Author Archives: Paul Horwitz

The importance of scientifically literate citizens

At the Concord Consortium our goal is to prepare students to ask questions and use mental models to answer them. Students who develop this habit of mind early on will, we hope, become engaged and scientifically literate adults. And surely they will not lack for important questions to ask!

Here’s an example: According to a recent study published by the National Academy of Sciences, global sea level has increased by more than two inches in this century alone! Why is that happening? People who live on low-lying islands or in coastal cities around the world would really like to know.

Representative Mo Brooks (R-AL), a member of the House Science and Technology Committee and Vice Chair of its Subcommittee on Space, recently proposed a model for this phenomenon. He offered the opinion that a significant cause of the rise in sea level is falling rocks and other erosion, pointing specifically to the California coastline and the White Cliffs of Dover. This debris “forces the sea levels to rise because now you have less space in those oceans because the bottom is moving up,” he explained.

Is he right? Can erosion be causing the rise in sea level? Most important: do you have to be a scientist to address that question?

Actually, anyone can do it. All it takes is a little physics, a little math, and Google.

First, the physics:

  1. When you throw a rock in the ocean, the volume of the ocean goes up by exactly the volume of the rock because…
  2. the water displaced by the rock pushes on the surrounding water and ends up being spread evenly across the surface of the ocean (remember, water is a liquid!).
  3. So the increase in sea level ends up as a thin layer of water—a layer whose volume is equal to that of the rock itself. And the volume of that little layer of water is the vertical rise in sea level times the surface area of the ocean, and that equals the volume of the displaced water, which is just the volume of the rock itself.

Let’s write that up as an equation:

(volume-of-rock) = (surface area of ocean) x (increase in sea level)

Now we need to know the surface area of the ocean. We could estimate it (4/5 of the Earth’s area is ocean, the radius of the Earth is 4000 miles…), or we could Google it.

From Google: The surface area of the Earth’s oceans is 510 million square kilometers.

So to make the sea level rise by one inch we would need to throw in a lot of rocks—or one REALLY BIG rock—whose volume is 1 inch times that surface area. How big is that? Here comes the math!

Let’s put everything in feet, so we can compare. A kilometer is 1000 meters and a meter is about 3.28 feet, so a kilometer is 3280 feet, which makes a square kilometer roughly 10.8 million square feet, which makes 510 million square kilometers, which works out to 5500 million million or 5.5 X 1015 square feet.

So the volume of rock required to raise the ocean level by one inch (1/12 of a foot) is

(5.5/12) X 1015 cubic feet or 0.46 X 1015 cubic feet or 460 trillion cubic feet

How big is 460 trillion? Is it a mountain or a molehill? Turns out, more like a mountain.

Back to Google: The volume of Mount Everest (starting from its base, not from sea level) is 2.1 trillion (2.1 X 1012) cubic feet. So to cause a one-inch rise in sea level you would need to push into the sea

460 / 2.1 = 220 Mount Everests

That’s a lot of rock! Can erosion possibly account for the equivalent of 220 Mount Everests in just a few years? Back to Google…

There’s not a lot of information concerning falling rocks, it turns out, but topsoil erosion is a major concern to a lot of people, so we do know something about that. The European Commission’s Joint Research Centre on Sustainable Resources estimates that 36 billion tons of soil are washed away, worldwide, every year. A cubic foot of rock weighs 150 pounds so all those Mount Everests (2.1 trillion cubic feet’s worth) weigh over 300 trillion pounds. At that rate, it would take almost 9000 years for soil erosion to raise the ocean level by an inch. If rocks and other non-soil debris contributed a similar amount it would still take thousands of years.

The Next Generation Science Standards call for students—and that really applies to all of us!—to learn to use models to answer questions. When we do that it becomes clear that erosion isn’t to blame for the rise in sea level. And the best part? We don’t need to rely on experts, we can figure it out for ourselves!

Needed Math for the 21st Century STEM Workplace

There are three kinds of mathematics: the math that’s taught, the math that’s learned, and the math that’s needed in the 21st century STEM workplace. With support from the Advanced Technological Education Program at the National Science Foundation, Michael Hacker, Co-Director of the Center for STEM Research at Hofstra University, and I organized a conference to study why those three “maths” are not the same.

Held in Baltimore from January 12th through the 15th, the conference attracted 46 attendees drawn from three groups: math educators, STEM content instructors, and STEM employers. Three fields of STEM employment were represented: Information and Communication Technology, Biotechnology, and Advanced Manufacturing.

There is ample evidence (see, for example, “Still Searching: Job Vacancies and STEM Skills”) that companies in these and other STEM-related fields are finding it difficult to find qualified employees for entry-level jobs. This is due in part to the poor math skills of prospective candidates, and – perhaps even more telling – their lack of confidence in their ability to “do” math. In this context, the objective of the meeting was to solicit from employers examples of problems that prospective recruits often could not solve. The meeting would then collectively examine those problems, identify the underlying relevant mathematical concepts and skills, and explore possible explanations for why high school and even two- and four-year college graduates find the problems so challenging.

A complete reporting of the findings of the conference must await our analysis of the data we collected over the course of two days of intense discussion. However, it is already evident that real-world challenges, such as those described by the employers, differ from the math problems that most students encounter in formal school settings.

An example – one of many – may illuminate these differences.

An employee in a communications technology firm is tasked with providing a commercial space, consisting of several offices as well as other rooms, with wireless Internet access. The tools available consist mainly of access points and routers, the former connecting multiple devices using radio frequency communication, the latter directing information between those devices and an Internet service provider (ISP). Access points have a limited range and their locations must be selected so that those ranges overlap, providing connectivity to every device on the network as well as to one or more routers. Routers, in turn, require connectivity to the ISP.

At first glance, the problem seems simple enough: just place the access points close enough to one another so that their ranges overlap. But real-world complications soon arise.

To save money, the number of access points should be minimized. Further, they require power so installing them in some locations may result in wiring expenses. The range of each access point may be affected by the materials used for interior walls or by metallic structures such as elevators or vaults. Privacy and security concerns dictate that access to the network be restricted, as much as possible, to the premises of the customer. Some locations within those premises – e.g., conference rooms – may require greater bandwidth than others.

These and other real-world considerations are not, strictly speaking, mathematical in nature, but insofar as they constrain the set of acceptable solutions, they require mathematical skills – e.g., modeling – that may be foreign to many would-be network technicians. Moreover, although the calculations required consist primarily of arithmetic operations on numbers (signed integers, decimals, and fractions), the semantics behind these calculations – unit conversions, use of the Pythagorean Theorem to compute point-to-point distances, algorithms for computing overall costs – are not explicitly called out in the statement of the problem.

Thus, even though the problem appears to require no more than middle school math and Algebra 1, it differs from the problems commonly encountered in traditional classes in those subjects.

  • The statement of the problem does not contain all the information required to solve it and may in fact contain irrelevant information.
  • The mathematical concepts and skills required are not spelled out (in contrast to the problems found at the end of the chapter in a math textbook, all of which involve the specific concept covered in that chapter).
  • The problem is multi-step and involves multiple variables.
  • The problem may have many solutions of varying utility, rather than a single “right” one.

A major finding of the conference was that the kind of mathematics encountered in each of the three domains represented (ICT, biotech, and manufacturing) involved contextualized problems similar to the one described above. Thus, an important barrier to success in these fields may arise from the features of such problems that we have identified.

Are there ways in which educational technologies such as those pioneered, deployed, and investigated by the Concord Consortium could help students to acquire the relevant, contextualized problem-solving skills? A major outcome of the conference may turn out to be a number of proposals aimed at answering that question.

Early start in educational research

New York Times Clipping

Paul Horwitz, senior scientist, got his start in research earlier than most — when he was three! We’ve enjoyed his stories for many years. This one was too good not to share. One day at lunch we decided to follow up on his memories and dig a little deeper. We contacted Lindsey Wyckoff at Bank Street College, who sent us this story from their archives. Here is Paul’s story:

It’s July 1942. Hitler’s armies have conquered most of continental Europe and are about to unleash their fury on the Russian city of Stalingrad. England has survived the “blitz” but thousands of frantic British parents have allowed their children to be evacuated, some as far away as Canada. In New York the Bank Street Nursery School, under the auspices of the Office of Civilian Defense, has embarked on an ambitious experiment. Forty-five preschool children, ages two to five, will be “evacuated” for six weeks to Lake Waneta in upstate New York in order to evaluate whether the trauma of being separated from their parents outweighs the risk of exposing them to possible attack.

I was one of those children.

I was three and a half, far too young to understand what was happening to me, much less why, but the weeks I spent at “camp” that year are among my earliest memories. And the memories, by and large, are good ones.

I remember being introduced to a special kind of photosensitive paper that could record the silhouettes of objects placed upon it. I remember kicking my legs in shallow water, thinking guiltily that I had tricked my parents into believing I could swim. I have a hazy memory of a newsreel crew with a huge camera that moved back and forth on wheels.

I have no recollection of the battery of psychological tests that must have been run on me, though I do remember my answer to one question: in a race would you rather be first or last? (I chose last, on the basis that that way I wouldn’t always be looking behind me to see whether someone was catching up.)

I have since learned that the experiment was a success: given proper care, including cuddle time as well as meals, young children proved unexpectedly resilient. So no permanent damage was done, though I very much doubt that one could attempt this kind of thing today.
In the end, as we know now, no evacuation of New York or any other American city was deemed necessary. The broad Atlantic and the absence of aircraft carriers from the German fleet offered protection enough in that long ago time. But today, as we learn to cope with sporadic and unpredictable violence resulting from a protracted “war on terror,” it is perhaps instructive to remember that we have survived much worse.

How not to Learn from Games

They’re the in thing, especially for teaching science. Everyone, it seems, is fascinated by the potential of educational games. They’re interactive and “multimedia,” they can adapt to individual students, they promote “authentic learning.” In short, they match the outsize expectations of a digital world. They’re definitely cool, but do they teach, and if so, what do they teach?

Full disclosure: I am an enthusiastic proponent of educational games. I created one called “ThinkerTools” so long ago that it ran on a Commodore 64 computer and had to be programmed in machine language to make it run fast enough. And, yes, I have no doubt that kids learn from such games. But do they learn what we think they’re learning? And how would we know if they were? Is it sufficient that they get better at the game? Surely not, else chess masters would be good at logic, and athletes would be physicists.

It is tempting to imagine that we can design educational games so cleverly that it would be impossible for a student to get good at the game without acquiring a deep understanding of whatever it is the game is trying to teach. Unfortunately, it doesn’t always work that way, as I learned from my experience with another educational game called GenScope.

GenScope was a multi-level genetics game. It linked processes at all different levels, from molecules to ecosystems, and we used it to create a bunch of engaging challenges for students. Our species of choice was dragons. We would show a dragon’s chromosomes, for instance, and ask students to figure out how to change its genes to make the dragon breathe fire. Later on, we would challenge them to breed a strain of blue dragons, or try to find two parent dragons that could only have two-legged offspring (hint: neither parent can have two legs).

We used the GenScope games in several high schools. We compared students who had used the games to others who had learned genetics by conventional means. To do this we designed a clever test that assessed precisely the reasoning skills we were trying to teach—and that we naively assumed were necessary to succeed at the games. Each time we did this, we found that the GenScope classes did no better on the test than the control group. Sometimes they did worse!

In the jargon of the trade the Holy Grail is “transfer,” and we weren’t getting much. Knowledge gained in one context is often difficult for the novice to apply to another one, even though to an expert the two situations appear very much alike.

To us, the researchers, the genetic principles behind the GenScope games were obvious, and their relevance to the questions on the test equally so. Clearly, that was not the case for the students, who became expert GenScope players but failed to apply what they learned to genetics.

There are ways around this impasse, of course, and I will describe a few in a future blog post. For the moment, though, let’s just keep in mind: there are lots of ways of getting good at an educational game. Only one of them involves learning what the game is supposed to teach.

What would it take to disprove Intelligent Design?

Scientific theories differ from other belief systems in that they are testable; in other words, they can be disproved. Imagine reading, for instance, any of the following headlines:

  • “Modern Chicken Fossil Found Side By Side with Dinosaur Bones”
  • “Chimpanzee DNA Radically Different From Human”
  • “New Data Shows Earth Only 10,000 [or 100,000 or 10,000,000] Years Old”

What do you think would happen to the theory of Evolution if any of those things occurred (assuming, of course, that the observations were replicated and confirmed)? It would certainly have to be radically modified, and might have to be rejected entirely, because according to the theory it just can’t happen that chickens and dinosaurs ever co-existed. And if human and ape DNA were found to differ by more than a few percent it would be very difficult, if not impossible, to reconcile that with present-day views of how these creatures evolved (relatively recently, from a common ancestor). And if the earth were really “only” ten million years old (much less ten thousand!) there wouldn’t have been nearly enough time for living cells, much less human beings, to have evolved.

In contrast, can you think of any way to disprove the theory of Intelligent Design? I used to think I could. Why not, I thought, look for imperfections in the design, instances where certain creatures seem less than ideally designed for their purpose. (As it happens, there are many examples of such suboptimal design.) But that approach doesn’t work. All the inefficiencies can ever prove is that the designer “works in mysterious ways,” or has a different aesthetic from ours about such things. So what appear to be botched designs may tell us something about the designer but they do not discredit the theory of Intelligent Design itself.

Intelligent Design doesn’t make predictions–other than the trivial one that living creatures should look as though they were designed. But that’s not a prediction; it’s just an explanation for a set of observations. It’s kind of like saying “Lightning looks as though Thor is throwing thunderbolts at us, therefore that’s what it must be.”

The Thor model of lightning is unscientific not because it’s wrong but because it’s untestable. There is no way, short of looking around for Thor and not finding him (and he could, after all, be hiding somewhere), of checking out the theory. Like Intelligent Design, it makes no predictions and, therefore, cannot be disproved. In contrast, the theory that lightning is caused by electric currents, while considerably harder to understand and at first blush a lot less plausible than the Thor model, predicts, among other things, that if Benjamin Franklin flies that kite in a thunderstorm one more time he’s liable to get fried.

Science is all too often taught as though it were merely a collection of facts. What we should be teaching is the process by which we have come to trust those facts, what evidence backs them up, and, most important, what new information could get us to change our minds. We need to teach kids that the hallmark of every scientific theory is that in addition to explaining known data it makes predictions about data that hasn’t been seen yet. Which means that every scientific theory is in constant danger of being disproved if those predictions fail to come true.

Until the Intelligent Design proponents can point to some finding–anything!– that might in future cause them to revise or abandon their theory, that theory is no more scientific than the once widespread belief that the plague was God’s punishment for our sins. If we allow such theories to be treated as science we might as well go back to curing disease by whipping each other to atone for those sins.

Evolution Readiness Progressions

The basic concepts of evolutionary theory are contained in the National Science Education Standards (National Research Council, Washington, DC, 1966) as well as those of the various states. For example, in the table below we show the alignment between the “big ideas” of evolution and the science standards for three states: Massachusetts, Missouri, and Texas. The “learning progressions” in the second column are adapted from the Atlas of Scientific Literacy (American Association for the Advancement of Science, 2007), while the quotes in the third column are from the Massachusetts Science Framework; we also index in that column the corresponding standards from the Missouri “Show-Me” Standards and the Texas Essential Knowledge and Skills Standards.

Beginner Level

Big Idea Learning Progression MA Science Framework
Basic needs of organisms Plants and animals need air and water; plants also need light and nutrients; animals also need food and shelter. “Identify the ways in which an organism’s habitat provides for its basic needs.” See also MO Science K-4: VII.B.2, TEKS Grade 4:5.A&B
Organisms and their environment For any particular environment, some kinds of organisms survive well, some less well, and some cannot survive at all. “Identify the structures in plants and animals that enable them to survive in an environment.” See also MO Science K-4: VII.A.2, TEKS Grade 4:8.A
Interspecific differences Plants and animals have different life cycles that include being born, developing into adults, reproducing, and dying. “Classify plants/animals according the physical characteristics that they share.” See also MO Science K-4: VII.C.1
Basic needs of species Groups of organisms can survive even though every individual in the group eventually dies. “Give examples of how changes in the environment have caused some organisms to die.” See also TEKS Grade 4:8.B
Interactions between species Organisms with similar needs compete with each one another for resources. “Investigate how invasive species out-compete native ones.” See also MO Science K-4: VII.A.2, TEKS Grade 4:8.B
Intra-specific differences Individuals of the same species may differ. “Observe differences between organisms.” See also MO Science K-4: VII.E.2, TEKS Grade 4:8.A
Heritability of traits Offspring are usually very much, but not exactly, like their parents. “Differentiate between inherited and other characteristics.” See also MO Science K-4: VII.D.2, TEKS Grade 4:8.C

Intermediate Level

Big Idea Learning Progression State Learning Standards
Basic needs of species For a species to survive, the individual organisms in it must reproduce fast enough to replace the ones that die out. “Describe how organisms meet their needs by using behaviors in response to stimuli received from the environment.” See also MO Science 5-8: VII.C.1&2, TEKS Grade 5:5.A
Interactions between species Every animal species depends on another species, plant or animal, for food. “Give examples of how organisms can cause changes in their environment to ensure their survival.” See also MO Science 5-8: VII.E.2, TEKS Grade 5:5.B
Intra-specific differences Differences between individuals in a species may give some an advantage in surviving and reproducing. “Give examples of how inherited characteristics may change over time as adaptations to changes in the environment that enables organisms to survive.” See also MO Science 5-8: VII.E.1&3, TEKS Grade 5:10.B
Heritability of traits Some traits of organisms are inherited from their parents; others are learned or acquired. “Recognize that every organism requires a set of instructions that specifies its traits.” See also TEKS Grade 5:10.A

Advanced Level

Big Idea Learning Progression State Learning Standards
Heritability of traits Heritable characteristics can affect the likelihood that an organism will survive and reproduce “Relate the extinction of species to a mismatch of adaptation and the environment.” See also MO Science 5-8: VII.E.4, TEKS Grade 7.10.C
Genetics Heritable traits are transmitted from parents to offspring via different forms of genes, called alleles. “Recognize that hereditary information is contained in genes located in the chromosomes of each cell.” See also TEKS Grade 7.10.C
Survival of fittest individuals in an ecosystem Offspring of advantaged individuals are more likely than others to survive and reproduce, increasing the proportion of organisms that have advantageous traits. “Recognize that biological evolution accounts for the diversity of species developed through gradual processes over many generations.” MO Science 5-8: VII.E.4, TEKS Grade 7.10.B