Welcome back math geeks! I have a problem, and I need your help. I went down a rabbit hole and what I found bothered me. Help me make sense of it. (Update: This post is the first 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 second post. Click here to read the third.)
I’m giving a workshop next week helping some 3rd, 4th, and 5th grade teachers plan out some fun summer school math and science stuff. They’re using some cool science curriculum and they wanted to have some engaging and enriching math activities for students as well. My brain immediately went to the amazing #MTBoS resources including Graham Fletcher’s 3-Act math lessons, Andrew Stadel’s estimation180 activities, John Steven’s “would you rather” prompts, and some number talks. So I have the bulk of what I wanted to do for the latter two-thirds of the workshop.
To help frame the workshop, I plan to use the following image that I’ve used dozens of times the past few years.
I love how how simply this diagram shows the overlap between the 8 Standards of Mathematical Practice, the 8 NGSS Science and Engineering Practices, and the 7 “student capacities” for ELA. It tidily shows that “argument” based on “reasoning” and “evidence” is the lynchpin that holds a lot of our work together as teachers. I’ve found it to be an effective tool to frame a lot of workshops where I’m asking teachers to think more broadly about the scope of their work and to seek connections between content areas. I usually build on this diagram by sharing Andrew Stadel’s video about Math Practice 3 and this Marzano article.
But this time I wanted to go deeper with teachers and explore more about what NGSS had to say about the overlap between math and science. More specifically, I wanted to find more details about the connection between Math Practice 3 (construct viable arguments and critique the reasoning of others) and Science and Engineering Practice 7 (engage in argument from evidence).
So I went looking. And fell into a rabbit hole.
The Rabbit Hole
I came across the California Science framework that included an overview of the NGSS (Chapter 1). I clicked on that, and I skimmed through the document looking for what I wanted. I came across this on page 71: “The level of mathematics and computational thinking in science should develop in parallel to the mathematical skills and practices expected by the CA CCSSM. Appendix L of the CA NGSS provides a discussion and examples of the connections between the content and the practices of the CA CCSSM and the CA NGSS.” (Special thanks for Chrissy Newell for shoving me in to this rabbit hole in the first place.)
Appendix L sounded like just the thing I was looking for so further on (down?) I went.
And then I read this on page 4 of Appendix L (emphasis mine):
About CCSSM practice standard MP.3: None of the connections boxes include a link to CCSSM practice standard MP.3, which reads, “Make viable arguments and critique the reasoning of others.” The lack of a connection to MP.3 might appear surprising, given that science too involves making arguments and critiquing them. However, there is a difference between mathematical arguments and scientific arguments—a difference so fundamental that it would be misleading to connect to MP.3 here. 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. As Isaac Asimov wrote “Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever.” Blurring the distinction between mathematical and scientific arguments leads to a misunderstanding of what science is about. For more information about argumentation in science, see the NGSS Science and Engineering Practice of “Engaging in argument from evidence.”
I went further down the rabbit hole to read more about “engaging in argument from evidence.” And then I looped back and read the paragraph above about a dozen times really trying to examine the meaning of the sentences I’ve emphasized.
Throw Me a Rope…
…and help me get out of this hole. Here are a few rambling thoughts.
- I’ll buy that science arguments are almost always based on evidence. (Although, let’s please remember that data can be massaged or poorly collected and evidence can be misinterpreted.) Scientific conclusions are based on reliable experiments that produce repeatable and predictable results. That said, didn’t Einstein make a scientific argument prior to collecting his evidence about gravity’s ability to curve space?
- I struggle to embrace that mathematical arguments are never based on evidence. I suppose that all math truths are fundamentally based on logic and proof and do not rely on external evidence.
- However, how do I argue for the merits or limitations of my mathematical model if I am not also paying attention to the accuracy of its predictions of the real world and citing that as evidence for my argument?
- Or if I’m teaching fractions and one student uses a diagram to show that 1/2 plus 1/3 is 5/6 and another student uses an number line, is that two different types of evidence that support the argument?
- If the line between “sound mathematical reasoning” and “scientific evidence” is worth drawing, when do students need to worry about drawing that line? Does keeping the line “blurred” do detrimental harm to the understanding of what science is?
- Does clarifying this line even produce a clearer understanding of what science is and the role that math plays as the universal language of science? Furthermore, does clarifying this line make math “just a bunch of boring proofs” that aren’t worth arguing about?
- Are math truths really eternal? Wasn’t there a kerfuffle about the discovery that the square root of 2 was irrational? Didn’t the discovery of non-Euclidean geometry turn our “eternal” understanding of math (and the shape of space in the universe) on its head?
- And why are NGSS documents (like the Venn Diagram and Appendix L) suggesting some conflicting evidence about the overlap around argument in math and science?
Help me out here. What sense do you make of all this? Where have I gone awry?
Most importantly, what should be the enduring understanding that teachers (at all levels) should take away from this discussion about argument in math and science classes?