In this Newsletter:

1. Math resource: MathGraph32
2. Prime numbered cicadas
3. Relieving test anxiety
4. Math puzzle
5. IntMath Poll
6. Final thought:

Once again I apologize for the long gap since the last IntMath Newsletter. I’m involved in several large projects (including developing online modules for a math course and 2 academic integrity courses). Once those are done, hopefully I’ll be able to get back to a regular schedule of writing.

1. Math resource: MathGraph32

MathGraph32 is a great free tool for exploring 2D and 3D math concepts. MathGraph32

2. Prime numbered cicadas

The 17-year cicada is due to emerge in north-eastern parts of USA in Spring 2013. What is their connection to prime numbers? Prime numbered cicadas

3. Relieving test anxiety

Annie Murphy Paul writes the Brilliant Report for Time magazine, It’s an interesting collection of research about the brain and how to squeeze more out of it.

A recent article, How to Eliminate Test Anxiety gives some good pointers, which as she says, are reasonably simple, inexpensive and, as recent studies show, effective."

Here’s some short quotes from her list of suggestions:

1. Unload on paper. Spend ten minutes writing about your thoughts and feelings immediately before taking a test.

The practice, called "expressive writing," is used by psychologists to reduce negative thoughts in people with depression. They tried the intervention on college students placed in a testing situation in Beilock’s lab, and in an actual Chicago school, where ninth-grade students engaged in the writing exercise before their first high school final. In both cases, students’ test scores “significantly improved,” according to an article they published last year in the journal Science.

2. Affirm your values. Apprehension over tests can be especially common among minority and female students. That’s because the prospect of evaluation poses for them what psychologists call "stereotype threat"—the possibility that a poor performance will confirm negative assumptions about the group to which they belong ([this posits] that girls can’t excel in math and science; blacks and Latinos aren’t college material).

3. Engage in relaxation exercises. Younger kids aren’t immune from test anxiety. As early as first and second grade, researchers see evidence of anxiety about testing. Their worries tend to manifest in non-verbal signs that adults may miss, [like] stomachaches, difficulty sleeping, and a persistent urge to leave the classroom to go to the bathroom.

See more details on these anxiety-reduction techniques here:

From The Brilliant Report: How To Eliminate Test Anxiety

4. Math puzzles

The puzzle in the last IntMath Newsletter was about expressing the number one using nines, a minus sign and dots.

Correct answers were given by dalcde, Christopher, Bonnie, Andrzej, Dineth, Nicos and Thomas.

Math symbols were a challenge: I knew it was going to be tricky to type in the answers for this puzzle. I started to write some pointers about how to do it and stopped, because that would have given the answers away! (There is a "Preview" button on the response box for the blog. You can use it to make sure your math looks OK before posting.)

1. Superscripts (powers): You can create a superscript by typing it like this in the "Respond" box of most blogs:

   9^9-9 = 1

   It will look like this:

   9^9-9 = 1

2. Dot above a number: This one is a bit trickier, as there is no HTML solution that works nicely in all browsers. The best way is to create an image using a tool like Codecogs Equation Editor. It has a visual interface, but requires some knowledge of LaTex. You need to enter:

   0.\dot{9} = 1

   At the bottom of the equation editor page there is an "embed" box with code. Copy that code into the "respond" box in most blogs and it will look like:

   

Note 1: I included the "0" before the decimal point in my answer above. A lot of people don’t notice the dot, and misread decimal numbers. (Yes, I know the question specified "one nine" only, but it’s worth mentioning.)

Note 2: I was aware the question was somewhat country-specific. The Europeans write a decimal number using a comma, not a dot.

And as Andrzej pointed out in his response regarding the recurring part:

There are three symbols for recurring fractions: dash, dot (UK and USA) and () in Poland. So the solution is slightly tricky; it depends on nationality.

It would be really nice if we had consistent math notation around the world, especially when for something as simple as writing numbers!

New puzzle: 216 cubes of side length 1 cm are arranged to make a cube with side length 6 cm.

A sphere of diameter 6 cm is inscribed in the large cube such that the center of the sphere is the center of the cube. How many complete unit cubes is contained in the sphere?

You can leave your responses here.

5. Final thought: Watching

Donald Trump is a real estate millionaire who became quite famous as a result of his role in the TV series The Apprentice. He once said:

Work hard. Someone’s always watching.

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: Prime cicadas appeared first on squareCircleZ.

(It’s been around for a while, but I only recently discovered it.)

There is a download version as well as an online Java-based version.

According to the site, MathGraph32 is an:

Open source cross-platform software of geometry, analysis and simulation.

It was developed by French-speaking mathematician, Yves Biton. Some of the examples are in French, but it’s quite easy to see what is going on. (There is an English version of the program).

Screen Shots

Here are some examples from MathGraph32 (images by them).

You can explore several examples (some for “teaching purposes”), and there are tutorials that help explain the use of the applet.

It’s well worth checking out!

The link again: MathGraph32

The post MathGraph32 appeared first on squareCircleZ.

Millions of Magicicada cicadas will emerge after 17 years of living underground. They will fly around for 3 weeks eating a lot of leaves, then mate, lay their eggs and die. (So you could say only around 0.3% of their lives would be called "fun".)

Magicicada [Image source]

There’s another group of cicadas that have a 13-year life cyle, and these ones tend to live in the southern US states. (The next appearance of these little guys will be 2014.)

The cicadas provide a feast for birds, fish and many animals. Some people eat them, too!

Why 17 years? Why 13?

It’s intriguing these cicadas have life expectancies that are prime numbers.

Some scientists believe this may be due to an evolutionary adaptation that secured their continued existence.

Even though the cicadas emerge in huge numbers, it’s possible they could be wiped out by a parasite.

Let’s consider the life cycle of this alleged parasite. One possibility is the parasite could have a one-year life cycle and eventually after waiting some years, they would find some cicadas to infest. The trouble with this is they wouldn’t have anything to feed on for many years so would probably die out first.

If the parasite has a 2-year life cycle, it will very rarely coincide with the cicada’s. Similarly for 3-year, 4-year and any number of years up to 17.

Compare this to the case of a 12-year cicada. It would coincide with parasite that had 1-year, 2-year, 3-year, 4-year, 6-year and 12-year cycles, so would be very vulnerable.

So for our prime-numbered life-cycle cicadas, the parasite would need to have a life cycle that exactly coincided with the cicadas’. No such parasite has ever been found, so it appears to have been a successful strategy.

[See more in this 2001 article in Science.]

The next appearance

Of course, like many plants and animals, the number of these cicadas is dwindling due to habitat loss, soil disturbance and abundant use of insecticides.

In 2030, when the Magicicada are due to emerge again, what kind of world will they see?

The post Prime numbered cicadas appeared first on squareCircleZ.

In this Newsletter:

1. Math resource – 3D-XplorMath-J
2. Math resource – Mathics
3. National STEM Video Game Challenge – on now
4. ComputerBasedMath partners with Estonia
5. Math puzzle
6. IntMath Poll
7. Final thought: Passing civilization on

Welcome back to the IntMath Newsletter, after quite a long break. I’ve been very busy with several projects lately, but I hope to be able to find some time for writing again soon.

1. Math resource: 3D-XplorMath-J

3D-XplorMath-J is a delightful Java-based 3D math visualization program. It runs on Windows, Mac and Linux.

There is a large range of 3D and 2D objects that you can manipulate, explore and change parameters. You can create animations of 3D objects as they morph from simpler shapes, and there are even animations of ordinary differential equation solutions.

Unfortunately there is nowhere to enter your own functions, but even so, this is a valuable 3D visualization tool, kindly provided free by Richard Palais at University of Califormia Irvine.

Get your copy here: 3D-XplorMath-J

2. Math resource: Mathics

A lot of math software is very expensive, but there are some good free alternatives.

Mathics is a "free, lightweight alternative to Mathematica". You can download it to run on your on own computer, but there’s a much simpler alternative: see the online version.

When you get there, click on the gallery link (at the bottom) to see some of the cool things Mathics can do (give it a few moments to process everything, and use a real browser like Chrome, Safari or Firefox.)

You can do your own math calculations with this tool – use the gallery as your starting point, or click the Documentation link at the top right of the screen.

3. National STEM Video Game Challenge – on now

The 2013 National STEM Video Game Challenge is accepting submissions right now. I think this is a great idea – learn some math and science while learning another valuable skill – game design.

From the site:

Game design is promising area of innovation in STEM learning. Research suggests that empowering youth to create their own video games promotes learner independence. Moreover, it encourages youth to take ownership over STEM knowledge, rather than viewing it as belonging to others, thereby ushering them into STEM communities of practice.

Entries will be accepted until April 24th.

It’s a shame this challenge is only for the USA at this point. How about a worldwide challenge?

4. ComputerBasedMath partners with Estonia

Computerbasedmath.org is the brainchild of Conrad Wolfram, creator of Mathematica and Wolfram|Alpha. His view (which I largely agree with) is that we should change math education in the light of all the math software that is now available. We spend far too much time on theoretical algebra and not enough time using computer math tools to solve meaningful problems.

ComputerBasedMath recently announced they partnered with the first country ready for "this new kind of maths education": Estonia.

This collaborative project will address school statistics, using computer-based techniques to write a completely new and innovative curriculum. Focusing on equipping students with the skills and abilities needed for a twenty-first-century economy, the Minister of Education and Research in Estonia, Jaak Aaviksoo, states that now is the time for computer-based maths: "We believe in the enthusiasm and potential of the Internet generation–they are ready for computer based mathematics. It will also give them a competitive advantage in the labour market." [Source]

For more information on this movement, see the ComputerBasedMath FAQs.

5. Math puzzles

The puzzle in the last IntMath Newsletter was about Peter, the enthusiastic student who runs to school.

Correct answers with reasoning were given by Ibrahim, Peter Hunter, Thomas, Guido, Christopher and Nicos.

New puzzle: Professor Chng says he can represent the number one using (a) threes nines and a minus sign, and (b) one nine and two dots. How is that possible?

You can leave your responses here.

6. IntMath Poll

The recent IntMath Poll asked readers how important it is for them to know the historical background to the math they are learning.

Here is the result

Very important
47%

Not important
27%

Important
25%

Total votes: 1800

It’s obviously important for the majority of students that they know where the math comes from and why it was developed. This helps find meaning in what they learn.

New poll: The current poll asks readers about the method used by the teacher in class for delivering math questions. You can respond on any (internal) page of IntMath.

7. Final thought: Passing civilzation on

This is from Lee Iacocca,Ford and Chrysler executive (b. 1924), and it speaks about the value of teachers:

In a completely rational society, the best of us would be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and the highest responsibility anyone could have.

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: XplorMath, Mathics, STEM challenge appeared first on squareCircleZ.

In this Newsletter:

1. Gong Xi Fa Cai
2. Counting game and Subitizing
3. Jacob’s Staff
4. Creating a Google-like classroom climate
5. Math puzzles
6. Friday math movie
7. Final thought – Perfect days

1. Gong Xi Fa Cai!

Chinese New Year falls on the second new Moon after the December solstice (unless a leap month messes things up). The earliest possible date is Jan 21st, and the latest is Feb 21st (but that won’t happen until 2319).

Read more in (the extensive) The Mathematics of the Chinese Calendar

2. Counting game and subitizing

Next up is a short article on why I created the game, and some early observations:

Counting Game and Subitizing

Then, for some interesting background, see this article:

Number Sense

3. Jacob’s Staff

Here’s a similar activity based on historic math: Noon Day Project. This activity re-creates Eratosthenes’ famous experiment 2000 years ago where he produced a good approximation of the size of the Earth based on measuring shadow lengths at noon in different parts of Egypt.

4. Creating a Google-like classroom climate

Google is well-known for being one of the best companies in the world to work for. Around 7,000 people per day apply to work there. The company offers excellent benefits and a relaxed, almost playful atmosphere that keeps creative juices flowing.

They put a lot of effort into finding the right people, and because Google doesn’t want to lose anyone, they create conditions that encourage success and fulfillment.

The article, Inside Google’s Culture of Success and Employee Happiness, got me thinking about why so many students drop out of school (often because of acute math anxiety, or because they don’t have a feeling of belonging, or a host of other reasons).

Good classroom "climate" is very important for effective learning, especially in math!

Here are some of the things Google does that could apply in class:

Hire (and reward) good teachers! I enjoy living in Asia where education is generally held in high esteem, and teachers are, for the most part, regarded as valuable members of society. Good teachers almost always leads to better learning outcomes. Keep the good ones by giving them purpose, recognition, and reward.

Collect data on (and share) what works: I don’t mean what works for standardized tests, here. I mean quality class activities, different approaches to explaining things, or different resources. What works? What needs changing? Teachers aren’t given enough time to think about what they’re doing, nor enough opportunities to learn from each other. (Google collects huge amounts of data to improve its processes, and creates many situations where workers mingle and share ideas.)

Warm greetings make a huge difference: Google found their best managers were warm and accepting of input from their team, leading to a "15% increase in productivity". Sadly, I see a lot of classrooms where there are no greetings at the beginning of the day, and no friendly talk with students about mutual interests. It’s often no more than "OK, today we’re going to do the quadratic equation…". A sense of belonging is important for learning and for sticking around.

Too much bureaucracy: Google has a "flat" structure where middle managers have quite a bit of responsibility, compared to many other companies. That’s not always the situaion with teachers.

Too many distractions: Many classrooms have too many interruptions. Are those announcements about sport buses – or uniforms – really that crucial? There are already too many things trying to grab everyone’s attention. Focus is good.

In summary, we need to create the best working & learning environment we can, whether it’s in school, or in a company.

5. Math puzzles

The puzzle in the last IntMath Newsletter involved finding the time taken to fill a bath.

Correct answers with reasoning were given by Christopher Buchanan, Christian Mills, Nicos Mavrommatis, S. Nickerson, Thomas A Buckley, Bonnie, Hamid Zandi, Lachezar Borisov, Dineth, Ramesh Babu and Mawanda Ismail.

It was interesting to see the variety of thought processes used to solve the puzzle. Some of you went straight for the algebra, and that’s fine. My approach with this kind of puzzle is to assume a reasonable amount of water for the bath (say 100 liters) and then work the rest of it from there. It makes it easier (for me) to check if my "rate in" and "rate out" values make sense.

Please note: As usual in math, the best answers are those that include reasons!

New puzzle: Peter is an enthusiastic student and runs to school each day, but usually gets tired and walks the rest of the way. His running speed is twice his walking speed. He notices on Monday that he walks twice the time he runs and it took 20 minutes to get to school. On Tuesday, he runs twice the time he walks. How long does it take to get to school on Tuesday?

You can leave your responses, with reasoning, here.

6. Friday math movies

7. Final thought: A perfect day

I enjoy helping people, and I hope you do, too. I think this sums things up nicely:

You have not lived a perfect day, unless you’ve done something for someone who will never be able to repay you. [Ruth Smeltzer]

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: Subitizing, Jacob, Fractals and Donald Duck appeared first on squareCircleZ.

Jacob’s Staff was invented in the 13th century as an aid in finding heights and distances. It was not until the 15th century that the German scientist Johannes Mueller popularized the instrument.

This was the age when the Europeans began exploring beyond the Mediterranean Sea. They needed better navigation tools so they could more accurately determine their latitude and longitude, and not run into too many rocks.

Jacob’s Staff was shaped like a cross-bow. It had a main rod with a scale, and several cross pieces which were used to determine angles, either from the horizontal looking up (the “angle of elevation”), or perhaps between 2 objects.

Jacob’s Staff. Image source]

We need to use half angles and half distances when measuring with this tool.

“The tangent of half the desired angle is given by taking half of the length of the crossbar, and by deducting the main rod length divided by the distance from the eye to the cross bar.”

That’s quite a mouthful, so let’s look at an example.

If we know the height of a tower h, and measure the angle using our Jacob’s Staff to be θ, then a good approximation for the distance to the tower is given by:

Here is a detail from a woodblock print from 1523 by Peter Apian, showing a person measuring the height of a tower using a Jacob’s Staff.

You can read more about Jacob’s Staff and an example involving finding height when you know the distance to an object in the German version of Wikipedia: Jakobsstab. (You can use Google Translate for your own language.)

Below is the full Peter Apian woodblock print. We can see people measuring heights and astronomical distances. Note the spherical representation of the Earth, when this concept was still rejected by many in society.

Image source

Math teachers could design an interesting practical lesson based on Jacob’s Staff, by getting students to make some of them and then use them for measuring. Students would learn some history, as well as some math.

The post Jacob’s Staff appeared first on squareCircleZ.

Donald Duck learns how math arose out of music (the Pythagoreans), art and architecture (the Greeks), nature (the Golden Ratio gets a good plug), games (vectors, angles) and of course, science and engineering.

As the commentator says when taking conic sections:

“No matter how you slice it, you always get mathematics.”

And a bit later:

“These are the doors to the future. And the key to the doors is mathematics.”

In the 1950s and 1960s, science had a higher profile than it does today. I can’t imagine too many animation houses producing such a video today.

So here is Donald in Mathmagic Land (27 min).

The post Friday math movie: Donald in Mathmagic Land appeared first on squareCircleZ.

When we count objects, there are several things going on. First, our brains need to distinguish between "object" and "non-object" (this can include having to differentiate size, color, texture and so on).

Next, a curious thing happens depending on the number of things we are trying to count.

If we count a small number of objects, say 5 or less, we don’t tend to count them individually. We just "see and know" that there are 3 boxes, or 3 cats or 3 circles in front of us. This process of "knowing" the number of objects is called subitizing. We share this ability with many animals who appear to "know" how many objects there are with just a glance.

But if the number of objects climbs beyond 5, we need to use more of our brain since we need to count items individually. We need to vocalize (at least in our head) how many we are up to, and we need to remember which items we have already counted and which ones there are to do. We may also use some strategies like counting in 2s or 3s to make it easier, but which also add to the cognitive load.

All this extra brainwork for counting higher numbers takes more time. But how much?

BBC’s Subitizing Game

The BBC’s article, Can you count faster than a chimp?, is accompanied by a "Count the Dots" Flash game, which instructs users…

"See how fast you can count. … As soon as you have counted the [dots], press the space bar to move on".

I felt there were problems with the way this game was designed. I could press the space bar at any time (pretending that I had already finished counting), which seemed a bit odd. Also, what if my count was incorrect? I may think I’m counting at a certain speed, but incorrect attempts were also included (and the game didn’t even "know" they were wrong).

The BBC was obviously not trying to be scientific, but their game didn’t hit the target for me.

My subitizing game

So I set about writing a counting game where you would actually need to count and it would keep track of your correct answers and times.

Try it out here: Counting Game

Related article

I wrote a related piece earlier called Number Sense, which starts with a video of Maggie the dog illustrating her counting ability.

Observations

Some interesting observations so far (with an admittedly small data set to date):

1. There are slight increases in counting times as you go from 1 up to 5 objects.
2. There is a noticeable increase in the time taken to count 6 objects and above. This is not the same finding as Dehaene’s in his book The Number Sense – How The Mind Creates Mathematics (his research indicated the time increase was almost linear for 3 or more objects. You can see his chart in the Number Sense article mentioned above.)
3. Counting 9 objects takes a bit less than 8 (perhaps we see "a lot" of objects and just assume it must be 9).
4. Your keyboard skills, including your knowledge of where the number keys are on your keyboard makes a difference. Also, if you have a number pad, it’s a lot easier and quicker (most laptop keyboards have a pseudo-number pad arrangement. Check your user manual).
5. It’s very hard to beat 10 correct in the allotted 10 seconds!

For the geeks (technical bit)

The game looks simple but has quite a bit going on behind the scenes. I don’t develop in Flash any more (since it doesn’t work on most mobile devices.)

The game uses:

1. jQuery for the "game" controls
2. JSXGraph for the dots (I wanted to use HTML5′s canvas, but the game wouldn’t work in IE8 or less)
3. Ajax, PHP and MySQL for sending the data to the server and for retrieving the average times for attempts

Conclusion

Our "number sense" is crucial for our success in math. One of the simplest manifestations of this lies in our ability to count.

I’m interested to see how people from different language groups perform on this task (and how it would be affected if different number symbols were used). See Math tests and rice paddies for some background on the cultural influences of number.

For convenience, here’s the link to the game again: Counting Game

The post Counting game and subitizing appeared first on squareCircleZ.

Mandelbrot’s contribution is particularly interesting. This mathematician wasn’t very interested in equations. Instead, he dealt with pictures.

The post Friday math movie: Fractals – Hunting the Hidden Dimension appeared first on squareCircleZ.

In this Newsletter:

1. Interactive Bezier Curve Graphs
2. Kinect Math
3. IntMath polls
4. HOTmaths revisited
5. Math puzzles
6. Friday math movie
7. Final thought – Questions for learning

Happy New Year everyone!

In case you missed it, 2013 is year of Mathematics of Planet Earth. Events will include public lectures like "The challenge of sustainability and the promise of mathematics" by Princeton University professor Simon Levin in Melbourne, Australia on 29th Jan 2013.

In math class we should be making better use of math students’ time and energy to plan for a more sustainable, and happier future.

1. Interactive Bezier Curve Graphs

Here’s an interactive graph where you can explore straight lines, parabolas, cubics and Bezier curves. You’ll find it towards the bottom of this Math of Vector Art page. Interactive Bezier Curve Graphs

2. Kinect Math

Many students prefer learning via hands-on activities. Kinect Math is an interesting approach using Microsoft’s Kinect. Students interact with various graphs and other math concepts using their hands. See some videos about it here: KinectMath – A Kinesthetic Learning Experience

3. IntMath Poll: Math homework per week

The latest poll asked readers about the amount of math homework they do per week during semester. Here are the results:

More than 6 hours
36%

Between 2 and 4 hours
16%

Between 1 and 2 hours
14%

Between 4 and 6 hours
12%

Less than 1 hour
12%

None
9%

Total votes: 2,900.

The surprise here is that over 1/5 of respondents do one hour or less For most math courses, this would be inadequate.

New poll: The latest poll asks readers whether it’s important to know the history of each math topic. You can respond on any page in IntMath.

4. HOTmaths revisited

HOTmaths is an interesting online math resource. I take another look after a few years. Do you use it? How do you find it? HOTmaths revisited

5. Math puzzles

The puzzle in the last IntMath Newsletter involved finding the number of possible positive integer solutions for a given equation.

Correct answers with reasoning were given by Thomas A Buckley and Tom Barrett.

Tomas Garza fired up Mathematica, produced a 100 x 200 x 200 item table, and then used brute-force counting (which took over a minute) to find the correct rows, and from there deduced the correct answer.

A systematic approach is to consider the smallest value of y + z which occurs when y = z = 1, and if so, 2x is then n − 2. Then we can let x = 1, 2, 3 … (n − 2)/2. Then y + z = n − 2, n − 4, n − 6, … to which there are n − 3, n − 5, n − 7, … solutions in positive integers. We then use the sum of an arithmetic progression formula to obtain the solution (n − 2)²/2.

New puzzle: A certain bathtub with the plug in and the tap on will fill at a uniform rate in 12 minutes. The bath will empty with the plug out at a uniform rate in 15 minutes. How long will it take to fill the bathtub if the plug is out, the tap is on, and it was originally empty?

Leave your responses here.

6. Friday math movie: Teach statistics before calculus!

This short talk argues that people should know about probability and statistics before calculus. Do you agree with him? Friday math movie: Teach statistics before calculus!

7. Final thought: Questions for learning

Many students do not maximize their learning outcomes. If we turn up for class with no idea what’s going to happen, we just make it harder for ourselves.

Here are some things to think about before your next math class.

How have I prepared for class today? (Revised last week’s key points? Understood what the formulas mean? Done the homework? Looked over what we are going to do today?)

What’s the best way for me to prepare for a class like this one? (Sharing ideas with others can be a good way to pick up tips.)

What questions do I have? (If you have a good clear list of things you want to find out or understand better, it helps focus your learning. Sharing questions and answers with classmates can help to understand what is going on.)

Why did I get these exam questions wrong? (Many students just worry about the exam grade, but don’t put effort into understanding what went wrong. Talking through the common errors is a good way not to repeat them.)

What do I need to do to avoid missing questions like these on the next exam? (Thinking about this will also help you focus in class. If your tests are mutliple choice, try hiding the choices before doing the problem. Too many students try to work backwards from the most likely correct answer!)

(Based on Teaching Metacognition to Improve Student Learning)

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: Bezier curves interactive, KinectMath, puzzles appeared first on squareCircleZ.