Student Voice in the Mathematics Classroom

I have mentioned in earlier posts, some of the problems with the way students typically view themselves in relation to mathematics and mathematicians.  This is a deep problem, which pops up throughout our culture.

Let me illustrate:  How often have you heard an adult say (or said yourself) to a kid, “You don’t like (or are struggling with) math class, oh that’s ok, I am not a math person”?  Have you ever heard someone make a parallel statement about reading…”Oh me too, I’m not one of those people who reads”.  Is it surprising that students think math is inaccessible and reserved for the elite.

Even those who view themselves as good at math are spending the majority of the time repeating processes and rules that they only partially understand (which would be alright if they were also being encouraged to ask questions about those partial understandings).  Students don’t speak about mathematics.  Teachers speak about mathematics, textbooks speak about mathematics, the state and the principal might speak a little about mathematics.  These actors have authority.  Go to any classroom (mine included) and you will almost certainly find a version of a pattern in which a teacher asks a question (which presumably he or she already knows the answer to), a student answers the question, and the teacher pronounces the answer right or wrong.  Sometimes the textbook takes the place of the teacher as the pronouncer.  But isn’t the beauty of mathematics that the right answer is right without pronunciation? Aren’t their deeper ideas to be explored?  Do students and teachers have a sense that many of the things we declare right or wrong in mathematics are actually choices (negative times negative is positive, for example)?

Its not always bad to tell students what is correct and incorrect, or to ask questions you already know the answer to, but it is bad to always do these things.  It creates a culture where you do mathematics they way you do it because that’s what you were told to do.  Then we all wring our hands with surprise when students are bored, angry, alienated, or simply defeated by the exhausting list of procedures and rules.  Standardized tests obviously accelerate the problem since the least risky way to prepare for these tests is to compartmentalize each type of question and then try to execute the correct procedure when that question comes up.

In what is sadly a very typical exchange, I was observing a classroom where a student showed me her attempt to add fractions.  In her work, she had simply skipped the common denominator, adding “tops” and  “bottoms”, a mistake that quite common.  I am not sure if she sensed from my face that something was out of place, but she quickly stated, “that’s not right is it”?  I broke the mold a little by asking her why she thought it wasn’t right, to which she replied innocently, “because that’s how I do it on tests and they always tell me that I am wrong”.  This was not a young girl, but a student who had probably been receiving lessons on this topic for years.  Its clear that she did not feel she knew mathematically why her work was wrong, or that she had any sense that this type of knowledge might be important.

In my research, I have worked with teachers who want to try to address these types of problems.  The good news is that there are lots of things to try.  The bad news is that none of them work very well.  It is very difficult to overcome an entire culture.  Students can get nervous, teachers can get nervous, parents can get nervous, administrators can get nervous.  And if your test scores should happen to go down, then what?  Still, I would urge any math teachers to think about these difficulties.  We should expose and consider the problems in our practice.  How can we ensure that our classrooms are places where teachers and students do mathematics?  How can we promote and encourage our students to have a sense of control and authority in their mathematics education?

If you have anything good, let me know.

Donut Points

As a first year teacher, I am not sure how good I was, but I was certainly enthusiastic.  I had discovered a talent for mathematics while at college, and a passion for interacting with young adults even earlier.  But my first attempts at combining the two were necessarily clumsy.

In math classrooms, it is sadly too common for students to view the material in a passive, receptive way.  Mathematics, to many, is a foreign language, or perhaps a foreign religion.  You can, through trial and error, memorization and repetition, or the perception of social cues, start to guess what the priest of mathematics at the chalkboard wants you to do or say.  Sometimes the high priests of mathematics will pass on answers to the odd questions so you can see if you are wrong or right, but you don’t imagine that you are supposed to understand each step–that every line is a choice, and that every choice must be valid and should be helpful.  You know things are right or wrong, but you don’t spend too much time worrying about if they make sense.

In my first few weeks, students rarely asked questions, and never corrected me.  I was confused because I am fairly inconsistent with arithmetic, and I know that I had always loved nothing more than pointing out to my teachers some mistake they hadn’t caught.  Slowly, it dawned on me that my students were not in the habit of questioning the teachers mathematical authority.  I was the priest, they were the catechumens (and they were more or less expecting the mathematical equivalent of the Baltimore Catechism).

In an attempt to remedy this, I stole an idea from one of my Professors.  I hung a crude poster next to the board and explained to each of my classes that every time I made a mistake and it was caught, by a student, I would mark a tally under their periods section of the poster.  When the class had reached 10 tally’s , I would buy them donuts.

Market-obsessed education reformers go on and on about incentives, but rarely bother to ask what kind of behavior they are trying to promote.  With Donut points, I did not reward students for scores or for grades, but for a simple behavior (calling me out) which, I believed, would have a huge impact on how students engaged with mathematics and viewed themselves as students.  I wanted to give them freedom and responsibility, and, after some tweaks to the policy (no shouting, no crying, no changing my work and trying to set me up) it was very successful.

As the year progressed, we had many fruitful discussions about whether a particular mathematical statement was a mistake or not.  If it was, what type? Should it count? Does a misspelling count? Does a poorly drawn graph count?  Does a hardly visibly minus sign count? If these things count, how do they get treated on exams?  I can’t tell you how “proficient” this made everyone in standard x,y,z but I am confident that my students learned, albeit slowly, that they had the potential to speak and therefore be mathematical thinkers.  I’m not sure it was enough, but it was a start.

An Introduction — Nice to textually meet you.

“The only purpose of education is Freedom, the only method is experience” – Leo Tolstoy

Every educator I have known has stories of how they were inspired and impacted during their time as a student.  Mine tend to focus on issues of authority, freedom, and trust.  I love working with young adults and trying to help them realize their power and responsibility in the world.  I believe that schools should be intellectual places and vibrant communities, built on strong relationships and that policies and curriculum are best when developed together and locally, allowing teachers and learners maximum trust and freedom.

I was first introduced to the education philosophy of Leo Tolstoy by a dear friend (and Tolstoy descendant) while I was a research student at Oxford University.  My thesis focused on attempts to provide students with mathematical authority in the context of the secondary classroom.  The statement that you read at the beginning of this post illustrated to me how simple the motivation behind education should be, but how complex and difficult the practice of education is.

If the purpose of education is freedom, then surely we must educate by allowing and encouraging freedom.  If the method is experience, then we must empower students and teachers to make use of their experience.  Thus, true education cannot be imposed or controlled but must involve choice, experiment, interaction, messiness—each context will be different, each problem different, each solution different, and any standardized, scripted, approach ultimately doomed to failure (almost by definition).

In this blog, I hope to share my musings, some of my work, and the thoughts and experiences of some teachers that I know personally.  I hope to interact with friends, colleagues, students, and strangers…so read and comment and hopefully we can produce something invigorating.