teaching philosophy

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.

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.

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.

Take Time to Tech – Perspectives after a Flip

Yesterday my calculus students reaped some of the benefits of a flipped class situation – I made some videos on differentiation rules and asked that they watch the videos sometime between our last class and when we met yesterday. We spent nearly the entire period working with derivatives rules for the first time. The fact that the students were getting their first extended period of deliberate practice with peers and me around (rather than alone while doing homework later on) will hopefully result in the students developing a strong foundation what is really an important skill for the rest of calculus.

They were using Wolfram Alpha to check their work, something that I paid lip-service to doing last year but did not introduce explicitly on the first day of learning these rules last year. There was plenty of mistake-catching going on and good conversations about simplifying and equivalent answers. I needed to do very little in this process – good in that the students were teaching themselves and each other and being active in their learning.

It was also interesting doing this so soon after discussing the role of technology in helping students learn on the #mathchat Twitter discussion. There were many great points made regarding the content of technology’s effective use across grades. It made me think quite a bit about my evolution regarding technology in the classroom. Many comments were made about calculator use, teaching pencil and paper algorithms, and the role of spreadsheets and programming in developing mathematical thinking. I found a lot of connections to my own thoughts and teaching experiences and it has me buzzing now to try to explain and define my thinking in these areas. Here goes:

Developing computational and algorithmic fluency has its place.

In the context of my students learning to apply the derivative rules, I know what is coming up the road. If students can quickly use these rules to develop a derivative function, than the more interesting applications that use the derivative will involve less brain power and time in the actual mechanics of differentiation. More student energy can then be focused in figuring out how to use the derivative as a tool to describe the behavior of other functions, write equations for tangent and normal lines, and do optimization and minimization.

There was a lot of discussion during the chat about the use of calculators in place of or in addition to students knowing their arithmetic. I do think that good arithmetic ability can make a difference in how easily students can learn to solve new types of mathematical problems – in much the same way that skill in differentiation makes understanding and solving application problems easier. Giving the students the mental tools needed to do arithmetic with pencil and paper algorithms empowers them to do arithmetic in cases when a calculator is not available.

Technology allows students to explore mathematical thinking, often in spite of having skill deficiencies.

One of the initiatives my colleagues took (and I signed on since it made a lot of sense) when I first started teaching was using calculators as part of instruction in teaching students to solve single variable linear equations. There was a lot of discussion and protest regarding how the students should be able to manage arithmetic of integers in their head. It wasn’t that I disagreed with this statement – of course the students should have ideally developed these skills in middle school. The first part of the class involving evaluating algebraic expressions and doing operations on signed numbers were done without calculators in the same way it had been done before.

The truth, however, was that the incoming students were severely deficient in number sense and arithmetic ability. Spending a semester or two of remediation before moving forward to meet the benchmarks of high school did not seem to make sense, especially in the context of the fact that students could use a calculator on the state test. So we went forward and used calculators to handle the arithmetic while students needed to reason their way through solving equations of various forms. They did learn how to use the technology to check the solutions they obtained through solving the equations step-by-step using properties. There were certainly downsides to doing things this way. Students did not necessarily know if the answers the calculators gave them made sense. They would figure it out in the end when checking, but it was certainly a handicap that existed. The fact that these students were able to make progress as high school math students meant a lot to them and often gave them the confidence to push forward in their classes and, over time, develop their weaknesses in various ways.

I have seen the same thing at the higher levels of mathematics and science. I used Geogebra last year in both pre-Calculus and Calculus with students that had rather weak algebra skills to explore concepts that I was taught from an algebra standpoint when I learned them. Giving them tools that allow the computer to do what it does well (calculate) and leave student minds free to make observations, identify patterns, and test theories that describe what is happening made class visibly different for many of these students. If a computer is able to generate an infinite number of graphs for a calculus student to identify what it means for a function graph to have a zero derivative, then using that technology is worth the time and effort spent setting up those opportunities for students.

Using skill level as a prerequisite for doing interesting or applied problems in mathematics is the wrong approach.

Saying you can’t drive a car until you can demonstrate each of the involved skills separately makes no sense. Saying that students won’t appreciate proportional reasoning until they have cross-multiplied until their pencils turn blue makes no sense. Saying that learning skills through some medium makes all the other projects and applications that some of us choose to explore in class possible does not make sense. It makes mathematics elitist, which it certainly should not be.

Yes, having limited math skills is a limit on the range of problem solving techniques that are available to students. A student that can’t solve an equation using algebra is destined to solve it by guess and check. Never underestimate the power that a good problem has to entice kids to want to know more about the mathematics involved. Sometimes (and I am not saying all the time) we need to work on the demand side in education, on the why, on the context of how learning to think in different ways applies to the lives of our students.

Emphasizing algorithms without providing students opportunity to develop context or some level of intuitive understanding (or both) has significant negative consequences.

I don’t mean to suggest that teaching algorithms on their own can’t result in students performing better on a type of problem. The human brain handles repetition extremely so well that learning to do one skill through repetition is not necessarily a bad way to learn to do that one thing.

One problem I see with this has to do with transferring this skill to something new, especially when the depth of available skills is not great. Toss a weak student ten one-step equations of the form x + 3 = -8, and then give them something like 0.2 x = 25, and chances are that student won’t solve it correctly without some level of intuition about the subtle differences between the two. Getting this right takes practice and feedback really good opportunity for students to be reflective of their process.

It is also far too easy when applying an algorithm to stop thinking critically about intermediate steps. I spoke to a colleague this week about his students learning long division and we both questioned the idea that the algorithm itself teaches place value. We looked at a student’s paper that was sitting on the desk and instantly found an example of how the algorithm was incorrectly applied but through a second error resulted in a correct answer. If we teach algorithms too much without giving activities that allow students to show some sort of understanding of some aspect of how the algorithm fits into their existing mathematical knowledge, it’s undercutting a real opportunity to get students to think rather than compute. I like the concepts pushed by the Computer-Based Math movement in using computers to compute as they do best, and leave the thinking (currently the strength of the human brain) to those possessing one.

As often as we can, it is important to get students to interact with the numbers they are manipulating. Teaching the algorithms for multiplying and adding large numbers does provide students with useful tools and does reinforce basic one digit arithmetic. I get worried sometimes when I hear about students going home and doing hundreds of these problems on their own for various reasons. If they enjoy doing it, that’s great, though I think we could introduce them to some other activities that they might see as equally if not more stimulating.

I do believe to some extent that full understanding is not necessary to move forward in mathematics, or any subject for that matter. I took a differential equations course in college trying to really understand things, and my first exam score was in the seventies, not what I wanted. I ended up memorizing a lot after that point and did very well for the rest of the course. It wasn’t until a systems design course I took the following year that I actually grasped many of the concepts that eluded me during the first exposure. This same thing worked for me in high school when I took my first honors track math class after being behind for a couple years. My teacher told me at one point to “memorize it if I didn’t understand it” which worked that year as I was developing my skills. Over time, I did figure out how to make it make sense for myself, but that took work on my part.

Uses of technology to apply/show/explore mathematical reasoning comprise the best public relations tool that mathematics has and desperately needs.

I really enjoyed reading Gary Rubenstein’s recent post about the difference between “math” and mathematics. I read it and agreed and have been thinking a lot along the lines of his entry since then.

Too many people say “I’m not good at math.” What they likely mean is that they aren’t good at computing. Or algorithms. Or they aren’t good at ________ where __________ is a set of steps that someone tried to teach them in school to solve a certain type of problem.

On the other end of the perceived “math” ability spectrum, parents are proud that their children come home and do hundreds of math problems during their free time. These students take the biggest numbers they can find and add them together or multiply them and then show their parents who are impressed that their normally distracted kids are able to focus on these tasks long enough to do them correctly.

It makes sense that most people, when asked to describe their experiences in math, describe pencil and paper algorithms and repetitive homework sets because that’s what their teachers spent their time doing. This, unfortunately, is the repetitive skills development process that is part of mathematical learning, but should not be the main course of any class. We show what we value by how we spend our time – if we spend our time on algorithmic thinking, then this is what students will think that we as teachers and as thinkers value as being important in mathematics.

This fact is one of the main reasons I started thinking how to change my class structure. My students were talking about not being good at a certain type of problem (“I don’t get this problem…I can’t do problems that need you to…”) rather than having difficulties with concepts (“I don’t get why linear functions have constant slope…I don’t get why x^2 + 9 is not factorable while x^2 – 9 is).

If we as teachers want students to value mathematics as more than learning a set of problems to be solved on a test, then we have to invest time into those activities that allow students to experience other types of mathematical thinking. This is where technology shines. The videos of Vi Hart, Wolfram Alpha, the antics of Dan Meyer, the Wolfram Demonstrations Project, the amazing capabilities of Geogebra – all of these offer different dimensions of what mathematical thinking really is all about.

We can share these with students and say “check these out tonight” at the end of a lesson and hope that students do so. Sometimes that works for a couple students. That isn’t enough.

I think we need to invest in technology with our students with our time. We need to deliberately use valuable class time to take them through how to use it and why it makes us excited to use it with them. It’s really the only way students will believe us. Show that it’s important, don’t just tell your students it is. That’s right – that valuable class time that we often plan out too carefully and structure so that they reach the well-defined goals we have for them – that time. Plan to use a specific amount of class time, and enough time, to let students play around with a mathematical idea using any of the amazing technology tools out there. Show them how you play with the tools yourself, but don’t make this the focus of this time – do so afterwards, perhaps.

To be clear – I am not saying do this all the time. Students need to learn algorithms, as I have already stated. Students also need to be looking at interesting problems. We should not wait to show them these problems until after students have demonstrated automaticity because it gives students the impression that the algorithms came before the thinking that went into them.

I am saying that balance is key.

The only way we are going to change the perception of what mathematical thinking really looks like is by living it and sharing it with our students.

Modeling anyone? Fans on carts edition.

After reading a lot about the success that others have had with teaching physics using the modeling method, I’m giving it a shot as I start Newton’s laws with my physics class. When I taught this with my AP physics previously, I did a traditional development of Newton’s laws describing (I admit it – lecturing) about Newton’s understanding of what caused acceleration. We talked about acceleration being proportional to net force and inversely proportional to mass, and then went from there exploring what it meant for net force to be zero through a series of problems involving net forces, components, etc.

What I did seemed to work in so far as students were able to solve the problems I gave them. The undying assumption of course is that what I did was efficient and made me feel that I had got across the material to students, but along the way I wasted an opportunity for students to SEE the principles in action and try to figure things out on their own. Since my students this year are not taking the course at the AP level, I see no reason not to try this and see how it compares in the long run to student understanding and enjoyment of the exploration of physics concepts. It is the sort of thing that I can see doing even in the Physics B curriculum, as dense as it is, given the fact that students really need a chance to play to connect the mathematics of the equations to the fact that physics describes the real world, not just idealized situations.

Here’s where I’d love to get some input though – I am giving my students a test in the first half of the 85 minute period tomorrow, and then my plan is to let them spend the rest of the time watching some videos that I took this afternoon of toy fans attached to cars on an air track. The students will get to play with the actual air track, but I want to introduce to the way I want them to play by seeing these videos that I created.

I have posted the series of videos here at my wiki site. The general instructions for what I want them to do are there, but I might as well run through them here as well.

First, I want them just to watch all the videos. No physics, just observation. After they have done this, I’ve posted a number of questions I want them to use to classify, analyze, and predict based on constant velocity and non-uniform velocity cases. I plan to have them sketch what effect a single fan would have on the motion of the cart. My plan in the end is to have them construct a situation with the fans that results in a given scenario. For example: arrange the fans on the cart so that The cart has zero initial velocity and an acceleration to the left. Draw position, velocity, and acceleration graphs, and then use Tracker to confirm/refute what their models suggest will happen.

Let me know your thoughts either here or through Twitter (@emwdx) – I am excited to try this, and excited to give the students a chance to get some first hand experience testing their own ideas. I had a blast playing with it this afternoon, and while I do have a different standard for what is ‘fun’ at times, I don’t think this is one of those times.

Wiki site: http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/52698/Unit_2__Carts_with_Fans.html

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