## What can I do when I don’t know (remember) what to do?

My Calc BC classes had a test yesterday. We are deep in the midst of thinking about infinite series and I threw a question on the test that I thought would be a respite in the middle of some heavy lifting. I asked for my students to write a repeating decimal as a fraction in reduced terms. My two classes had slightly different numbers, I’ll concentrate my conversation here on this repeating decimal: 0.217217217217…

Many of my students remembered an approach where we called this number x and then created another number with the same repeating block. In this case, it would be 1000x as 217.217217217217…

A simple subtraction yields 999x = 217 and we have our desired fraction of x = 217 / 999

At least one student rewrote this as 0.217 + 0.000217 + 0.000000217 + … This student recognized this as an infinite geometric series whose first term is 217/1000 and whose common ratio is 1/1000. Remembering a nice formula gives us the same answer.

Some students fumbled on the problem not remembering either of these strategies. What I want to concentrate on here is the work of three students who all presented their reasoning essentially in the form of short paragraphs explaining how they zeroed in on this fraction. I won’t quote them directly and I will probably mix up their reasoning a bit after a long Friday.

One student presented the fraction 5/23 = 0.2173913 as a starting point. He admitted that this was after fumbling around with a couple of fractions all of which were ratios of primes. He reasoned that having no common factors would likely create ‘ugly’ decimals of some form. From there he mixed and matched some interesting reasoning. He showed that 2/9 = 0.222222… so he reasoned that 9s in denominators might be a nice thing. He showed something like 57/99 = 0.57575757… and this gave him confidence in targeting 217/999 for this problem. A different student started by pointing out that 1/5 < 0.217217217… < 1/4. This gave her an idea of where to start trying fractions that might zero in on the desired target. A third student presented some of these same ideas and concluded, somewhat apologetically, that he combined past knowledge and logic to arrive at his correct conclusion.

There is an awful lot to unpack in that particular explanation and it reminds me that one of my missions is to consistently champion such an approach to math. I displayed one of the papers on my document camera and made sure to publicly commend all three of these students. I pointed out that the ability to create this logic, to tie together past skills and ideas is FAR more meaningful than remembering some technique or formula. This is especially impressive in the time pressures involved in taking an in class test.

I had shared an old blog post in class recently when a student made a suggestion involving integration. The suggestion he made reminded me of a post I wrote a couple of years ago (you can find it here https://mrdardy.mtbos.org/2020/01/05/more-bragging/ ) and this prompted my students to notice that I am not blogging as much as I used to. They jokingly suggested that they were not worthy of blog posts. I need to make sure I use this space more frequently than I have been. I hope this is a kick start to more writing in 2022.

Oh yeah – a fun note about that post I linked above. I showed it to my students and one of them took a screen shot and forwarded to screen shot to the student I wrote about back in 2020. He replied while we were still in class together that he remembered that conversation from when he was in Calc BC!!!

## More Bragging

Right before the Christmas break my Calc BC class was up tot heir elbows in new integration techniques. One of the old skills that comes into play often in this unit is the idea of long division as a way to rewrite rational expressions. All of my students have known how to do this but without having exercised that skill in some time, there is a bit of grumbling about it. I presented the following problem (or one very similar to it)

My intent was that my students recognize that they want to ‘reduce’ the fraction so that the numerator is a lower degree than the denominator. Long division makes this magic happen and then logs come into play in actually evaluating the integral. At least one of my students did not quickly recall long division. But, rather than ask me for a refresher, Allen rewrote the problem as follows:

He explained his reasoning and I will paraphrase as well as I can from a conversation that happened more than two weeks ago. He recognized that the numerator had a leading term that could be expressed as a multiple of the leading term in the denominator. So he introduced a term in the numerator (and then subtracted the term he introduced) so that he had a leading piece of the integral that is a simple polynomial. The 12x^2 term now served as a multiple (again!) of the leading term in the denominator. Simply adding and then subtracting 48 x allowed him to simplify the next piece of the integral. One more iteration (subtracting and then adding 191 this time) got him to exactly where he would have been if he remembered long division.

I have to admit that I would love it if he remembered long division, I think it is a useful skill and probably saved him a little time. However, after listening to him explain his reasoning I realize that he displayed a pretty deep level of understanding here about how this expression can be rewritten in steps. The level of analysis and understanding here (I think) far exceeds simply remembering an algorithm learned in precalculus days. This has me debating how I want to approach the instruction of long division of polynomials moving forward. I’d love to hear your thoughts on this here in the comments or over on the twitters where I am @mrdardy

## Problem Sets

For quite a while now I have been writing problem sets for my AP Calculus BC students. I scour old books, math competition files I have, problem sets from Exeter and other schools. I cobble together odd, open ended sets of problems intended to give my students the opportunity to grapple with novel problems in a manageable time frame. I encourage the students to confer with each other, to talk to me, to play with GeoGebra, Desmos, WolframAlpha, etc. In a way this is intended as a grade buffer, but mostly it is a way to get them to play with fun students. This year, I am also writing problem sets for my Calculus Honors and Precalculus Honors students. I want to write about something cool that some of my Precalc Honors kiddos presented. Here is the question I presented:

1. Consider the graph of the function f(x) = 5/x  from the point (1,5) to the point (5,1). Explain a way to approximate the length of the curve between these points and arrive at some numerical approximation. You can describe your process in words, with a graph, or a combination of the two.

Now, my goal with this is to prime the pump for important calculus notions of infinite sums, Riemann sums, etc. I hoped that some students would suggest plotting a couple of points along the curve and adding the distances. One student in particular kept pressing me on this question which, admittedly, is probably more open-ended and formless than it should be. I already have ideas about improving this for next year. Anyway, I asked this student to draw the curve on the board and nudged her in the direction I wanted. I probably gave away my thoughts and she probably shared this idea with a bunch of other students. That’s alright, they’ll earn points and they have a seed planted that might come to bloom. However, a few students presented an argument I did not anticipate at all. A GeoGebra sketch will help:

A few students observed that the arc in question seems pretty similar in length to one quarter of the circumference of the circle in the diagram. They concluded that 2*pi would be a decent approximation. Calculus tells me that 6.1448 is the length. This a fantastic approximation and it is pretty fantastic thinking. These students knew that they did not have a formula for the length of the arc along f(x) = 5/x but they do know how to find the length of an arc on a circle. I am pretty proud of this line of thinking and I want to brag about them here tonight and in class tomorrow.

## Trig Identities

The students are presented with the above image and the very simple facts that this is a rectangle and that the two yellow triangles are similar. The point on the right side of the triangle is movable. A few things right off the bat struck me as wonderful here. We talked about WHY we could know that the yellow triangles were similar. So, we had the opportunity to remember the AA postulate. A student in one of my classes knew that the upper angle is alpha + beta because it is the alternate interior angle of the lower left corner angle. Super sweet! I was going to present a boring conversation about 90 – alpha and 90 – beta on the top. So, I liked that aspect right off. I also LOVED the aspect of how open this construction is AND the fact that it was not at all obvious to my students what we were about to discover. Pretty cool.

I ran this first for my Precalculus Honors kiddos and had each small group discuss where the wages need to go then we put our thoughts together. After a (very) gentle reminder of the structural properties of rectangles, we realized that we had discovered the angle addition formulas for cosine and for sine. An interesting response followed. One of my more curious and driven students asked ‘Don’t we have to prove this?’ I think that this speaks volumes about the natural response to the idea of ‘proof’ in our students. This exercise seemed clear and concise. Couldn’t qualify as a proof, right? Now, I am not fooling myself here. There is still a great deal of simply committing these formulas to memory at the end of the day. But I am convinced (CONVINCED!) that this feels more meaningful now. My kids were able to see and derive for themselves these relationships. They stopped and thought about similarity, about ratio definitions for the cosine and sine functions, and about the structural requirements of calling something a rectangle. I went on to tell them that they do not need to commit to memory double angle formulas because they come straight from here. Most students don’t take my advice on things like this, they feel safer simply consuming memory space with formula after formula, but that is another issue entirely.

After this went SO well with my precalc honors kiddos I unveiled it in my Calculus Honors class. We were just getting to the point where we were dealing with derivatives of trig functions and I knew that the chain rule was about to be laid on top of this. I guessed that this would be a great exercise to jog their dormant trig memories from last year. Again, in each section of Calc Honors, small group conversations led directly to sharing of ideas and a quick dissection of the diagram. I am pretty sure that these conversations woke up some sleeping facts in their brains and I hope it pays off in the form of quicker recall and comfort when we lay the chain rule on top of the standard trig derivatives soon.

Many thanks to Tim and to all the others who shared out ideas when I sent out my call for help. My students don’t really understand how much better their education in my room is due to the network of supportive, smart, creative folks out there. I do make an explicit point of telling them when I am using ideas/activities from others to help make all of this clear. The subtext I hope sinks in is this – If you have an interesting questions, send it out to the world. You’ll get some interesting feedback.

## A Fun Rabbit Hole

Last week – I know, it’s taken too long to write about this – my Precalculus Honors class started the day with a brief quiz. One of my PCH students named Max finished the quiz early and started sketching on his scrap paper. He showed me a diagram like this:

He described the problem this way – I have a square and a quarter circle coming across it. I also have a circle inscribed in the square. What is the area of these little regions? (I clumsily sketched in those regions on GeoGebra)

Well, it turns out the the topic of the day in AP Calculus BC that day was to be trigonometric substitution for integrals and this problem would be a lovely introduction to the need for this skill. AP BC was meeting for the 90 minute block and I decided that I would introduce Max’s problem, spend about ten minutes dissecting what we could and then hit a bit of a wall where I would introduce this new skill. I was pretty proud of myself and feeling very fortunate that Max thought of this question. Well, as we all know, life doesn’t always work out the way we want it to in school. I presented this problem and told them that it came up in Precalculus Honors. My BC kiddos started dissecting it right away. They concentrated on the lower left corner, they decided we should agree to a side length for the square and off they went. We decided the square should have a side length of 2 so the inscribed circle would have a radius of 1. Avoiding fractions until we HAVE to deal with them is a good plan in general, right? So, the lower region is 1/4 of the difference between the inscribed circle’s area of pi and the square’s area of 4. Good start. Next we convinced ourselves that the two remaining squiggly areas are congruent. It would have been nice if we could drop a line from the point of intersection to divide that region in two but it’s not symmetric. The different radii of the circles intersecting prevents that from being true. So, here is where I figured I would introduce this new technique. I mentioned this idea but the feeling in the room was that we should be able to answer this question using tools that a precalc student should be able to use. I was sitting in the back of the room at this point with my laptop on and a GeoGebra sketch projected on the front wall. Ideas and questions started flowing and students asked for a Desmos sketch like the one below:

So, this sketch is pretty impressive in its detail but, more importantly, this sketch happened about 20 minutes after lunch began and after I excused myself to run an errand during lunch. During the 90 minute class, my colleague David from across the hall wandered in a couple of times asking kids to explain what they were doing. He told me that Nancy, Andy, and Michael worked for at least a half an hour of they hour long lunch debating this problem. The other thing that happened while I was gone was that Andy, Kelly, and Michael had modified my Desmos sketch on my laptop pursuing their idea. Their modification is here –

I was feeling pretty great about their perseverance, their engagement, and the amount of geometry and trig that was being remembered in the service of this curious problem proposed by one of my students.  I was also more than happy to amend this week’s Calc test by taking off the one problem that relied on the trig substitution technique. I had one more class after lunch (one of my Honors Calculus sections) so I sadly erased some of the work on the board and I described the problem to that group. Some of them had already heard about it during lunch! My BC kiddos were still talking about it even after they left. At the end of the day one of our Differential Equations students wandered into my room. He said ‘I heard there was a good problem today.’ He, Owen, then proceeded to discuss the problem with Andy and Nancy who had come back to the room to discuss this. Owen dove in to the problem debating with Andy and Kelly and he produced these sketches – (the first one got rotated in translation)

I tweeted the problem out, like I do, and a former student jumped in and offered this sketch –

Another colleague, Adam, came by when he overheard this conversation and he attacked the problem using Google sketch up to find the ratio that Nick wanted – it was smaller than his proposed 10% neighborhood.

There is no real ending to this story, the weekend came, life moved on. On Monday my BC class was more focused on asking questions about this week’s test. My Precalc Honors kids were impressed by my enthusiasm in talking about all of this but they did not share Max’s curiosity about the question. I went home feeling pretty great about the sense of play and sense of curiosity of many of my students and my colleagues. While I cannot let everyday roll this way, I need (NEED!) to make sure to create spaces where this kind of magic can happen. I think almost all of the credit for this adventure lies with my students who are interested, motivated, curious, and persistent. I hope that I have helped them along by modeling curiosity and by being willing to let this kind of free range play happen in class.

## 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 27 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:

• 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.