Mathematical Musings

Last week, I jumped the gun a little bit and wrote about the differences between traditional and reform mathematics, not knowing that this week’s readings were going to address that difference in a much more direct manner.

In particular, I spent a couple of paragraphs explaining the difference between the two, and this week, I came across the real crux of the matter in one of the articles I was reading:

The reform mathematics movement contends that learning mathematics requires that students understand mathematics content through mathematical inquiry rather than through a passive reception of rules. (Suurtamm, Lawson, and Koch, p. 32)

This is the real beating heart, and a single-sentence that sums up the philosophy I’ve been using to teach for the last twelve years.  It is not the job of a teacher, in my never-humble opinion, to simply spoon out knowledge to be devoured and assimilated in the classroom.  At least, not in the twenty-first century.

In the age of Wikipedia, YouTube, and a thousand other resources for factual information, there is no point in asking students to memorize facts and processes.  It is more useful for students to develop skills in researching, problem-solving, and analysis – that’s how they’re going to succeed in the future.

An analogy I’ve had some success with in the past when trying to explain this to people who are not math teachers uses D-Day as an example.  If a student knows that a massive allied invasion hit the beaches of Normandy during the Second World War, and that invasion resulted directly in the liberation of Western Europe, eventually pushing the occupying Axis forces back into Germany and resulting in the fall of the German government and military, is it really important that they know this invasion happened on June 6, 1944?  In an environment where they have access to wikipedia and other online resources on their ever-present cell phones?  Generally, I’m able to convince people that this is only relevant for those who are training for appearances on Jeopardy.

Are the ideas and minutiae important?  Of course they are, and despite how you might read that last paragraph, I’m not advocating otherwise.  But the days of spending history classes memorizing dates and names of events are long gone.  So too are the days of spending math class memorizing formulae and using them, free of context and real-world meaning, over and over again.

Where do we go with this?  In an ideal world, students would just explore those areas of math that appealed to them, and the teacher could serve as a single resource among many.  Unfortunately, in a world of government-mandated curriculum and university program requirements, we have to do the best we can to make inquiry a viable method of instruction.

As the delivery of the math curriculum has changed over the past twenty years, the role of the teacher has changed with it.  In educational terms, we usually mark the divide between the traditional math curriculum, and the reform math curriculum.  The former is characterized by teachers in the, well, traditional role.  The teacher stands at the front, delivers information, and then the students practice those ideas up until they’re given a test.

Much of the research that has been done over the past thirty years or more values the reform curriculum, in terms of its effectiveness for student learning.  This style of curriculum is designed behind the idea that students develop their own ideas around math, and how it works – often, they can be presented with investigations, or mathematical experiments, or information, and can learn mathematical concepts through pattern recognition or guided exploration.

The latter is how I have always taught – to my mind, it’s never enough to just give them the mathematical information.  They have to understand the underlying ideas, how those ideas fit together, and how they influence each other.  All of this amounts to the philosophy that if they construct their own understanding of the mathematical ideas, it results in deeper and better understanding of the important concepts.

Group work is an important part of this process.  Certain people are thoughtful or inspired enough to be able to follow through a process like this on their own, but the vast majority benefit from having others around them working at the same pace, to use as sounding boards in developing ideas.  This collaborative process can be critical in developing the understanding we seek.  However, choosing a task that is best accomplished as a group is a challenging one.  Enter: open-ended questioning.

Chizhik (2002) wrote:

…variable-answer tasks may allow for a lot of interaction on any group decision as there are no correct answers. With each step of the solution being open to question, more access could be provided for each group member to give
input, thereby increasing and possibly equalizing verbal participation of all group members. On a single-answer task, however, once a perceived correct suggestion has been offered, then the group has to move on without much discussion. (p. 182)

Open-ended questions are problems posed where there is more than one solution – the very definition that Chizhik gives earlier in the above paper.

One of the most challenging things about assigning group work is in attempting to assign work that gives all students an opportunity to contribute.  Often, one or two group members can be domineering, taking control of the larger group and “solving the problem,” leaving the rest of the group with nothing to do but transcribe answers and ideas.  With an open-ended question, it leaves the door open for other students to come up with alternative methods and even alternative solutions that are just as valid.  This can ensure that those students who are not as assertive or not as quick to reach conclusions are still able to contribute, and not be shut out of the thinking process.

At my school, there is a constant conversation between students, teachers, guidance counselors, parents, administrators, and pretty much anyone else who is even tangentially involved in students’ applications to university.  A lot of the time this conversation brings to light the (relatively) common perception that certain university programs require completion of a calculus course not because the skills are necessary to their future studies, but rather because it’s a way to “weed out” the weaker students.

I can’t speak to the truth of this perception; I’ve never worked at a university, and I’m not in regular contact with people who work there either. But it still feels… plausible.

Regardless of the veracity, it was interesting to find this thought reflected in one of this week’s readings:

In purely functional terms, mathematics has long been recognized as a critical filter. Course work in mathematics had traditionally been a gateway to technological literacy and higher education.  (Shoenfeld, p. 13)

Of course this is an equity issue – if students aren’t learning the fundamentals of mathematics in grade five, then they’re advancing through school in a way that is going to make success more difficult in future years.  And that lack of success is going to translate to missed opportunities in… well… life.

I recently watched the documentary Waiting for Superman, and of striking poignancy was the story of one of the families profiled in the film.  One of the students from Los Angeles, Daisy, had dreams of medical or veterinary school, but she and her parents believed that she needed to enter a charter school in order to be successful enough to make it to college.  However, math and reading were particularly called out as problems that would emerge and persist if she went to her local junior high schools.

Math is a gateway.  In recognizing it as such, it also recognizes that students who struggle in math (or those who do not achieve a solid understanding of math) may not be able to pass through those gateways.

I’ve been a teacher for twelve years, now, and a lot of my attitudes have changed in that time.  Contrary to what you might expect (and contrary to perhaps the general trend of teachers as they gain experience), I am a lot more mellow than I used to be.  I think, in hindsight, that I was perhaps one of the most rigid new teachers that ever taught; in terms of classroom management, in terms of my expectations, and in terms of treating students the same.

This was even visible in the way that I felt about the use of math symbols and terminology; I was precise, and picky when grading students’ achievement on their submitted work, trying to ensure that they used the right words at the right time.

More recently, though, my attitude has completely changed.  I attribute this in large part to my overseas experience, where I was teaching a class of students who were functionally bilingual, but whose first language was usually Spanish.  This issue was compounded by the fact that their path, generally, took them from grades seven through nine in an English-language math classroom, to Spanish-language classrooms for grades ten through twelve, and then to Colombian Universities, where the language of instruction was almost certainly Spanish as well.

As their grade nine teacher, it was hard to justify to myself (or indeed, to them) that they needed to know the English terminology for mathematical concepts.  So many of the words weren’t even similar between languages (slope and pendiente).  In addition, the terminology doesn’t tend to be explicitly used in problem-solving situations either.

Generally, the important thing about mathematical concepts is that you’re able to internalize an understanding of those concepts, explain them to another person, and use them in appropriate situations.

Alder’s discussion

this kind of explicit teaching can result in a language-related dilemma of transparency with its dual characteristics of visibility and invisibility…For mathematics teachers, it is not simply a matter of going on too long but of managing and mediating the shift of focus between mathematical language and the mathematical problem (which of course are intertwined). (Alder, 63)

I read something today as part of my course readings that really jumped out at me:

Insert here the story of the teacher who seemed shocked about the suggestion that we might deal with students differently based on their background. (de Abreu and Cline, ##)

I think the teacher’s reaction in this quote is a perfectly natural one.  Of course we, as teachers, think that the way that we treat students should be equitable – we want to ensure that every student, regardless of their background, is set up for future success in math.

The position that de Abreu and Cline take in their article seems counter intuitive at first.  Should we teach students according to their culture?  Their ethnicity?  Most of the teachers that I have spoken with will say “no, of course not.”  As teachers, we think that someone’s heritage – their life outside of school – shouldn’t have any impact on their ability to succeed at mathematics.  And of course not!  Why should it?

It does, of course.  A lot of a student’s world view comes from their experiences outside of school – they do not exist in a sterile bubble where what they learn at school is the only context in which they understand the world.  And if we’re doing our job correctly, then the math that they learn should be intrinsically linked with that world context.  So, logically, we need to be aware that different students understand math in different ways.  We may not need to adjust our teaching explicitly, but being aware of the cultural context in which students learn is a necessary part of how we teach.

What does this look like?  On the simple end of the scale, I think it’s important to acknowledge that sometimes students learn processes in different ways.  For example, I can think of a time from my teaching in South America when I was working with students and solving rational equations.  A big part of this process, naturally, is dividing fractions.  In Colombia, the way that this process is taught is using a method my students called “La Oreja” – literally, “The Ear.”  It’s difficult to explain how this works in text, so I’ll try to add an image or video later.  In North America, we tend to “Flip and Multiply.”  Both methods work, and both methods have a logic and mathematical understanding to them – why should they be forced to conform to a different method when the one they’ve got works, and is mathematically sound?

So… how does this affect my practice?  It doesn’t, to a large extent.  Or rather, it doesn’t inspire me to do much that I’m not already doing – because I’ve always tried to be open-minded when it came to prior knowledge and how it’s impacted in the class.  But it’s cause to think too: How many times have I inadvertently disparaged someone’s prior knowledge, or the context that they bring from their home environment – and what impact has that had on student learning and enthusiasm for math?

Reference: de Abreu, G., & Cline, T. Chapter 8 Social Valorization of Mathematical Practices: The Implications for Learners in Multicultural Schools