What’s Galileo got to do with how we teach math?


Turns out that Copernicus’ new theory (Earth revolves around Sun) had more trouble getting accepted than is commonly known.

According to Alan Chalmers, a philosopher of science and author of What is this Thing Called Science, his ideas faced opposition not only from religious , orthodoxy but from some pretty stubborn facts that the new theory could not explain.  For instance, if the Earth was really spinning, why didn’t rocks and mud spin off the Earth?  And why did blocks of lead fall straight down from the Tower of Pisa, instead of landing farther away (or closer) when the Earth spun beneath them as they fell?

These facts, and a host of others that contradicted the Copernican view, could not be adequately accounted for by scientists for dozens, even hundreds, of years, yet Galileo abandoned the Ptolemaic theory (Earth is center of Universe) in spite of these facts while many others held fast to Ptolemaic theory.


Chalmers suggests that both the Ptolemaic and the Copernican view were almost hopelessly complex, convoluted and confronted with dozens of facts that they couldn’t easily explain, but that the main attraction of the Copernican theory lay in the “neat way it explained a number of features…which could be explained in the rival Ptolemaic theory only in an unattractive, artificial way.”

Neat…attractive.  One of the single largest leaps of scientific progress in history came about in part because of “evidence” that belongs more to the realm of poetry than science.

Is science no more different from poetry than a sonnet is from free verse?  And if the criteria are unclear for abandoning the Ptolemaic theory in favor of the Copernican, then how can we decide what evidence we will need in order to determine whether to abandon or defend any of our current theories?  What do we do when faced with such a situation, where the facts line up on both sides of the debate?

This conundrum seems to be even worse when we consider that this is the situation as it exists in the hard sciences, and our debates in the social sciences are likely to be even more difficult to conclusively resolve.  Can we ever make progress?  Can we ever really learn anything?

A possible solution comes from Karl Popper, generally regarded as one of the greatest philosophers of science.  Popper offers a neat three-step logic model for learning that he then expands to four steps and uses to explain everything from the behavior of plants to natural selection to the scientific method and the progress of scientific theories. In three steps, the common sense method of learning through trial and error goes Problem – Attempted Solutions – Elimination of unsuccessful attempts.

Here’s a quick example: a tree needs water – sends out roots in all directions – roots that don’t reach water wither and die, those that reach water grow longer and stronger.  Popper then elegantly uses this schema to explain Darwin’s revolutionary theory: change in environment causes danger for a species – lots of genetic mutations – most of these mutations kill the organism, but perhaps one results in a successful adaptation.

The scientific method improves on this innate procedure by adding circularity and conceptualizing attempted solutions as theories: Old Problem – formation of tentative theories – attempts at elimination (or we could say falsification) of these theories – New Problems.

So maybe this model gets us out of the problems Chalmers raised.  What do we do when we are faced with a situation where the facts line up on both sides of the debate?  We propose theories and see if we can falsify them.  In other words, we engage in research!  That’s what Galileo did.  He saw that some facts, and a very few thinkers, seemed to support this new Copernican theory.  He decided to open his mind, and test out these two competing theories with a series of experiments.  Galileo risked the wrath of the Inquisition, but his work eventually changed the thinking of an entire culture.

You might think that this solution would be encouraging to a budding researcher such as me, but I’m not sure I feel encouraged.  Instead, I find myself wondering if this solution even applies to educational research, or if, perhaps, we are stuck back in 1543, when ideological orthodoxy, rather than scientific evidence, decided what could be accepted as truth.

Let’s look at a current problem and see if we can find out.

Problem: kids in the U.S. aren’t learning mathematical problem-solving skills like they should be.

Tentative Theories:

  • Teachers ought to require students to think up their own solutions to problems and make connections across concepts.


  • Teachers ought to teach students to apply a certain procedure in a given situation.

Attempts at elimination: At least twenty years of research suggests that when teachers require creative solutions and making connections, students learn more.  The National Council of Teachers of Mathematics (NCTM) believes this.  The folks who wrote the Common Core standards believe this.

On the other hand, rank and file teachers generally focus on teaching students to apply a certain procedure in a given situation (Hiebert and Grouws, 2007).  Textbooks demonstrate one procedure and then provide lots of practice in applying that procedure.  Testing reinforces this status quo by pressuring teachers to cover so many standards – one teacher recently told me she “does critical thinking” with her students only after the standardized tests in the spring.

As I read the research evidence, it appears to line up in favor of eliminating, or at least curtailing, the procedural approach to teaching. I don’t have the space to adequately present all the evidence right here, but let’s pretend I’m right.  Why don’t teachers follow what the research evidence recommends?

Even if you don’t agree with me on the evidence regarding this particular question of math instruction, every time a new curriculum is adopted, every time teachers are sent to a new training session, they face this question: should I change the way I’m teaching, or continue with what has gotten me this far?  How should they make these decisions?  On what expertise should they rely?  And why do they often not seem to rely on the evidence that is trusted by researchers?

There are at least two possibilities:  First, perhaps they don’t know of this evidence. This seems plausible.  After all, the line of communication between researchers and teachers is more like a string between two paper cups than a high speed internet cable.

Second, perhaps teachers don’t recognize the research consensus as authentic expertise.  This lack of trust could come from two sub-sources: some teachers, isolated in their classrooms, might not accept evidence from some ivory tower disconnected from their daily reality, and choose instead to place their faith in what they see working for them.  Others might be following the lead of textbooks, tests, and district pacing guides, implicitly assigning authority to these sources rather than to researchers.

What other possibilities am I missing?  What are your theories?

The line between science and dogma can be blurred.  Sometimes the facts don’t line up clearly on one side.  But sometimes they do, and we still don’t follow them.  The story of Copernicus’ great advance should remind us that progress is a tenuous prospect, often impeded by legitimate debates that are intertwined with personal biases and political intrigue.

Popper contends that “the application of…the critical method alone explains the extraordinarily rapid growth of the scientific form of knowledge,” but if the field of education is to come anywhere close to making kind of progress that we see today in the hard sciences, we will all, researchers, teachers, administrators, and policy makers, need to come together to advance, defend, and critique our tentative theories, to begin to form an open-minded scientific community of experts.

Keep following this site.  Contribute your thoughts.  Invite your friends.  We can build this community together.

– Kevin


6 thoughts on “What’s Galileo got to do with how we teach math?

  1. Hey, Kevin. What a lovely post. Well thought through, nicely structured, but I gotta push back just a bit…

    I’m totally with you up until here: “On the other hand, rank and file teachers generally focus on teaching students to apply a certain procedure in a given situation (Hiebert and Grouws, 2007).”

    After that, we part ways… ☺ Let me explain.

    I don’t think the situation is as simple as “most math teachers are doing a pretty poor job,” and, as much as I’ve totally got mad love for Hiebert and Grouws, I don’t think their 2007 work demonstrates this conclusively. I also don’t know if it’s a good idea to vilify math teachers, especially on the weight of that one citation.

    I would frame the argument (from the part at which we parted ways, onward) slightly differently.

    You list two options: one, teachers don’t know about the research; two, teachers don’t respect the research. I think it’s your authorly duty to list a third: three, there are too many external pressures for teachers to courageously buck the system of testing and accountability and teach in a more authentic way, a way that they know to be more conducive to authentic learning and critical thinking. You certainly touch on this idea a few paragraphs back (“Testing reinforces this status quo…”), but I think it’s a central point that bears… centrality.

    Your opinion and your argument in the piece are cogent, so I guess I’m respectfully disagreeing, rather than saying “your point is invalid, dude.”


  2. Just wanted to offer a fourth option: it’s scary! As a teacher I would love to devote more time to just allowing kids to problem solve and struggle in an organic, imaginative way however…as with Galileo’s theory, it is not ‘neat’ and may very well be unattractive at times as discussions become debates that continue pass the scheduled block and experiments become messy, complicated tasks (all of which are fine by me, but may not be in the eyes of an admin doing my formal observation).

    My fear of letting go not only stems from the pressure and critique that may come from the higher ups, but also because out of all my education coursework, only one class even addressed what it would be like to teach and manage students in a setting in which they are allowed to come up with their own solutions. After all, as G mentioned, we are accountable for solutions that will guarantee “proficient” or “advanced” on the standardized test. Even in elementary schools, we are beholden to benchmarks as evidence of student growth and these benchmarks do not hold creative, student-invented solutions in high esteem (hopefully just not yet). In short, due to all the things being asked of us it is not with an upturned nose that we do not trust the research; it’s that despite the research there are very real children who we are responsible for and without the resources to help teach us how to teach them in these new ways, we rely on what has worked so as not to jeopardize them. Without the support of admin to say deviating from the scope and sequence for a few days to try out this experiment, it’s hard to take that risk. Without instructional strategies that have been proven to work in low-income, ELL populated classrooms with a class size of almost 30 kids, it’s hard to take that risk.

    Be assured that we teachers are trying. We do read the research. We are sneaking in critical conversations during days when the specials teachers are out sick or when we finish the required less early. It happens in math small groups and guided reading. We start little by little and as time goes on and the research grows, perhaps it will be built into the curriculum and we won’t have to steal time. I’m looking forward to the day when teachers can all be a little more like Ms. Frizzle whose motto is, “Take chances, make mistakes, and get messy!”

  3. The instructional shift you describe is a great deal more complicated and difficult than it would appear from your description. And teachers are rarely provided the supports — the ongoing, job-embedded professional development, the teaching materials and the time to work with other teachers to develop lesson plans — they would need to make that shift.

  4. AWilliams and L –
    Fear, complexity, institutional pressure, and lack of support are certainly a big part of this problem. Probably central, as gt pointed out. The lack of support that L pointed to is deeply intertwined with the lack of communication among researchers, district administrators, and teachers – we in education don’t make decisions based on a consensus view of all the available evidence. Instead, we each blindly examine our own small part of the elephant, and then make the best of what evidence we have.

    The responsibility for opening up lines of communication lies with all of us, but I certainly didn’t mean to put the blame at the feet of teachers. As long as we have a system that forces teachers to “sneak in critical conversations during days when the specials teachers are out sick,” then making this change will be an uphill journey. But opening up “critical conversations” is why we started this blog, and forging connections among researchers, policymakers, and teachers is why I got into this research and policy world after teaching elementary school for nine years. Thanks for beginning this journey together.

  5. I would like to voice my hearty agreement with AWilliams’ comments. My response to K’s question — i.e., why adoption of what I would call “problem-based” models of teaching isn’t more widespread — was that it’s pretty frickin’ intimidating.

    Fellow commenters who know me from class have probably sensed that I fall on the side of teacher advocacy more often than not, so you know I’m not implying teachers are lazy when I say that the less procedural, more open-ended way of teaching is really hard to do, and even harder to do well. It’s hard to plan, hard to assess, and hard to grade. And I don’t think our system scaffolds teachers in how to successfully implement this approach (not in our schools of education nor in our litany of one-off PD sessions). So while I completely agree with K that lack of information and research dissemination is a fundamental issue here, I think there is also an underlying fear of the unknown at play. Great lip-service is paid to fostering critical thinking in our education system, yet teachers often operate in risk-averse environments that inhibit this kind of approach.

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