tt_icon_170Have you ever really listened to the sound of a bouncing ball? There’s some elegant mathematics to be had in this simple thing. In this episode of my Science Teaching Tips podcast, staff educator and physicist Tom Humphrey takes us to the most perfect bouncing ball I’ve ever seen (or heard) — an exhibit at the Exploratorium. The platform the ball is bouncing on is a huge chunk of heavy marble, bolted to the floor. (What does that have to do with anything? Think about conservation of energy and momentum). You hear some surprising things as a small metal ball bounces on that surface. Even without the exhibit, this is something you can do with your students, and integrate science and math into your curriculum.

Listen to the episode – Follow the bouncing ball

…for anyone who hasn’t seen this one yet…

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I love this… from GraphJam.

http://graphjam.com

Pie I Have Eaten: http://graphjam.com

The NSDL has pulled together some classroom resources for teaching about voting and polls, voting technology, and the history of voting.  These are taken from the NSDL Expert Voices blog.

Annenberg/CPB: Cast Your Vote
From the NSDL Middle School Portal: Math and Science Pathways

Multiple polls claim to know how public opinion shifts day-to-day during political campaigns. This web site offers a ficticious look into an election campaign at the math behind the polls. Concepts such as random sampling, margin of error, confidence intervals, and ways in which surveys can go wrong are reviewed.

Majority Vote: What percentage does it take to win a vote?
From the NSDL Middle School Portal: Math and Science Pathways

Understanding national election results is complicated. This classroom activity helps students think carefully about how percentages are used mathematically to determine voting outcomes. The importance of understanding the meaning of percentages in media and marketing is also noted.

Voting Rights
From NSDL Teachers’ Domain: Digital Media Resources Pathway

The Fifteenth Amendment to the Constitution was passed that prohibited racial discrimination in voting was passed in 1870. The Voting Rights Act, however, was not signed into law until 1965. Find out what happened in the nearly one hundred years between 1870 and 1965 to ensure that everyone has the right to vote in this multimedia resource from Teacher’s Domain.

Election 2000: A Case Study in Human Factors and Design
From the NSDL Engineering Pathway: Engineering Education Resources

The goal in presenting this case based on controversies surrounding the November 2000 presidential election, specifically the difficulties encountered in interpreting imperfectly punched ballots, is to help college-level students recognize how engineering solutions can be brought to bear in solving problems of national importance.

In the latest episode of my Science Teaching Tips podcast, you can hear (the wildly funny) children’s book author David Schwartz talk about how he used kids’ skepticism to get them to do some good measurement problem.   A class disagreed with the numbers in one of his math books, and set out to prove him wrong by surveying the heights of all the students in the school.  What a teacher’s dream!

Hear the episode — Then YOU measure it.

Richard Hake and Jeffry Mallow have compiled over 700 research papers on how males and females learn — and are taught — science and mathematics.  Wow!

You can download the PDF of their work here. If that link stops working at some point, the permalink is in Reference 55 here.

The first page reads:

This 12.8 MB compilation of over 700 annotated references and 1000 hot-linked URL’s provides a window into the vast literature on Gender Issues in Science/Math Education (GISME). The present listing is an update, expansion, and generalization of the earlier 0.23 MB Gender Issues in Physics/Science Education (GIPSE) by Mallow & Hake (2002). Included in references on general gender issues in science and math, are sub-topics that include:
(a) Affirmative Action;
(b) Constructivism: Educational and Social;
(c) Drivers of Education Reform and Gender Equity: Economic Competitiveness and Preservation of Life on Planet Earth;
(d) Education and the Brain;
(e) Gender & Spatial Visualization;
(f) Harvard President Summers’ Speculation on Innate Gender Differences in Science and Math Ability;
(g) Hollywood Actress Danica McKellar’s book Math Doesn’t Suck;
(h) Interactive Engagement;
(i) International Comparisons;
(j) Introductory Physics Curriculum S (for Synthesis);
(k) Is There a Female Science? – Pro & Con;
(l) Schools Shortchange Girls (or is it Boys)?;
(m) Sex Differences in Mathematical Ability: Fact or Artifact?;
(n) Status of Women Faculty at MIT.

In this Part 1 (8.2 MB), all references are in listed in alphabetical order on pages 3-178. In Part 2
(4.6 MB) references related to sub-topics “a” through “n” are listed in subject order as indicated above.

On a related note, here is a post from Swans on Tea about the recent discussion on instituting Title IX in Science.

The issue here, though, is whether the comparison to sports is an appropriate one to make. It’s not.

Men and women don’t compete with and against each other in these sporting events. Title IX has been very successful at expanding womens’ participation in sports, because it focused on equality of opportunity and did not assume equality of ability — women are not fighting for a roster spot on a single football, soccer or baseball team, etc. …The lack of opportunity for women that prompted Title IX was the lack of teams on which they could compete, and one could (and did) create and fund these teams. The situation in science is very much different in the difficulties that exist and the solutions that can be proffered.

REVISED 9/11/08

I just recently got an email from my father that showed to me once again that the apple doesn’t fall far from the tree. I guess geekgirl is truly the progeny of geekdad. (He’s a retired physical chemist, BTW). He saw an interesting article in Science about how we calculate fuel efficiency in quite a misleading fashion. We calculate miles per gallon, which tells us if we’ve got a gallon of gas in the tank then we can go X number of miles. But, says Richard Larrick of Duke University, this ratio should instead be turned on its head. If we want to go, say, 1000 miles, how many gallons will it take us? After all, the amount of gas consumed by a car does not decrease linearly as the mpg of the car increases. My Honda Civic (40 mpg) will take me twice as far on the same amount of gas as, say, a Ford Explorer (20 mpg). And if I want to drive 1000 miles, my Honda Civic will take me there on half the amount of gas that it would take the Explorer. Great, that all makes sense. But now if I start looking at making improvements to the mileage of either car, there’s where the “mpg” measure is misleading. It turns out that adding 10 mpg to the low-mileage car will save you a lot more gas and money than a 10 mpg improvement in the high-mileage car.

So he went and graphed it (gotta love geek dad) by just dividing 1000 miles by the miles per gallon and multiplying by $4 per gallon. The 1000 miles part isn’t important for the argument, it just scales up the final answer.

You can see that improving the efficiency of a 10 mpg car to 20 mpg has a much larger effect in the cost (and the # of gallons used) than does improving the efficiency of a 40 mpg car to a 50 mpg car. I’m not feeling so bad about not getting a hybrid car now.

The essential message is that we can’t do calculations like 1/x – 1/y in our heads. In Science Magazine’s podcast, Larrick says:

And, to kind of understand why MPG tricks people it’s useful to do a little bit of math. And so, you could think about a problem that a, a family might face of deciding whether to get rid of an SUV that gets 10 miles per gallon, or a sedan that gets 25 miles per gallon. And let’s say that they’re both driven about the same distance, roughly like 100 miles a week, and with the SUV they need another big car, so they’re thinking about a minivan that gets 20 miles per gallon. And with the sedan they’re thinking about replacing it with, let’s say a hybrid sedan that might get 50 miles per gallon. Well people are very attracted to the idea of replacing a car that gets 25 miles per gallon with one that gets 50 – that’s a big jump of 25 miles per gallon. And, getting rid of the SUV that gets 10 miles per gallon to a car that gets 20 miles per gallon – that just
isn’t as big of a jump, it doesn’t look as impressive….

So, let’s just think about how many gallons the SUV uses – the car that gets 10
miles per gallon. So, if we’re driving a hundred miles, that’s going to use 10 gallons to
go the hundred miles. If we replace that with the minivan that gets 20 miles per gallon,
we’re only going to use 5 gallons to drive the same hundred miles – we’ve now saved 5
gallons just by replacing the SUV with the minivan, going from 10 MPGs to 20 MPGs.
Let’s do the same calculation for the other car that could be replaced – which was the
sedan, that does get 25 miles per gallon – replace it with a small hybrid that gets 50 miles
per gallon. Well, at 25 miles per gallon that car’s only using 4 gallons to go a hundred
miles, and the hybrid’s only going to be using 2 gallons to go a hundred miles. That’s
just a 2-gallon savings. The big savings comes from getting rid of the most inefficient
car, the SUV that gets 10 miles per gallon with one that’s more efficient – the one that
gets 20 miles per gallon.

Or, as my dad says, in more abstract language:

The dependence of the cost (or gallons consumed) is not a linear function of mpg with a constant negative slope, rather it is a reciprocal function of mpg with a decreasing negative slope as mpg increases.

The cost is not a linear function of the mpg as most people think and base car buying decisions on. Instead, as the simple calculation shows, it is a curved function. What is clear from the graph is that small gains in mpg for a low mpg vehicle have a relatively large effect on the cost to drive 1000 miles. The effect becomes much less important as the mileage of vehicle improves. Bottom line, you save much more money if you ditch the SUV getting 15 mpg for a car getting 25 mpg (savings $107 for a 10 mpg increase) than if you change from a vehicle getting 25 mpg to one getting 45 mpg (savings only $71 for a 20 mpg increase!). Our dependence on
foreign oil would be greatly reduced if we were to focus on improving the mpg of the very low mileage vehicles on the road.

I also learned recently that European’s rank their vehicles according to how many liters of gas are required to drive 1000 km rather than rating them according to kilometers per liter (or mpg as in the U.S.). The European’s ranking is a more realistic way to compare vehicles and we should adopt it in the U.S..

You can see more on this at the Everyday Scientist, who says:

The real problem with MPG is that the same change in the MPG correspond to a huge change in fuel used at the low MPG end, and almost no change if a car already has a high MPG rating. Going from 20-25 MPG saves more gas that going from 35-50 MPG; going from 12-14 MPG saves more than either. This isn’t intuitive, and you really need to calculate the savings per mile in order to make a rational decision.

The take-away message is that we can’t do calculations like (1/a – 1/b) in our heads.

Richard Larrick tells us more about his research on the topic:

So, our actual research posed a series of questions about if you wanted to replace one vehicle with another one, which change is going to be most beneficial, in terms of reducing – and we, we couched it largely in environmental terms – the gas that’s used and therefore the effect on the environment. And people rate, for example, a change from 42 to 48 as being more beneficial then a change from 16 to 20 miles per gallon. And, without working through that math I hope that it’s obvious now that that 4 MPG improvement on 16 really reduces the amount of gas used quite a bit, and 42 to 48 doesn’t make, it’s still beneficial, but it isn’t nearly as large a change.

Interviewer – Robert Frederick
Right, they’re thinking it as a linear scale.

Interviewee – Richard Larrick
Exactly. And it really is a curvilinear relationship where the steep drops, in gallons that are actually used, occur among the MPG in the teens, and it gets flatter and flatter as you approach the kind of high end of what we see now – which is about 50 MPGs. So, the small MPG steps, on inefficient cars, have a big impact on reducing the amount of gas that’s burned.

UPDATE 11/24/08 – I just heard from Rick Larrick, who saw this post and wanted to share his websites with us:

I just ran across your sciencegeekgirl blog.  Great stuff.

And thanks for the mention of our Science article.  I’m definitely trying to get the word out, and, to be completely honest, really want to see the EPA, consumer reports, or both change to GPM.  That has become my mission!

I wanted to let you know about two webpages I’ve been running where I keep updates on the GPM argument:

http://faculty.fuqua.duke.edu/~larrick/bio/Reshighlights.htm

http://www.mpgillusion.blogspot.com/

Thanks again for the mention.  Best, Rick

It can be tough to get K-8 students engaged in math, or to really get across the idea of size and scale.  My latest podcast features a talk by math enthusiast David Schwartz talking about some real-world size comparisons that can make size and scale relevant to children’s lives.  Give it a listen!

David Schwartz’s website is at www.davidschwartz.com

Title: Science Teaching Tips
Episode: 46.  If you could hop like a frog…

Enjoy!

[[SESSION: SYMBOLIC CALCULATORS AFFECT EPISTEMIC FRAMING, TOM BING]]

This post is primarily for college teachers, but K-12 educators might get something from it too.

This researcher (Tom Bing… here’s his dissertation) has done some very interesting work on the paths that students take as they solve physics problem. He gave one particular example of students who, as they worked a problem, lost sight of the goal of hte problem. They used some very high level mathematical skills and computer programming, but that wasn’t particularly helpful in light of the goal of solving the particular problem. He interprets this in terms of how they thought about what this problem was about.

After all, when we start to solve a physics problem, the first task is to decide what sort of work we have to do to get to the answer. He calls this Framing. Essentially, the frame is the answer to the question, “what is this activity about?”‘ In particular, he’s interested in what kind of math they bring to the fore when trying to go about solving the problem once they’ve framed it. He calls this epistemic. So, he works on epistemic framing.

I’m struck by the idea of framing in this context since there is a fairly controversial science blog called Framing Science which specifically urges scientists to “frame” their research in order to (sort of) sell it to the public and the media. Bing’s work is a completely different context, but this idea of a “frame” is, I think a very compelling one, and one that scientists don’t tend to think about. It’s sort of a journalistic or social science kind of way of thinking about something. How something is framed doesn’t define the content of the item in question (be it scientific research, a math problem, or a conversation) but it does define how that content is interpreted. It puts the activity in question in a particular context, and gives it a particular color or light. I believe this is an overlooked part of education. Bing suggests that someone’s interpretation of a situation affects their actions. For instance, if you walk into a library, that triggers your whole idea of what a library is and what you do there. You have certain expectations of what you’ll find and what sorts of activities you’ll engage in there. It’s the same way in a physics class.

For Physics students, he says, a particular part of what they know about physics is sparked depending on how they understand the problem and what it’s asking. He looks at what they argue, mathematically, in order to probe what their frame is.

In the example he gave, a set of students were trying to solve the expectation value for the problem of the square well in quantum mechanics. They do some sophisticated calculations, but they frame the problem as a calculation problem. They sort of lose sight of what they are doing and even when their calculation doesn’t work, they keep working on it, and don’t go back to the underlying physics. In this case, they mistakenly plug in the limits of integration as going from negative infinity to infinity, thus getting an integral that diverges. Rather than go back to the physics problem to recognize that one only needs to integrate over the width of the well (from 0 to L, because the wave function is zero outside of the well), they continue to use Mathematica and other calculational tools to try to solve the integral in different ways. They are stuck in the frame of “this is a calculational problem.”

Then, one student realizes that the limits should only be over the width of the well. The other students laugh as they realize their mistake and laugh at themselves. There is a complete change in tone in the group, and one student who wasn’t listening to the conversation clues in that something has changed, just by the very tenor of the discussion, even though he hasn’t heard what they have said. Bing notes that this change shows a shift in their thinking and in how they are framing the problem.

Interesting stuff!

I just had to repost from the Deep Sea News blog, which points out an alarming 300% increase in the number of shark attacks in the last year in a particular town in Mexico:

Aren’t statistics wonderful things? That’s why when you read something in the medical news about “50% fewer heart attacks” or some such due to XYZ drug, your first question should be “but what was the number to begin with”?

In this case, a 300% increase means 3 attacks instead of one. That’s hardly a statistically robust difference. But the local papers surmised that Sharks are hunting humans. Blogger CR McLain writes in Deep Sea News:

Thankfully the Mexican Navy has been called on to track down and kill these death wielding beasts.

PLEASE PEOPLE! Although tragic, three attacks and two deaths is not extraordinary that searching for pattern or cause is necessary. You don’t see people freaking out about pigeon related deaths. An increase of one to three is hardly a pattern. In four tosses of a penny this morning I just got 1 head and 3 tails…it happens. The fact that the media is in a frenzy combined with Mexico actually spending money on searching for causes and using sophisticated Naval ships to exterminate the sharks is nothing short of absurd. Let’s get a bit of perspective…

Americans killed by guns in the U.S. each year: 30,000

Americans killed by tobacco in the U.S. each year: 418,000

Americans killed after being struck by police Tasers in 2004: 40

U.S. murder rate: 5.9 per 100,000

U.S. traffic fatalities each year: 39,000

People injured/killed by lightning each year in the U.S.: Struck: 700, Killed: 70

Deaths from obesity per year in the USA: 112,000

I will take a shark any day over a Twinkie, lighting strike, the flu, a tsunami, Taser, cigarette, hand gun, war, or a car any day.