Follow-Up: Letting Students' Curiosity Steer a Lesson

In an earlier post, I wrote about exploring a student's question during class, and letting the subsequent questions of the whole group drive the remainder of the lesson.  We were looking at critical points and inflection points with the first and second derivative, and my students had noticed that for several of our examples, the function's inflection point was the midpoint between the two critical points.

A couple of weeks after that lesson took place, I found a day for the class to just explore that question - when is the inflection point the midpoint of the critical points?  We split the board into two sections and began recording functions that satisfied that condition and ones that didn't.  My students set to work with their graphing calculators to create and explore functions.  Here are some questions that arose during their work:

• What do you have to do to get two turning points?
• Is it just true for all cubic functions?
• Is it possible to find a quartic, or quintic function with an inflection point at the midpoint of two critical points?

Eventually, most of the class gave up on quartic and higher degree functions, and the focus shifted mainly to cubics.  Several times a cubic function was added to the "doesn't fit" column, but after double checking these we found that all of them had inflection points directly between their midpoints.  So, by the end of the class period, we had a white board full of cubic functions that showed this critical-inflection point relationship, and no counterexamples.  We were inclined to assume it was true for all cubic functions.

In retrospect, it seems like a pretty obvious result, at least as far as the x-values are concerned.  A cubic function's first derivative is quadratic, and the vertex of a parabola (where the zero of the second derivative will occur) is always halfway between its zeros.  I wasn't certain about the y-values, though.  To be sure, I wrote up a proof (here's a link to that proof).  I began with a generic, standard form cubic function and carried out the differentiation to find the first and second derivatives, and then used algebra to find the zeros of those derivatives and the accompanying y-values for the function.

Unfortunately, I didn't have a class period to spend working through a proof of this result with my class.  The work is pretty accessible for students who have had Algebra II or more and some basic Calculus -- derivatives using the power rule, quadratic formula, and some binomial theorem for plugging numbers back in to the original cubic function.  I made my proof available to my class for anyone who was interested.  They were satisfied to know that their hypothesis was true at least for cubics, but they were still making conjectures about when the same relationship might be true in quartics and higher degree functions, which is great.

Maybe I didn't word it well, but when I did a Google search to see if there was anything out there about this relationship between critical and inflection points, I didn't find much.  I found at least one exploration worksheet that guided students to notice the relationship for some specific examples, but not much else.  Does anyone out there know if this is a commonly known relationship?  Maybe it's too "obvious" to be worthy of mention in most textbooks.  I didn't notice it right away.

I'm still curious, and so are my students, if this ever happens in higher degree functions.  And if so, when?  If anyone can point me in the direction of a resource that might help me learn more about it, I'd be grateful.  Thanks!

Instructional Dialogue - Math Anxiety

One of our readings for Math 629 struck me as good material for my instructional dialogue.  Jackson and Leffingwell's (1999) The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College left me asking myself whether or not I exhibited any of the anxiety producing behaviors their article listed.  They wrote about overt and covert behaviors that teachers display that contribute to students' anxiety or otherwise send negative messages to students.  While I could tell myself that I was innocent of many of the behaviors described, there were some that I had to think twice about, like relying on prerequisite knowledge, and even saying things like, "You have done this before in Algebra I."  Even if that's not said with a condescending tone, it's disconcerting for any student who doesn't remember how to do whatever "this" is.  Overall, the article left me concerned that I might, even if its usually unintentional, be doing or saying things that add to students' math anxiety, or make them feel worse about their ability level.  After reading the article, I'm worried about an ill-timed sigh or furrowed brow might have a big impact on my students' comfort and confidence levels, or their willingness to ask questions.

With these things in mind, I asked one of my fellow math teachers to observe a class period.  Before the observation, I asked him to be mindful of this list of questions and concerns:

• Does Matt display any behaviors that might increase students' math anxiety? (Examples might include expressed frustration at repeated questions, gestures or mannerisms that suggest annoyance, avoidance of certain questions, lack of eye contact, etc.)
• Does Matt do anything that might empower one gender or another to be more or less vocal and participatory during class. (Does he call on one gender more often?  Does he not do enough to get responses from a variety of students, allowing two or three to dominate?  Does one gender seem to dominate discussion?)
• Do you see anything in general that Matt could improve upon or that he is doing well?

The lesson my friend observed was on graphing rational functions.  We were taking generic graphs with asymptotes and intercepts plotted (no scale or numbers) and sketching in the shapes of the graphs following some graphing "rules" that we had discussed for rational functions (for example: the curve can only pass through the x-axis at an x-intercept; the curve cannot pass through a vertical asymptote, but instead must go to positive or negative infinity as it approaches one).

The lesson went well, unusually so.  A wider variety of students than usual were volunteering to answer questions than usual, and just about an even balance between male and female students.  There was a lot of discussion, and even some arguing about the mathematics.  Students were asking insightful questions about the behavior of the graphs and what causes it.  It was a fun class period.  I was reminded again the next day how well things had gone, because the class seemed kind of flat by comparison.

My friend didn't notice any of the behaviors or tendencies I had listed in my questions for him.  He kept a tally of responses from different genders, and again, it was almost an even split.  He also noted that my questioning of all students was consistent in difficulty, and that I wasn't guiding with my questions.  He said it appeared as if I have great rapport with my students and that they seemed very comfortable with me, asking questions and offering responses.  Another thing that he made note of that he liked was that I took students' suggestions of how to sketch a portion of a graph, drew it that way on the board whether it was right or wrong, and then asked the class whether they agreed or disagreed.  On some days my class might have gotten frustrated with me for not being direct about right and wrong answers, but on that day they seemed to embrace it and liked the added discussion.

While I was relieved that my friend didn't notice any of the behaviors I was worried about, since things went so well I am left wondering why they did, in hopes that I could have more class discussions like that.  I don't know that I can take any credit for how well it went - most of that is probably due to my students and their interest and willingness to discuss - but here are some things that I think may have helped:

• Having another teacher in the room, especially someone who's there just to watch you teach, really makes you bring your "A game", I think.  I'm not sure exactly how that affected me, but it probably made me relish the good discussion that was happening, and be more thoughtful about the questions and answers I was offering.
• Rational functions was a new topic for my class, one that they hadn't had much prior exposure to in earlier classes.  I think this leveled the playing field a bit and contributed to a wide variety of students taking part in the discussion.  The graphing rules that we had may have also empowered them to argue with each other a little more, too, rather than just waiting to see if I said the answers were right or wrong.
• The list of questions I had given my friend to look for was fresh in my mind, and as such I was especially mindful of how I was asking and answering my questions, the tone of voice I was using, and the mix of students I was calling on.  I was even thinking about my use of eye contact when I asked questions.  I think you can sometimes draw out a response to a question from one student or one section of the room by directing your eye contact at them while you ask, and looking at them during the wait time.  Or maybe it just makes them nervous.
• It might have just been one of those days when things are going to go well, and I was lucky enough to have another teacher there to witness the good discussion.

After all, I really appreciated the instructional dialogue process.  I think just picking something for one of my peers to look at and thinking about that ahead of time improved my teaching a bit.  Having someone else in the room to observe my teaching helped me to remember what my A game looks like, and left me challenged to try to put forth my best effort every day, whether another teacher is watching or not.  Even though the lesson that my friend observed went well, his observation notes left me with more to think about and some things to keep working on and building on.  I'm hoping I can do more instructional dialogues with my colleagues in the future.

Math 629 Project Update - Improving Transfer

For my Math 629 project, I am doing some work that will hopefully contribute to my master's project.  What I am hoping to produce is a working draft of the literature review portion for my project.  In my foundations and curriculum development courses at Grand Valley, I have done some work addressing the problem of poor transfer ability in mathematics students, and higher order thinking skills in general.  I am planning to do the same for my master's project.

To transfer learning is to take what you have learned and apply it to something new or different.  I am often surprised at the difficulty my students have with math problems that are only slightly different than ones they have encountered previously.  It's an issue that is very disconcerting to me, too, because I want their learning to have value.  If they can't do anything with it beyond a narrow set of examples, it's not very valuable.  Another immediate concern is the Smarter Balanced Assessment, which will (likely) replace the Michigan Merit Exam.  It will demand a lot more flexibility and transfer ability from students.

To address this problem in my master's project, I have in mind a capstone unit that would fit at the conclusion of an algebra II course.  The unit would engage students in problem-based learning and cooperative group work, with a lot of metacognitive reasoning structured into the activities.  I do think that all of these things would be best applied throughout a course, but for my project a capstone unit seemed appealing to me, since it offers more opportunities to build connections between a variety of algebra II topics.

I have found a number of good primary research articles to support those three elements (problem-based learning, cooperative group work, and metacognitive reasoning) as a means of addressing the problem of low transfer ability.  One in particular has shaped my approach to the problem more than any other one article.  It addresses the problem, and ties together most of the ingredients I am applying in my proposed solution.  Kramarski and Mevarech (2003) studied the effects of cooperative group work and metacognitive instruction on secondary students' mathematics reasoning and ability.  Their study compared a control group to three other treatment groups, one of which underwent cooperative learning, another received metacognitive instruction, and another underwent a combination of the two.  The results showed better outcomes in the group that received both treatments than in any other group.  This group provided more correct explanations for their reasoning, and they outperformed their peers on tasks designed to assess transfer ability.  In an earlier study, Kramarski, Mevarech, and Arami (2002) found that metacognitive instruction improved student performance on both standard mathematics tasks and authentic tasks.

Both of these studies are well designed with decent sample sizes, and they look at secondary math students, which is my focus.  Their main drawback with respect to my project is that they are Israeli studies, which raises the question of external validity.  But they are more robust than most of the other studies I have found on cooperative learning and/or metacognitive instruction, and their control-group design also makes me more confident that I could extend their results to my own setting.  As I look for and gather more sources, though, I am hopeful that I can find a few more domestic studies touch on the same topics.


Referenced Sources:

Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom:  The effects of cooperative learning and metacognitive training. American Educational Research Journal, 40(1), 281-310.

Kramarski, B., Mevarech, Z.M., & Arami, M. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks. Educational Studies in Mathematics, 49, 225-250.

Letting Students' Curiosity Steer a Lesson

A few weeks ago in my calculus class, I was introducing the use of the first and second derivatives to find a function's critical points and inflections points.  For our first example, we took a look at .  We first graphed the function using Desmos, and then set about finding its derivatives. The first and second derivatives are and , which are pretty easy to work with when finding the zeros.  The critical points of the function are (0, 4) and (2, 0), and the inflection point is (1, 2).

I was about ready to move on to a second example, when one of my students asked, "Is the inflection point always going to be the midpoint of the critical points?"  I acknowledged that it was a great question, but started to explain right away why I was pretty sure the answer was no, knowing that a higher odd degree function could be created to have two critical points with any variety of different behaviors in between.

What was I thinking?!  Here I had a perfect opportunity, a student-initiated reason to do and pay attention to more examples, and the first thing I did was to dismiss the possibility that the conjecture was true.  Halfway through my explanation I realized that I was making a mistake, but fortunately for me, my students were undeterred.  Instead of dropping the point, another student refined the conjecture, and suggested, "Maybe it only happens when the function displays its maximum number of possible turning points," like a cubic function with two turning points, or a quartic function with three turning points, etc.

I really didn't know whether that conjecture was true or not, but I had also finally realized how valuable this student inquiry was.  We let the question be the drive for our next few examples.  We took a look at another cubic function next, fairly similar to the first, and once again the inflection point was the midpoint between the two critical points.  Since our class time was running short, we decided to move on to a quartic example.  We used Desmos to set up a quartic graph with sliders for the coefficients,  which made it easy to manipulate and create a function with three turning points.

The quartic function we looked at was , which did turn out to be a counterexample for our class' conjecture.  Class was just about over, so we didn't have time to look at more examples, but even as they were packing up, a few of my students were still throwing out possible modifiers on the conjecture.

That class period reminded me, and demonstrated to me in new ways, how valuable it can be to run with students' questions.  The question about the critical and inflection points' relationships to one another had most of the class curious.  It provided an "intellectual need" for more examples.  It even added some suspense to the remaining examples we covered.  The whole class was engaged in exploring whether the conjecture was true, or otherwise when it might be true.  This in turn meant they were thinking critically about the topic, and asking questions that got beyond the plain mechanics of how to find and plot critical points and inflection points.

Seeing the value in that experience, I have tried to be more open to exploring student questions, particularly with my calculus class.  They have always asked more questions than most of my other groups, but it seems they have been asking even more, as we take more time to consider them in class.  I am finding that I have to be selective again, though, about which bigger questions we take time for.  There are some that have more potential to lead to enhanced learning than others, and sometimes the "others" need to be left for another time for the sake of covering a new concept.  But maybe my priorities still need some adjusting?

As for the question of when inflection points coincide with the midpoint of two critical points, I still don't know the answer, but we have noticed it in many examples since that class period.  Maybe it's a commonly known theorem, but I don't want to Google it yet.  I'm hoping to find a day yet this year for my class to explore the question to see if we can find some commonalities in the functions that behave that way.  It would be a good exploratory math experience for them.

Polar Functions in a Daily Desmos Challenge

I was intrigued by Desmos when it was introduced in class, so I was interested in spending a little more time with it in my math work.  Desmos is an online graphing calculator with a lot of neat features.  We took a look at dailydesmos.com in class, which is a website that posts Desmos-generated graphs as challenge problems for other Desmos users to try.  On that particular night of class, Daily Desmos #285 was up on the website, and caught my eye as an interesting graph and a fun challenge.

Daily Desmos #285 (Advanced)

By the look of it, I assumed that this was probably a polar function.  In our precalculus class, we go over some of these – I think the ones we do are called cardioids and limaçon curves, and spirals as well – but I had never seen one that looked quite like this.  I played around with some polar functions on Desmos to try to refresh my memory on how they work.  I triedand then and then , the last of which produces a circle centered at the origin, with radius 1.  I decided this circle equation was the one to begin manipulating, since it does not pass through the origin, or pole.  Having each of those terms squared seemed like it would be a key feature for this rule no matter how it turned out, since r looks to stay positive throughout the graph.  If  r  did ever change from positive to negative, the graph would have pass through the pole.

The Daily Desmos graph begins at about.  Since it goes out about that far, and just a little farther (maybe 2 units?), when , I figured I would not add a constant to my rule.  I thought that would make the graph lop-sided, and while it's not actually symmetric, it's close (can you be close to symmetric?).  Instead, I decided to try a multiplier of 17 on the term of my rule.  Made a similar decision for the term of my rule.  Since the term would drop out to 0 on the y-axis whenever  is a multiple of , the would be the main player at those points.  The first of these points on Daily Desmos #285 is about , but I decided that the multiplier on should be just a little bit lower than 4, due to the spiral effect going on in this graph.

From teaching precalc, I knew that a function of the form would produce a spiral, so I knew I would need a term like this as well.  Looking at the successive passes of the curve on one axis at a time gave me the idea that this term should be increasing r values by a little bit less than 5 for each full rotation.  I decided to try 4.6 as my first estimate.  To figure out what k should be for this term, I solvedfor k to get .

With these guesses and estimates, I put together my first attempt at really matching the graph: 

Here’s the result:

My Attempt

The Daily Desmos Challenge

I was pretty satisfied that I had at least discovered the right form for my polar function.  All it would take to get things to match more accurately would be to adjust my constants a little bit.

In retrospect, I didn’t take advantage of Desmos as an exploratory tool very well.  If you type in a function in Desmos with letters in place of numbers, it can automatically create sliders for the would-be constants.  It might have made more sense to set up a rule with sliders for the constants, once I had decided on a form for the rule.  That is, I would have entered into Desmos.  That would have made it easier to investigate different values for each constant.  At the same time, it was good to do some analysis of the points and make educated guesses.

This was a fun way for me to review some polar functions concepts, but it would also be a great activity for my precalculus students in the future.  Providing students with laptops, or going to a computer lab, and then presenting the with graphs like this would be a great way for them to test and expand their understanding at the close of a polar functions unit.  Something similar could certainly be done in other classes and units as well, with other types of functions.  If computers aren't readily available, this could be done with graphing calculators as well.  Taking the time to do this activity deepened and reinforced my understanding of polar functions and how they work, and got me thinking about the features of several types of polar graphs.  I think it could do the same for my students.

Using "The Letter Game" to Introduce Deductive Systems and Proof in Geometry

A new trimester will begin in just a couple of weeks, and I will have a new bunch of geometry students.  Before long, we'll be digging in to geometric proof, which is a concept/activity/process that we will develop and use throughout most of the course.  It is always a tough concept to develop with students, and I am always wondering how I can make it easier for them to grasp.

One really nice, fairly accessible activity that we have used with students at my school is "The Letter Game", which was published by Don Gernes in his article The Rules of the Game (full citation below).  The game amounts to a very simple deductive system, in which students are asked to "prove" various statements.  It gives students an opportunity to practice working within the bounds a deductive system, and to get a sense for the structure of proofs and the process of proving something.  This comes before they have to confront the added complexity of geometry, which is the big deductive system they ultimately will be learning to work within.

Before getting to the "The Letter Game", Gernes suggests having students think about games that they are already familiar with: monopoly, basektball, soccer, etc.  Each of these games has undefined terms, defined terms, postulates, and maybe even some theorems.  These are the makings of a deductive system.  Gernes presents basketball as an example, and here's what he lists in each of these categories:
______________
Undefined Terms
Ball
Player
Court
Baskets

Defined Terms
Field Goal
Foul
Free Throw
Traveling

Postulates
If a player is fouled, then the player gets to shoot a free throw.
If a player travels, then the other team gets possession of the ball.
If a player makes a field goal, then the player's team gets two points.

Theorem
The referee objectively applies the rules of the game to each play.

This obviously doesn't represent a complete list of all of the terms and rules that make up the game of basketball, but it is enough to have a discussion about what the undefined terms, etc. are, and how each of them contributes to the structure of a deductive system.

The Letter Game is where students get a chance to try their hand at some proofs.  Gernes keeps the system simple, so students don't have too much to keep track of as they begin proving "theorems".  Here's the deductive system he establishes:
______________
Undefined Terms
Letter M, I, and U

Definition
x means any string of I's and U's.

Postulates
1. If a string of letters ends in I, you may add a U at the end.
2. If you have Mx, then you may add x to get Mxx
3. If 3 I's occur, that is, III, then you may substitute U in their place.
4. If UU occurs, you drop it.

With these four postulates, students take given strings of letters and then "prove" another string of letters, using one postulate at a time to manipulate the string.  For example - Given: MIII   Prove: M.

Even though many of my students end up groaning about geometric proofs, most of them end up enjoying the letter game.  It is accessible enough that most of them are able to make some headway, and after a quick example or two, most of the class is able to work independently.  I usually don't get any students reverting to "shut down" mode because they can't get it.  Sometimes I even have students finished early who create their own new letter "theorems" and challenging one another to prove them.  While the activity doesn't make them masters of geometric proof, it does reinforce the concept of a deductive system, and what it is to work in and prove new theorems in a deductive system.  Every step that is taken or statement that is made needs to be backed up by a definition, postulate, or theorem.

Opening up the much more broad deductive system of geometry, with all of its definitions, postulates, and theorems, always makes geometric proof more difficult for my students.  The Letter Game hasn't eliminated those difficulties for my geometry students, but it has been a good first experience with proof for them.  Gernes' activities build a nice segue from concepts that students are familiar and comfortable with to the way geometry works as a system.  His article is definitely worth a look:

Gernes, D. (1999).  The rules of the game.  The Mathematics Teacher, 92(5), 424-429.

An Introductory GeoGebra Activity for Calculus Students

My calculus students had never used GeoGebra before this activity, so I wanted to do something fairly simple with them.  This activity takes about five steps on GeoGebra, so it wasn't too demanding for first time users.  At the same time, it provided some nice visual support for the concepts we had been learning in class, and provided them with a tool to explore those concepts further.  I've tried to make the instructions below friendly for first time GeoGebra users, and I'm a novice anyway.

Setting Within the Course

When I brought my class to the computer lab for this activity, my calculus students had already been using the limit definition of a derivative for three days.  We had also been discussing the derivative at a point as the function's instantaneous rate of change, and the slope of a line tangent to the function at that point.  When we had opportunities to do so, we were taking special notice of times when the derivative was zero and the tangent line was horizontal.  We had also spent just a little bit of time more generally discussing the relationships between graphs of functions and the graphs of their derivatives, including a few mentions of concavity.  Furthermore, after working with the limit definition for several days, my students had noticed and we had informally discussed the power rule.

The Activity

With these concepts in mind, we did the following together on GeoGebra:

1) First, we entered a function in the Input bar at the bottom of the window.  The function we started with was y = x^3+3x^2-1.  In the algebra pane, GeoGebra relabels this as f(x).  This function provided opportunities to talk about horizontal tangent lines (critical points) as well as changes in concavity (inflection points).  At this point I also showed my students how to use the Move Graphics tool to center their graph in the window, and to adjust the scale on each axis to get a good picture of the function.

2) With our function graphed, we added a new point on the curve by selecting the Point tool and clicking on the curve.  GeoGebra automatically labels this point A.

3) We then put a tangent line on the curve at point A.  GeoGebra has a tool for this under the fourth box from the left.  Once the Tangents tool is selected, click on the curve and click on the point of tangency (point A for us), in no particular order.  This is a good time to show students how to use the Move tool if they haven't discovered it already.  It's the cursor at the upper left of the tool bar (or pressing the Esc key will select this tool).  If students click and hold point A with the move tool, they can slide it along the curve and watch the tangent line change.

4) Now we'll have GeoGebra measure the slope of the tangent line (the value of which is that of the derivative).  GeoGebra has a variety of measuring tools.  Once the Slope tool is selected, click on the the tangent line and GeoGebra will add a rise/run triangle and display the slope.  The value of the slope gets added to the algebra pane as m.

5) We finally add one more point to the sketch, using the Input bar again.  Type in (x(A),m) to define this last point, which GeoGebra will call point B.  This tells GeoGebra to use the x-value from point A (the point on the curve) and to plot the slope of the curve as the y-value.  This point displays the value of the graph's derivative at any x-value.  By right-clicking on point B, Trace On can be selected.

With trace turned on, the path of this point will be traced out on the sketch as point A is moved along the curve, revealing the shape of the derivative function.  (I also changed the color of point B using the pull-downs at the top of the graphics view.)

Discussion

The completed sketch opens up a lot of good talking points.  Students should be able to see that the local maximum and local minimum of  f(x) line up with the zeros of the derivative function.  If inflection points have been discussed, it's also nice to see that this point lines up with the vertex of the derivative.  As point A is moved back and forth across the inflection point, the tangent line makes a nice rocking motion that shows the slope is beginning to increase/decrease.  In general, the tangent line in the sketch provides a good visual connection between the curve of  f(x) and the sign of the derivative.

Of course, students can also try to predict the shape of the derivative function, or find the derivative function algebraically and plot it on the sketch as well.  Wherever possible, it seems good to keep students' visual and algebraic understandings closely intertwined. (Check out Aspinwall and Shaw's (2002) article, When Visualization is a Barrier to Mathematical Understanding, in Mathematics Teacher vol. 95, no. 9)

This GeoGebra sketch is also a good tool for students to explore derivatives for functions that they don't yet know how to differentiate algebraically.  Helping my students work through this activity took less time than I had anticipated, so they had about ten minutes of class time in the computer lab after we were done with the first function.  After I explained that they could change f(x) by double-clicking it in the algebra pane without altering any of the other features of the sketch, most of them started exploring other functions.  Students were discovering on their own that cosine is the derivative of sine, and one of them was looking at the graph of tangent, and noticing similarities (but important differences) between the sketched derivative and the secant function.  Having a tool available to explore derivatives increased their curiosity.  I may pull up the same sketch again in class as we learn about the derivatives of specific types of functions.

If you would like to go to the completed sketch, here is the GeoGebra Tube link.