I teach two sections of Honors Algebra II and two sections of Algebra II. Today, I introduced the concept of finding roots of rational functions. I did this with a discovery-based task.
The student response to the task contained a lot of learning for me, and not just in the form of what they wrote down. Many mathematics students seem caught up in anxiety about class that is rooted in their past experiences in the classroom. I’ll discuss that at the end; if you want to just go straight there, look for the last section title.
Bellwork and Beyond
I began with an exercise:
Because it’s been a while since we studied polynomials, a necessary concept for rational functions, I wanted to see where the student gaps were.
Students were nearly unanimously correct for (1) (a polynomial) and (6) (not a polynomial). Students were also largely correct for (2) (not a polynomial) and (3) (a polynomial). And students were nearly unanimously incorrect for (4) (a polynomial) and (5) (not a polynomial).
Indeed, over the course of the day (about 90 students), only two students correctly sorted all six expressions.
However, I was initially mistaken about why many students were wrong about (4). This was a reminder to me that our own preconceptions as teachers can lead us astray: It’s not enough to know that students are incorrect, because if we “fix” the wrong misunderstanding, we’re not using our time efficiently.
My thought: Students will forget that the expression 4 is a function (on x). This is a topic I’ve covered repeatedly. Since y = 4 for all values of x, the basic definition of a function is satisfied.
While that may have confused some students, though, many said they thought that it violated the definition of polynomial because it didn’t involve any actual adding:
Polynomial: A sum of monomials.
I only learned of this misunderstanding by first starting with the definition of polynomial and then discussing why each expression either satisfied or violated it.
Because (5) “looks like” a sum of monomials and (4) is only a single term, students got confused.
Indeed, looking at my list of expressions from this perspective, the “is there adding?” test results in precisely the most common answer set given.
Based on this, in later hours, I modified my definition of polynomials:
Polynomial: A sum of one or more monomials.
Of course, I then proceeded to remind them of the definition of monomials:
Monomial: A term that can be written in the form axⁿ, where a is any real number and n is a non-negative integer.
Students continued to struggle with identifying why I specify “non-negative” instead of “positive.” This is one of many concepts where, as a teacher, I in turn struggle with understanding their confusion. They’re aware of 0 as a numeric value, and understand when explicitly told that 0 is neither positive nor negative, but when asked to call up when prompted, “Why would I specify ‘non-negative’ instead of ‘positive’?”, or leadingly, “What number is neither negative nor positive?”, an answer is rarely forthcoming.
I think part of this is rooted in mathematical notation itself. x⁰ is nearly never something that is written, except to make the point that it doesn’t need to be written (even though it’s not technically correct that x⁰=1 for all values of x).
Task: Exploring Roots of Rational Functions
Once we were done with this task, I moved on to the discovery task.
In my first hour (Honors Algebra II), I simply handed the worksheet out, not providing any instructions or context beyond “Please answer the questions on the sheet. I want to see what you can do without explicit guidance.” Here is the first side:
Students resisted doing this at all until I provided context, which I started with in my other sections. I got the best engagement in my second Honors class, enough that I was hopeful about my other two hours… but that commentary will come.
The context I provided compared mathematics to science. When I asked students to explain the scientific method, they could do so, in all four sections:
- Observe a natural phenomenon
- Create a hypothesis
- Test it
- Adjust it
- Repeat the last two steps
In the non-Honors classes, there did seem to be some sentiment that the failure of data to satisfy a hypothesis ought to mean getting rid of the hypothesis entirely instead of adjusting and trying again. This may be an area for future exploration.
I explained that mathematicians follow a similar method when developing new ideas:
- Observe a mathematical event
- Create a conjecture
- Test it
- Adjust it
- Repeat the last two steps until it seems to work for all cases
- Strive to generate a proof or a “why” for the conjecture
Before students get to college, mathematics education tends to be focused far too much on learning existing mechanics and procedures and not enough on how mathematicians develop new concepts.
Once students were satisfied that they were being mathematicians, they were more willing to engage in the task.
Even so, they fell back on what were clearly survival strategies. I have found that, by the time students get to Algebra II, they have come to rely on a set of techniques intended to make sure that whatever makes it to paper is what the teacher wants it to be.
The goal for far too many students, in other words, is not understanding but compliance.
So when it came to question (2), for instance, some students graphed y = (-1 + 1)/(-1) because they thought that they were supposed to substitute first, and then graph.
And then there was the refrain familiar to math teachers who ask deliberately vague questions: “What kind of answer do you want?”
The question is often asked with a sense of anxiety or fear. Through one-on-one conversations, I have learned that many of my students have had deeply traumatic experiences at the hands of former mathematics teachers: Teachers who demand specific procedures, even when the instructions aren’t clear and even when other interpretations and methods are possible.
“Write whatever comes to mind for you,” I say.
“So I can write anything?” is the response.
“If it’s something that you see happening in the graph, yes.”
Here are some responses I got for 2a:
- Slopes down
- y = 0: They all cross the x-axis at -1
- A graph and (-1, 0)
- y = 0
- When x is -1, y is 0
These answers differ in rigor and relevance, but they all reflect valid student thinking.
Two of my non-Honors students expressed the concern that the answers were too obvious, so they must be doing something wrong. This is part of the mathematical trauma that I’ve seen before: If the answer seems obvious, it’s wrong.
This myth is the exact opposite of my own experience with mathematics: If you have a full understanding of what’s going on, most answers are obvious. The challenge for my own learning of mathematics is getting to that level of understanding.
I was pleasantly surprised that many students were willing to predict before graphing, rather than going straight to the calculator for questions 3 and 5 (and 7, shown below):
One of the students in my last hour answered (7) with y = 0 for each. I had been anticipating the x value, as most students provided. I praised this answer because, while it wasn’t what I had been intending, it accurately answered the question as I had phrased it.
As a less experienced teacher, I may have gotten frustrated with the student, or accused them of being a smart aleck. But it was an important learning point for me, another reminder that my perceptions as the person who wrote the question and as an experienced mathematician, is different than those of a student.
Perhaps I should have been more explicit (“What is the value of x as it crosses the x-axis?”, for instance), but perhaps it’s just fine as is. After all, as is, it allows for a discussion to take place, a discussion that’s stifled by leading students through deliberate doors.
I did not get to discuss question 8 with any students today, and the whole-class discussion of the entire sheet will happen on Monday.
Most of the students in my non-Honors classes resisted doing this task, as they do with most open-ended tasks. A few students wouldn’t even look at the paper until I sat with them and encouraged them personally.
This one-on-one consultation is a time-consuming activity that feels difficult to justify when we have a thick amount of curriculum to get through. I am exploring it now in part because this particular chapter has less variety of topics because the topics themselves are more difficult to grasp.
Trauma and the Mathematics Classroom
Many of my students are comfortable with completing worksheets where they have been given a model and just need to repeat the steps. Based on reports from around the country, this is a common behavior: Tell me what to do, let me puppet it, and give me points for having done so.
For a student, this is a safe route. There is no risk, and if school (and mathematics class more specifically) has been established as a risk-based endeavor, it is more compelling for students to develop survival strategies rather than true learning strategies.
We speak, rightfully, of trauma-informed teaching where we consider the classroom-external traumas that students encounter: Poverty and homelessness, death and illness of family members, abuse and neglect, peer rejection and bullying, drug abuse, and so on.
How often do we reflect on the traumas that have been inflicted on our students by ourselves and our fellow teachers?
I thought about this question often today, as students responded in various ways to this task. I thought about it when I saw the plug-and-chug survival strategy for question 2, and even in the low response rate for question 1.
I hear it in the litany of “Is this right? Is this right?” insecurely repeated by students who want to have faith in themselves but have had the rug of confidence been pulled out from under them one too many times.
It is difficult to truly learn when you are afraid of the learning environment. The classroom could be, and for many students is, a safe place to explore.
But for many students, the classroom is a place where the tasks serve as a reminder that say, “You’re stupid. You’re inferior. You will never understand this. Only smart people get this.”
With that message in mind, it’s no wonder many students give up and dig in with their cell phones. The cell phone is a safe friend. So much of the cell phone experience is explicitly designed to keep the user happy: So much of the classroom experience seems designed to discourage the student. The choice of distraction over scholarship, in that light, is not just obvious, but natural and even the saner choice.
We won’t get higher student engagement through edicts about distractions. We won’t get it through more complaints about “lazy” students. If we don’t adequately and honestly address not just the classroom-external but also the classroom-internal sources of trauma, we’ll continue fighting an unwinnable battle.