Cross-Country Mathematics: Spending Some Time Off the Beaten Track

There is so much talk about Explicit Instruction vs. Inquiry Based Learning, I’m actually becoming turned off discussions about it. I now think there is a different debate arising from the pedagogy battlefield: Mutually Inclusive vs. Mutually Exclusive. What I mean by this is that some people believe that both forms of instruction have their place (either at different points of a student’s learning journey or in different situations), whereas others have a firm belief that either Explicit Instruction or Inquiry Based Learning has no place at all in their practice. I am slowly making my way into a third debate… There is a best practice vs. There is no best practice. Partly because I’m tired of people being over-critical of teachers whose contexts are significantly different from their own, and also because I firmly believe that what works for some students may not work for others and vice-versa.

What I find is a common thread amongst conversations about mathematics education is the pressure of time. I’m going to be quite general in this next little bit for the sake of being succinct. Supporters of Explicit Instruction comment on the benefits of being able to teach students a method or procedure and have them working on exercises quite quickly, usually because there is less questioning and discussion at each step in an explanation. This can sound quite enticing, as adopters of a primarily inquiry-based approach may be accustomed to having quite a large portion of the lesson taken up by rich mathematical discussion, with not a lot of time left for students to consolidate their learning or practice their newly learnt or developed procedures. Personally, I find that the mathematical discussions and questioning is how I develop positive cultures towards learning and doing mathematics in my classroom, so I am reluctant to completely cut that out of my methodology for the sake of getting more time. So, is there a sweet spot somewhere in between? There might be, but could an attempt to do both justice undermine the aim of each? It’s a tricky one, but I think there’s merit in not rushing to carry out procedures – here’s why…

Regular readers of this blog, of which I think there are about three, would know that I’ve been engaging with Craig Barton’s podcast. I’ve particularly liked how he has broadened my sphere of mathematical thinkers, similar to the feeling I had when I first joined Twitter. Some guests have challenged my thinking, while others have spoken to my current belief set in a really nice way. A few, however, have really inspired me – Jeremy Hodgen was one of those. He recommended a paper written in 1990 by Magdalene Lampert titled When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. There are a number of very interesting elements to this paper, but the essence of it (as described by Lampert) was to examine whether and how it might be possible to bring the practice of knowing mathematics in school closer to what it means to know mathematics within the discipline. A notion raised in the paper talks about some of Imre Lakato’s work:

Mathematics develops as a process of “conscious guessing” about relationships among quantities and shapes, with proof following a “zig-zag” path starting from conjectures and moving to the examination of premises through the use of counterexamples or “refutations”.

This idea of “conscious guessing” is something I became more accustomed to when I started to engage in some of Polya’s work. In his film, Let Us Teach Guessing – A Demonstration With George Polya, he says to the class, “Test your guess. It’s the difference between a savage and a scientist”. This was recommended through Anne Watson and John Mason’s interview on Craig’s podcast, in which Anne and John spoke at length about how important it was for students to be conjecturing in maths, and how willing they needed to be to disprove their own conjectures. This process, whether it’s coined as conscious guessing, testing your guess, or making conjectures, is such a critical phase of the mathematical learning experience as it helps students to learn about what it really means to do and know mathematics. Lampert’s paper sheds light on doing and knowing mathematics in school (remember, this was in 1990);

Doing mathematics means following the rules laid down by the teacher; knowing mathematics in school means remembering and applying the correct rule when the teacher asks a question

For me, this connects with a previous post I wrote on what impacts how teachers teach. I think how you teach is largely affected by your purpose as a teacher and how you want students to see mathematics, which I think is also impacted by how you see mathematics.

Bringing this back to what it means for the classroom, I think there needs to be a balance between Explicit Instruction and Inquiry Based Learning, but not necessarily an equal split for each lesson. Personally, I would like my students to come to know and do mathematics in a way described by the NCTM and MSEB in Lampert’s paper, before honing their skills to do well on a test:

Mathematics students should be making conjectures, abstracting mathematical properties, explaining their reasoning, validating their assertions, and discussing and questioning their own thinking and the thinking of others

Through these opportunities, teachers can surface students thinking through dialogue and are better able to embed formative assessment (check out the book Jeremy Hodgen and Dylan Wiliam put together, Mathematics Inside the Black Box: Assessment for Learning in the Mathematics Classroom).

This type of mathematics is described quite nicely by Henry Pollak as “cross-country mathematics” (also quoted in Lampert’s paper). I think, as a teacher, I value the time students spend zig-zagging around and I actively try to keep them in a space where they are testing their conjectures and generalising and formulating rules as they move through. I do find that this takes quite a bit of time and, on reflection, I’m not so good at tying things back together into something meaningful for my students before the end of a unit – it’s kind of like there’s a heap of unfinished documents in their long-term memory which need tidying up and sorting. I think that’s where and when Explicit Instruction can really come to the floor. In a similar way to how Cognitive Load Theory enthusiasts are reluctant to let students loose on open-ended problems before the automation of essential skills, I am reluctant to delve into the refinement of knowledge until students know and understand the mathematics for what it is, rather than what it ends up as. Hopefully, this is far better described by Pollak, which I’ll close this post with:

In contrast to walking on a well-marked path, the cross-country terrain is jagged and uncertain; watching someone traverse it is a key to learning how to traverse it yourself… [As a student, I] had very interesting time watching him [his teacher] struggle, inventing proofs and trying to think about the right way to do it. I learned a lot more mathematics that way than I might have if it had been a perfectly polished lecture; and I think already at that time I developed my feeling that I like cross-country mathematics. Mathematics, as we teach it, is too often like walking on a path that is carefully laid out through the woods; it never comes up against any cliffs or thickets; it is all nice and easy. (Albers & Alexanderson, 1985)


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