What is Argument? And Emerging from the Rabbit Hole

(Update:  This post is the second in a series about my learning and thinking about argument and how it relates to our work in the math classroom.  Click here to read the first post.  Click here to read the third.)

Welcome back math geeks!  Last week, I was preparing for a workshop facilitating the learning of a dozen elementary teachers as they prepared for a 6-week interdisciplinary math/science summer academy.  I chose to focus their thinking on “argument.”  More specifically, I wanted teachers to internalize how making arguments based on reason and evidence is a lynchpin that holds our instructional work together across academic disciplines.

To narrow the focus of this workshop on NGSS and CCSS-M content, I decided to dive deeper into the framework of NGSS to find more explicit ways to connect grade level math learning (and instruction) with grade level science content.  And that dive ended with me going down a rabbit hole where I found some troubling language defining how a math argument and a science argument differ.  Much of my understanding about “argument” was turned on it’s head.  I wrote about it here.  (It may be useful to read that post and the comments it generated.)

With the help of the #MTBoS community and after some independent research, I emerged out of my rabbit hole and I’d like to share my learning with you.  It’s my hope that other teachers, coaches, principals, and leaders might find this content useful for future professional learning opportunities.  As always, feedback and input welcome.  Let’s get better together.

Quick Backstory

Traditionally, I’ve used this image to help teachers see how argument is a lynchpin that holds our work in math, science, and English together.  This image is useful to frame conversations with teachers about the importance of asking students to explain their thinking and justify their conclusions.  We must embed questions such as “How do you know?” and “Can you convince me with evidence?” into our instructional practice in all classes.

But Appendix L from the California NGSS Framework seems to contradict the role argument plays in this model.  An excerpt:

…there is a difference between mathematical arguments and scientific arguments—a difference so fundamental that it would be misleading to connect MP.3 [to S.7]. The difference is that scientific arguments are always based on evidence, whereas mathematical arguments never are. It is this difference that renders the findings of science provisional and the findings of mathematics eternal….Blurring the distinction between mathematical and scientific arguments leads to a misunderstanding of what science is about.

Yikes!  That’s some strong language.  And it was language that troubled me because it didn’t fit into my schema around the role of “argument,” “reason,” and “evidence” in both math and science classrooms.

What is Argument?

Researching this question inevitably (and often immediately) leads to a conversation about deductive and inductive reasoning.  We use deductive reasoning to create a proof based on logic that results in a conclusion that is always true.  Forever and ever.  (Think of those classic geometry proofs you may have done in high school or the proof that derives the quadratic formula from ax² + bx + c = 0.)

We use inductive reasoning to make a conclusion based on evidence (data from observation) and that the conclusion is only probably true as a model for generalizing (predicting) future results.  In other words, new evidence and more data may contradict (or change or support) existing conclusions. Therefore inductive conclusions are provisional and subject to revision when new evidence (data) suggests a need to change conclusions.

It’s my understanding that this is what science is fundamentally about and what the NGSS authors are striving to clarify.  Mathematical arguments don’t require evidence to be proven; science arguments do.  A scientific argument without data is, by definition, not a scientific argument.

Here’s my problem:  To frame mathematical arguments only as deductive proofs leads to a misunderstanding of what mathematics is about and leads to boring classrooms for math students.

An Example

Let me explain my concerns using “proportions” as an example.  There’s a proof that shows why the “cross multiply and divide” algorithm works for solving any proportion.  It makes logical sense.  And it’s truly effective at navigating the reality that “real-world” proportion problems often don’t involve tidy numbers that can be calculated mentally or with visual models.

However, if we approach the topic leading with the “cross multiply proof” and having students “set up proportions” and “solve them procedurally,” we rob students of the opportunity to derive the proof for themselves through inspection and exploration.  Furthermore, we deny them an opportunity to share the different ways we can reason proportionally.  Students will create a misunderstanding of what learning and doing math is about.  (And more importantly, students may mistakenly identify as a “non-math person” because they misunderstand about what it means to do math and be good at it.)

The CCSS-M and the progression documents are clear that the instructional focus should be on “reasoning about proportional relationships” rather than “solving proportions.”  Indeed, the CCSS-M doesn’t even choose to define “proportion” explicitly.  A proportion is simply a true equation.  There’s no instructional need or justification to lead with the proof first as a way to reason about solving them.

Instead, when we invite students to investigate, inspect, and inquire about mathematical problems, we invite them to think more inductively and intuitively.  From this inspection and guided conversations, students can begin to generalize their observations, formulate strategies, and refine those strategies for more efficiency.  Then, if appropriate, a deductive analysis of why an algorithm is true can be offered.

This pedagogy aligns with the NCTM’s position that  “procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving.”

Why This Matters

I can appreciate what the authors of Appendix L are trying to express.  All of scientific research is based on inductive reasoning.  Scientists conduct experiments to gather data to answer a statistical question.  They generalize results from specific observations, generalizations that may be revised in the light of new data.  Scientists rarely engage in deductive reasoning because deductive reasoning does not require evidence to be true, and if there’s no evidence, there’s no science.  (Hence, the debate about string theory.)

(Aside:  Einstein’s work is an interesting case.  His theories about the relationship between mass and gravity were mathematical conclusions years before they were scientific conclusions.  It was not until evidence of the theory was collected years later that his ideas went from the realm of the mathematical to the scientific.)

But, as a math teacher, I really struggle with the suggestion that “mathematical arguments are never based on evidence.”   Statistics is the mathematics of evidence.  It seeks to explain patterns in data.  All statistical arguments are inductive.  You could argue that scientists make statistical arguments and mathematicians do not.  You could also argue that the mathematics of statistics is in the proof of the formulas that are used and not in their actual use.  I would concede both points to you.  But I don’t think the sanctity of either discipline is harmed by “blurring the line” here, at least not in a K-8 classroom.  I think it’s fair to suggest that mathematical reasoning can be both deductive and inductive.

In pedagogic terms, a math class based on deductive reasoning alone is probably going to be really boring and tedious.  (Show me otherwise!)  Deductive reasoning is a difficult thing to do and requires advanced thinking skills and a sound foundation in logic.  At it’s core, mathematics is a philosophical and logical structure based on deductive reasoning and proofs that hold true.  But to teach math from only that perspective doesn’t create good mathematical thinkers; it creates bored non-thinkers.

Instead, I think students should be encouraged to inductively reason about evidence and posit their own rules and generalizations of the patterns they are seeing.  That’s where the juice of learning exists in math class.  Teachers should be encouraged to structure math lessons that invite inquiry and investigation, foster conjecture and student discourse, and put the onus of answering “Is this true?  Is this right?  How do we know?” on the shoulders of classroom community of learners.

But the reasoning and the conclusions will most often be at the inductive level.  And that’s OK.  To me, anyway.  Learning to be an inductive thinker is fun and engaging and rewarding.  Maybe you disagree?

I’d love to hear what you think.

Next up, I want to tell you about my new favorite term:  abductive reasoning.

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