Monthly Archives: November 2011

Math Portfolio - Sharing my own story.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Teaching Proofs in Geometry - What I do.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Graphical Systems in Geogebra and crashing LEGO robots in Algebra 2

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

I used this as my warm-up activity today:

a) Estimate the solution of the system.

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

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

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

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

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

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

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

The rules:

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

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

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


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

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

UPDATE: This is the video of the next day's class when students solved their functions. I set the robots apart from each other and the students did the rest.

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.