that. The real reason this doesn’t happen as often as I want it to is that it doesn’t necessarily feel productive. I force myself to push through this because I’ve had the idea of clear learning goals and measurable objectives drilled into my head from the moment I started teaching. The problem is that real learning doesn’t look like this. When we figure things out, it isn’t with the end goal in mind. Unstructured time to just be in the presence of an idea that motivates itself is enough to get students to think as they do during play.

The biggest tool that we have at our disposal though is the use of technology. I’ve seen students discover by accident that when you hit the equal sign on some calculators, it repeats the previous operation with the answer. What do most students do when they discover this after multiplying? They hit it a bunch of times until the calculator overflows. Sometimes they will do the same thing with multiplying a decimal, and the number of zeroes to the right of the decimal point increases. Is there a lesson on place value or exponential functions there? Sure. The moment you tell them that though, it suddenly ceases to be exploration and starts becoming Math Class.

It’s also easy to create a Geogebra sketch of a quadrilateral with measured angles and tell students to “play” with it for five minutes. The goal is not to have them discover the sum of the interior angles is 360 degrees, though they might observe that. The goal is instead to give them a chance to interact with a mathematical object and have an experience that is all their own. Then start the lesson. See what happens. This is exactly what Noah Podolefsky from the PhET physics simulation project at the University of Colorado recommends students be allowed to do for 5 – 10 minutes before telling students what you want them to use the simulation to do.

The other aspect of this is in decorating my classroom. I don’t want so much on the walls that students will be continually distracted. I do want things that create interesting learning experiences without much effort. I hung a spring between two corners of the room as an example of a catenary curve – students don’t care about that. They do walk by it all the time and make it bounce up and down. Sometimes they see how long it takes for the vibrations to die down. Sometimes they hang things on it to see how it changes the droop in the overall spring. I have a bicycle wheel that normally is used as a demonstration of conservation of angular momentum. Students have instead spun it and observed that it stays upright like a top on the table. I have a checker board with checkers, the game Set, little metal puzzles, and a bunch of other things that don’t require a whole lot of explanation to be interesting. It’s amazing to see how the students use their down time to interact with these objects and with each other – it makes my classroom the same safe learning environment of a museum. The dream is to create this environment during every single lesson.

Looking back, the kid that stepped in front of my at the Great Lakes Science Center wasn’t learning what the exhibit designers intended him to learn. He was, however, constructing his own knowledge when he spun the prism as fast he could. He might have gotten some notion of what feels different about a force and a torque. He might have seen that the rotation only increased in speed while his hand was in contact with it – an intuitive concept related to Newton’s 2nd law.

Or not. It was pretty hypocritical of me to judge and potentially hamper his learning process when so many others (including my mom, who had many good, flammable reasons to do so) did not. He wasn’t using the museum wrong – I was. He was just doing what came naturally.

Entirety of Frank Oppenheimer’s speech to the American Association of Museums, 1982 can be found here.
I also find myself going back to this article written by Oppenheimer about teaching as a quick reminder of all sorts of important ideas.

Exploring Point Slope Form through Geogebra

In geometry we are studying parallel and perpendicular lines and the theorems that can be proven about them. In thinking about how to present the connection between algebra and geometry for this unit, I wanted to include an exploration of what exactly makes lines (and linear functions) so special: constant rate of change, or slope.

We did not get a chance to do this entire exploration in class, but I am expecting students to look at it at some point over the weekend. I know they have seen slope-intercept form, but point-slope form is convenient in many ways, especially in the way it applies the concept of slope between points located on a line.

Download my Geogebra file here.

Comments welcome!

Lessons from the CME project – Verbal Systems

In contrast to what I wrote in a previous post about disliking word problems relating to solving systems, I found myself returning to the topic with a new approach that I really liked. I’ve read through the presentations of the Center for Mathematics Education (CME) project, and have gotten an idea of what they do through the examples they present. I’ve been very interested in getting actual copies of their textbooks, but haven’t gotten around to it both because of my location (no shipping to China as far as I know) and because, well, other things have occupied my time.

I really like the general theme of how mathematical thinking is closely aligned with the sort of logical thinking we already do. The concept of ‘guess-check-generalize’ makes sense especially in the context of what I always find my students doing anyway when I present them with a word problem. See this post on the CME blog to get an idea of what it’s all about. In the past I have tended to use verbal problems, especially in the context of systems of equations, as a way of reinforcing solution methods of these systems. I have also found many students will naturally use a brute-force guess-and-check method of trying to solve them. I was consistently impressed when kids with low levels of number sense and arithmetic ability would fall upon a solution to a system of equations after a period of deliberate and focused trial and error. Why were these students so willing to spend ten minutes trying a bunch of solutions while being unable to sit and listen for five minutes on how to solve it methodically? Was what I was presenting so abstract and disconnected that the obvious method that made sense to them but was a lot more work was clearly the better choice? Clearly so.

My response early on in my teaching career was then to give systems of equations that they would NOT be able to solve by brute force. Systems with solutions that were decimals and fractions were much less likely to be figured out. Doing this though felt so arbitrary. If I have to modify the questions I was asking in a contrived way in order for my algebraic method to finally become the better solution to this group of students, there was something wrong with MY presentation and application of the mathematics, not with the students’ method.

This is part of the reason I would get frustrated teaching verbal systems of equations as part of solving systems of equations. The situations that came up (in most textbooks that I read) were made in a way that they fit the solution methods that were simple to solve using elimination or substitution. A person could know almost no English and still figure out a system of equations that most likely solved the given problem.

The thing that seems different about the guess-check-generalize framework though is that it encourages the type of self-aware mathematical thinking that we want students to do. This was the first time I really presented a problem this way, but it seemed to work well, particularly in the case of some of the students that have demonstrated both weaker math skills and/or a limited English proficiency. I gave them a problem of this type:

A store has a sale on sneakers and shirts. Tyrone buys three shirts and two pairs of shoes for $225. Maria buys two pairs of sneakers and five shirts and pays $325. What are the prices for a pair of sneakers and a shirt?

When I asked students to guess a solution to the problem, one student immediately ‘guessed’ that the answer was “2x + 3y = 225”. It was a great moment telling the student that if he went into a store and asked a salesperson how much a shirt and pair of shoes was, and the salesperson started spouting off an equation, that salesperson would most likely be smacked in the head and fired for being unhelpful. It makes no sense to respond to a verbal question with an equation, but that is what students (including mine, unfortunately) have been conditioned into doing. With that expected response out of the way, we could move on with the guess-check-generalize model.

I decided to call the answer a “model answer” instead of a guess – I have an ongoing battle with students about how much I hate the word “guesstimate” because people tend to use it to make a true guess sound more authoritative by connecting it to the very different word estimate. I asked a student what a possible answer could be to the question.

What was pretty interesting was that the rest of the creation of the mathematical system came naturally from this guess. What were the variables? Since what the question was asking us to find was the unknowns, the quantities found in the model answer were what we would probably model in a system of equations. There was no argument this time about how X was not equal to “SHIRTS” and Y equal to “SHOES” – instead it was plainly obvious from the model answer that we were guessing a price of a shirt and a price of a pair of shoes. Here was how the legend appeared:

The system of equations came just as easily. No teaching the formulaic way I once did that “for a cost type equation, you multiply the x-cost by the x variable, add it to the y-cost multiplied by the y-variable, and set it equal to the total cost.” Instead, we just found what the cost would be using our model answer:

If the model answer had been correct, then the cost would have been $70. I targeted this question toward one of the students that I was more concerned might not understand the whole process, and it seemed to come naturally. Clearly the $70 was wrong, but the students were actually thinking about this fact rather than blindly putting together an equation. The calculation using the model answer not only did this for them, it screamed out to us what the actual equation had to be. Smooth as silk.

It was exhilarating seeing this work with my group. Granted, they are generally a pretty strong group, but verbal problems like these (especially given the international make up of this class) tend to make them all visibly uncomfortable. This worked much more smoothly than any of my previous lessons. I certainly have tried to get students to think this way before, but never explicitly used a guess to generate the rest of the equation. For those ESOL students, it seems like a non-threatening first step to come up with an example of what an answer to the question might look like. This idea could help all students that have a tendency not to read questions all the way through and guess what they are being asked to do.

It is very possible that I’m just late to the guess-check-generalize party and teaching using this method is obvious. If that is the case, I apologize to my students for getting it wrong for so long. I see a lot of the similarities between this and modeling, which I’ve really enjoyed using with my students through exploration of Newton’s laws. Maybe the parity between them is why I’m suddenly so excited about the overall concept.

In the end, I do have some more tricks up my sleeve for how I want to use some actual, interesting, realistic, and authentic problems with this group. The robot crash went really well and the students enjoyed that activity. I go back and forth as to the benefit of giving them word problems like the one we worked on. They exist in math world, the world of math textbooks, but not so much in the reality of us as math teachers trying to teach what authentic mathematical thinking looks like.

Math Portfolio – Sharing my own story.

In Calculus, I use the third edition of Finney, Demana, Waits, and Kennedy. I love the selection of activities and explorations that are used to get students where they need to be for the Calculus AB exam. A colleague recommended that I check out Dan Kennedy’s website as a treasure trove of resources both mathematical and philosophical about teaching. One of the things I found there that I decided to bite the bullet and do this year is having students put together a math portfolio detailing their work over the year.

The reasons for doing this are many, some of them more selfish than others, but they include the following:

  • By having a record of student work, I can easily look back and remind myself of some of the major mistakes and misconceptions that students have at a particular moment in time.
  • I like reading and seeing how students respond to their own work. I often have students reflect on their work on short time scales (“I should have studied X or Y to do better on the unit test”) but don’t do as much over long periods of time (“I’ve become much better at graphing lines in comparison to when we first met linear functions in class.”) Part of this is because my students don’t tend to hold on to their papers for very long. I take partial responsibility for this, never holding them accountable for it, though I do occasionally remind them that the easiest way to study for a final exam is to look at old exams.
  • I think students selecting what work represents their progress often means things that are very different than what teachers see as their best work. Sometimes students are afraid of sharing their failures, though we as teachers see those as being the most meaningful learning experiences. Whichever is right, having students actively evaluating their own work and thinking about their own learning process is valuable for being able to identify how they learn best.

My introduction to the concept of the portfolio took a lot from Dan Kennedy’s document describing them, and I am incredibly thankful for his decision to publish his document online. My own document describing the content of the portfolio and how it is integrated into the grade is here.

At the beginning of the year I introduced the idea, and the response wasn’t applause. It was, incidentally, very similar to the introduction this year of true student-led conferences. The students wanted to know why we were demanding they do much more work just for parents and teachers that see their work anyway on the report card. My responses, fully sincere, included the ones I gave above: portfolios are opportunities to highlight not the grade that was received, but the learning process that it describes. Conferences, however, went extremely well as reported by teachers, parents, and most impressively, the students. Since requiring students to also produce the portfolio, I have been equally impressed by some of the thoughts shared by students about what they do and do not understand, the mistakes they tend to make, and also some of the things that go through their minds when thinking about learning.

One of my requirements is that students write a reflection and scan in their skills quizzes any time they want to retake a quiz. This is my current implementation of standards-based-grading, though I am considering expanding it significantly soon. This raises the bar somewhat for what students have to do to retake, but I don’t object to this requirement at all. Sometimes I have to tell them to do the reflections a second time – in this situation, they usually look something like “I didn’t get it but now I studied and I get it” without any detail as to what it is, what “not getting it” means, or what “studied” actually looks like. Once I get them past this point to do some serious thinking about what they have difficulty understanding, I am very pleased with the responses.

I tried handling the start of the portfolio myself since I wanted to make sure they all looked similar in case these did become official school documents at some point. This was a lot of work keeping track of quiz retakes, reflections, scanning them in, etc – I finally turned over the files as they were last week and have given them to students to keep up to date. Some strong students, however, have nothing in their portfolios because they weren’t retaking quizzes, and the only thing I had time to really check up on before the end of the first quarter was that the bios were in place.

What I decided to do to show ALL students what I was looking for in the math reflection portion (with the mathematics exploration to be added soon) is to share my own portfolio with some artifacts from high school that I still happen to have. I’ve always guarded my test, quiz, and project papers from high school as really authentic sources of material not only to use with my own classes, but also to show students that might not believe I ever had any difficulty in math.

Here is my own math portfolio, complete with biography and student (namely my own) work:  Weinberg portfolio example

I shared this today with students and had some really interesting responses:

  • “This is really your work from high school? Why in the world did you save it?”
  • “You had a 63 on a math test?”
  • “That looks like really hard math”

I got to tell them (1) to read it all the way through to see my comments and (2) that I was proud to show them some of my work along the way to becoming the math student that I was when I left high school. If nothing else, I am hoping that they will read it first because of the inherent fascination students have with their teachers as actual people (I love when they say things like ‘It’s cool to know you are a real person) and second to get some inspiration for the sort of thinking and reflection I want them to put together.

I know it is difficult to expect reflection to be a perfect process when it is new – it takes time and effort and it doesn’t immediately pay dividends. I want students to understand that reflection is not only a really beneficial process, but that over time becomes enjoyable. It shows that learning is a continual process, that you don’t just suddenly “get it”. This is the same process that I am enjoying about writing on this blog. It takes time, I have to make time to do it – in the end, I really enjoy looking back at my thoughts and holding myself to the commitments I make to my own practice and my students.

So I am leading by example. This group of students continues to really impress me when I expect great things out of them – here’s just one more way I am hoping to help them grow.

Teaching Proofs in Geometry – What I do.

This is the second year that I’ve had a standard geometry class to teach. The other times when I’ve taught some of the same topics, it has been in the context of integrated curricula, so there wasn’t too much emphasis on proof. When the time came last year to decide how teaching proofs fit into my overall teaching philosophy, it was a new concept. I’ve seen some pretty amazing teachers have great success in teaching it to students who subsequently are able to score very highly on standardized exams. I’m in the fortunate position of not having to align my proof teaching to the format on an exam. As a result, I’ve been able to fit what I see as the power of proof-writing to the needs and skills of my students in the bigger context of getting them to think logically and communicate their ideas.

As a result, my general feeling about writing proofs is as follows:

  • Memorizing theorems by their number in the textbook is less important than being able to communicate what they say.

I’ll accept ‘vertical angles theorem’ but fully expect my students to be able to draw me a quick diagram to show me what the theorem really says. This is especially important for international school students who may move away to a new math classroom in another part of the world in which ‘Theorem 2-3’ has no meaning. I won’t ask students to state a theorem word for word on an assessment either, but they must know the hypothesis and conclusion well enough to know when they can apply it to justify a step in their proofs.

  • Being clear about notation and clear connections between steps in proofs is important.

Since the focus of my geometry class is clear communication, correct use of notation is important. If angle A and angle B are congruent, and the measures of these angles are then used in a subsequent step of the proof, it needs to be stated that the measures of angle A and measures of angle B are congruent equal. I’m not going to fail a student for using incorrect notation in a proof if the rest of the logic is sound, but a student will not receive a perfect score either if he/she uses congruent angles interchangeably with their measures.

  • Struggling and getting feedback from others is the key to learning to do this correctly.

I don’t want my students memorizing proofs. I want them to understand how logic and theorems applied step by step can prove statements to be true. Human interaction is key to seeing whether a statement is logical or not – I like taking the ‘make-it-better’ approach with students. If a student says angle A and angle B are congruent, and that statement is not given information, then there needs to be some logical statement to justify it. In all likelihood, there is another person in the classroom that can help provide that missing information , and it won’t necessarily be me. As I wrote in a previous post, it was tough letting the students struggle with proofs in the beginning, but they helped each other beautifully to fill in the gaps in their understanding. This makes it hard for students that are used to being able to see a thousand examples and get it, but since that isn’t my intent for this course, I’m fine with that.

My progression for teaching proofs starts with giving the students a chance to investigate a concept and predict a theorem using Geogebra or a pencil and paper sketch. I like using Geogebra for this purpose because it instantly lets students check whether a property is true for many different configurations of the geometrical objects.

As an example, the diagram at right is one similar to what my students made during a recent class. The students see that some angles are congruent and that others are supplementary. They can make a conjecture about them always being congruent after moving the points around and seeing that their measures are always equal. This grounds the idea of writing a proof in the idea that they know that if parallel lines are cut by a transversal, then alternate exterior angles are congruent. There’s no failing in this if the activity has been designed correctly – students will observe a pattern.

The work of writing the proof doesn’t start here – usually some work needs to be done to get a complete conditional statement to be used as a theorem. When students suggest hypotheses for the statement, and it isn’t as complete as it needs to be, I (or even better, other students) play devil’s advocate and construct diagrams that might serve as counterexamples for the entire statement NOT to be true. Students might suggest ‘if two lines are intersected by a third line, then alternate exterior angles are the same’. If I’ve done my job correctly, students will (and at this point are) catching each other on using congruent rather than the same, and not saying that angles are equal. This is a great spot for the students that love catching mistakes (though often don’t catch their own). Until students are comfortable writing the theorems using precise and correct mathematical language based on their observations, writing the proofs themselves is a huge challenge.

I balance the above activities with another introductory step in writing proofs. I’ll provide the statements in order for the proof and ask students to provide the reasons. This works well because students seem to see coming up with the statements as the tough part, and the reasons come from a menu of properties and theorems that we’ve put together previously in class. I don’t like doing too much of this as it doesn’t require as much social interaction aside from “is this the right reason?” from students as the rest of proof writing.

The final step to writing proofs comes in the form of returning to a diagram like that above. If students are proving the statement “if parallel lines in a plane are cut by a transversal, then alternate exterior angles are congruent”, I expect them to draw a diagram (on paper or on Geogebra) showing parallel lines cut by a transversal. I tell them to pick an exterior angle and give it/find its measure. Then they need to go step by step and find the other measures of the angles using only theorems we know, and NOT using the statement we are trying to prove. (We call this the ‘cheap’ way.) My way of prompting this development is by asking questions. If a student is sitting and staring at angle 3 in the diagram, I can ask about another angle he/she knows is congruent to that angle. A student will invariable state a correct angle just from having a correct diagram, but this is the important part: the student MUST be able to identify in words what theorem/postulate allows the student to say that the other angle is congruent, either verbally or in writing.

The key thing to show students at this point is that there are MANY ways to make this process happen. Some will see vertical angles right away, and say that angle 3 is congruent to angle 2 because of the vertical angles theorem. This then leads to seeing that angle 2 is congruent to angle 1 by the corresponding angle postulate, and then the final step of using transitivity to prove the theorem. Some students will jump from angle 3 to angle 2 (vertical angles theorem), then angle 2 to angle 4 (alternate interior angles theorem), then angle 4 to angle 1 (vertical angles theorem again). Having students share at this point the many ways of doing this is crucial – letting them justify which angles are congruent using concrete values for the angles, and justifying each step with another theorem, definition, or postulate is the important part. Once they have done this, I let them work together to write the full proof using the concrete road map. They don’t get it right the first time, but having the real numbers as an example grounds the abstraction of the idea of proof enough for students to see how the proof comes together.

The weaker students in the group need one extra step sometimes. I let them fill in all of the angles in the diagram first using what they know – this part, they tend to be pretty good at, and I don’t flinch when they use the calculator to do the arithmetic since some need that to be successful. Then we hop from angle to angle and the student must explain using the correct vocabulary why the angles are congruent or supplementary. In keeping this as an exercise in concrete numbers, I’ve had some success in these students (and the ESOL students) using the correct vocabulary, even if they are unable to write the proof completely on their own.

I started to see the dividends of this progression this week, and I am really pleased to see how far they have come in being able to justify their statements. The only thing we did need to work on was how to structure an answer to questions that ask students to make a conclusion based on given information and the theorems they know. This was in response to using converses of the parallel line theorems to show that given lines are parallel. To help them with this process, I gave them this frame and set of examples:

I was very impressed with how this improved the responses of all students in the class. We had some great conversations about the content of student conclusions using this format. In diagram (b), two students had different conclusions about why lines CF and HA are parallel, and there was some really great student-led discussion in explaining why they were both correct. I forced myself to listen and let their thinking guide this discussion, and I was really happy with how it worked.

This year’s group still is not super thrilled about having to write proofs, but they are not showing the outright hatred that the last group was showing at this point. I have been emphasizing the move from concrete to abstract much more with this group, as well as showing that the proof is really a logical next-step from reasoning how two angles with definite measures relate to each other in a diagram.  If nothing else, students are already better communicators of their math thinking in comparison to the first day of class when there was plenty of wild gesturing and pointing to ‘that thing, yeah’ on the whiteboard. Continuing to develop this is, I believe the real goal of a geometry class and not the memorization of theorems. My next step is to include the statement writing process as the first step in solving an algebraic problem. Many students are still throwing the dice and either setting algebraic expressions equal to each other, or adding them together and setting them equal to 180 because that’s what they did in their other classes. I am trying, trying, trying to get them out of this habit.

I hope that in sharing my process, others might get ideas on how to either make a certain geometry teacher better (me) or to enhance what is already going on in other classrooms. If any readers have suggestions on how to improve this as time goes forward, I am most thankful for any and all advice you can provide.

Graphical Systems in Geogebra and crashing LEGO robots in Algebra 2

In the Algebra 2 class, we started our unit on solving systems of equations. From a teaching perspective, this provides all sorts of opportunities for students to conceptualize what solutions to systems mean from a graphical, algebraic, and numerical perspective. Some students seem to like the topic because it tends to be fairly straight forward, is algorithmic, and has many ways to check and confirm whether it has been done correctly.

I used this as my warm-up activity today:

a) Estimate the solution of the system.

b) Write an equation for each line in standard form.

c) In Geogebra, select CAS view and type the following using your two equations: Solve[{7x+3y=6,3x-4y=12},{x,y}]

d) Use your calculator and convert these values to decimals. How close are these to your estimate?

We had some great discussions about the positives and negatives of graphical solutions to equations. Weaker students got some much needed practice writing equations for lines. For all students, this led to some good conversations about choosing two points that the lines clearly pass through for writing equations (if possible) rather than guessing at the y-intercept. The students also got the idea of how Geogebra can solve a system of equations exactly as a quick check for their algebra, an improvement over substituting (which is at times more trouble than it’s worth for students with poor arithmetic) and slightly faster than solving for y on a graphing calculator and finding the intersection.

I also like the unit, though I don’t tend to like the word problems. It’s hard to convince students about the large scale importance of coin problems (especially in an international school with everyone used to different currency) or finding how many tickets were sold at the door or advance since anyone with a brain would just ask the person tallying the tickets.

I also found myself thinking about Dan Meyer’s post over the summer about how many word problems are made up for the purposes of math, rather than using mathematics to analyze cool situations and create problems out of the situations. Getting students to figure out how to use the math to do this is ultimately what we want them to learn to do anyway. Figuring out when trains pass each other is not exciting to students, but I realized this morning while brushing my teeth that doing this problem with real robots either crashing into each other or racing adds a neat dimension to this problem. The question of figuring out both when they will crash or catch up to each other, and also where they will do so is a clear motivation for finding a solution to a system of equations describing their positions as functions of time.

So I gave the students the two robots (videos of them posted at http://bit.ly/vIs0lu and http://bit.ly/u9jSPB) . I told them I was going to set them apart a certain distance that was tentatively 80 centimeters, but said I wanted the ability to change that at any time. I wanted them to predict when and where they would collide.

The rules:

No, you can’t just run the experiment and see where they crash. That not only defeats the purpose of this exercise, but we will be doing this sort of activity in a couple different ways during the unit, so being able to do this analytically is important. You also can’t run both robots at the same time – that’s for those of you that are going to try to be lawyers and break that first rule.

You can measure anything you want using any units that you want using either robot individually.

At some point, you should be able to show me how you are modeling the position of each robot as a function of time.


And I set them off to figure things out. Despite the fact there were only two robots, the 12 kids naturally divided themselves up into a couple teams to characterize each robot, and there was some good sharing of data amidst some whining about how annoying it was to actually measure things. In the end, most students at least had some idea of how they were going to put together their models, and some had actually written out what they were. As one would hope for these types of activities, there were plenty of examples of students helping others to understand what they were doing. The engagement was clearly there, as confirmed by students visibly excited to run the robot and time how long it took for it to move around.

It was a fun exercise that I plan to return to in a few ways during this unit – perhaps some interrobo-species interaction (my iCreate robot is charging up as we speak). Fun times.

UPDATE: This is the video of the next day’s class when students solved their functions. I set the robots apart from each other and the students did the rest.
[youtube http://www.youtube.com/watch?v=As1JNGVPh0M&w=420&h=315]

Using #Geogebra to Predict and then Verify

Last year’s class introducing logarithmic and exponential differentiation was a bust. I tried to include it as an application of implicit differentiation, but I knew afterwards then, and still believe now that doing so was an incredibly horrible idea. There’s no way students are going to ‘see’ an application of an abstract concept like implicit differentiation better…by using it in another abstract concept. I’ve accepted that, and vowed this year to do a much better job.

I also had a shocking moment yesterday when a Calculus student came to me after school and asked me ‘what is the derivative?’ We had started the unit with a conceptual development of the derivative using limits and average rate of change, and had since moved to applying differentiation rules, so we were deep in that process – power rule, quotient rule, product rule, chain rule…really the primary ‘rules’ section of any Calculus course. I was taken aback by the comment – had I really stopped emphasizing the definition of the derivative in our class activities? In a way, yes. We had been writing equations for tangent lines and graphing them, but we hadn’t seen the limit definition (which I’ve been impressed by students remembering) in a little while. This proved that not only did I need to do a better job with logs and exponential functions, but that a little conceptual basis in that process would be useful.

I always like using Geogebra as a tool to pre-load information I am about to give students – what is about to happen? What should my result look like when I do this on pencil and paper? The graphing capabilities make it really easy to do this and set this up – I created this file and made it look the way I wanted in a few minutes.

You can direct download the file here.

These were the instructions I gave students:

Sketch what you would expect the derivative of y = 2^x to look like. Then click the ‘Show Derivative Function’ to graph the actual derivative. How close were you?

How would you expect your sketch to change for the derivative of y = 3^x?

Graph and make a prediction of the graph of the derivative of y = 2^-x. Check and see how close you were using the Geogebra tool.

Can you adjust the slider value for a so that the derivative is the same as the function itself? Use the arrow keys to adjust the slider more precisely.

Go through this same process to sketch the derivative of y = ln(x) in a new Geogebra window. Create this by going to the ‘File’ menu and selecting ‘New Window’.

It was really great seeing students predicting what the derivative would be, and then using the applet to confirm what they thought. There were lots of good conversations about scale factors and reflections, and some of them pretty much nailed what the general forms were going to be. This made the algebraic derivation a piece of cake – they knew where it was headed.

I also sprung this on them:

I’ve been really getting into the idea of standard based grading, and have been doing a form of it through my quizzes for a while, but it is still a small component of the overall grade calculation. While their grades aren’t being calculated any differently at the moment, I shared that this list would make a really good tool as we prepare for the unit exam on derivatives next week, and most started going through on their own and deciding what they needed to work on.

I’m still getting caught up after a couple very busy weeks, but I really like how this group in Calculus has been developing and maturing as math students in only a couple months. Their questions are more directed: ‘I don’t understand this application of the chain rule’ compared to ‘I don’t get it’. Their written work is detailed and clear, making it easy to locate errors. As a group, they get along really well, and class periods are filled with moments of furious productivity and camaraderie as well as humor and smiles throughout.

It was raining hard all day. I watched some students walk into class, look outside at the afternoon sky, and sink into their chairs, clearly feeling a bit down. I told them it was perfect Calculus weather – why not sit inside and do some differentiation?

Probably not what they had in mind. By the end of class, everyone left the classroom looking much more positive than when they walked in, and at least feeling good about the work they had in front of them.

Dare to be silent.

I made a promise to myself today – I was going to force the physics class to speak. It isn’t that they don’t answer questions and participate, it’s that usually they seem to do that to please me. Sometimes they will explain ideas to each other and compare answers, but it never works as beautifully as I want it to.

So today I told them I wasn’t going to talk about a problem I gave them. They were. And then I sat on an empty table and waited. It was really difficult for me. Eventually someone asked someone else for an answer. I stayed quiet. Then another person nodded and agreed and then said nothing. I stayed quiet. Then someone disagreed.

Full disclosure – at this point I gestured wildly, but still stayed quiet.

After about five minutes of awkward silence punctuated with half explanations that trailed off, something happened – I don’t know what the trigger was because if I did I would bottle it and sell it at educational conferences – a full discussion was suddenly underway. I was so amazed that I almost didn’t think to capture it – thankfully I did get the following part:
[youtube http://www.youtube.com/watch?v=YVSSjQVpB70&w=420&h=315]

Especially cool to see this knowing that English is not the first language of the students speaking.

I’m going to try to do this more often, though I again must point out that it was incredibly difficult working through the silence. The students in the end decided they had something to say, so they shared their thoughts with each other. I did nothing but wait for it to happen.

Build the robot the way you want…No, you’re doing it wrong!

I teach an exploratory class for middle school students in robotics. The students rotate between robotics and some other electives during each quarter of the year, and there are no grades – just an opportunity to learn something interesting while doing. I like the no grades part, particularly because assessing progress in robotics is quite hard to do. My usual model is saying you get x points for doing the bare minimum (a D) and then incremental increases in the grade for doing progressively more challenging tasks.

It works, but I really like getting the opportunity to not have to do it. There are so many things you could measure to assess the students in their building and programming skills, but in my experience, the students don’t tend to explore or tinker as much in that situation.

Today while working on the day’s challenge on using sensors and loops, a student was fixated on building the following:

There weren’t any instructions to do this – he just started putting things together, liked what he had created, and continued building it today.

I had to stop myself for a moment because teacher Evan started to come out and remind him to stay on task and contribute toward his team’s solution to the challenge. Thankfully robotics Evan intervened and let it happen.

This is an exploratory class. It’s supposed to expose the students to new situations that might interest them later on. Why in the world would I stop the exact sort of thinking and exploring the class was designed to provoke? It also turned out that this student, along with the other two in the group, were all taking turns in the programming and building so that each would have a chance to play in this way. In spite of their taking time to free build, this group actually solved the three challenges before the rest in the class.

It connects to a great TED talk on doodling by Sunni Brown. One idea from her talk was that doodling contributes to “creative problem solving and deep information processing.” I think that these students (and all of us that ‘play’ with building toys like LEGO) are engaging in a similar process by free building. The connections students are making in figuring out how the tools work through play are not easy to measure. This is a horrible reason not to provide them time to do so. I do think there is an interesting connection between the tendency of these students to play and their ability to figure out the subtler parts of the class challenges.

After all, as David Wees pointed out, giving explicit step-by-step instructions on how to use creative tools like LEGO takes the creativity (and much of the fun) out of it. There has to be time to experiment and learn by doing how the tools work together, and that’s exactly what this student was doing.

My final reaction during the class today was that I told them all that I wanted to take a picture of every off-topic LEGO design they create. Document it all. If it’s cool enough to engage you for the time it takes to create it, I want a gallery of those designs to celebrate them.

The sad part? This made some of them stop. I can’t win!

Are there too many people on this thing? (#anyqs)

Two images I want to share for the purposes of #anyqs. For those unaware, this hashtag means I want you to look at these images and let me know what mathematical questions jump out at you right away.

Once you’ve taken a moment to think about these, please send me a tweet (@emwdx) with the hashtag #anyqs letting me know what you think this is all about. That would be great if you could take a moment.

What would be even cooler, however, is if you could also take pictures of your own of these signs in elevators (or other places you might find them) as well as a little information about where it is being taken. For example, the left picture is taken in the elevator in my apartment building while the right is from the gondola that the 9th graders and I took to get down from Mount Tai on the rainy second-to-last day of the China trip.

I am especially curious to see how these signs vary between buildings, countries, and even between elevator banks within a building. I’d like to share these with students tomorrow, so please snap a quick picture and send it (or a link) my way. It hopefully goes without saying that I will give proper credit for photos that make it into the materials I use with students, and will share what comes of it with you all.

So, for the sake of an interesting idea that I think will start some cool discussions with students – skip the stairs today. Snap a picture in the elevator, and then reward yourself for your generosity (and for decision to postpone exercise for unselfish reasons) by eating a doughnut or other craved food of your choice.

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