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Math terms – learning with humor

August3

“There is a fine line between a Numerator and Denominator
but only a fraction of people will find it funny!”

How can you tell the difference between a Numerator and Denominator?
Answer: make up something silly. For example, the Denominator starts with a “D”, so it is the “down” part of a fraction. Or, there are more letters in the word “Denominator” so it is heavier than “Numerator” and falls to the bottom of a fraction.

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3 Principles of Mathematics

August1

Mathematics is true: students know this, since there’s always a correct answer.
The philosophers ask, “What is truth?” but the mathematician replies “Mathematics is Truth.”

Mathematics is orderly: Algebra requires arithmetic, Trigonometry requires algebra, Geometry requires both.

Mathematics teaches us about infinity: For any number, no matter how big, you can always find a bigger number by adding one. Likewise, no matter how small a decimal or fraction is, you can always find a smaller number by dividing it in half. Infinity is also involved in repeating decimals, sequences, series, and geometry (lines extend infinitely). Infinity gets even more attention in calculus (limits) and sizes of sets (cardinality). Of the liberal arts, Mathematics delights in teaching us about infinity – that is, teaches us about something that is bigger than ourselves. [Ed: Incidentally, the God of the Bible is infinite (Psalm 147:5)]

 

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Crime fighting, etc. with Inverse Math

July27

Inverse problems are mathematical detective problems. An example of an inverse problem is trying to find the shape of an object only knowing its shadows. Is it possible to do this at all? What sort of errors are we likely to make, and how much extra information is needed?

Other examples of inverse problems are remote sensing of the land or sea from satellite images, using medical images for diagnosing tumours, and interpreting seismographs to prospect for oil. Another example is perhaps not the first thing that springs to mind when you think of maths: fighting crime.

When a crime has been committed police must look at all the evidence left at the crime scene and work backwards to deduce what happened and who did it. Often, the evidence is a result of a physical process that is well understood — like a speeding car causing skid marks. So to find out the exact cause of the evidence — the speed of the car — the maths that describes the physics needs to be run backwards. This means solving an inverse problem.

Let’s step into an ordinary day in the life of a police unit and see how mathematics can help fight crime. We are investigating a car accident and need to answer the question: was the car speeding?

The evidence available is the collision damage on the vehicles involved, witness reports, and tyre skid marks. Examining the skid marks can help reconstruct the accident. The marks are caused by the speed of the car as well as other factors such as braking force, friction with the road and impacts with other vehicles.

Mathematically we can use mechanics to model this event in terms of $s$, the length of the skid, $u$, the speed of the vehicle, $g$, the acceleration due to gravity and $\mu $, the coefficient of friction times braking efficiency.

The model links the cause (the speed of the car) to the effect (the distance of the skid). As long as we have an accurate estimate of $\mu $, the value describing friction and braking efficiency, we can solve the problem and determine the speed of the car from its skid marks. Read more on this fascinating application of Mathematics in the real world here at plus math.

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Problem-Solving Problems

July18

Check out the sets of problem-solving questions on the NZMaths website. Copymasters are included and there are excellent problems for all levels, covering Algebra, Measurement, Geometry, Statistics, etc. Here is one example, from Level 2:

“There are 2 pirates and 4 treasure chests on an island.

The pirates have 1 small boat to take the treasure to their ship.

The boat can take 2 pirates or 1 pirate and 1 chest of treasure.

How many trips do the pirates have to take to get all the treasure and both pirates onto the ship?”

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Math is about solving problems!

July18

There is a dilemma in Mathematics – the pressure to get students through coursework versus the joy of problem-solving and mathematical investigation. As sure as one plus one equals two, it happens year after year. Kids who have been bringing home A’s in chemistry and acing AP Calculus arrive at college with visions of STEM careers dancing in their heads. Then they hit an invisible, but very painful, wall.

According to research from the University of California, Los Angeles, as many as 60 percent of all college students who intend to study a STEM (science, technology, engineering, math) subject end up transferring out. In an era when politicians and educators are beside themselves with worry over American students’ lagging math and science scores compared to the whiz kids of Shanghai and Japan, this attrition trend so troubles experts it has spawned an entire field of research on “STEM drop-out,” citing reasons from gender and race to GPAs and peer relationships.

So why do even the most accomplished students burn out of STEM programs when they hit college? One recent article in the New York Times explored possible reasons — from the alluring grade inflation in the arts and humanities, to what one engineering professor characterized as the boring, largely theoretical “math-science death march” of first-year requirements.

That may explain the phenomenon, at least in part. But math experts around the country point to another culprit.

Richard Rusczyk, a former Math Olympiad winner and the founder of the online math program Art of Problem Solving,  is part of a group of math educators who sees the mystery of the disappearing STEM major from a different angle. It’s not that kids aren’t getting enough math, they say, but that we’re teaching K-12 math all wrong.

Rusczyk’s insight is based on a phenomenon he witnessed firsthand when he arrived at Princeton University and began studying math alongside kids who had attended the most prestigious high schools in the country. “These were kids who had never gotten anything but 95s and 100s on their tests and suddenly they were struggling and were getting 62s on tests and they decided they weren’t any good [at math],” he explains.

Call it the mathematical reality check. Suddenly, Rusczyk recalls, formerly accomplished students were faced with a new idea: that math required more than rote learning — it required creativity, grit, and strenuous mental gymnastics. “They had been taught that math was a set of destinations and they were taught to follow a set of rules to get to those places,” he recalls. “They were never taught how to read a map, or even that there is a map.”

Indeed, traditional math curriculum is to teach discrete algorithms, a set of rules that elicit a correct answer, like how to do long division, say, or how to use the Pythagorean theorem. Then students “learn” the material by doing a large quantity of similar problems. The result, says Rusczyk, is that students are rarely asked to solve a problem they are not thoroughly familiar with. Instead, they come to think of math as a series of rules to be memorized. The trouble is kids don’t necessarily learn how to attack a new or different kind of equation.

Rusczyk watched many of his fellow students, long accustomed to being “quick studies,” as they soured on math after experiencing what they perceived as failure. They quit — transferring their hopes and dreams to a less numerically challenging field like sociology or graphic design.

Rusczyk, in contrast, felt far more prepared when faced with a problem he didn’t know how to solve. Despite having attended what he characterizes as an average public school without a lot of advanced math classes, he had participated in math clubs and contests. In math clubs, he’d become accustomed to facing harder, multifaceted problems where the right approach wasn’t immediately apparent.

Math as problem solving

Instead of just learning how to follow rules, he explains, “In math competitions, I learned how to solve problems that I hadn’t seen before.” Instead of math becoming something he accomplished in return for a perfect score, he came to see math as problem solving — an exciting pleasure that was a distant relation to the rote drudgery of memorizing algorithms.

When Rusczyk looked around him, he noticed a pattern. His classmates who had experienced this kind of difficult problem solving — usually in after-school math clubs — could survive the transition to college math.

Read the full article here.

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Another Mathcamp bytes the dust

July16

Mathcamp is an immersive summer experience for mathematically talented students ages 13‑18 from all over the world.

Mathcamp was founded in 1993, with just two students, and has since grown into an amazing community of 120 students each summer and 1500+ alumni who share a love of mathematics. Mathcamp is an intensive 5-week-long summer program for mathematically talented high school students, designed to expose these students to the beauty of advanced mathematical ideas and to new ways of thinking.

More than just a summer camp, Mathcamp is a vibrant community, made up of a wide variety of people who share a common love of learning and passion for mathematics. At Mathcamp, students can explore undergraduate and even graduate-level topics while building problem-solving skills that will help them in any field they choose to study. The 2019 Mathcamp has just about finished but this is a great time to plan to attend Mathcamp in 2020!

Check out more about Mathcamp here.

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The Privacy Puzzle Improved

July14

A team of Reed Statistics students won first place in a prestigious national competition for an innovative algorithm that helps researchers glean information from datasets—without compromising individual privacy. That right – a technique for ensuing your privacy while allowing statistical analysis of data!

Zeki Kazan ’20, Kaiyan Shi ’20, and Simon Couch ’21 (seen here) won the Undergraduate Statistics Research Project Competition for their project, “A Differentially Private Wilcoxon Signed-Rank Test,” which outlines a new algorithm for hypothesis testing that upholds the privacy of the underlying data. In fact, their technique is twice as powerful as the standard private method, meaning that it requires less than half as much data to achieve the same statistical power.

Simply put, the problem is that big databases hold immense promise for answering scientific questions, but many organizations won’t allow researchers access to them because of the risk of an inadvertent breach of privacy—even when obvious markers like name and address have been stripped away. In 2014, for example, the New York City Taxi and Limousine Commission released a giant database of taxi rides in response to a freedom-of-information request. The commission attempted to anonymize the data, but enterprising journalists were able to piece together various clues to identify rides taken by celebrities.

To understand the Reed project, you need to know that statisticians often compare two sets of data using a tool known as a hypothesis test. Each hypothesis test requires a certain amount of data before it can detect a relationship between the two sets—the less data it needs, the more statistical power it has.

Now to go deeper …

There are many different types of hypothesis tests. The Reed team focused on the Wilcoxon Signed-Rank Test, which is commonly used when there is paired-sample data—where there is a natural association between the two sets (e.g. a patient’s blood pressure before and after watching a horror movie). It compares the sets in an attempt to determine whether there is a statistically significant relationship.

The team reworked the Wilcoxon test to ensure privacy, and employed an innovative technique to reduce the amount of data it required. With these two seemingly simple tweaks, the enhanced algorithm turned out to be much more powerful, yielding significant real-world implications. When tested, their model had a statistical power that was much closer to public-setting tests: achieving the same statistical power with only 40% of the data required by the earlier private-setting model. Because of this increased efficiency, the Reed algorithm can be used on smaller datasets, whereas previous models required enormous quantities of data. (source: here). No wonder this group of math geeks look happy in the picture above!

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Math Menu at Reed College

July13

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When dividing by a fraction the answer gets bigger?

July13

That’s right. Here is a typical problem that asks you to divide a number by a fraction:

Think of it this way –

how many halves are there in 30?

Answer: Well, there will be 60 halves in 30. So the answer will be 60 + 10 = 70.

Note that we do the division part first since division is more powerful that addition – this follows the BEMA Rules that H3 recommends for all Math students. Most classrooms teach that division by 1/2 is the same as multiplying by 2. Of course, this is correct, but it may be easier for some students to view this problem as “divide 30 into halves and then add 10”.

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Chicken question leaves math students scratching their heads

July10

A 13-year-old boy from Texas recently won a national math competition with an answer rooted in probabilities — and a dash of farming. The boy, Luke Robitaille, took less than a second to buzz in at the Raytheon Mathcounts National Competition with the correct answer.

The question? In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?

For Luke, a seventh-grader from Euless, Texas, who is home schooled, the victory meant beating a teammate from Texas in the final round. As national champion, he will receive a $20,000 college scholarship and a trip to Space Camp in Huntsville, Alabama.

You get to think about things and move logically toward solving problems,” said Luke, who came in second place at last year’s competition.

The contest, which was open to students in the sixth, seventh and eighth grades, took place at a hotel ballroom in Lake Buena Vista, Florida, before an audience of 1,000 people.

Now, back to that pecking problem. What is your answer? Check out Post Support to see if you are correct, but no pecking/peeking yet!

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