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.

My Teaching Situation

I am currently teaching Geometry, Algebra II, and Calculus (non-AP).  Aside from my Calculus class, my classes are all fairly sizable, at 26, 28, and 30 students.  Calculus, on the other hand, has only 10 students in it, so I get to know those students especially well.  There were more signed up at the beginning of the year, but I think many of those students made the jump to AP Calc before the term began.

We are on trimesters, with 5 class periods a day (we call them "macros").  Each class period is 72 minutes long.  We have a great building and a lot of resources available.  I have a document camera and projector in my room.  We do not have one-to-one technology, but there are laptop carts and computer labs available when I have a GeoGebra activity to do with my students.

One aspect of my teaching situation that has been a little bit challenging at times is the distribution of our math department throughout the building.  There are two other math teachers in my wing of the building, but the other eight of them are spread out in other locations.  When I am teaching the same class with those other teachers, I have to be deliberate about checking in with them and collaborating with them.  There would be  a lot more impromptu collaboration if the departments in our building were arranged by wing.  I know others in Math 629 have mentioned much more real isolation, though, so this is only a minor challenge.

Another challenge, although I know it's not the least bit unique, has always been time.  I feel this particularly in Geometry.  Both halves of our Geometry course fit very tightly within a trimester.

Euclid's Elements - Book III

At this point, I am finished with my project - a translation of Book III of the elements with links to GeoGebra manipulatives (no proofs at this point, however).  Please take a look, and feel free to leave feedback.

You can use the link to the right under Pages, or click here.

If you'd like to just take a look at the collection of my GeoGebra worksheets for the project, click here.

Big Take-Aways from Math 641

As the semester wraps up, here are some of the big things I will take away from Math 641, Modern Geometry:

Get students asking questions.
As a result of our work in class and our readings for Math 641, I am trying to give my students more opportunities to ask questions, develop curiosity, and explore ideas.  On a day to day basis, I am asking my students "why" more often, and asking them to defend their answers.  I have increased my wait time, too.  On a broader scale, I am mindful of the value of letting students explore and discover new ideas, rather than having everything new presented to them directly.  I want to eventually work more exploratory days into the classes that I teach at opportune points in the curriculum.  This can be a step toward helping students develop arguments to defend their thinking.  Having them reach their own conjectures and then supporting their reasoning could be a more natural way to introduce them to proof.

GeoGebra is an excellent tool and resource.
I knew what GeoGebra was before Math 641, but I had not taken the time to try it out.  Especially once we took a look at GeoGebra Tube, I was hugely impressed by how broadly applicable GeoGebra is to all math classes.  I have spent days in the computer lab with my geometry classes using GeoGebra this year, but I plan to do the same with my calculus class as well.  It can be used to illustrate and explore such a wide variety concepts.  It is also simple enough to be accessible to students - the learning curve is not nearly as steep as that of a program like Maple.

There are far more ways to present and explore mathematical ideas than I am currently aware of.
That is an obvious statement.  What Math 641 opened my eyes to is some of the better ways to learn about new ways to teach concepts and new resources.  I had never used Twitter before this class, and I had never done any blogging.  I wasn't thrilled to have to get a Twitter account at the start of the semester, or start a blog.  But I've grown to appreciate what a resource Twitter is as a forum for discussions about math ed, among other things of course.  Twitter and blogging are great means for sharing ideas, and there are lots of people with far better ideas than me maintaining blogs.  Rather than sticking only to publications like Mathematics Teacher, resources like the mathtwitterblogosphere are another great place to find new ideas.

Geometry and mathematics in general are far more broad than most students get a chance to see.
Working with new topics week to week in Math 641, and exploring those topics from a variety of angles reminded me that there's always a big and interesting math horizon to explore.  As a high school teacher, it's easy to remain relegated to the standards and the narrow math sphere they encompass.  Modern Geometry was a refreshing contrast.  There were many times during the semester when I thought, if I could spend three days doing this with my students, they would be way more engaged, and a few more of them might even decide they like math.  I know I can try to work some of those things in to my regular classes, but in general I want to give my students more glimpses of how many different things are mathematical.  Maybe I'll have to keep up a good bulletin board for once.

Learning GeoGebra - Part 2

Working on my project for Math 641 has forced me to learn a lot more about GeoGebra.  Some of the manipulatives for the propositions in Book III were quite simple, but others had a lot going on.  Here are some of the new things that I learned:

How to define a figure made up of two or more other objects
I first ran into this issue when I was trying to make a picture for a circular segment.  I couldn't find a tool (and maybe there is one?) to make a segment that I could fill and color.  After a few Google searches, I found instructions for how to create a list.  In the input line at the bottom of the GeoGebra window I used set braces to enter the following, for example

     {Segment[A,B], Arc[c,B,A]}

This would define the figure bounded by line segment AB and the arc on circle c between B and A.  It showed up as a list in the algebra pane, which I could select and give a color to for emphasis.  Of course, different types of figures could be included in a list, but I used these mainly to create segments, since my work was with circles.


More varied ways of assigning conditions to show objects
In an earlier post, I explained how I had learned to use check-boxes and sliders to show and hide figures.  For many the manipulatives that I was making for the propositions, I wanted things to change color or have tick marks added to them when lengths or angles were the same.  I learned how to put measures other than a slider value in the conditions to show object line in object properties.  I used the distance between points,

    Distance[A,B]

frequently.  I also learned how to include measures of objects that were in a worksheet by using their letter names, which required using the "alpha" button on the right side of the dialogue box for angles.  If I wanted measures to be within a certain range of one another, I used the absolute value of their difference, for example

    abs(a-b)<0.1

As I mentioned, I often used these to make segments change color when they were congruent, or to have tick marks added.  Most of the time, I had constructed completely separate segments with the different colors or marks.  It wasn't until I was almost done with the project that I realized if color was the only thing I wanted to change, I could put conditions in the "Dynamic Colors" settings right below "Condition to Show Object" (so obvious!).  However, it was still nice to be able to change line styles and thicknesses in addition to color.


I understand that these are very simple things for more seasoned GeoGebra users, and they do seem simple to me now.  They were new ground for me as I worked on my project, though, and each one was a hurdle that I had to work through to make each of my manipulatives work the way they were supposed to.

If you know of easier ways to do the things described above, I'd love to know!  Please leave a comment.

An Interesting Idea

The basic idea is this: How to motivate students to help one another improve their understanding.

In our most recent department meeting, one of my colleagues described a group work system that he has been trying in his classes.  Here are some basics about the structure of our homework that I need to mention before I describe the system further:

- We have two-day homework.  That is, there is a day in between the lesson and the due date to allow for questions to be asked in class.
- Actual homework (what is done out of the book) is not part of students' calculated grade.  On the due date for an assignment, they take a short half-sheet quiz that we call a daily assessment, and their grade is based on the score they earn.  They may use their homework as a reference on daily assessments.
- If students have completed a short list (10-12 problems, usually) of "required" homework, they have the opportunity to retake their daily assessment to earn back lost points.  They have to complete some additional practice problems first.  We call the retakes "reworks".

At the start of a unit, my colleague creates groups of 3 to 4 students.  He puts one top student and one low-achieving student in each group, and tries to pick the other two students based on who might work well together.  On the day between the lesson and the daily assessment, he has these groups work on a few practice problems based on the previous day's lesson.  While students work on these, they often get their homework out to reference and compare, and many of the homework questions get answered within the groups.  So, while more time is spent in groups, there is a reduction in how much time is spent on homework questions as a class.

The especially interesting part of this group system is the incentive my colleague attaches to it.  He calculates each group's overall average grade at the start of the unit, and again at the end of the unit.  If a group's average rises between the start and close of the unit, he awards all of the members of that group extra credit points on their unit tests.

The great thing about this incentive is that it gives excelling students a good reason to help out the ones who are having difficulty.  After all, the students with the lowest grades are the ones that have the most room for improvement.  Alternatively, it's hard to bring a 98% up a whole lot.  Helping a struggling group member is the most promising way of earning those extra credit points.  Teaching is the great way to learn, so there's a benefit to the excelling students also.  My colleague says he has seen students taking an interest in each others' progress, and that group members begin holding one another accountable for doing their homework and doing reworks as needed.

I'm not a big fan of extra credit points in general, and I might be even less comfortable with adding them to test scores.  In some form or another, though, this sounds like a great idea.  Does anyone else out there have a good method for getting students to invest in each others' learning?

Project Update

For my Math 641 project, I am working on a "translation" of Book III of Euclid's Elements.  As my source for the Elements, I am using Joyce's Elements webpage.  It is obviously in English already, so by translate I mean that I am trying to use more modern terminology for the things described in Book III.  It has been a challenge though, particularly for some of the longer, multi-faceted propositions.  It is difficult to know how to state some of those more clearly, without getting much too wordy (as I'm prone to do).

For visuals to accompany the definitions, and for manipulatives to accompany each proposition, I am using GeoGebra.  I am still a rookie when it comes to GeoGebra, so it is probably taking me longer to develop these things than most, but I have been surprised at how time consuming making these manipulatives has been.  I've been learning a lot about how to set conditions to show objects, and having to think carefully about what conditions I can set in a diagram to emphasize each proposition.  I had originally imagined that I would make an illustrated step-by-step GeoGebra file for the proofs of each proposition as well, but that might be an ongoing task for some time in the future.

To organize and display my work, I am adding an additional page to this blog per Dr. Golden's suggestion.  I had considered using Google Sites, but after looking that over it seemed just as well to lay out the definitions and propositions in the blog, with links to the GeoGebra pages for each manipulative.

Compass and Straight-Edge Constructions

Compass and straight-edge constructions are a part of my school's curriculum for Geometry A.  The constructions that we teach and assess include:

- copying a segment

- copying an angle

- perpendicular bisector of a segment

- bisecting an angle

- constructing a line perpendicular to a given line through a point not on the line

- constructing a line parallel to a given line through a given point

I personally enjoy doing and teaching these constructions.  It's hands on, and I enjoy seeing how they work out.  I'm intrigued by all that can be done with just a compass and straight-edge, especially when it comes to things like quadrature and more complicated constructions.  I like the history of it, too, and it seems amazing what early geometers and mathematicians were able to figure out with these tools.

I wonder sometimes what value these constructions have for my students, though.  Some of the basic constructions can be used to draw other things that we study, like a triangle with certain lengths or angles, or the incenter of a triangle, although we usually have them explore concepts with difficult constructions like that on GeoGebra.  I'm not sure whether these types of constructions really enhance students' visual-spatial reasoning relative to the things they are constructing either.  I know I could do a better job emphasizing the fixed distance feature of arcs, but even if I did that, I'm afraid most of my students would still learn these as mechanical processes.

Eric Pandisico, in his article Alternative Geometric Constructions: Promoting Mathematical Reasoning suggests that students can take the most away from constructions when they have the chance to try them with a variety of tools.  Rather than stick primarily to compass and straight-edge constructions, Pandisico promotes the use of several tools to do the same constructions, including Miras, a simple right angle source, like a notecard, and even a two-edged straight-edge.  He suggests that because each tool has the potential to emphasize a different feature or property of each construction, doing one construction three different times with different tools can give students a deeper and more full understanding of the geometry underlying each construction.

We only teach these constructions with a compass and straight-edge.  We don't use any other methods for creating these things.  After trying these constructions and others with patty paper in Math 641, I could see the value in using more than one method.  The patty paper made the symmetry of the constructions much easier to see, and even though the results were the same, the thinking I was doing about each construction was much different, and for some of them more intuitive.

Classmate Trevor Kuzee did a nice job of comparing and contrasting patty paper and compass/straight-edge constructions in this blog post.  I'm not necessarily drawn toward one method over another, but I will try to give my students a chance and using more than one method the next time I teach constructions.

To help my students review the compass and straight-edge constructions that they had learned, I made the GeoGebra worksheet pictured below.   This didn't represent much new GeoGebra learning for me.  It was a lot of assigning values to show objects again, and there were lots of hidden circles underneath those arcs.  I did set points to which I fixed the text boxes for the instructions, which allowed me to keep those well aligned even between constructions.

Reference:
Padisico, Eric A. (2002) Alternative geometric constructions: Promoting mathematical reasoning. The Mathematics Teacher, 95(1), 32-36.

Reflective Properties of Conic Sections

One of the things I find most interesting about conic sections is their reflective properties.  Here are those properties, as I understand them (for the sake of a common description, I'll suppose a "beam" is bouncing off each conic section):

1) Any beam parallel to the axis of a parabola will be reflected to its focus.  Or, any beam leaving the focus of a parabola will be reflected to a path that is parallel to the axis.

2) Any beam that passes through or leaves one focus of an ellipse will be reflected to pass through the other focus.

3) Any beam directed toward one focus of a hyperbola will be reflected toward the other focus.

These are a nice talking point when teaching students about conic sections, since they provide some interesting reasons for locating the foci or finding equations for sections with specific foci or focal distances.  Flashlights and satellite dishes are familiar objects for kids, and it's always nice to have an answer to the question "Who uses this stuff?"

One of these days I'd like to team up with the engineering teacher at my school and make a model that demonstrates these properties.  I imagine setting up pieces of conics with shared foci, as sketched out below.   Our technical education department has a computerized lathe, so I think we could get the appropriate grooves cut into a piece of particle board and then put a thin strip of reflective material in each one.  There would be a track for a laser pointer that runs perpendicular to the axis of the parabola, so the beam could move back and forth and always be parallel to the axis.  If it were set up correctly, the beam would always hit the last focus (the second focus of the hyperbola) even as the laser pointer is moved back and forth.  I'm not sure how tight the precision would have to be, but I guess that last focus could just be as big as necessary.

Maybe students wouldn't get as excited about this as I would, but I think it would be a pretty cool demonstration for class.  Here's my attempt at creating the same in GeoGebra, but I'm not sure how to plot only part of a conic section yet.

Teaching Proofs

As I mentioned in my first blog post, on Van Hiele levels,  I have some misgivings about the way proofs are presented in the curriculum from which my geometry classes are taught.  The work that we have been doing in Modern Geometry this semester has opened my eyes to the value of conjecturing.  Ideally, proof in my students' geometry classes would develop naturally out of discussion and defense of their own conjectures.  Instruction on how to construct proofs more formally could follow.

As it is, my students face formal geometric proof head on.  They start by writing "algebraic proofs," which basically entails setting up and solving equations with justifications (maybe a theorem or two, mostly properties of equality) for each step.  Then they practice proofs that are more based in geometry concepts.  Early on most of their work is filling in blanks in two-column proofs (which is the format we stick to, for the most part), providing missing statements and reasons.  The next chapter in the course includes parallel and perpendicular lines, and proofs are included again.  The big finish for proofs in Geometry A (although there are more in Geometry B) is triangle congruence proofs.

Proofs are always a struggle, for the majority of my students anyway.  To help them along, I try to point out common patterns in the structure of the proofs we write in class, such as "the final statement is whatever you were asked to prove" or "to change a word into an equation or an equation into a word, use its definition."  I try to remind them to keep the end result in mind, and sometimes we even start at the end of a proof and work backwards a few steps.  A lot of these things that I do to help students "get through" proofs don't really get at the meaning involved, and to a large extent I often feel I'm just teaching them the rules of a game.

It was hard not to be somewhat encouraged though, when by the end of our chapter on triangle congruence, many of my students were able to put together a pretty solid triangle congruence proof on their own.  It made me wonder if teaching formal proof head on might have some value.  The big question, of course, was whether or not my students could transfer any of the skills they were demonstrating to more varied proofs.  I got a partial answer to that question in my class' response to a bonus question on one of their unit tests.

The bonus problem asked them to write a proof of the Triangle Sum Theorem (the sum of the angle measures is 180 degrees).  Not every student attempted the proof.  Among those who did, only two gave fairly conceptually solid proofs, and theirs weren't entirely rock solid as far as our formal standards from class went.  Below is a copy of the problem on the test (I think more could have been said of what is "Given:"), and the two best proofs from my class.

For only two students to come this close took some wind out of my sails again.  But at the same time, it is still the slightest bit encouraging that a couple of them were able to go "off pattern" and prove something fairly different from the triangle congruence cases we had been working with (maybe my hopes for students' proof fluency are way too low).

So I still think that a heavily conjecture-based lead in to proof would serve more of my students better.  But having gone through the usual proof sequence with my students once again, I don't believe what they've done will have no value for them as they continue in mathematics.  If they aren't able to transfer their proof writing to more novel proofs now, might some ground work still have been laid for future proof writing?  It seems like writing proofs formally will always be somewhat awkward the first time, even if it is preceded by lots of practice with informal arguments.

I will have many of the same students again next trimester in Geometry B, and the first unit is on quadrilaterals and their properties.  We are planning to give students a few days to hopefully conjecture some of these properties on their own and then try to prove them with what they already know.  This will be another opportunity to see how much they were able to take away from their education in proofs this trimester.