## Persistence and Creativity

I spent six hours yesterday watching students take final exams. Three two-hour shifts. Sigh…

My last group taking an exam was my AP Calculus BC team. They had 24 multiple-choice questions and four free response questions, so they had to be efficient in their problem-solving. One of my students asked me about a multiple choice question that troubled him. It was an infinite geometric series question. This student joined our school, and our country, last year as a freshman. He place tested so highly that he started in AP Calculus AB. However, it has become apparent that there are a few facts/skills that he does not have at his command. This is rarely a problem since he is so creative. On this question, he did not have the formula in his brain for calculating the sum of an infinite geometric series. He could have listed out a handful of terms to look for convergence. He could have shrugged his shoulders and guessed since it was one of twenty-four questions. Instead, he wrote a short program on his TI-84 that gave him increasingly good approximations until he saw one of the answer choices emerge. He did this in the middle of a two-hour exam! He asked me afterwards and just laughed when I showed him how easy this problem could have been. I am convinced that he will remember this formula forever now, but I am also convinced that I will remember this story forever now. His ability to problem-solve in this situation is SO much more powerful than having a formula at his command. It is interesting that this happened on the same day that I sent our a question on twitter about the use of formula sheets on assessments. I am disinclined to use formula sheets, but I could be convinced otherwise. Here is a story that would have never happened if a formula sheet was present.

I am going to cross post this over at the One Good Thing blog space.

## Another Fun Problem Debate

A super quick post this morning. On my last problem set for AP Calculus BC I included the problem below:

A 10 m rope is fastened to one of the outside corners of a house, which has the form of a rectangle, 6 m long and 4 m wide. A dog is fastened to the rope. What is the perimeter of the region that the dog can access?

I have asked a form of this question a couple of times over the years. One year I did not mention that the rope was on the exterior of the house and I had a student assume that the dog was tied to a leash inside the house. I fixed that mistake.

A GeoGebra sketch below shows the image I have in my head for this problem.

My answer to this question, and the answer that 11 of my 14 students had, was that the perimeter is 20π. Two students argued that the answer should be 20π + 20. Their argument is that the borders of the house, the sides of the rectangle in the drawing, are also part of the perimeter. I loved the debate that ensued and most students migrated to this point of view. I reflexively thought of perimeter as exterior, while these two students argued that perimeter is boundary. I think I agree with them and I LOVE the fact that they cared enough to debate this on a problem that counted for one point out of about 400 something for the term. I also LOVE that students who got their answer marked correct started arguing against the answer that they arrived at.

## AP Calculus BC – A Lovely Debate

It has been WAY too long since I posted. Energy has been scattered in a number of directions lately…

Yesterday in AP Calculus BC their sixth problem set of the year was due. I heard quite a bit of debate among my young scholars about one question in particular. Here it is:

Now, the curve with the integral from 0 to 12 highlighted:

Finally, the curve with the integral from 1 to 13 highlighted:

We have a test today, so I pivoted the class conversation toward some practice problems but I felt that there was probably still more to discuss about this problem. Days like yesterday remind me of how spoiled I am that I get to do math with a group of scholars that are willing to engage in energetic debate about math. They are rarely interested in just knowing the answer, they seem to genuinely enjoy to process that gets them there. I am not going to pretend that all 15 students were actively engaged joining in with their ideas. I will say that there were all at least attentive and willing to let the conversation flow without pointing out, anxiously, that we need to talk about the upcoming test.

We decided a number of years ago to teach BC as a second year Calculus course at our school. We know that many of our students can accelerate through the BC curriculum in one fast year or by buying some Precalculus Honors time, but conversations like this one would be hard (not impossible) to find time for. Every year I hear a scholar say something along the lines of ‘I knew how to do this before but now I think I know why it works.’ Statements like that one, and conversations like yesterday’s make me feel good about this curriculum decision we made.

## Checking in on a new Policy

I recently blogged about a commitment made by our math department. As a reminder to you, dear reader, here is the statement from one of my course syllabi:

Beginning in the 2017 – 2018 academic year, our math department is adopting a policy of expecting test corrections on all in-class tests. The policy is described below.

• When grading tests initially each question will get one of three point assignments
• Full credit for reasonable support work and correct answer.
• Half-credit for minor mistakes as long as some reasoning is shown.
• Zero credit (in very rare cases) when there is no reasonable support shown or if the question is simply left blank.
• When grading tests, I will not put comments, I will simply mark one of these three ways.
• You will be allowed to turn in corrections. Corrections will be on separate paper and will have written explanations of errors made in addition to the correct work and answer. This work is to be in the student’s words but can be the result of consultation/help. These corrections will always be due at the beginning of the second class meeting day after the assessment is returned. You will return your original test along with your correction notes. I will remind you of this every time I return a graded test to you.
• It is not required that you turn in test corrections.
• The student can earn up to half of the points they missed on each individual problem.
• This policy does not apply to quizzes, only to in-class tests.

There are  a couple of items in the learning curve to report on here. I will lead with the positives –

• We have an after school conference time that many students take advantage of for extra help. My room has been more crowded this year than it has been for years. I take this as a plus sign that students are committed to seeking help and investing in their math studies. Many of them are there talking with me and with their classmates about test corrections.
• I have had a number of students turn in their corrections the same day that they received their tests. It is a rare thing for students to turn in work days early, it is happening regularly right now.
• I walked out of my classroom today with a new student who was talking excitedly about how she is really thinking carefully about her work and she is sure that she’ll remember material better because of this.
• It is faster to return tests since I am simply marking them 100%, 50%, or 0% for each problem.

• I anticipated that there would be very few zero scores on problems. There have been more than I thought but I hope that is a factor of students learning to show some work. This may be a positive as a zero stings a bit and they may be more inclined to be careful in their explications.
• Some students feel stung by relatively minor mistakes that initially result in a 50% on an individual problem. These minor mistakes turn into 75%. I am trying to point out that relatively major mistakes can also end up at the 75% level but some students feel a bit cheated.
• Early in the year averages fluctuate quite a bit anyway, but the fluctuations are exaggerated in this system. I see already that overall averages are a bit higher than normal and there is much less variance in scores. However, some students are scared since they have a hard time seeing the long game as clearly.
• Explaining this to folks outside the department has been a bit of a challenge.

We have midterm grade comments looming and as department chair I know I will have some questions coming my way about our reasoning and the long-term impact on grades. I will try to steer the conversations to long-term impact on learning and self-sufficiency. So far the policy has exceeded my hopes in my classes. I will be checking in with my department to get other points of view soon and I plan on sharing some of those conversations as well.

## Should My Preferences be my Students’ Preferences?

At the beginning of every year I am reminded that there are little quirks I have about how I would like to see answers presented and I have this silly assumption that my brand new students should just know what I want. I wish that I remembered this in August and prepped myself and my students with some conversations. I am writing this brief post this morning while one of my classes is taking a test. I would love some feedback here in the comments or over on twitter where I am @mrdardy

These are tiny, tiny problems to focus on. I recognize this. But I also know that I will spend many months with this group of students and I want them to understand my thoughts since I am asking them to routinely explain theirs on paper to me. I would love to hear points of view about how important any of these habits are for kids of this caliber. I am completely open to recognizing that these are just weird little quirks of mine, but I also hope that there is some mathematical logic underlying all of this. Please drop a line to let me know what you think and how you have responded to students in your efforts to help them develop a clear and consistent strategy for communicating their mathematical ideas.

## So, What Kinds of Change?

In my last post I wrote about our department’s terrific two day workshop with Henri Picciotto. One of the major decisions we made based on the time we spent together is that we have decided, as a whole department team, is that we will allow test corrections on all tests in our department. Before I dive into the format of the decision we made, I want to include a couple of important links here with other points of view about assessment policies. The first comes from a new twitter contact Steve Gnagni (@Steve_Gnagni) who shared this interesting document written by Rick Wormeli (@rickwormeli AND @rickwormeli2 for reasons I am not sure I understand!) called Redos and Retakes Done Right and the second is a link Henri shared gathering together some of his ideas about assessments.

Or, I should say I was totally excited about it. I know that there are different ways to view this process and the meaning of it. I know that we decided that events that we call tests are subject to this correction policy. We decided (for a number of reasons, some more ideologically defensible than others) that short quizzes were not subject to this policy. I know that I will be balancing this with graded take-home problem sets and on these problem sets I always encourage collaboration. So, when Steve Gnagni shared the article above, I found myself doubting some of the decisions we made. I found old reactions about grades being really seriously challenged and I began to doubt whether our decision on process is ideologically pure enough. I also know that this is progress. I will be sharing Rick Wormeli’s article with my team in the fall and we will be checking in with each other on how we feel about the impact of this new process.

I want to thank Henri again and to thank my new twitter pal Steve Gnagni for sharing their ideas. As long as we are all willing to keep questioning ourselves we can continue to help our students grow.

## My Students are Making Some Smart Guesses

This conversation was a wonderful way to end our day on Friday. I am delighted that my students are comfortable enough to make these guesses out loud and even more delighted that they are making such good guesses right now. I pointed out how helpful it is to play with GeoGebra to check these guesses and I hope (I hope hope hope!) that some of my students are making a habit of this.

## A Delightful Conversation

Last week in my Geometry class we had a fantastic conversation about a homework problem. Here is the problem in question –

I wish that I could take credit for having written this, but I am certain that I ‘borrowed’ it from somewhere. Likely from the fantastic resources shared with me by Carmel Schettino (@SchettinoPBL)

If you recognize the above problem as your own, feel free to claim it and let me know. Know in advance that I am very grateful for such a rich problem to tie together ideas of distances, slopes, line equations, properties of squares, and triangle congruencies all into one tidy package!

## Hands-On Geometry

I’ve been at this high school math gig for a good long while now but I periodically have to remind myself of a couple of important facts. The most important one is that not everybody’s mind works like mine. Just because I like a certain way of thinking, or dislike a certain way of learning, I should not assume all my students will agree. In fact, I can be pretty certain that all of my students will not agree, there’s too many individuals for that to work.

When I studied Geometry I did not like physical drawings and constructions. In part because I am a bit inept when it comes to controlling something like a compass, but also because getting my hands engaged does not seem to fire too many of my neurons. So, when I wrote my Geometry book a couple of years ago I did not include much in the way of hands-on manipulations. The past couple of years of working through the text with our students has pointed out the weakness of this approach. So, I put my head together with one of my talented colleagues to try and make an activity that would trigger some neurons for those students who come to life when they get their hands busy. I had been using a pretty cool activity I ran across from Jennifer Silverman but I made pretty flimsy paper copies to work with on a pipe building activity where kids had to manipulate bent angle joints with different pipe lengths. It’s a great activity but using simple paper copies dragged the activity down. We invested in some packs of AngLegs this year and my colleague wrote a pretty cool activity modeled off of our pipe building activity. You can find his document here.

I was impressed as each of the seat groups in my class played with the AngLegs making some discoveries about combinations that worked and those that would not. We discussed, without naming it yet, the triangle inequality theorem to explain why some combos did not work. But the real fun, and the clever heart of my colleague’s activity, was when I asked one student from each group to come to the front of the room. When they left their group the remaining group members were given the following task – I slightly modified the original document on the fly – I asked them to make and measure a triangle. Find six measures, the three side lengths and the three angles. They then put the triangle away where it could not be seen. I sent the volunteers back and their teammate gave them three pieces of information. I left it to each group to decide what information to share. Once given three clues the volunteer student needed to manipulate the AngLegs to copy the triangle described. What ensued was a terrific conversation about what information is necessary to guarantee that I have to make the same triangle. We used this as a launching pad to discuss congruence theorems for triangles. I have some great links in the text to some wonderful GeoGebra activities up on the GeoGebraTube site but I know that many of my students do not do these explorations.  I also know that some just need to get their hands dirty, so to speak. Some kids were able to recreate the triangle but admitted that it was a bit of luck. Some stumbled upon the ambiguous case of the Law of Sines without being told that this is what happened. Some realized that they had no choice but to create the correct triangle.

I was really pleased by the level of engagement and I am now thinking about ways to use the AngLeg sets again soon when we start talking about side and angle bisectors. I want to have tables create and draw their own triangles before we stumble into discoveries about concurrence of these bisectors. This will feel, I hope, a little more authentic than me just giving them a prescribed triangle which may feel a bit like I am just luring them into some pre-prepared trap. I think that this activity we ran benefited my students and we have referred to it on a number of occasions already. The grouping of three or four students together at a time helps and allowing them to get their hands busy has helped. Looking forward to loosening up a bit more and letting my students be more tactile in their approach to Geometry. I’ll still show them the GeoGebra and introduce them to Euclid the Game  but I need to remind myself that they are not a bunch of mini Dardys in the room.