Welcome to H3 Maths

Blog Support for Growing Mathematicians

Putting a spring into your Math journey


If your class did well on their last math test, does that mean you are a smart class, or does it mean that the test was easy? Well, that might not be easy to answer, but here is a video that will explain how forces work with springs, and that might put a spring in your step as you head into equation land – and, after all, don’t we love to soak in hot springs? Professor Walter Lewin explains …

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Girls = Boys in Math, at least in the early years


There’s new evidence that girls start out with the same math abilities as boys (source: npr.org).

A study of 104 children from ages 3 to 10 found similar patterns of brain activity in boys and girls as they engaged in basic math tasks, researchers reported Friday in the journal Science of Learning.

“They are indistinguishable,” says Jessica Cantlon, an author of the study and professor of developmental neuroscience at Carnegie Mellon University.

The finding challenges the idea that more boys than girls end up in STEM fields (science, technology, engineering and mathematics) because they are inherently better at the sort of thinking those fields require. It also backs other studiesthat found similar math abilities in males and females early in life.

“The results of this study are not too surprising because typically we don’t see sex differences at the ages assessed in this study or for the types of math tasks they did, which were fairly simple,” says David Geary, a psychologist and curators’ distinguished professor at the University of Missouri who was not involved in the research.

But there is evidence of sex differences in some exceptional older students, Geary says.

For example, boys outnumber girls by about 3 to 1 when researchers identify adolescents who achieve “very, very high-end performance in mathematics,” Geary says, adding that scientists are still trying to understand why that gap exists.

Finally, a standardized test of mathematics ability found no difference between boys and girls.

So why are fields like mathematics and computer science so dominated by men?

Cantlon suspects the answer involves the societal messages girls and young women get, and the difficulty of entering a field that includes very few women. “You can look at ratios of women and men participating in different activities and you can get the hint,” she says.

But Geary says an international study he did with Gijsbert Stoet at the University of Essex suggests a different explanation.

Using an international database on adolescent achievement in science, mathematics and reading, they found that in two-thirds of all countries, female students performed at least as well as males in science.

Yet paradoxically, females in wealthier countries with more gender equality, including the U.S., were less likely than females in other countries to get degrees in fields such as math and computer science.

Geary thinks the reason may be that women in these countries are under less pressure to choose a field that promises an economic payback and have more freedom to pursue what interests them most.

A study of gender achievement gaps in U.S. schools found that the gaps varied widely depending on whether the school was in a wealthy area.

When all school districts are pooled together, “there isn’t really a gender achievement gap in math, but there is in reading,” says Erin Fahle, an assistant professor at St. John’s University and an author of the gender gap study. Males were about two-thirds of an academic year behind females, she says.

But when the researchers focused on more affluent school districts, “boys tended to do better than girls in math,” Fahle says.

That research, along with the new study, makes a compelling case that factors other than biological differences explain why girls are less likely to pursue degrees and jobs in math and science, she says.

Reflecting on Pi


Learn more about Reflection here!

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A Tale of 21 Cities – H3 on Tour


In what turned out to be a traveling whirlwind, H3 recently completed a world tour in 78 days. Here are the numbers:

13 flights, 5 trains, 3 cars, 1 bus
21 main cities, including: Auckland, San Fancisco, Seattle, London, Brussels, Rome, Florence, Venice, Trieste, Zagreb, Split, Hvar, Dubrovnick, Barcelona, Gibraltar, Lisbon, Montpellier, Cannes, Dubai, Perth, Brisbane.
41 meetings with friends including math-focused conversations with some!

Which explains why we are a little tired and have been slack in posting regularly.
A favorite picture is this one, taken in Montpellier. Note the two drain covers, two crossings, two opposing curves, etc. Yes, the mathematical thinking influenced this picture, plus some patience and a timely cyclist! Which leads to the main point of this post …

Whatever you are doing, your background in Mathematics will influence your thinking – plane and train times, ticketing, composition in photo-taking opportunities, admiration for beauty in landscapes and the design of cities and building, a love for art and culture, etc.

You see, Mathematics is inherent in the world around us – the patterns in plants, the patterns in the weather, the shapes and curves of hills and mountains and rivers. And, even in chaos, Mathematics makes a grand appearance.

Galloping galaxies – Pt II


The universe doesn’t look right. It suddenly looks . . . out of whack.

The universe is unimaginably big, and it keeps getting bigger. But astronomers cannot agree on how quickly it is growing – and the more they study the problem, the more they disagree. Some scientists call this a “crisis” in cosmology. A less dramatic term in circulation is “the Hubble Constant tension.”

Nine decades ago, the astronomer Edwin Hubble showed that the universe is orders of magnitude vaster than previously imagined – and the whole kit and kaboodle is expanding. The rate of that expansion is a number called the Hubble Constant.

It’s a slippery number, however. Measurements using different techniques have produced different results, and the numbers show no sign of converging even as researchers refine their observations.

No one is panicking. To the contrary, the theorists are intrigued. They hope the Hubble Constant confusion is the harbinger of a potential major discovery – some “new physics.”

“Any time there’s a discrepancy, some kind of anomaly, we all get very excited,” said Katherine Mack, a physicist at North Carolina State University who co-wrote a recent paper examining the issue.

The Hubble Constant is a central feature of any theory about the evolution and ultimate fate of the universe. This number may have zero effect on daily human existence, but there’s a lot at stake cosmologically.

“Where’s it all going to go? How’s it all going to end? That’s a big question,” Mack said. One widely supported estimate of the cosmic expansion uses the background radiation that permeates space – light emitted when the universe was young.

That gives a Hubble Constant of about 67 kilometres per second per megaparsec. (A parsec is a distance of a bit more than three light-years. According to this estimate, a galaxy one million parsecs from Earth is receding at 67 kilometres per second, and a galaxy twice as distant is receding at 134 kilometres per second.) All these speeds are faster than H3’s Camry!

But another carefully calibrated measurement, based on light emitted from exploding stars – supernovae – has come up with a Hubble Constant of 73.

This isn’t horseshoes or hand grenades: Close doesn’t count. People want the actual, real, universe-expanding Hubble Constant, and no one is eager to round it up to the nearest 10.

This northern hemisphere summer, as leaders in the field assembled in Santa Barbara, California, to discuss the “tension,” physicist Wendy Freedman of the University of Chicago presented a new estimate of the constant that was based on examination of red giant stars.

Her number: 70. But the advocates for 67 and 73 held their ground. The tension remained. Freedman told The Washington Post, “There can’t be three different numbers.”

There are more than that, actually. On October 23, researchers at the University of California at Davis published a paper that looked at three gravitational lenses – in which massive galaxies function like magnifying glasses for things behind them in deeper space. Their number: 77.

It could be simply that some of the measurements are based on erroneous assumptions. Imagine two speed guns giving strikingly different measurements of a bowled fastball. One obvious, boring explanation is that one of the speed guns needs to be recalibrated.

It’s conceivable, for example, that astronomers haven’t fully factored in the way cosmic dust can interfere with observations, which wouldn’t be the first time that has happened.

But the more delicious possibility is that there’s something new to be discovered about the way the universe evolved.

One idea floating around is that there could have been something called Early Dark Energy that skewed the appearance of the background radiation.

“New physics might be that there’s some form of energy that acted in the earliest moments of the evolution of the universe. You’d get an injection of energy that’d then have to disappear,” Freedman said.

“If it’s new physics, it’s so exciting,” says Jo Dunkley, a professor of physics at Princeton. But, she added: “I’m just not willing yet to jump into the opinion that it’s new physics. I’m more sceptical of our ability to understand our measurement uncertainties.”

Meanwhile, estimates from the team behind the Planck space telescope, which studied the cosmic microwave background radiation, continue to centre on 67.

So the disparity persists. That leaves open the tantalising possibility that “No one’s wrong. Something else is going on in the universe.”

Article first posted in The Washington Post. See also Galloping Galaxies, Pt I


Phew – a 1000 posts!


A milestone of sorts – H3 has completed a thousand posts.
Just over 10 years and some 33,000 views later, it does seem significant that this number has been reached. Thank you all for following H3, however briefly, and our hope is that you have found many useful connections between Mathematics in the classroom and Mathematics in the real world. Not that the classroom is not real, but you probably know what we mean?

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Superposition and the Quantum Race


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The Wisdom of Earl


H3 was enjoying meeting up with Earl in Montana. Montana is Big Sky country and Earl has a big heart for helping students with their math – which is the subject we naturally focused on after a while (there was hay to feed to the horses and some garden crops to pick first). I mentioned to Earl that many students ask why they have to do difficult topics in math, such as Algebra.
Earl looked out across the distant peaks and pure math wisdom flowed from his lips. “Well, ya know, I ask my students whether they want to make their own career choices, or have someone else decide for them. Most want to be in charge of their own destiny and make their own career pathways. So, I tell them that Algebra (or some other learning challenge) must be conquered first. It is just a small step to getting to make their own choices for the future.

How true! Now, if I can just persuade Earl to write a book of his remarkable career – one that was made possible thanks to him jumping over a few learning hurdles too!

Branch out with some Fractals


Breathe. As your lungs expand, air fills 500 million tiny alveoli, each a fraction of a millimeter across. As you exhale, these millions of tiny breaths merge effortlessly through larger and larger airways into one ultimate breath. These airways are fractal.

Fractals are a mathematical tool for describing objects with detail at every scale. Mathematicians and physicists use fractals and related concepts to understand how things change going from small to big.

You and I translate between vastly different scales when we think about how our choices affect the world. Is this latte contributing to climate change? Should I vote in this election?

These conceptual tools apply to the body as well as landscapes, natural disasters and society.

Fractals everywhere

In 1967, mathematician Benoit Mandelbrot asked, “How long is the coast of Britain?”

It’s a trick question. The answer depends on how you measure it. If you trace the outline on a map, you get one answer, but if you walk the coastline with a meter stick, the result is quite different. Anyone who has tried to estimate the length of a rugged hiking trail from a map knows the treachery of the large-scale picture.

That’s because lungs, the British coastline and hiking trails all have fractality: their length, number of branches or some other quantity depends on the scale or resolution you use to measure them.

The coastline is also similar —it’s made out of smaller copies of itself. Fern fronds, trees, snail shells, landscapes, the silhouettes of mountains and river networks all look like smaller versions of themselves.

That’s why, when you’re looking at an aerial photograph of a landscape (like this NASA one of Cape Farewell in New Zealand), it’s often hard to tell whether the scale bar should be 50 km or 500 m.

Your lungs are self-similar, because the body finely calibrates each branch in exact proportions, making each branch a smaller replica of the previous. This modular design makes lungs efficient at any size. Think of a child and an adult, or a mouse, a whale. The only difference between small and large is in how many times the airways branch.

This fractality appears in art and architecture, in the arches within arches of Roman aqueducts and the spires of Gothic cathedrals that mirror the forest canopy. Even ancient Chinese calligraphers Huai Su and Yan Zhenqing understood the fractality/branching of summer clouds, cracks in a wall and water stains in a leaking house in 722.

Scale invariance

Self-similar objects have a scale invariance. In other words, some property holds regardless of how big they get, such as the efficiency of lungs.

In effect, scale invariance describes what changes between scales by saying what doesn’t change.

Leonardo da Vinci observed that, as trees branch, the total cross-sectional area of all branches is preserved. In other words, going from trunk to twigs, the number of branches and their diameter change with each branching, but the total thickness of all branches bundled together stays the same.

Da Vinci’s observation (right) implies a scale invariance: For every branch of a certain radius, there are four downstream branches with half that radius.

Earthquake frequency has a similar scale invariance, which was observed in the 1940s. The big ones come to mind—Lisbon 1755, San Francisco 1989—but many small earthquakes occur in California every day. The Gutenberg-Richter law says that earthquake frequency depends on the size of the earthquake. The answer is surprisingly simple. A tenfold bigger earthquake occurs roughly one-tenth as often.

From understanding how humans are made up of tiny cells to how we affect the planet, self-similarity, fractals and scale changes often help translate from one level of organization to another.

Adapted from an article in Physics.org

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Be a Number Wizz with Scott Flansburg


Where did we get our number system from? Watch Scott Flansburg amaze with his addition skills and be equally transformed as he explains that our number system is based on angles. Yes, Trigonometry is the foundation of Number, or is Number the foundation of Trigonometry? Are you getting confused?
Find out more in this video, suitable for all students studying Mathematics, whether in Elementary or College.

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