# Tag Archives: reflection

## Exponent rules and Witchcraft

I just received this email from a student:

I FINALLY UNDERSTAND YOUR WITCHCRAFT OF WHY 3 TO THE POWER OF 0 IS ONE.

3^0 = 3^(1 + -1) = (3^1)*(3^-1) = 3 * (1/3)

Talk about an accomplished summer.

This group in Algebra 2 took a lot of convincing. I went through about four or five different approaches to proving this. They objected to using laws of exponents since 30 is one of the rules of exponents. They didn’t like writing out factors and dividing them out. They didn’t like following patterns. While they did accept that they could use the exponent rule as fact, they didn’t like doing this. I really liked that they pushed me so far on this, and I don’t entirely believe that their disbelief was simply a method of delaying the lesson of the day.

Whatever it was that led this particular student to have such a revelation, it makes me incredibly proud that this student chose to follow that lead, especially given that it is the middle of summer vacation. Despite labeling the content of the course ‘witchcraft’, I’m marking this down in the ‘win’ column.

Filed under algebra 2, teaching stories

## 2012-2013 Year In Review – Standards Based Grading

This is the first in a series of posts about things I did with my classes this year.

When I made the decision last fall to commit to standards based grading, these were the main unknowns that hung at the back of my mind:

• How would students respond to the change?
• How would my own use of SBG change over the course of the year?
• How would using SBG change the way I plan, teach, and assess?

These questions will all be answered as I reflect in this post.

### What did I do?

In the beginning of the year, I used a purely binary system of SBG – were students proficient or not? If they were proficient, they had a 5/5. Not yet proficient students received a 0/5 for a given standard. All of these scores included a 5 point base grade to be able to implement this in PowerSchool.

As the semester went on, the possible proficiency levels changed to a 0, 2.5, or 5. This was in response to students making progress in developing their skills (and getting feedback on their progress through Blue Harvest but not seeing visible changes to their course grade. As much as I encouraged students not to worry about the grade, I also wanted to be able to show progress through the breakdown of each unit’s skills through PowerSchool. It served as a communication channel to both parents and the students on what they were learning, and I could see students feeling a bit unsatisfied by getting a few questions correct, but not getting marked as proficient yet. I also figured out that I needed to do more work defining what it meant to be proficient before I could really run a binary system.

By the start of the second semester, I used this scheme for the meaning of each proficiency score:

• 1 – You’ve demonstrated basic awareness of the vocabulary and definitions of the standard. You aren’t able to solve problems from start to finish, even with help, but you can answer yes/no or true or false questions correctly about the ideas for this standard.
• 2 – You can solve a problem from start to finish with your notes, another student, or your teacher reminding you what you need to do. You are not only able to identify the vocabulary or definitions for a given skill, but can substitute values and write equations that can be solved to find values for definitions. If you are unable to solve an equation related to this standard due to weak algebra skills, you won’t be moving on to the next level on this standard.
• 3 – You can independently solve a question related to the standard without help from notes, other students, or the teacher. This score is what you receive when you do well on a quiz assessing a single standard. This score will also be the maximum you will receive on this standard if you consistently make arithmetic or algebraic errors on problems related to this standard.
• 4 – You have shown you can apply concepts related to this standard on an in-class exam or in another situation where you must identify which concepts are involved in solving a problem. This compares to success on a quiz on which you know the standard being assessed. You can apply the content of a standard in a new context that you have not seen before. You can clearly explain your reasoning, but have some difficulty using precise mathematical language.
• 5 – You have met or exceeded the maximum expectations for proficiency on this standard. You have completed a project of your own design, written a program, or made some other creative demonstration of your ability to apply this standard together with other standards of the unit. You are able to clearly explain your reasoning in the context of precise mathematical definitions and language.

All of the standards in a unit were equally weighted. All units had between 5 and 7 standards. In most classes, the standards grade was 90% of the overall course grade, the exception being AP Calculus and AP Physics, where it was 30%. In contrast to first semester, students needed to sign up online for any standards they wanted to retake the following day. The maximum number of standards they could retake in a day was limited to two. I actually held students to this (again, in contrast to first semester), and I am really glad that I did.

Before I start my post, I need to thank Daniel Schneider for his brilliant post on how SBG changes everything here. I agree with the majority of his points, and will try not to repeat them below.

### What worked:

• Students were uniformly positive about being able to focus on specific skills or concepts separate from each other. The clarity of knowing that they needed to know led some students to be more independent in their learning. Some students made the conscious decision to not pursue certain standards that they felt were too difficult for them. The most positive aspect of their response was that students felt the system was, above all else, a fair representation of their understanding of the class.
• Defining the standards at the beginning of the unit was incredibly useful for setting the course and the context for the lessons that followed. While I have previously spent time sketching a unit plan out of what I wanted students to be able to do at the end, SBG required me not only to define specifically what my students needed to do, but also to communicate that definition clearly to students. That last part is the game changer. It got both me and the students defining and exploring what it means to be proficient in the context of a specific skill. Rather than saying “you got these questions wrong”, I was able to say “you were able to answer this when I was there helping you, but not when I left you alone to do it without help. That’s a 2.”
• SBG helped all students in the class be more involved and independent in making decisions about their own learning. The strongest students quickly figured out the basics of each standard and worked to apply them to as many different contexts as possible. They worked on communicating their ideas and digging in to solve difficult problems that probed the edges of their understanding. The weaker students could prioritize those standards that seemed easiest to them, and often framed their questions around the basic vocabulary, understanding definitions, and setting up a plan to a problem solution without necessarily knowing how to actually carry out that plan. I also changed my questions to students based on what I knew about their proficiency, and students came to understand that I was asking a level 1 question compared with a level 3 question. I also had some students giving a standards quiz back to me after deciding that they knew they weren’t ready to show me what they knew. They asked for retakes later on when they were prepared. That was pretty cool.
• Every test question was another opportunity to demonstrate proficiency, not lose points. It was remarkably freeing to delete all of the point values from questions that I used from previous exams. Students also responded in a positive way. I found in some cases that because students weren’t sure which standard was being assessed, they were more willing to try on problems that they might have otherwise left blank. There’s still more work to be done on this, but I looked forward to grading exams to see what students did on the various problems. *Ok, maybe look forward is the wrong term. But it still was really cool to see student anxiety and fear about exams decrease to some extent.

### What needs work:

• Students want more detail in defining what each standard means. The students came up with the perfect way to address this – sample problems or questions that relate to each standard. While the students were pretty good at sorting problems at the end of the unit based on the relevant standards, they were not typically able to do this at the beginning. The earlier they understand what is involved in each standard, the more quickly they can focus their work to achieve proficiency. That’s an easy order to fill.
• I need to do more outreach to parents on what the standards mean. I thought about making a video at the beginning of the year that showed the basics, but I realize now that it took me the entire year to understand exactly what I meant by the different standards grades. Now that I really understand the system better, I’ll be able to do an introduction when the new year begins.
• The system didn’t help those students that refuse to do what they know they need to do to improve their learning. This system did help in helping these students know with even more clarity what they need to work on. I was not fully effective in helping all students act on this need in a way that worked for them.
• Reassessment isn’t the ongoing process that it needs to be. I had 80 of the 162 reassessment requests for this semester happen in the last week of the semester. Luckily I made my reassessment system in Python work in time to make this less of a headache than it was at the end of the first semester. I made it a habit to regularly give standards quizzes between 1 or 2 classes after being exposed to the standard for the first time. These quizzes did not assess previous standards, however, so a student’s retake opportunities were squarely on his or her own shoulders. I’m not convinced this increased responsibility is a problem, but making it an ongoing part of my class needs to be a priority for planning the new year.

I am really glad to have made the step to SBG this year. It is the biggest structural change I’ve made to my grading policy ever. It led to some of the most candid and productive conversations with students about the learning learning process that I’ve ever had. I’m going to stop with the superlatives, even though they are warranted.

Filed under reflection, teaching philosophy

## Angry Birds Project – Results and Post-Mortem

In my post last week, I detailed what I was having students do to get some experience modeling quadratic functions using Angry Birds. I was at the 21CL conference in Hong Kong, so the students did this with a substitute teacher. The student teams each submitted their five predictions for the ratio of hit distance to the distance from the slingshot to the edge of the picture. I brought them into Geogebra and created a set of pictures like this one:

After learning some features of Camtasia I hadn’t yet used, I put together this summary video of the activity:

I played the video, and the students were engaged watching the videos, but there was a general sense of dread (not suspense) on their faces as the team with the best predictions was revealed. This, of course, made me really nervous. They did clap for the winners when they were revealed, and we had some good discussion about modeling, which videos were more difficult and why, but there was a general sense of discomfort all through this activity. Given that I wasn’t quite able to figure out exactly why they were being so awkward, I asked them what they thought of the activity on a scale of 1 – 10.

They hated it.

I should have guessed there might be something wrong when I received three separate emails from the three members one team with results that were completely different. Seeing three members of one team work independently (and inefficiently) is something I’m pretty tuned in to when I am in the room, but this was bigger. It didn’t sound like there was much utilization of the fact that they were in teams. I need to ask about this, but I think they were all working in parallel rather than dividing up the labor, talking about their results, and comparing to each other.

• I need to be a lot more aware of the level of my own excitement around activity in comparison to that of the students. I showed one of the shortened videos at the end of the previous class and asked what questions they really wanted to know. They all said they wanted to know where the bird would land, but in all honesty, I think they were being charitable. They didn’t really care that much. In the game, you learn shortly after whether the bird you fling will hit where you want it to or not. Here, they had to go through a process of importing a picture, fitting a parabola, and finding a zero of a function using Geogebra, and then went a weekend without knowing.

While it is true that using a computer made this task possible, and was more enjoyable than being forced to do this by hand, the relativity of this scale should be suspect. “Oh good, you’re giving me pain meds after pulling my tooth. Let’s do this again!”

• A note about pseudocontext – throwing Angry Birds in to a project does not by itself does not necessarily engage students. It is a way in. I think the way I did this was less contrived than other similar projects I’ve seen, but that didn’t make it a good one. Trying to make things ‘relevant’ by connecting math to something the students like can look desperate if done in the wrong way. I think this was the wrong way.
• I would have gotten a lot more mileage out of the video if I had stopped it here:

That would have been relevant to them, and probably would have resulted in turning this activity back around. I am kicking myself for not doing that. Seriously. That moment WAS when the students were all watching and interested, and I missed it.

Next time. You try and fail and reflect – I’m still glad I did it.

We went on to have a lovely conversation about complex numbers and the equation $x^{2}+4 = 0$. One student immediately said that $\sqrt{-2}$ was just fine to substitute. Another stayed after class to explain why she thought it was a disturbing idea.

No harm done.

P.S. – Anyone who uses this post as a reason not to try these ideas out with their class and to instead slog on with standard lectures has missed the point. I didn’t do this completely right. That doesn’t mean it couldn’t be a home run in the right hands.

## Why computational thinking matters – Part I

My presentation at 21CLHK yesterday was an attempt to summarize much of the exploration I’ve done over the past year in my classroom into the connection between learning mathematical concepts and programming. I see a lot of potential there, but the details about how to integrate it effectively and naturally still need to be fleshed out.

After the presentation, I felt there needed to be some way to keep the content active other than just posting the slides. I’ve decided to take some of the main pieces of the presentation and package them as videos describing my thinking. I’m seeing this as an iterative process – in all likelihood, these videos will change as I refine my understanding of what I understand about the situation. Here is the start of what will hopefully be a developing collection:

I want to express my appreciation to Dan Meyer for his time chatting with during the conference about my ideas on making computation a part of the classroom experience. He pushed back against some my assertions and was honest about which arguments made sense and which needed more definition. I think this is a big deal, but the message on the power of computational thinking has to be spot on so it isn’t misunderstood or misused.

With the help of the edu-blogging community, I think we can nail this thing down together. Let’s talk.

Filed under computational-thinking, reflection

## Four (not so) easy pieces – 21CLHK Debrief

I have had an amazing time over the last couple of days at the 21st Century Learning conference in Hong Kong. It’s easy for a technology conference to dip into the red zone of using technology for its own sake. The presenters and attendees here though remained really focused on creating meaningful experiences for students through the tools as a fundamental principle, not an afterthought.

I’m tired, but I think it’s important to note down a few important points that I want to remember to put into action when I return to school. These points, likely by design by the organizers, are framed nicely by the keynote speakers and their messages.

### Teach students how to manage device anxiety. (Dr. Larry Rosen)

This needs to be done explicitly and modeled by teachers. This will not get better by accident – instead, we must make an effort to show students how to avoid losing focus through deliberate practice.

### Help students form identities as producers of media. (Dr. Nichole Pinkard)

Hoping that students will produce excellent work in a context that does not extend beyond the classroom doors is a sure way to expect less from them. We must expect students to define their place in the digital media world through work that they find meaningful.

### Connect students to the rest of the world. (Dr. Jennifer Lane)

With all of the expertise available through the internet, there is no excuse for limiting students to finding out information in isolation from people that use that information to do their jobs. I want to find feasible ways to put my students in contact with people that are solving interesting problems and doing real world work.

### Share what I find perplexing with students, and help them do the same. (Dan Meyer)

Though it might feel good to say that I want my students to find things that interesting, I need to model this process if I want students to adopt it for themselves. Curating my own list of perplexing ideas and helping students maintain their own list is a perfect way to make this more than a pipe dream.

Here’s to meeting you all again soon. Safe travels!

Filed under reflection

## A tale of two gradebooks – my SBG journey continues

I realized this morning that I could look back at the assignments from my PowerSchool gradebook from a year ago and see the distribution of assignments I had by the end of the semester:

My grades were category based – 5% class-work, 10% homework completion, 10% portfolio, 60% unit tests, and 15% quizzes. This comprised 80% of the semester grade, and was the grade that students saw for the majority of the semester. A semester exam at the end made up the remaining 20%.

While I did enter some information about the homework assignments, my grade was just a reflection of how they completed it relative to the effort I expected them to make while working on it. No penalty for being wrong on problems, but a cumulative penalty developed over time for students tending not to turn it in. This, however, was essentially a behavior grade, and not an indication of what they were actually learning. The homework was the most frequent way for students to get feedback, and it did help students improve in what they were learning, but the completion grade was definitely not a measure of what they were learning at all. There were six quizzes that fit into my reassessment system. Not important enough to matter, I realize now with 20/20 hindsight.

The entire Standards-based-grading community shoots me a look saying ‘we told you so’, but only momentarily and without even a hint of snark. They know I am on their side now.

Here is a screen shot of the assignments in my grade-book as of this morning:

There is a clear indication of what my students have been working on here. With the exception of the portfolio, a student can look at this (and the descriptions I’ve included for each standard) and have a pretty good idea of what they did and didn’t understand over the course of the semester. They know what they should be working on before the semester exam next week. The parents can get a pretty good idea of what they are looking at as well. I knew making the change to standards based grading (SBG) made sense, but there have been so many additional reasons I am happy to have made the change that I really don’t want to go back to the old system.

I’ll do more of a post-game analysis of my SBG implementation in PowerSchool soon. I will be making changes and enhancing parts that I like about what I have done so far. I have to first make it through the busy time ahead of marking exams, submitting comments, and getting my life ready for the extended winter break that is peeking its beautiful head over the piles of reassessments on my desk. It is really satisfying to see that my students have weathered the transition to SBG beautifully. Their grades really do emphasize the positive aspects of learning that a pure assignments & points system blurs without thinking twice.

Filed under reflection

## 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.

Filed under reflection, teaching philosophy

## 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.

Filed under physics

## 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.

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.

Filed under reflection, Uncategorized

## Nice work when you can get it….

I graded my first set of physics tests today. With the group deciding not to take the class at the Advanced Placement level, we’ve been able to slow down and spend time experimenting and really engaging with kinematics and projectile motion. I assigned them problems and helped them learn, but I was more impressed with the experiments they worked on and their engagement level during those activities. I was concerned about what would happen when we returned to solving problems, but I was very pleasantly surprised.

I’m interested in sharing student work, so this is the first time I will be doing it. When I asked the student if it would be OK to share, the student agreed and was really excited that I would want to show the work to other teachers.

This student started out solving problems in a very scattered way: calculations here, sketches there, units nowhere to be found. When I showed the structure I wanted students to use to solve problems, it was initially a burden. The student didn’t like doing it. Upon grading, I was very happy to see this:

The degrees vs. radians issue is one that I always battle, made especially difficult this year because students have me for physics (when I insist on degree mode almost exclusively) and then calculus (when I change my insistence to radian mode) right in a row. Yes, the student should have noticed that multiplying by the ratio should not have resulted in a negative sign for initial y-velocity. Yes, it should have again become obvious that something was up when he found he needed to ‘add’ a negative sign in the answer at the end to make the sign of the answer make sense according to his own sign convention.

The fact that the student can notice these things (and that I can see where the errors are) would not even be something I could discuss if it wasn’t for the structure I put in place. By learning to use the structure to organize thoughts, this student became able to solve problems in a logical manner rather than with calculations all over the page. I don’t like teaching procedures, but this is an example of where it pays off.

We like students exploring and experimenting and constructing their own knowledge. These are really good ways for them to spend time in our classroom. I include using correct mathematical notation, showing steps, labeling axes, learning terminology, and other things of that nature as part of my class expectations – at times a battle that seems unimportant in the context of what I really want students to know how to do in five or ten years. There is room for procedural knowledge, however, and this student’s success is evidence of why we do it.

The main point (and the thing I’ve been working to change compared with how I did things for a while) is that these procedures should not be the meat of a lesson or the main focus of instruction all the time. These things CAN be taught by computers or videos and don’t necessarily need a human in the room. It is important for students to have skills and have access to resources that help them develop those skills.

But getting answers is not the point – this is the tricky part that we have to do a better job of selling to students, at least I do. Clear communication of reasoning and sharing the logic of our ideas are some of those “21st century skills” that students should have when they leave our classrooms. If a student needs to learn a structure to help them with this process, it is worth the time needed to help them learn it.