How Do We Help our Students Ask Questions?

I have told this story to a number of friends and colleagues over the years. One of my favorite former students – he graduated in 1994 – gave me what I value as the best compliment I think I ever received about my teaching. He was a brilliant kid, school seemed effortless to him. I taught him for four years in a row culminating in AP Calculus BC when he was a junior. He took a math class at UF his senior year. About ten years ago I was living in Jersey and he was living in NYC so I had the chance to see him a few times there. Once we were having lunch and he told me a story. He worked in a small business doing financial analysis and he was frustrated by a problem he had been working on. He told his boss that he was going to take a long lunch to clear his head. When he came back his boss had left some notes for him on the file he had been working on. He told me that his boss reminded him of me. “He asks questions that I would not think of asking myself.” I walked away so happy about this. He did not remember a trig graph or a derivative or the fundamental theorem of calculus (although he probably did remember these things!), instead he remembered that I asked him questions that he did not think of asking himself. I felt SO good after that conversation. I was thinking of it today after school during our afternoon conference time built in to our day. All of our teachers are expected to be in our rooms for about a half hour after the end of the school day and many students make a habit of dropping by to ask questions. I was talking to two of my  Calculus Honors kiddos. This is our non-AP class that takes a deep year-long dive into Differential Calculus. We were looking at some problems on one of the problem sets I wrote and these two young women were saying that they understood the problems when we talked in class but they did not know how to start them on their own. I pointed out that almost all of the ideas in class came from the students, I rarely flat out TELL them how to solve a problem, we work through the question together. What I work really hard on is to ask questions of the students that prompt them to see connections and realize what they know about the problem. I want them to feel the power of being the ones who generate the answer. One of the girls said that she does not know what questions to ask herself when she is home working on these problems. So, the challenge is to figure out how to help her, and others, across that bridge. Is it enough to simply model an inquisitive mindset? Is it enough to be a good role model in persistently asking questions? How can I explicitly help my students develop that instinct and ability to push themselves along a solution path by asking meaningful questions? I would love to hear any wisdom on this front. I am going to share a meaningful quote that I ran across in my days as a doctoral student studying curriculum and instruction:

Genuine enquiry is an important state for students to recognize and internalize as socially valid. Consequently it is an important state for teachers to enact. But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry. For if students are never in the presence of genuine enquiry, but always in the presence of experts who know all the answers, then students are likely to form the impression that there is an enormous amount to know, and that experts already know it all, when what society wants (or claims to want) is that each individual learn to enquire, weigh up, to analyse, to conjecture, and to draw and justify conclusions.


John Mason

Why Do We Know What We Know?

In my AP Calculus BC class today I presented an activity from Christopher Danielson that I found through the MTBoS Search Engine

You should check out Christopher’s post. I asked the questions he proposed, but I had a new first one. I asked my students to discuss in their small groups what assumptions they were making about this function pictured here. I heard some good stuff. They talked about continuity, they talked about it being a polynomial function, hopefully one with an even highest power. They talked about the fact that circle C could not have a root since it is entirely above the x-axis (although one student raised the question of complex roots and this prompted a conversation about use of the word root versus calling them x-intercepts), they talked about the minimum number of critical values. In general, just some great recall. At our school, BC is a second year Calculus class so we were talking about ideas from last October/November. This led me to raise a question that I was a bit worried about. Earlier, we had made mention of the fact that polynomial functions with an odd highest power have all real numbers as their range. Sure, we know this. But why? Do we really have an idea why this is true? I was worried that this was too vague a question. I was worried that they would waive it away. We know this is true, Mr. Dardy. Why talk about this? Instead, we got some great GREAT conversations. I was told to think about limits, to think about derivatives. I jokingly asked if we should think about area between curves or optimization or some other time honored Calculus ideas. I was told to consider the limit as x grows without bound both for a positive leading coefficient and for a negative one. We discussed how all terms in the polynomial eventually become insignificant compared to the highest powered term. We talked about the derivative being even powered and what we know about those graphs. Man, I was just so pleased that they were willing to travel down this sort of hazy questioning path with me and reinforce what they know and WHY they know it. I say this every year, but this class absolutely spoils me.


Reflections – While thoughts are Fresh

In my two morning classes today we tackled the problem that I just blogged about (link here)

Some interesting observations first and then some questions that came from my learners.

When data is not presented as a table, there is a distinct extra layer of processing that has to happen. One student went straight to desmos to graph the points his group had. I liked that he wanted a visualization. A number of them, when I asked how to find the AROC between two pictures were flummoxed. Let me give you an example of what I saw from a number of students. Look at the picture below:

A number of students divided 1381.5 by 34.18 (more on that in a moment) and arrived at 40.418 as an answer. This matched the given information but they did not seem to notice that this did not answer any part of my question about AROC from point A to point B, from point B to point C, or from point A to point C. [This conversation makes sense (I hope!) if you have read the first blog post that is linked above]

I had to poke to get them to recognize that any time we talk of an average rate of change, we are talking about more than one data point to consider. A number of them were happy to enter a time like 30:56 as 30.56. This disappointed me a bit. For smaller minute data points I can see the mistake a little more clearly. For something as close to a full hour as 56 minutes, the willingness to enter .56 seemed more clearly wrong.

Once we ironed out the fact that we need to see time and distance as coordinates of a data point, then the slope idea for AROC fell into place more comfortably. Before talking about my challenging final question, I want to share a few questions and observations from the classes.

A student asked about the clock on the dashboard. Does it still count when the car is sitting? When I am sleeping? Great question, the answer was no. It only moved when the car was moving.

Conversations came up about the geography of Florida where I was driving. They guessed that some snapshots were taken in city traffic, others after highway travel.

Discussions came up about why the average on the dash did not change even when intervals had different AROCs. I relied on a baseball analogy. Late in the year a 0.250 hitter might have a great day and go 3 for 4 while not changing his overall average at all. That seemed to make some sense to them.

The final question I asked was more difficult for them than I had anticipated. I asked each group to consider the following situation. How long would I have to travel at 60 MPH to raise the trip meter average velocity to 42? In my mind I simply wanted to use the last data point as the jumping off point and add an unknown time ti to the x value (the time input) while adding a distance of 60t to the y variable (the distance output.) This idea did not organically appear in class and I pushed a bit more than I wish I had as I saw our 45 minute time together elapsing. I am often comfortable with questions being unresolved at the end of a class period. However, with this question at this time of the year (this was only our fifth class together) I really wanted a conclusion to the mystery. I will definitely revisit this experiment as we work more deeply on our ideas of rates of change and I will remind them of this conversation on a number of occasions. An unexpected bonus idea that came through loud and clear was the MVT even though we don’t have a name for it or a formula describing it yet.

Pretty pleased, I must say.


Exploring Rates of Change

One of the courses I teach this year is a course called Honors Calculus. It is a non-AP course and we made a decision about 6 years ago to make this a course in Differential Calculus. While the AP AB course completes a college semester in a high school year, this course completes an AP AB semester in a high school year. This allows us to remedy some bad habits, fill in some gaps in understanding or mechanics and, most importantly, really slow down and think about what we are exploring. I have an activity that I do on the first day of class where we explore motion. This year we went in the hallway and rolled a whiffle ball, a softball, and a lacrosse ball down the hall a fixed distance. We tried to roll each with approximately the same force and had a good conversation about what the data told us. We made some physics based observations that I did not plan for and we talked about what we knew and what we did not know about the rate at which the balls were rolling. My goal was to arrive at the conclusion that we could talk about average rates of change but not so much about the rate at a particular instant. The conversations went reasonably well, but we got distracted a bit by conversations about bounciness of balls and air resistance. In any event, I think I planted some decent ideas to consider as we embark on a conversation about  average rates of change of a function on an interval (our text calls this the AROC) and we are about to wrestle with the limitations of being able to know much about the instantaneous rate of change (the IROC)

This summer I rented a car whose dashboard gave information that I knew would work well for this class. A picture below will prompt your teacher brain as well I think.

I took six such photos during the course of my trip. Tomorrow, I intend to give each of my groups of students (I have them in groups of three) three of the pictures. I’ll scramble them up a bit so different groups should have different subsets of the data. I intend to ask them some pretty simple questions that should generate some good conversations. I want to ask them the following questions:

  1. What was my average speed between any of the two pictures? (So each group should have three answers for this)
  2. Can you determine my maximum velocity in that time interval?
  3. I want to raise the average to 42 MPH. How far would I have to travel at 60 MPH for this to happen?


These are not terribly deep questions, but they feel rooted in an example of real world data (I was inspired by Denis Sheeran’s wonderful book Instant Relevance for this data driven experiment) I also think that this will continue to scratch at the itch that will make the breakthrough of being able to find the IROC feel more meaningful.

I have all the photos together in a WORD doc on my dropbox. You can find that file here. I would love to hear any clever ideas about how to play with these images/this data.