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.)

## Quick Backstory

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.”

**For more information about argumentation in science, see the NGSS Science and Engineering Practice of “Engaging in argument from evidence.”**

*Blurring the distinction between mathematical and scientific arguments leads to a misunderstanding of what science is about.*Um, what?!??!

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?

Having taught both NGSS and CCSSM, as well as served on hiring committees for science education faculty, I think the biggest thing I took away is the importance of phenomena to the NGSS. At every point in making an argument in an NGSS aligned curriculum, the argument should be based on or around the phenomena being observed. This fits very well with what is written above, namely that science arguments are always based on evidence.

I never bring up the idea of phenomena with my math pre-service teachers (PSTs), but I focus heavily on the term with the science PSTs. This is an essential and necessary condition for teaching the NGSS.

Thanks Glenn! What are your thoughts about this issue in elementary (or even middle school) classrooms? Much of the math in younger grades encourages students to make inductive conclusions based on observation and patterns. Rarely are they “proving” something in a formal, deductive sense. What enduring understandings should teachers be building for students around different arguments?

Naive question: Is the math argument vs science argument equivalent to deductive reasoning vs inductive reasoning? And if so, don’t mathematicians and scientists use both? Perhaps my struggle is in the absoluteness in the language NGSS uses here…that scientific argument is “always” based on evidence and math “never” is. The language seems to me to drive a misleading wedge between the two disciplines rather than clarify anything. But I’m no expert in this. Definitions are important.

Chase,

I have been thinking about the difference between inductive and deductive arguments, and I think it misses the mark. As math educators, logic and logical argumentation play a role because of the nature of proof and how proofs play a role in mathematics. However, there is no ‘proof’ called for in science education. The role of argument is always to support your claim with evidence. So, for example, if I claim that summer occurs because the Earth is closer to the sun in summer (a very common misconception in science), as the teacher you would ask for the evidence for that claim. I would go to an astronomy book and look for evidence and disconfirm my claim.

Yes, math educators recognize this form of argumentation as inductive in nature, but this is not a term used in science education. Ever. In science ed, the requirement is to have learners ground all of their argumentation in the evidence, the natural phenomena, that occurs. Bringing this type of distinction to science educators creates a, “sure, there is deductive reasoning, but why would you do that in science?” kind of response. I know. I tried. This is a case where the science educators and the math educators speak past one another. The word “argument” exists in both spaces, but means different things to each.

The NGSS is referring to the practice of science education in the K-12 space. That practice is one where “argument” really is an inductive or abductive argument. Always. What Corey, rightly, refers to below is the practice of science at a post-graduate, professional level. At that level, yes, the role of deductive, inductive, and abductive reasoning plays a strong role in science. However, at the K-12 level, there is not a space for deductive reasoning in science. There are no ‘first principles’ that form the role of theorems, from which all other science deductions are made. In that sense, there is no role for deduction in science ed.

I can’t speak to the elementary question very well. My experience is all at the secondary level. I believe that deduction requires a certain level of development in a brain, and where that occurs is far, far beyond my knowledge.

Honestly, even though I took several proofs classes in college, it wasn’t the math classes that ever created a deep understanding of the differences between the three types of proofs. It was my philosophy courses, and the logic courses in particular. The math classes just said, “do this type of proof, given these theorems.”

Glenn! This really helps my thinking. I’m embarrassed to admit that I hadn’t heard of abductive argument until now and a quick read makes me lament not discovering it earlier. Could be very useful language for the younger grades, in particular when students have to estimate quantities and lengths. I appreciate the time and thoughtfulness. I’m going to use some of your language (citing you) in a follow-up post.

Quick question: Is “abductive argument (or reasoning)” a common phrase in the science education world? Never heard it used in my years of math…and wondering about the depth and scope of my ignorance and it’s impact on my own thinking.

I think the more general term is Occam’s Razor. Given any two explanations, the simplest is the correct. Honestly, outside of a geometry course or logic course I haven’t ever really used the terms. Well, not true. I used all 3 all the time when I was speech and debate coach. But in math classes, only in geometry.

As a followup, I just asked my two colleagues who are purely science educators (and educators of preservice teachers) about the use of deductive reasoning in science. They both just gave me funny looks.

So my first thoughts are that we are talking about the difference between inductive and deductive reasoning. Anyone else getting this sense?

And if that is it, then i am thinking that mathematics is being misunderstood. Just like science, inductive reasoning often inspires our investigations. And am I wrong in thinking that many science arguments depend heavily on deductive thinking?

I would like to hear a discussion between leaders in science and math ed on this topic, applying both math and science argumentaion standards.

Thank you for this thought provoking post.

I disagree with the distinction being between mathematical and scientific reasoning. The difference is between inductive and deductive reasoning and both have a place in mathematics and in science.

Inductive reasoning is drawing conclusions from observations. We make observations and discover patterns in mathematics and in science.

Deductive reasoning is logic. It is using things we know and rules and operations to discover new things. This is also done in both mathematics and science. Theoretical sciences are pretty much all deductive. Theoretical physics is a pretty huge branch of science and uses deductive reasoning almost exclusively.

We can see patterns in mathematics (inductive reasoning), then try to explain why they appear and verify that they always hold, or find exceptions where they do not (deductive reasoning).

Theoretical scientists use deductive reasoning to make predictions, then applied scientists try to verify those results experimentally. I think they’re making the wrong distinction above.

Thanks Corey! This comment is well crafted, and I like the induction/deduction distinction you’re making here. They are different and perhaps one (deduction) may lend itself to math more than the other (induction) and vice versa for science. Perhaps we should be focusing more as teachers on the distinction you’re suggesting rather than on the distinction between math and science. This is good thinking that helps me. Thanks you!

We look for patterns (inductive) when we’re wondering how things work. But a pattern is not a proof. We need deductive reasoning for that. Yes, kids looking for patterns is great. I wonder if any of the pattern breaking examples are at a level elementary kids could think about. (https://researchinpractice.wordpress.com/2010/05/07/pattern-breaking/ and https://researchinpractice.wordpress.com/2010/07/12/pattern-breaking-ii/)

I wrote a paper on my thinking in solving a mathematical problem. I definitely used both kinds of reasoning. http://scholarship.claremont.edu/jhm/vol1/iss2/7/