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.

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:

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.

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.

Physics #wcydwt - Indirect Measurement

While cleaning up after robotics class today, I noticed a statics problem involving an object hanging from a couple wires that was poking out from under one of my many piles of papers. We had looked at this question earlier during the week in class. A couple students were out for a volleyball tournament in Beijing, so I wanted to do something hands on and multimedia-esque that the missing students wouldn't feel too upset about missing, but could somehow still be involved and connected with the class work from today.

I realized that we hadn't yet used the spring scales during our discussion of forces. My obsession with #wcydwt lately has been on using the novelty of a minimum amount of information to get students to see a problem jump off the page/screen. I also wanted the students in class to get the joy of holding back information from their classmates to see if they could figure out the missing info. Lastly, I wanted there to be a simple physics problem that would serve to assess whether all of the students understood how to solve a 2D equilibrium problem.

So I grabbed the spring scales, some string, slotted weights, and told the students to put together a few pictures using these materials. We briefly discussed what information could be given, and what they wanted to leave out for the athletes to figure out on their own. I admit - I pushed them along, and given more time I would have given them more choice, but I don't think my selfishness and excitement in doing this was too much. The other factor - the vice principal had given us an extra pizza to share - they were also really pushing for efficiency. It wasn't all me.

And thus the spring scale picture project was born, thanks to one student's iPhone and Geogebra:

The complete link of the assignment is at http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/e495b/Unit_2__Spring_Scale_Challenge.html.

I'm sure I am not the first to do this, but it was so simple to execute that I had to give it a shot, and I am sharing it because I'm trying to share everything I can these days. We will see what happens when the results come in next week.

Scheming with Schema...

When teaching physics before, I found the process of building free body diagrams with students to be a fairly smooth process. It took a lot of feedback to get there, but they way I introduced the topic was along the lines of the chart below:

This chart was based on one I had from my own physics notes taken during class with Mr. Bob Shurtz who influenced me both as a student (helping me explore the love of physics and engineering I didn't know I had beforehand) and then as a colleague while designing my own AP Physics course in the Bronx.

I held students to the requirement in the beginning that every time they constructed a FBD they must make one of these charts because my feeling was it would help both in identifying the important forces acting on a single object and in discussions of Newton's 3rd law. The students grumbled as they tend to do when we expect them to use organization scaffolds like this that they feel they don't need. As time went on and FBDs were drawn correctly, I would loosen that requirement to the point that students were drawing diagrams and, minimally, felt guilty if they weren't at least thinking to make sure all of those forces could be identified. Those charts were admittedly annoying, but I felt they at least got students in the right mindset for drawing free body diagrams, so it was a good thing to require.

When my fans on carts exploration with the students went long last week, I decided to push the introduction of FBDs to this past Monday. We did have time last week to talk out different types of forces (normal, gravity) so they at least had some ideas of what different forces could be included in the chart. This extra time gave me the weekend to take a closer look at Modeling Instruction, and more specifically, at the concept of drawing system schema. I had never heard this term, but it appeared all over the modeling literature, so I decided to take a closer look at the Arizona State University site on modeling where I found an excellent paper that details using them as part of the FBD development process. It seemed harmless enough. Worst case, it would be a scaffold like the chart I mentioned earlier, used in the beginning and then taken away over time.

It was especially lucky that shortly after reading this, Kelly O'Shea had posted an excellent guide on how she introduces the Balanced Force Particle model to her class. It seemed like such a natural way to analyze problems, so I introduced it to the class as part of drawing free body diagrams for the first time on Monday.

Some really interesting things happened during that class and during Wednesday's class that deserve to be shared here. First, I was impressed how naturally students took to the idea of drawing the schemata. Not a complaint in the room.
They shared with each other, pointed things out, and quickly came to an agreement of what they should look like.
It was incredibly natural for them to then draw a dotted circle around the object they were analyzing and see the free body diagram nearly jump out at them. The discussions about directions and what should be in the diagram were matter of fact and clear with virtually no input from me. Score one for the schema.

The second thing that came up during class on Wednesday was in discussing a homework problem about a bicycle moving down a hill at constant speed due to a drag force of magnitude cv. The schema that one student had put together looked something like this:

The students were wondering how they would combine the friction from the ground and the air drag force into one to use the given information.

I was floored - after giving this problem for four years in a row, this was the first time the students even thought to think of anything about the friction on the ground. They decided to neglect this force after we thought about whether drag force had anything to do with the ground, but the fact that we even had this discussion was amazing and really shows the power of the schema to get students to think about what they are doing.

The final thing the class pointed out was an inconsistency that had again never even occurred to me. On Wednesday, we were looking at the following sketch as part of a kinetic friction problem:

The block was moving at constant velocity across a surface with coefficient of friction of 0.7. I asked the students to draw a schema, FBD, and figure out what the magnitude of the force F must be. They started working on their schemata, but then had these uncomfortable looks on their faces shortly afterwards.

Looking at the diagram, they had no problem identifying the effects of the entire earth and the ground, and they were fairly sure based on the situation that drag was not an important part of it. The thing they really didn't know how to handle was that disembodied force F.

What object was causing it? Where was it coming from? How in the world could they include it in the diagram if they didn't know what interaction was governing its presence?

At this point in previous years, students didn't generally mind that random forces were being applied to blocks, spheres, or other random shapes - they just knew that they had to do a sum of the net force in x and y and solve for unknowns in the problem. In the context of the schema, however, the students were clearly thinking about the situation in exactly the way I had taught them to do and were genuinely concerned that there was no clear source of this force. This goes back to the fact that they were seeing the system schema as a representation of real objects, which is really what we want students to be doing! I had never thought about this before, but it was so amazing to know that they were having these thoughts on the second day of meeting the free body diagram.

We agreed on the spot, given my omnipresent power as a physics teacher, that any time a force appeared in a problem diagram that had no clear source, that it had to be because of an interaction with me, and they could include me in the schema to indicate that interaction. For the purposes of satisfying their newly found need for a source for every force (a possible catch phrase for schemata?) they now have permission to do this in their schemata.

I admit that my students in the past have gotten away with abstracting the process of equilibrium problems into barely more than a math problem. That capability has still gotten them to analyze some interesting situations and pushed them to explain phenomena that they observe in their own lives. Still, the way using the schema changed our conversations over the past couple days is an impressive piece of evidence in favor of using them.

In short? I'm sold. I'll take twenty.

What do you do when they don't need you?

I've tried an experiment over the last two days - my advanced algebra students and geometry students each had some challenging tasks that I sort of left to them to figure out. Last year, I taught them very explicitly how to do the tasks at hand, modeled some examples along side their own work, and then gave them time in class to practice. For homework, I gave them more problems that were similar to those we did in class, giving them more chances to practice what I had assigned them.

This year I turned it around. In geometry, we are starting proofs. I gave them a couple relatively simple ones, and asked them in groups of two to construct some sort of logical reasoning to go from a starting point to proving the statement I had given them. There was a lot of struggling, difficulty stating using facts why one logical statement led to another. Over time, they did start communicating with each other and sharing what they were thinking. I did occasionally poke one group in a certain direction, but didn't lead the whole group in that way. Eventually they were all thinking along the lines that I envisioned at the beginning. I could have modeled for them what I did last year, but I saw a lot of really good conversations along the way. By the end, they were much closer to making their own proofs than they had in the beginning. By the end, they were clearly seeing the connections between thoughts. This was only the second class period during which we had talked about proofs. While I don't think any of them would wager large sums of money over constructing geometric proofs, I think they at least see how the system can be used to make logical statements that are irrefutable.

I did something similar with the advanced algebra group which was to figure out graphing absolute value functions during our lesson last Friday. I gave them an exploration that was, in hindsight, confusing and didn't do much aside from frustrate them with Geogebra commands. I told them that I wanted them to use Geogebra, the textbook, Wolfram Alpha, and any other resources available to learn how to graph any arbitrary absolute value function by hand. At the end of the class I broke down and apologized for giving a poorly designed exploration. I told them I would put together a video on graphing functions, and I did - posted it on the wiki.

When we had class on Tuesday, I found out that none of the students had actually taken advantage of the video. They had looked in the textbook. They had graphed functions over and over. When I asked students to share what they had figured out, one student used a table of values and a piecewise function based on the sign of the argument of the absolute value function. Another student had graphed both the argument of the absolute value function and its opposite since that was what this student had observed, and then erased either the top half or bottom half of the graph. Another student broke down and did what the book said to do. By the end, all of them were graphing absolute value functions using their own method. I wasn't sure about understanding, but in the end, I admit that I didn't quite mind. They all had their own models for what was going on, and they were confident that they could use the technology to confirm whether what they were doing was right or not.

I have always wondered about what I would do in the situation where it was clear a group of students didn't need me to be there. In a way, this is part of my fear of tools like Khan academy - if there are others out there that are more engaging, better at explaining ideas, or better at coming up with really interesting questions that got students thinking about what they were doing, and these people happened to make videos: what would I do in my classroom if students got hold of these videos? Would I be mad? Or would I figure out a way to take advantage of the fact that students had figured something out on their own and use it to do something even more interesting or impressive?

I think I do a good job of engaging students - today we were talking about differentiability and we were joking like crazy about whether functions would be differentiable at a given point or not. Really? Were we really joking about this? It seemed like everyone was minimally entertained, but based on the questions I was bouncing around from person to person, it seemed like they also understood the concept. I know teachers that are more effective though at making kids understand and be entertained and do problem after problem until concepts are so painfully clear that they become automatic. These teachers could easily make a career in stand-up or television based on their comedic brilliance and presence - what if they decide to make videos?

What then? What do I do? If I get to that point, is it the end of my usefulness as a teacher?

Or is that just the beginning? Maybe that's the point where I can assume my students have a certain level of basic knowledge and I can then build off of that level to do even cooler stuff. Maybe that's where I can assume, for once, that my students have a base level of skills, and can then rise above to analyze bridges or patterns in nature or create a mathematical model that inspires a student to choose to be a doctor or engineer where he or she never would have if I hadn't done the right project or group activity or lesson. Maybe the fact that I entrusted my students to try to figure out something on their own is enough that they feel empowered to try things even if failing again and again is a possibility. Make mistakes and come back asking questions about why their theories were incorrect. I remember worrying at one point in my career what horrible things would happen if I somehow introduced students to a concept in a way that it caused them get a question wrong and cause them just one more failure in a line of failures. If I could teach in a way that makes students feel OK coming in the next day saying "I didn't get it, but this is what I tried" I'd feel pretty good about myself. That is, ultimately, the sort of resilience that a person needs to survive in this world.

If the students show us that they don't need us to show them how to solve a specific problem, then we as teachers should honestly accept this fact. Our goals, if they can do the problem in a way that is mathematically correct, should shift to applying that ability to doing something more profound and relevant, be it communicating that solution or applying the solution to a new situation that is different but connected in a subtle way. Our job puts us in contact with some really amazing minds that are eager to do what we say in some circumstances. In the cases that they demonstrate that they don't need us - that is when we must apply our professional judgment and teach them to expand their knowledge to something bigger than themselves.

Not sure why I'm waxing so philosophical today, but I've been really impressed with my students this week, and it's only Wednesday! After these few days, it feels like what I'm doing with this group is like using a super computer to do word processing. I only hope they are enjoying the process as much as I am.