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.