math

Time limits and test anxiety

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When the clock is going TICK TICK TICK, it can be hard to think clearly, because you’re anxious about the clock.

Math anxiety is well understood, and no small part of this comes from the pressure of timed tests. Ultimately, some people take tests faster than other people. I would hope that you want your tests to measure how much students have learned, not their ability to take tests under pressure. If this is the case, then everybody taking the test needs to feel that they have adequate time. Continue reading →

Science, math skills, and high school students

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There was a diversity of visceral reactions to EO Wilson’s op-ed piece, which argued that you can be a great scientist without being good at math. The lowdown can be found at Dynamic Ecology (with 15 updates as of this writing!) I wrote my own take on it here.

Before we go any further, I’m curious about all of you, what’s your take on the op-ed piece? Compel yourself to make a choice. (If you have caveats, put them in the comments section of the post, but please pick one or the other).

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One common reaction by scientists who teach and train students was, “Thanks, Ed, for making my job harder.” That was my first thought, as I regularly teach Biostatistics.

Another common reaction was, “That’s not the message to send students early on as they still are developing their skills in all aspects.”

With these ideas in mind, I brought the op-ed piece to a bunch of high school teachers. They read it and we discussed it for about half an hour.  How did the discussion go?

First, let me tell you more about the teachers. I regularly meet with this crowd as a part of an NSF-funded Noyce Master Teacher Fellow program that I run with education faculty. They all have their Master’s degree (most in education, some in science) and were competitively selected for this program as a result of their experience, excellence and continued commitment to teach in high-need urban schools in South Los Angeles. These teachers work in rough schools, with kids who have the deck stacked against them even before they enter the classroom. They were picked for this program because they are the ones staying at their schools even though most new teachers leave after a very short time.

These teachers are talented, dedicated, overworked, and mentors to new teachers. I tell them so often how much I respect and admire they work they’re probably sick of hearing it. (I have learned a lot from them about teaching over the past couple years, no surprise there.) One of the reasons I try to praise often is because they hear it so little elsewhere. The newspapers and the mayor and the school board and anybody who has a loud mouth will say that these teachers are the problem that need to be fixed. Let me tell you, that’s entirely backwards. These teachers are the solution to the problem. Free these teachers to do what they were professionally trained to do, with the resources to do it, and you’ll see the positive changes that have been so elusive. (Making this change, sadly, is politically complex).

These teachers know their stuff. Moreover, they teach exactly the population of students that NSF is trying to hard to recruit into the sciences: “underrepresented.”

The opinion of these teachers about the requisite math skills for becoming a scientist matters, more so than anybody else in the whole of the USA.

What did they say about Wilson’s piece? Immediately after we all read it, I did an informal survey: thumbs up or thumbs down, just like in this post. (Rest in peace, Roger Ebert.)

All I saw were thumbs up, or neutralish waves of whatever. I asked, why is that?

The general consensus was that being good at the process of science isn’t inherently mathematical. You don’t want to dissuade someone who is interested in science, after all. Of course, you need to use math, but that shouldn’t stop you from pursuing science and the math can come along for the ride. That was the initial response.

Then, one person (the only physics teacher in the bunch) disagreed, and a biology teacher joined in. They said that to be good at the practice of science, in real life, you have to be able to do math. You can’t really understand some fundamental principles in science unless you can grasp the math.  There were some disagreements, that this was endemic to physics, but then plenty of examples throughout the sciences were brought up. It was also raised that engineering is growing in importance and will be a key feature in new state educational standards soon to be adopted.

The discussion then turned to the fact that specific skill sets are required not just to be able to do science, but also to land positions, perform your job, and be able to adapt to evolving requirements of these jobs. Not all scientists can choose to work on whatever they want, even though E.O. Wilson has that option, and we need to train students to be prepared for the opportunities that rise before them and to be able to use their skills to create the opportunities that they want, or need.

If you’re E.O. Wilson, then you don’t need math, we decided. But if you’re not Wilson, with National Academy mathematicians available for collaboration, then sophisticated math is a very practical skill that will serve you well in the sciences more than almost any other resource. Especially if we are training students from disadvantaged backgrounds, we want to be able to confer upon our students every possible advantage, and being analytically and mathematically adept is key. It’s genuinely a key. It opens doors.

In the end, we agreed that Wilson was right on the fact: It’s possible to be a great scientist and not be great at math; this is a possibility.

We also agreed that this was a downright destructive choice to communicate such an idea.

Wilson’s article lamented that he had a hard time recruiting Harvard students to become scientists because of their math phobia. Nearly all of his students are archetypes of privilege, who also received strong preparation in high school before winding up at Harvard.

Meanwhile, the students in the classes of our master teachers who are lucky enough to graduate and then go to college, are likely to need remedial math. At my university, it’s been normal for a majority of entering students to require remedial math courses right off the bat because they don’t pass the stunningly basic placement exam. Do we want to tell them that math isn’t important to become a scientist? Should we tell them that this remedial math doesn’t matter, and that the calculus course required for our major is pointless?

Perhaps Wilson would like to visit us, and tell my students that they don’t need to worry too much about developing math talents to further their careers as scientists.

Far too often, my students have heard while growing up that they don’t need to work hard at something difficult. They have heard plenty enough that they should just slide into tasks suited to their inherent abilities, whatever they may be, rather than kick it up a notch and genuinely improve one’s talents. If you’re the first one in your family to go to college, expectations are paramount.

Maybe Wilson should limit his don’t-sweat-the-math message to his Harvard students. That way, our students will get jobs over their underprepared Ivy League competitors.

And then I woke up.

Tribalism in the sciences: empiricists vs. theoreticians

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In complex societies, tribes inevitably emerge when individuals with similarities band together, to promote and defend their own interests. I’m not going to go all Jared Diamond on you and pretend to be an anthropology scholar. But I can go so far as to claim that like individuals gravitate to like, and then things have the potential to get ugly.

Scientific tribes are based on ideas. These often track one’s scientific lineage, but ultimately your own ideas — and the people with whom you associate — become your tribe.

Like in any social group, membership in a tribe offers a blend of benefits and costs. Tribes can expand your influence and power, though mostly only as far as the reach of your tribe. The leaders of tribes might be propelled into a greater role beyond the tribe, but the rank-and-file members of the tribe are stuck in that group.

In science, you can join a tribe, but you don’t have to. If you’re research active and collaborate, it takes some work to avoid drifting into one.

The problem with these tribes is that most people haven’t learned how to play nice. What’s worse, is that people have trouble separating out criticism of ideas from personal attacks. Some people conflate the two together. Others use personal attacks when they aren’t necessary or warranted.

On an unsettled topic, I occasionally do enjoy me a good argument, if I think it’s going somewhere and I have the capacity to learn or make a difference.

That’s a rare opportunity, because it seems that most interlocutors are not entering into discussion to genuinely convince another person, or with a mind that is adequately open to change. Instead, people enter an argument to win. I’m open to being convinced, but instead of getting a convincing argument, I usually get an attack on my ideas rather than a sales job on more attractive ideas.

That’s no good. That kind of discussion isn’t worth my time. I’d rather be exposed to something that has the capacity for a positive change, either on myself or others.

Those polemics used to be something I used to like, I think, though it was a while ago. I went to one of those liberal arts colleges where it’s not uncommon to find yourself staying awake into the wee hours of the morning discussing politics, history, religion, science, sociology and the nature of existence, and where all of these ideas intersect. I loved it. At the time, the school was as diverse as a privileged expensive school could possibly be, so there was always someone to disagree with you. It was an intellectually challenging environment, and I loved it. I learned a lot about how to disagree with people but still maintain respect for, and from, others. I wasn’t always successful, but I learned that this respect this is a priority. One model for this kind of collegiality is the late Paul Wellstone.

It turns out that most people haven’t developed that skill, even scientists with PhDs. Perhaps they have the skill but not the patience to exercise it. Or, maybe, they have the skill but have decided that winning an argument is more important to establish social dominance within a tribe. Social dominance within a tribe is important, because in a tribal environment you can only get ahead unless you’re leading the tribe.

This is why scientists often engage in pointless arguments in which nobody changes their minds.

One example is the recent kerfuffle when E.O. Wilson was the author of a Nature magazine article with a complex population demography model that purportedly supported group selection over kin selection in the evolution of eusociality. (I have to admit that, despite a few careful reads, I mostly but not entirely understood the technical merits of the paper.) The massive backlash from the kin selectionist tribe was not based on the actual science in the article, but instead at the inflammatory (and factually incorrect) statements within the article directed at the other tribe. Wilson designed the paper to start a hissyfit, and it did. There were several letters published in response to the article, which essentially were designed to punish Wilson for offending the tribe which he used to lead.

Both sides wanted to win the argument. Meanwhile, in all honesty, I can’t think of a single person who was an author to any of the articles or rebuttals that has deliberately and publicly sought to reconcile the ample contradictory evidence that exists. I think most of the people involved really wish to understand the science of how eusociality has evolved so many times, and under what selective forces. But nearly everything published is tribal in nature. Why is that?

I suspect that the benefits of the tribe outweigh the costs and limitations. it’s easier to lead a tribe than forge your own way. It’s not only easier intellectually, it can be better for one’s career. Ecology is filled with a history of feuds among tribes, and I’m sure other disciplines are the same way. The leaders of these tribes now have named professorships, big salaries, and are revered as great elders within their subsubfields. That’s what you get for leading a tribe.

To ascend to leadership of a tribe, you have to have certain attributes. One prerequisite is that you need to have an academic position at well-known research institution. Since I work in a small pond, that rules me out of tribal leadership. Unless I pick up and move to a place where I have PhD students, a big lab, and larger grants, I’ll never get past the status of beta male.

Since I can’t ascend to tribal leadership, why would I want to join a tribe? There are benefits to being a member of the tribe, but there are also costs and limitations. The benefits are small enough for me that I don’t want to incur those costs. A few years ago, I stuck my neck out to publicly support a well-established member of a tribe who was attacked by a rogue journalist, and at the first opportunity he disavowed my support, by lying to me, in a major public diss. It seems I’m not able to join that tribe, after all. (I don’t mind bringing it up here because, after all, I was already totally dissed as insignificant by this guy.) You won’t see me doing that again.

As the proprietor of this blog, I have to be particularly conscious about how tribalism works, as heavily expressing an opinion here or there could easily shift me towards a tribal affiliation, even though I wouldn’t get much benefits from the tribe. I can’t think of many scientific issues on which I feel the need to choose one side or the other. (Of course, I’m not counting non-controversies that make it into the media as controversies.) On the other hand, I am inclined to call out the ridiculousness of arguments when both sides aren’t listening to one another well enough.

I’m a member of a few clearly defined social groups, reflecting who I spend my time with in the sciences. These mostly include social insect researchers and also those who work in tropical rainforests, mostly at one particular field station. That group numbers easily in the hundreds to a few thousand. It’s a good crowd. But I stay out of arguments, like the silly one about Wilson that I mentioned above. I’m not an ant tribalist, or a La Selva tribalist. But those are the people with whom I run.

Which brings me to the current events that prompted me to write this post.

The latest tribalist kerfuffle started this weekend, yet again with E.O. Wilson, the gentleman rabble-rouser. He wrote an op-ed piece run by Rupert Murdoch’s Wall Street Journal, in which he argued that you didn’t have to be good at math to become a great scientist.

I agree with his idea, with some caveats. His supporting arguments weren’t that robust, mostly trumpeting his own success and ability to collaborate with top-notch modelers.

This didn’t stop some people from taking huge exception, yet again, like clockwork. There’s a good discussion over at Dynamic Ecology about Wilson’s notion that math isn’t important for generating new ideas.

It’s no coincidence that Wilson’s position on math comes not too long after he used some very sophisticated math to make an argument that got aroused tribal anger. The incongruity of the position that “math doesn’t matter but complex math is the main support of my controversial stance” is sending some people into fits.

Of course, this had to have been entirely calculated by Wilson, who wanted to start this argument. I think it’s a good discussion for us to have, broadly speaking, about the attributes that we need to develop to make creative scientists. That’s where this discussion is going, I suspect he hopes, once the outrage passes.

What are the tribes engaged in this argument that was prompted by Wilson? It’s one as old as the field: the empiricists vs. the theoreticians.

Yawn.

The theoreticians essentially have ruled the roost for the last fifty years in ecology. There’s always been a place for work that is driven by empirical investigation, which in fact occupies far more pages in journals that the more math-intensive theoretical work. Despite being outnumbered, the theoretically-focused researchers are the ones who tend to fill up the editorial boards, publish in the highest-impact journals, and attract the biggest crowds at conferences. There clearly is a celebrity culture in the field, and the top theoreticians mostly rank higher than the top empiricists.

Keep in mind that this is an artificial dividing line. Few are wholly theoretical or empirical. However, for those that have clearly identified affiliations, the theoreticians are in charge. They’ll probably tell you that their status has emerged because their work is more valuable. When David Tilman received an award from the Ecological Society, the main point of his address was that you should do theoretical work just like he does because other work is less valuable. That’s gutsy.

If theoreticians are so in charge of affairs, then why are they so upset when someone says that mathematically-driven theories are a footnote to science rather than the heart of it? That op-ed piece, after all, isn’t going to change what the theoreticians allow to be published in the top journals in the field. Why get so upset?

They’re upset because it came from Wilson. This man built his fame, in part, using theoretical models using somewhat to very fancy math, with collaborators who were good at math. He essentially wrote that he was the ideas man and that the math collaborators are easy to come by. If he mentioned Robert MacArthur by name as one of the easy-to-come-by-theoretical-collaborators, then all hell would have broken loose, considering MacArthur’s status as a tribe leader before his untimely death.

For an empiricist who built his reputation with the help from more analytically minded coauthors who often did the rhetorical heavy-lifiting, it’s pretty brutal for Wilson to overtly suggest in the Wall Street Journal that his contributions were the important ones. He was the man with the vision and those other guys with the math could have been anybody.

Now that’s gotta hurt.

If I was a theoretician, you’re damn right I’d be pissed off.

I’d be pissed off because I ‘d have difficulty separating the logic of Wilson’s argument from the personal nature of his message. What’s Wilson’s argument? That you can have great ideas, and make those ideas come into reality and make scientific progress happen, without being particularly good at math. You need to be okay at the math, but you don’t have to obsess on it.

Is that true? Well, partially. But it’s not true if you’re going to become a theoretician.

So why are theoreticians so offended, if Wilson says that there’s another valid route to become a scientist that isn’t driven by math-heavy theories? I think it’s because many of the them think that the central ideas in the sciences nowadays are mostly mathematical.

Are there major progresses to be made without a lot of math? My initial thought is: hell yes there are. We’re still in the wild west of scientific discovery, with huge frontiers yet to be explored. Not everybody agrees with that, though.

That is an interesting debate, in my view.

As I’m not in the theoretical tribe, I can look at this with some distance. I can do that because my contributions weren’t directly insulted, and I am in a position to separate the concept of his argument from the people in the argument.

Wilson, in a rhetorically inelegant fashion, just reignited the ol’ empirical vs. theoretical fight. I think if he were rhetorically elegant, it would have passed unread. It would have been too intellectual for Fox News The Wall Street Journal. And it’s such an old saw that typical venues wouldn’t be interested in hearing it. I wonder if the WSJ was his first choice.

Here, is the essence of the disagreement:

Are the central concepts in science based on equations and mathematical relationships, or are they built on broader principles that do not have to be described by mathematical models?

Here is how I reconcile the disagreement: All relationships can be described with math. To fully understand any phenomenon, math is the language of nature and the language of science. Math is key to understanding patterns and relationships, as math essentially the only way they can be expressed in a specific form, other than using logic. However, in order to be able to write the equations that describe the patterns, we must first be able to know what the variables are, and how they might be able to relate to one another. Wilson’s point, though written inelegantly, was that many of the potential relationships that might exist haven’t even yet come to our attention. You can’t build the model without knowing which variables to put into the model.

The fundamental divide between empiricists and theoreticians is a disagreement about whether we know what the most important variables are. Empiricists are in search of the variables, and theoreticians are seeking to develop the specific patterns among variables. When empiricists do experimental and observational research, they’re testing whether specific things matter, and if so, how.

A few times in my career as an empiricist, I think I’ve come upon new variables, or shown in a very clear way that the relationships between a few variables matters in a way that wouldn’t compel theoreticians without theoretical evidence. I am not as personally interested in working out the specific relationships between key variables as I am sorting out which variables matter.

I think the same could be said about Wilson. He thought that the size of an island, and its shape and distance from the mainland (and so on) would be very predictive of the species richness on an island. Then, he buddied up with MacArthur who worked with him on the details. I think they both were important – perhaps essential – in the development of the Theory of Island Biogoegraphy. I don’t know the history enough to know whether this is something that MacArthur would have, or could have, done without Wilson. Wilson didn’t invoke this example in his piece. Instead he invoked George Oster, who worked on social insect caste theory with Wilson. In this case, Wilson was clearly the social insect ideas man and Oster was the modeling man. I do think that Wilson is correct in this case – that Oster couldn’t have done it without Wilson in particular, but that Wilson could have found many modelers to work with him on this monograph. It was inelegant for Wilson to point out this fact. I hope I’m more gracious when I hit that stage of my life.

To slightly rephrase, here’s where the divide lies: does the world still need people who are envisioning these variables in the broad sense, or do we all need to learn how to do the complex math to model relationships?

I think we all should learn the math, we all should learn how to model, and this would inform our world view. However, there are only so many hours in the day. It so happens that some of the most visionary people are the ones that have focused on things other than modeling. It also so happens that some of our visionaries are excellent modelers.

As David Foster Wallace has pointed out (stay tuned for a post later this week): what we learn in our studies is not how to think, but what to think about. Should we think about models, or should we think about what belongs in the models? These are somewhat mutually exclusive, I think. We do need people who think about the latter more than the former.

In my experience, when I spend to much time trying to model relationships, I lose sight of the forest – both in metaphorical and in actual terms. If my projects lead to developing and testing models, I’m all over it. But right now, I’m still trying to identify which relationships matter, because there is so much that remains unknown. (In the coming month, I’ll take the time to write another long post about how avoiding modeling led to a discovery, oddly enough in one of Wilson’s pet genera.)

So yes, I think Wilson is right. You can be a visionary without being a modeler.

Modelers themselves are also visionaries. That’s where Wilson is wrong.