Persistence and Patience

Neither of the qualities in the title of this post are apparent in abundance at this time of the school year (exams here start on May 21!!!) so I am especially pleased to be able to write about today in my two Geometry classes. They each took a quiz with me in their last class and I asked them to read ahead to the next section (more on that later) before meeting again today. After a brief warm up we reviewed the quiz and I returned papers. Then I presented them with this image from our Geometry text

 

Now, at this point we have established that the measure of a central angle in degrees is equal to the measure of its intercepted arc in degrees. We have proven that the measure of an inscribed angle in degrees is half of the measure of its intercepted arc in degrees. I told them that my dream for today was to derive some formula relating the measure of angle EFC to some combination of the arcs BE, EC, CD, and DB. I reminded them of what we already know and suggested that using what we already know is often pretty helpful when trying to learn something new. I then stepped back and let kids toss out ideas. In one of the two classes I took a walk to the water fountain and popped in on a colleague for a two minute chat. I came back into a room with people debating and waving their hands in the air to show what segments they wanted to draw. I wish I saw more drawing at their desks, but it was good engagement. In both classes they wanted me to name the vertical angle pairs at the hinge point F. I was a bit surprised that neither group wanted to name arcs yet, but it worked out just fine. I dropped a hint that another segment drawn would help and this spurred some lively debate about what segments to draw. BE and DC were popular but I pointed out that the vertical angles were not included really in these triangles that were formed. Some folks wanted some radii drawn. One girl was sad when we named our vertical angle pairs. She saw that BFE and DFC were clearly equal but that they faced arcs that clearly weren’t. I was pleased by this observation. There was a real desire in each class to assume that BD and EC were equal and I wish my diagram was more clearly designed to discourage that. We finally settled on drawing the chord BD which created two inscribed angles that we called x and y creating arcs called 2x and 2y. The heavy lifting and guessing were done by that point. Arriving at the conclusion that we now need a relationship between one angle and two arcs came in a couple of steps. In both classes I made a clear statement about what we just discovered and they seemed pretty pleased with themselves, if a bit tired from the exertion. In my morning class we were on our 90 minute block and I gave them some practice exercises that we will revisit tomorrow. In my afternoon class that ended the day we simply left the discovery on the board. I start tomorrow morning with that crew and I am excited to pick this up again.

In each of my classes I have 14 students. One was absent today, so I had 29 overall joining in on this conversation. One of them, when I mentioned my dream for the day, said ‘In the reading last night it said that this angle is the average of two arcs.’ [Thank you Niko!]

One student. Now, it is entirely possible that other students did the reading I requested. It is possible that some read the section and were confused by it. It is possible that some read it and forgot the conclusion. It is possible that some who read it simply did not feel like saying anything out loud. It feels more likely that few (maybe only 1?!?) did the reading.

I could dwell on that, but I am dwelling on the discovery we made. I am dwelling on the persistence and patience of my students today. I am dwelling on what went right to end a day that started poorly for me.

 

Later, I’ll dwell on how I can help overcome the likely habits of not reading that I am faced with. I’ll save that for a less beautiful day than today.

A Guest Post

The last time I wrote, I was talking about how spoiled I am by my students. I have two students who did an enormous amount of thinking about a random walk problem I proposed. Just for your reference, the problem was

Starting at the origin, a bug jumps one unit either left, right, up, or down. He jumps once each second. List the possible locations (and probabilities associated with those locations) that the bug could be after 6 seconds.

After working on it for awhile with one of my students I waived this one off and told my students to feel free to ignore it. A number did not. Two in particular, Bobby and Matthew (both of whom seemed happy to have me name them!) collaborated over a chat line of some sort and delivered a lovely presentation on their work. I asked them to guest post for me and below is their post. I cannot emphasize enough how impressed I am by their ingenuity and determination. There are two other files they asked me to make available.  A 3d representation and a 4d representation generated by their code.

 

Bobby: Last week, a blog post referred to a random walk problem that our Calculus class worked on, and two students that took a coding approach. We are those two students, and we’re here to discuss how we solved the problem, and the far more interesting work that came after it. For those of you who don’t know, the problem was a 2-dimensional random walk of 6 steps.

Matthew: When I first looked at this problem in class, I thought it would give way far more easily than it actually did. My first attempt was to rewrite the problem as choosing from the four cardinal directions a set of six steps, e.g.  {↑ ,→, ↓, ,←,}. By doing so, my hope was to reduce every position on the grid into a set of the bare minimum moves needed to reach it, and pairs of blank spaces which I could fill with a pairs of steps which added to the zero vector. Since the grid had 8-fold symmetry, I was unafraid of solving each point individually, however when the time came to actually do the problem, I found a number of oversights in my initial work which resulted in some pesky double counting.

Bobby: For lazy Calculus students (read: myself), the obvious solution for any random walk problem is to make a computer do it, so i set out to approximate probabilities by running a billion trials of a random walk and counting up the results.

However, this brutish method is relatively unsatisfying, and by the time I finished, Matthew had abandoned his set permutation approach and decided to make a program that iterated through each possible set of 6 steps once and counted up the destinations. This clearly being the better approach, I did the same thing, and we exchanged our results (which matched) and exchanged our code to make improvements: I took some of Matthew’s ideas to make the program easier to scale up and down, and I’d like to think he took some of my ideas and stopped naming variables “DONK,” but I’m sure he didn’t.

This grid shows the number of ways to land at any given point. After a short time looking over our grid of results, we noticed that each edge of the square was the 6th row of pascal’s triangle. A little more poking around and Matthew noticed that the grid was in fact a multiplication table of that row of the triangle with itself, meaning every entry could be written in the form (nCa)*(nCb), with n being the number of steps and a and b being the relative coordinates of the entry in the table. Finally, we made the observation that this is the 6th layer of Pascal’s square pyramid, a 3d structure much like the triangle where each number is the sum of the four above it. Recalling that a 1d random walk is the nth row of Pascal’s triangle where n = # of steps, a pattern seemed to emerge that we hoped might scale up in dimensions.

Bobby:  We did a lot more work, particularly in 3 and 4 dimensions. Our program logic started by defining a variable to track which number walk we were on, and keyed in on the base-four version of that number to define a set of moves for our bug. For example, test 5 translates to 000011 in base 4, and the bug reads each digit individually, right to left, using it as an instruction for motion: 0 = right, 1 = up, 2 = left, 3 = down, translating 000011 to U, U, R, R, R, R.

This logic makes it easy to change the number of dimensions or steps, so we went to 3 dimensions, expecting to get 3-dimensional cross-sections of some 4-dimensional solid for each particular solution. The results did not disappoint, as we got the expected octahedron as a result. Obviously, this output was a little tough to format, but I think what we settled on is easy enough to read. Each grid separated by the others from a row of asterisks is a 2d cross-section of the solution octahedron. LINK DATA Stack those layers on top of each other and the visualization is pretty easy, I think.

At this point, we had to do four dimensions, and had a good idea of how it would look. The 2d solution is the shape of the 1d solution stacked on itself, the 3d solution is a stack of the shapes of the 2d solution, so we the 4d solution would obviously be a series of 3d octahedrons in the same x, y, z, but translated along our fourth spatial axis. It took me a moment to wrap my head around this data, but I’ve become very comfortable with it. LINK DATA Each section under the rows reading “NEW 3D SOLID…” is a series of grids that are stacked into an octahedron, and then the octahedrons are stacked on each other in the fourth spatial dimension. This, of course, is also a 4d cross-section of a 5d Pascalian solid, each nth 4d layer of which is a solution to the 4d random walk of n steps, but that’s about as far as I can go envisioning these shapes. Matthew tells me the 5d structure is “Pascal’s Orthoplexal Hyperpyramid” but I’m not sure I can trust that.

Matthew: Of course, the idea of layers of Pascal’s square pyramid as multiplication tables of the rows of Pascal’s triangle is a fascinating one, and one I sought to prove. Though my initial effort to prove it algebraically did meet with success, I was unsatisfied with my clunky proof and did some quick googling on the subject. My search turned up a blog with a simple proof by induction, far more elegant than anything I had done. Still, I find that proofs by induction are rarely enough to understand a result intuitively, and so I continued working with Bobby, looking for a better way to interpret the formula in terms of the problem, as well as potentially a way of generalizing the result to higher dimensional walks.

Eventually, I took a step back from the symbol soup and searched for a more intuitive interpretation of the formula we had discovered in terms of the original problem. After some while, I had an epiphany while placing the problem back into the language of sets. My initial idea of viewing the problem as selecting a set of 6 steps along cardinal directions could be reformulated in a way which made the formula obvious. Instead of considering steps along the cardinal directions like the original problem had stated, I realized that every unit vector along a cardinal direction could be rewritten uniquely as the sum of two diagonal vectors of magnitude √2/2, e.g. {} = {↖,↗}. With this realization in hand, the original problem can be rephrased as a random walk in six steps of magnitude √2/2 along the line y=x, followed by another random walk of six steps perpendicular to the first. Or, to give an example in my initial interpretation of the problem as sets of steps, the two dimensional walk {↑ ,→, ↓, ,←,} is the same walk as {{↖,↗},{↘,↗},{↘,↙},{↖,↙},{↖,↙},{↘,↙}}, which is the same as the union of the two one dimensional walks {↖,↘,↘,↖,↖,↘} and {↗,↗,↙,↙,↙,↙}. Once the problem is reformulated in this way, the reason why the table obeys such a simple multiplicative relationship in two dimensions becomes satisfyingly clear. Though I am pleased that we were able to transform such an originally dense and unapproachable problem into a setting where it is obvious, we have unfortunately as of yet been unable to find an application of this method to higher dimensions.

I’m Spoiled

This post is inspired by my AP Calculus BC class. At the beginning of each of our seven day rotations, I give them a problem set. These problem sets are pretty wide ranging, some Calculus questions, but mostly wide open fun problems to explore. Here is a problem from the set that was due today (actually due yesterday, but we were snowed out):

Starting at the origin, a bug jumps one unit either left, right, up, or down. He jumps once each second. List the possible locations (and probabilities associated with those locations) that the bug could be after 6 seconds.

I have given them variations of this where the intrepid bug could only move left or right. I thought that this would be an intriguing extension. During my library duty nine nights ago one of my students spent about 15 minutes with me bouncing ideas around. We came up with the GeoGebra sketch below:  

This seemed pretty daunting and I admitted to him that I may have overreached with this problem. We shared our conversation the next day with the whole class, including showing this intimidating graph. Trying to figure out how many of the 4096 pathways could lead to each point felt overwhelming. A couple of students started tossing out ideas but I kind of encouraged them to let this problem slide and blame me for poor planning.

When the problem sets came in today three of my students discussed some coding attacks that they took. Two of them were collaborating late into the night working on an approach to the problem and the output from their attack is below:

They took over the class conversation today pointing out that they recognized a pattern based on a multiplication table where the column header AND the row header were each the 6th row of Pascal’s Triangle. The numbers in their printout above are the products of this table. A long explanation invoking a three dimensional Pascal’s Triangle (I thought of layers like oranges in a conical pile) was presented with great enthusiasm. Some of the kids in class kind of glazed out, but a number were fully engaged. I was engaged, awed, and slightly confused. I have asked the two who collaborated on this if they would be willing to be guest bloggers here and they seem amenable to that idea. I hope to have a follow up, in their words, in the next day or two.

I am SO spoiled to be able to sit back and learn from students who could have easily left this problem alone, but were unsatisfied with that notion.

 

Vocabulary

This post is inspired by a twitter exchange with the awesome Joe Schwartz (@JSchwartz10a) and by a running exchange with one of my Geometry students. Joe tweeted out the following picture: 

The picture was accompanied by the question : “Do 3rd graders know the answer to this question? Truly curious.” It just so happens that my Lil Dardy is in 3rd grade. I showed her the question and (briefly) explained the equation written. I replied to Joe that she was not surprised to see it written that seven sixes is the same a five sixes plus two more. However, she did not know any vocabulary word to describe this. Joe replied, succinctly, “And she doesn’t need one…” It made me smile. It also made me think when I was reviewing for a test with one of my Geometry classes. We just finished a chapter on triangle bisectors and centers. Loads of vocab in this chapter. Very few new skills, just new words describing relationships. Thinking back to the exchange with Joe I found myself questioning my decisions in writing the book and in teaching this chapter. During the test review a student asked if there would be any vocabulary on the test. This particular student has asked this question before just about every test. I answered the way I do just about every time. I told him that he needed to know what these words mean to accurately interpret the questions at hand. For example, if I ask about altitudes to a triangle, he needs to know what that means. However, there would not be a question where I simply ask him to replicate the definition of an altitude. Thinking back on this exchange, and this way that I answer the question, I have a ton of questions that I need to ask myself and I will start by posing some of them  my readers out there.

  • My guess (an uncharitable one) is that the student asking about vocabulary is looking to avoid committing anything formal to his short term memory before a test. Admirable in a certain way, but what does this question say about what he thinks his job on a test is? Why would students who have been working with words day after day express any serious concern about being asked what those words mean?
  • Real people have real vocabulary that they use in their studies, in their work environment, etc. I recoil at the suggestion that I should do something objectionable now because someone will do it to my students later. But, I am beginning to wonder whether I am cheating my students a bit. Should I be more emphatic in urging them to be careful about vocabulary now so that they will better understand what they read or hear later? Am I being lazy when I let them casually refer to the longest side of any triangle as the hypotenuse? [Note: I have written about this before. I DO correct them, but in a pretty gentle, nudging way. I remind them every time that the hypotenuse is a specific name, but this habit has settled in with my students for a couple of years now.]
  • What are we communicating to our math students if we mark points off or hold them accountable in some ways to formal language if they can get their mathematical ideas across through their work? Are these skills dependent upon one another? Is it okay that my students can swing into action and write the equation of an altitude of a triangle but be uncomfortable and vague if asked to write a definition for what an altitude of a triangle is? As someone who is so comfortable with these words, I struggle to understand how someone can write that line without being comfortable that they can write a definition, but I’ve been teaching long enough to know that this is a real thing.
  • Is this another instance where students have been trained to think that there is one right way to answer a question and their job is to make sure that they simply regurgitate (if they can decode correctly) what that correct answer is. I, of course, hope that my grading policies and the way that I communicate in class convinces my students that this is not the way life is in my classroom. However, I know that I am battling impressions that have formed over years.
  • More importantly – Does it matter that my students know things like the altitudes of a triangle intersect at the orthocenter? Is there ANY chance that they will remember this in a few months? In the past few years I taught the course, I pretty much only mentioned the word centroid and avoided talking about incenters, circumcenters, and orthocenters. I am not at all sure that I made the right decision then or that I made the right decision this year in explicitly defining them. In my text the words centroid and incenter are explicitly defined. Circumcenter and orthocenter do not even appear in the text. A mistake then? A mistake now? I’d love to hear some advice/opinions.

Gotta get dressed for school now. More thoughts swirling and I hope I am disciplined enough to get them down soon.

Thanks to Joe for prompting this post!

As always, you can reach me here in the comments section or over on twitter where I am @mrdardy

A Tale of Two Questions

This past week I had a quiz for each of my Geometry sections. The two sections are out of synch a bit due to our rotating schedule. They typically assess on different days with different versions of whatever quiz or test I recently wrote. This week’s quiz had two different forms of the final question. I present them below:

In the diagram below you see a triangle ABC and you see what are called the exterior angles of the triangle marked. What is the sum of the measures these exterior angles? Be careful to carefully show your reasoning. Mark any angles clearly that you want to refer to in your explanation.  

The problem above was presented to my class on Thursday.

In the diagram below you see a triangle ABC and you see what are called the exterior angles of the triangle marked. The sum of these exterior angles is 3600. Write a proof explaining to me why this is true. Mark any angles on the diagram that you refer to in your proof.

The problem above was presented to my class on Friday.

Both classes had the same first problem on their quiz. They were asked to prove that the interior angles of a triangle sum to 180 degrees. This proof was explicitly presented in class and in their text. My thought was that this challenging fifth problem should be a (somewhat) natural consequence of the first problem on the quiz.

The students who took the quiz on Thursday struggled on the first problem and it bled over to the last. They generally performed better on the last problem than on the first. In part, this is due to my decisions about partial credit. I was definitely more generous with partial credit on the problem at the end of the quiz since they had not seen any explicit proof of this fact. My colleague who also teaches Geometry felt that I might be reaching a bit with this last question. My Friday class performed better on the first proof than the Thursday crew and they did a MUCH better job on the last problem. I am trying to sort this out and there are too many variables at play. First, the class who took the quiz on Friday has performed at a slightly, but consistently, higher level overall during the first trimester of our year. Second,there is always the possibility that information about the quiz was discussed in a way that gave the Friday class some advantage. Finally, the problem presented to them gave an answer and asked for justification while the problem as presented to the Thursday class did not provide the conclusion. I was more strict with partial credit with the Friday quiz class since the conclusion was given to them and the whole burden of the problem was the explanation.

The main reason I am writing about this is that I am trying to make myself think clearly about what my goals are in a  problem like this one and to convince myself that I was trying to get at the same thing with both classes. Did I drastically change the nature of what was being assessed by presenting the conclusion already? I have thought out loud on this blogspace about a similar question here – https://mrdardy.mtbos.org/2017/09/22/a-quick-question-about-test-questions/

Did revealing the answer to the question fundamentally change the level of challenge inherent in the question? Is it THAT much easier to reason through the proof when you know what you are supposed to conclude?

Our Geometry course is the last course in our curriculum where there is no Honors option. Everyone who takes geometry takes the same course at our school. This means that there is a wider variety of interest and talent in this room than in my other classes. I think that there is a tendency in a non-honors math class to think that the students cannot tackle challenging or novel questions. I have heard several colleagues over the years say something along the lines of ‘I can’t ask that question if I haven’t shown them how to do it.’ These are terrific teachers saying this and they are coming from a good place, they want their students to succeed and they do not want them discouraged or dismayed by assessments. I think I am coming from a good place as well, it’s just a different place. I’d also say that in the case of the question above, especially in its first form, I do believe that I have shown my students how to tackle such a question. They know that the interior angles sum to 180 degrees. They see three supplementary pairs of angles so that sum is 540 degrees. The difference is the exterior angles. Half of the students in the Thursday group earned four or five points out of five on the problem. Those who earned four generally had sound logic with real flaws in the vocabulary explaining their answers. Maybe my docking them a point is an entirely different question about how I assess.

Another reason I am writing this is that I want to have a conversation with my department about questions like this one, questions that are not a simple transformation of what has already been practiced. I have students who imply that I am the first teacher they have who asks them questions that feel like they might be ‘from left field.’ I know that students (all people, really) will exaggerate their concerns in the face of feeling stressed. I think most of my students do a nice job of stepping up to challenges like this one, especially when points are riding on it on an assessment. But I also know that there is an instinct at times to simply dodge these situations. The same group of kids who took the quiz on Thursday were presented with a problem from Steve Wyborney’s website on Friday in class. I showed them the video of the duplicator lab problem.  When the video ended I asked them to begin talking about the problem with their neighbors – in this class everyone sits in groups of three that get randomly reassigned every fifth day. I was met with mostly silence. To be fair, this was about 8:10 in the morning. However, when I showed them the comments section with teachers talking about their fourth and fifth graders solving the problem, they suddenly started talking. So, I don’t know if they were shamed into action or they simply needed to suspect that they were more than capable of solving the problem before they moved. I have to feel that the struggle with the problem on Thursday and their reluctance to engage with a novel problem on Friday morning are related. I also fear that I have not done enough yet to create a culture where they jump into these problems. I am interested in how the conversation goes with my department on Wednesday morning and I would love to hear from any readers as well.

 

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:

I wish I remembered where I found this problem. I know I have used it in the past but I do not remember much debate about it. A little background about how I approach these problem sets before I relate the conversation we had. We are on a seven day class rotation in our new schedule. This rotation sometimes even includes two weekends. I assign a problem set the first day we meet on each rotation and it is due the first day of the next rotation. My students have time to wrestle with these problems, to ask each other questions, ask me questions, look up ideas/clues/data/etc. These are sets of ten problems and they only count ten points. All of the problem sets together count about as much as one test in a term. By the time the term is up, each problem set will end up weighing about 2 – 3% of a term grade. I grade them pretty easily, looking mostly for thoughtful process and I am willing to deal with half points which feels a bit silly but it honors thoughtful work rather than whacking them for some little mistake. Anyway, the point here is that these should be low stress and they should encourage collaborative thought and work. I have been clear that I encourage them to think together. Anyway, when they came into class yesterday there were a flood of questions for each other about fine points of interpretation for this problem and students were asking me about two distinct interpretations of part b of the problem. I loved the debate, I just wish it had been happening more than 2 minutes before the work was due. The primary debate was whether to answer part b by interpreting the year as the interval from 0 to 12 or whether to interpret it as 1 to 13. Our friend GeoGebra helped us a bit. Three screen shots below: 

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

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

Nice, right? The area is the same according to GeoGebra. What followed was such a highlight of my day. Intense debate about why these seem to be the same. One student notes that the obnoxious coefficient for x in the function yields a period of essentially 12. Not a surprise, right? Nice observation. A nice debate about the limits. Is 1 the end of January or the beginning? I feel it needs to be the end of the month the way the problem is written, so I came down on the side of 1 to 12 as the proper integral. An observation about the symmetry of right Riemann sums where some months have overlaps and others have underestimates. Another nice observation.  One student questions this model because he notes, as a northern city, Seattle should have more rain in the warmer weather months rather than have this mediterranean rainfall pattern. Wow. I pointed out that I know people who lived in Seattle and have visited there and that this is indicative of what I know about the area. I do not think that this would have ever occurred to me! A debate about whether we simply want to add f(1) + f(2) + f(3) + … + f(12) rather than integrate. One student mentions that he looked up the yearly rainfall in the Seattle area and the answer he found was pretty close to our integral. Good golly, that made me happy. A student questioned whether I wanted two answers to part a. I admitted that the word extreme, in this context, only made me think of a Maximum. I admitted that I should have been more careful than to make that assumption and I should include the minimum in the answer to part a as well.

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.

Now a few negatives, followed by some philosophical pondering about this whole endeavor.

  • 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.

 

When I was working on my doctorate I had a professor (my thesis advisor) who had a policy that every paper will be rewritten, not just every paper can be rewritten. The way he did this was to return our first drafts with no comments, just hash marks in the margin at certain points of the paper. These hash marks might be there to point out a flaw in our argument or our paper’s structure. They might also indicate a highlight. They might indicate a misspelling or a simple grammar problem. He was willing to discuss these hash marks in his office hours as long as it was clear that we had sound questions about them, in other words we had to prove to him that we had reflected on our writing. I have never thought so much about my own writing as I did in that class. I do not expect my 14 year olds to do this kind of self analysis, but I know that they ARE capable of careful reflection if they are given the time, space, and motivation to do so. What I am seeing when they come to me is that they have looked over their test, they have referenced notes and their HW. They have done careful thinking and they can usually explain their mistake on their own. This is not universal, but it is happening more often than not. Many students who come to my room to work on corrections have almost no question for me. They are berating themselves for ‘stupid mistakes’, they are laughing at silly things they wrote, they are even saying ‘I have no idea what this work means’ I am pretty convinced that this can be a huge growth opportunity for my students. They are being responsible for their own error analysis here and they are writing thoughtful reflections in the form of ‘On problem 4 I made this mistake, I should have done this instead’

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

I just graded my first problem set from my AP Calculus BC wizards. One of the questions asked for the point on the curve y = x^(1/2) that was closest to the point (4,0). Everyone correctly identified x = 3.5 as the critical x value and about half of them identified (3.5)^(1/2) or sqrt(3.5) [my LaTex skills are weak and I am in a bit of a hurry…] as their y-coordinate. However, about half of them gave me a three decimal approximation. For reasons I cannot completely justify it makes me nutty to see a less exact answer written as an extra step in their work. I know that their science teacher is not interested in radicals in their answers. I know that carpenters do not have radicals on their measuring tapes. I also know that all of my Calc BC kiddos had the exact value for y written at one point in their solutions and many chose to do a little extra writing to make their answer less exact. Am I being a crank if I make this a point of conversation? Along the same lines, I urge them not to rationalize but many cannot help themselves. I urge them to leave a line equation in point-slope form when that is one of the steps in their problem-solving process. I ask them to write x < 3 rather that 3 > x, especially in a piecewise function where I want to read through their domain in order.

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.