Author: IMC

How can parents best help their children with their schooling without actually doing it for them? This article is part of our series on Parents’ Role in Education, focusing on how best to support learning from early childhood to Year 12.

Before beginning official schooling, parents can give their young children a boost in learning mathematics by noticing, exploring and talking about maths during everyday activities at home or out and about.

New research shows that parents play a key role in helping their children learn mathematics concepts involving time, shape, measurement and number. This mathematical knowledge developed before school is predictive of literacy and numeracy achievements in later grades.

One successful approach for strengthening the role of parents in mathematics learning is Let’s Count, implemented by The Smith Family. This builds on parents’ strengths and capabilities as the first mathematics educators of their children.

The Let’s Count longitudinal evaluation findings show that when early years educators encourage parents and families to confidently notice, explore and talk about mathematics in everyday activities, their young children’s learning flourishes.

Indeed, children whose families had taken part in Let’s Count showed greater mathematical skills than those in a comparison group whose families had not participated. For example, they were more successful with correctly making a group of seven (89% versus 63%); continuing patterns (56% versus 34%); and counting collections of 20 objects (58% versus 37%).

These findings, among many others, are a strong endorsement of the power of families helping their children to learn about mathematics in everyday contexts.

What parents can do to promote maths every day

Discussing and exploring mathematics with children requires no special resources. Instead, what is needed is awareness and confidence for parents about how to engage.

However, our research shows that one of the biggest barriers to this is parents’ lack of confidence in leading maths education at home.

Through examining international research, we identified the type of activities that are important for early maths learning which are easy for parents to use. These include:

1. Comparing objects and describing which is longer, shorter, heavier, or holds less.
2. Playing with and describing 2D shapes and 3D objects.
3. Describing where things are positioned, for example, north, outside, behind, opposite.
4. Describing, copying, and extending patterns found in everyday situations.
5. Using time-words to describe points in time, events and routines (including days, months, seasons and celebrations).
6. Comparing and talking about the duration of everyday events and the sequence in which they occur.
7. Saying number names forward in sequence to ten (and eventually to 20 and beyond).
8. Using numbers to describe and compare collections.
9. Using perceptual and conceptual subitising (recognising quantities based on visual patterns), counting and matching to compare the number of items in one collection with another.
10. Showing different ways to make a total (at first with models and small numbers).
11. Matching number names, symbols and quantities up to ten.

Games to play using everyday situations

Neuroscience research has provided crucial evidence about the importance of early nurturing and support for learning, brain development, and the development of positive dispositions for learning.

Early brain development or “learning” is all about the quality of children’s sensory and motor experiences within positive and nurturing relationships and environments. This explains why programs such as Let’s Count are successful.

Sometimes it can be difficult to come up with activities and games to play that boost children’s mathematics learning, but there are plenty. For example, talk with your children as you prepare meals together. Talk about measuring and comparing ingredients and amounts.

You can play children’s card games and games involving dice, such as Snakes and Ladders, or maps, shapes and money. You can also read stories and notice the mathematics – the sequence of events, and the descriptions of characters and settings.

Although these activities may seem simple and informal, they build on what children notice and question, give families the chance to talk about mathematical ideas and language, and show children that maths is used throughout the day.

Parents are encouraged to provide learning opportunities that are engaging and relevant to their children. www.shutterstock.com

Make it relevant to them

Most importantly, encouraging maths and numeracy in young children relies on making it appealing and relevant to them.

For example, when you take your child for a walk down the street, in the park or on the beach, bring their attention to the objects around them – houses, cars, trees, signs.

Talk about the shapes and sizes of the objects, talk about and look for similarities and differences (for example: let’s find a taller tree or a heavier rock), count the number of cars parked in the street or time how long it takes to reach the next corner.

Discuss the temperature or the speed of your walking pace.

Collect leaves or shells, and make repeating patterns on the sand or grass, or play Mathematical I Spy (I spy with my little eye, something that’s taller than mum).

It is never too soon to begin these activities. Babies who are only weeks old notice differences in shapes and the number of objects in their line of sight.

So, from the earliest of ages, talk with your child about the world around them, being descriptive and using mathematical words. As they grow, build on what they notice about shapes, numbers, and measures. This is how you teach them mathematics.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Sivanes Phillipson, Ann Gervasoni

An artificial intelligence that can turn mathematical concepts written in English into a formal proving language for computers could make problems easier for other AIs to solve.

Maths can be difficult for a computer to understand

An artificial intelligence can translate maths problems written in plain English to formal code, making them easier for computers to solve in a crucial step towards building a machine capable of discovering new maths.

Computers have been used to verify mathematical proofs for some time, but they can only do it if the problems have been prepared in a specifically designed proving language, rather than for the mix of mathematical notation and written text used by mathematicians. This process, known as formalisation, can take years of work for just a single proof, so only a small fraction of mathematical knowledge has been formalised and then proved by a machine.

Yuhuai Wu at Google and his colleagues used a neural network called Codex created by AI research company OpenAI. It has been trained on large amounts of text and programming data from the web and can be used by programmers to generate workable code.

Proving languages share similarities with programming languages, so the team decided to see if Codex could formalise a bank of 12,500 secondary school maths competition problems. It was able to translate a quarter of all problems into a format that was compatible with a formal proof solver program called Isabelle. Many of the unsuccessful translations were the result of the system not understanding certain mathematical concepts, says Wu. “If you show the model with an example that explains that concept, the model can then quickly pick it up.”

To test the effectiveness of this auto-formalisation process, the team then applied Codex to a set of problems that had already been formalised by humans. Codex generated its own formal versions of these problems, and the team used another AI called MiniF2F to solve both versions.

The auto-formalised problems improved MiniF2F’s success rate from 29 per cent to 35 per cent, suggesting that Codex was better at formalising these problems than the humans were.

It is a modest improvement, but Wu says the team’s work is only a proof of concept. “If the goal is to train a machine that is capable of doing the same level of mathematics as the best humans, then auto-formalisation seems to be a very crucial path towards it,” says Wu.

Improving the success rate further would allow AIs to compete with human mathematicians, says team member Albert Jiang at the University of Cambridge. “If we get to 100 per cent, we will definitely be creating an artificial intelligence agent that’s able to win an International Maths Olympiad gold medal,” he says, referring to the top prize in a leading maths competition.

While the immediate goal is to improve the auto-formalisation models, and automated proving machines, there could be larger implications. Eventually, says Wu, the models could uncover areas of mathematics currently unknown to humans.

The capacity for reasoning in such a machine could also make it well-suited for verification tasks in a wide range of fields. “You can verify whether a piece of software is doing exactly what you asked it to do, or you can verify hardware chips, so it has applications in financial trading algorithms and hardware design,” says Jiang.

It is an exciting development for using machines to find new mathematics, says Yang-Hui He at the London Institute for Mathematical Sciences, but the real challenge will be in using the model on mathematical research, much of which is written in LaTeX, a typesetting system. “We only use LaTeX because it types nicely, but it’s a natural language in some sense, it has its own rules,” says He.

Users can define their own functions and symbols in LaTeX that might only be used in a single mathematical paper, which could be tricky for a neural network to tackle that has only been trained on the plain text, says He.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Alex Wilkins*

Credit: Can you find the center of a shape? You’ll be able to–even for the oddest oblong creation–with this simple science activity. No strings attached (okay, maybe one)!  George Retseck

A centering science activity

Introduction
With a little time, you can probably find the center of simple shapes such as circles and squares pretty easily. But how do you find the “middle” of an irregular shape such as a drawing of a dog or a cat? This project will show you how to do it using nothing but string and paper clips!

Background
How do you define the exact center of an object? One way to do this is to find the object’s center of mass. The center of mass is the point about which an object will balance if you try to rest it on your fingertip. Or if you hang an object, for example a picture frame from a nail, the center of mass will hang directly below the nail.

For symmetrical objects, finding the center of mass is relatively easy. For example, for a rectangular picture frame, you know the center of mass is in the middle of the rectangle and you can find that with a ruler. When you hang the picture frame, you will make sure it is centered on the nail—otherwise it will tip to one side and will be off-center. The same applies to other symmetrical objects such as a spherical basketball; you know the center of mass is in the middle of the sphere.

What about irregularly shaped objects such as a dog or cat or person? Now finding the center of mass is not so easy! This activity will show you how to find the center of mass for any two-dimensional shape you cut out of paper using a trick that has to do with the hanging picture frame mentioned above. If you hang a shape from a single point, you know the center of mass will always rest directly below that point. So, if you hang a shape from two different points (one at a time) and draw a line straight down from each point, the center of mass is where those lines intersect. This technique can be used for any irregular two-dimensional shape. Don’t believe it? Try this activity to find out!

Materials

• Paper (Heavier paper, such as construction paper, card stock or thin cardboard from the side of a cereal box will work best.)
• Scissors (Have an adult help with cutting if necessary—especially on thicker materials.)
• String
• Pencil
• Ruler
• Two paper clips or a pushpin and another small, relatively heavy object you can tie to the string (such as a metal washer)

Preparation

• Cut a piece of string about one foot long and tie a paper clip to each end. (Alternatively, you can use any other small object such as a metal washer on one end—this will serve as a weight—and any other small, pointy object like a needle or pushpin on the other end—this will be used to puncture the paper.)

Procedure

• Start with an easy shape: Cut out a rectangular piece of paper or cardboard. Can you guess where the center of mass of the rectangle is? If so, use a ruler to measure where you think it will be and mark this spot with your pencil.
• Punch several small holes around the edge of the paper. Make them as close to the edge as possible without ripping the paper. (This is important for the accuracy of this technique). The exact location of the holes does not matter but this technique will work best if you space them all the way around the edge (not just put two holes right next to each other).
• Now poke one end of one paper clip (or pushpin) through one of the holes to act like a hanging hook. Make sure the paper can swing easily from the hook and does not get stuck (Rotate it back and forth a few times to loosen the hole if necessary).
• Hold on to your “hook” and hold the paper up against the wall. Let the paper swing freely and make sure the string can hang straight down and does not get stuck.
• Use a pencil and ruler to draw a straight line on the paper along the string. Does this line go through the center of mass you predicted earlier?
• Now, hang your paper from a different hole and repeat the process. Where does this line intersect the first line?
• Repeat the process several more times with different holes. Do all the lines intersect at the same point?
• Now cut out an irregular shape. You can cut out a “blob” or draw something like a dog or cat and then cut out the outline. Make sure the shape you cut out remains stiff and flat. (That is, do not cut very thin sections that might be floppy.) Can you use a ruler to predict where the center of mass of your irregular shape will be? This is much harder!
• Punch holes around the edge of your irregular shape and repeat the activity. One at a time, hang the shape and the string from one of the holes and draw a line along the string. Where do the lines intersect? Does this match up with what you predicted?
• Extra: If you use a stiff enough material to cut out your shape (such as cardboard), can you try balancing it on your fingertip at the center of mass? What happens if you try to balance it about another point?

Observations and results
You should have found that the center of mass of the rectangle is right in the middle of the piece—halfway along the width and halfway along the height. You can easily locate this spot with a ruler. Then, when you hang the rectangle from a hole on its edge, the string should always pass through this point, regardless of which hole you use. Whereas it is much harder to predict the center of mass for an irregular shape, the same principle holds true.

Regardless of what point you hang the irregular shape from, the string will always pass through the center of mass. So, if you hang it from two or more points (one at a time), you can find the intersection of these lines—and that is the center of mass.

Note that due to small variables in the activity (such as friction on the hook that prevents the paper from rotating perfectly or the holes not being close enough to the edge of the paper), if you draw multiple lines, they might not all intersect in exactly the same place but they should still be fairly close to one another.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Ben Finio & Science Buddies

Credit: Heads or tails: Is it a 50-50 chance? Crunch some numbers and flip some coins to find out! George Retseck

A probabilistic science project from Science Buddies

Introduction
Have you ever heard anyone say the chance of something happening is “50–50”? What does that actually mean? This phrase has something to do with probability. Probability tells you how likely it is that an event will occur. This means that for certain events you can actually calculate how likely it is that they will happen. In this activity, you will do these calculations and then test them to see whether they hold true for reality!

Background
Probability allows us to quantify the likelihood an event will occur. You might be familiar with words we use to talk about probability, such as “certain,” “likely,” “unlikely,” “impossible,” and so on. You probably also know that the probability of an event happening spans from impossible, which means that this event will not happen under any circumstance, to certainty, which means that an event will happen without a doubt. In mathematics, these extreme probabilities are expressed as 0 (impossible) and 1 (certain). This means a probability number is always a number from 0 to 1. Probability can also be written as a percentage, which is a number from 0 to 100 percent. The higher the probability number or percentage of an event, the more likely is it that the event will occur.

The probability of a certain event occurring depends on how many possible outcomes the event has. If an event has only one possible outcome, the probability for this outcome is always 1 (or 100 percent). If there is more than one possible outcome, however, this changes. A simple example is the coin toss. If you toss a coin, there are two possible outcomes (heads or tails). As long as the coin was not manipulated, the theoretical probabilities of both outcomes are the same–they are equally probable. The sum of all possible outcomes is always 1 (or 100 percent) because it is certain that one of the possible outcomes will happen. This means that for the coin toss, the theoretical probability of either heads or tails is 0.5 (or 50 percent).

It gets more complicated with a six-sided die. In this case if you roll the die, there are 6 possible outcomes (1, 2, 3, 4, 5 or 6). Can you figure out what the theoretical probability for each number is? It is 1/6 or 0.17 (or 17 percent). In this activity, you will put your probability calculations to the test. The interesting part about probabilities is that knowing the theoretical likelihood of a certain outcome doesn’t necessarily tell you anything about the experimental probabilities when you actually try it out (except when the probability is 0 or 1). For example, outcomes with very low theoretical probabilities do actually occur in reality, although they are very unlikely. So how do your theoretical probabilities match your experimental results? You will find out by tossing a coin and rolling a die in this activity.

Materials

• Coin
• Six-sided die
• Paper
• Pen or pencil

Preparation

• Prepare a tally sheet to count how many times the coin has landed on heads or tails.
• Prepare a second tally sheet to count how often you have rolled each number with the die.

Procedure

• Calculate the theoretical probability for a coin to land on heads or tails, respectively. Write the probabilities in fraction form. What is the theoretical probability for each side? 
• Now get ready to toss your coin. Out of the 10 tosses, how often do you expect to get heads or tails?
• Toss the coin 10 times. After each toss, record if you got heads or tails in your tally sheet.
• Count how often you got heads and how often you got tails. Write your results in fraction form. For example, 3 tails out of 10 tosses would be 3/10 or 0.3. (The denominator will always be the number of times you toss the coin, and the numerator will be the outcome you are measuring, such as the number of times the coin lands on tails.) You could also express the same results looking at heads landings for the same 10 tosses. So that would be 7 heads out of 10 tosses: 7/10 or 0.7. Do your results match your expectations?
• Do another 10 coin tosses. Do you expect the same results? Why or why not?
• Compare your results from the second round with the ones from the first round. Are they the same? Why or why not?
• Continue tossing the coin. This time toss it 30 times in a row. Record your results for each toss in your tally sheet. What results do you expect this time?
• Look at your results from the 30 coin tosses and convert them into fraction form. How are they different from your previous results for the 10 coin tosses?
• Count how many heads and tails you got for your total coin tosses so far, which should be 50. Again, write your results in fraction form (with the number of tosses as the denominator (50) and the result you are tallying as the numerator). Does your experimental probability match your theoretical probability from the first step? (An easy way to convert this fraction into a percentage is to multiply the denominator and the numerator each by 2, so 50 x 2 = 100. And after you multiply your numerator by 2, you will have a number that is out of 100—and a percentage.)
• Calculate the theoretical probability for rolling each number on a six-sided die. Write the probabilities in fraction form. What is the theoretical probability for each number?
• Take the dice and roll it 10 times. After each roll, record which number you got in your tally sheet. Out of the 10 rolls, how often do you expect to get each number?
• After 10 rolls, compare your results (written in fraction form) with your predictions. How close are they?
• Do another 10 rolls with the dice, recording the result of each roll. Do your results change?
• Now roll the dice 30 times in a row (recording the result after each roll). How often did you roll each number this time?
• Count how often you rolled each number in all combined 50 rolls. Write your results in fraction form. Does your experimental probability match your theoretical probability? (Use the same formula you used for the coin toss, multiplying the denominator and the numerator each by 2 to get the percentage.)
• Compare your calculated probability numbers with your actual data for both activities (coin and dice). What do your combined results tell you about probability?
• Extra: Increase the number of coin tosses and dice rolls even further. How do your results compare with the calculated probabilities with increasing number of events (tosses or rolls)? 
• Extra: Look up how probabilities can be represented by probability trees. Can you draw a probability tree for the coin toss and dice roll?
• Extra: If you are interested in more advanced probability calculations, find out how you can calculate the probability of a recurring event, for example: How likely it is that you would get two heads in a row when tossing a coin? 

Observations and Results
Calculating the probabilities for tossing a coin is fairly straightforward. A coin toss has only two possible outcomes: heads or tails. Both outcomes are equally likely. This means that the theoretical probability to get either heads or tails is 0.5 (or 50 percent). The probabilities of all possible outcomes should add up to 1 (or 100 percent), which it does. When you tossed the coin 10 times, however, you most likely did not get five heads and five tails. In reality, your results might have been 4 heads and 6 tails (or another non-5-and-5 result). These numbers would be your experimental probabilities. In this example, they are 4 out of 10 (0.4) for heads and 6 out of 10 (0.6) for tails. When you repeated the 10 coin tosses, you probably ended up with a different result in the second round. The same was probably true for the 30 coin tosses. Even when you added up all 50 coin tosses, you most likely did not end up in a perfectly even probability for heads and tails. Your experimental probabilities thus probably didn’t match your calculated (theoretical) probabilities.

You likely observed a similar phenomenon when rolling the dice. Although the theoretical probability for each number is 1 out of 6 (1/6 or 0.17), in reality your experimental probabilities probably looked different. Instead of rolling each number 17 percent out of your total rolls, you might have rolled them more or less often.

If you continued tossing the coin or rolling the dice, you probably have observed that the more trials (coin tosses or dice rolls) you did, the closer the experimental probability was to the theoretical probability. Overall these results mean that even if you know the theoretical probabilities for each possible outcome, you can never know what the actual experimental probabilities will be if there is more than one outcome for an event. After all, a theoretical probability is just predicting how the chances are that an event or a specific outcome occurs—it won’t tell you what will actually happen!

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Science Buddies & Svenja Lohner

There are some great uses. Shutterstock

Despite the fact that mathematics is often described as the underpinning science, it is often not given enough credit when scientific discoveries are presented. But the contribution of mathematics and statistics is essential and has transformed entire areas of research – many discoveries would not have been possible without it. In fact, as a mathematician, I have contributed to scientific discoveries and provided solutions to problems that biology was yet to solve.

Seven years ago, I attended a lecture on some biological research that was taking place at Heriot-Watt University. My colleagues had an unsolved problem which related to the movement of bag-like structures called vesicles which move hormones and neurotransmitters such as insulin or serotonin around cells and the body.

Their problem lay in that vesicles were known to follow specific tracks along the cell skeleton which lead to special molecules which then caused the vesicle to release its contents into the cell. However, when the biologists themselves tried to find these tracks, they were not in the expected places.

A bag that carries hormones to their location. OpenStax, CC BY

It is important to understand how vesicles behave, or in fact misbehave, as they have been linked to conditions such diabetes and neurological disorders. The biologists were struggling to find a way to understand the vesicles – but I had a solution in my mathematical toolkit.

Maths can beat biology

After two years of collaboration I told my colleagues: “my model and computer experiments are better than your microscope!”

What I meant by this rather confident statement was that by using mathematics to model how molecules move in a cell we could predict and run multiple experiments on a computer at a smaller scale and faster rate than a microscope. It could allow us to uncover things that the biologist’s resources could not, and might even point us in the direction of target molecules for future treatments of diabetes and neurological disorders.

The mathematical model allowed us to recognise that the movement of vesicles requires energy – and the maths models it through an energy landscape. It imagined a vesicle to be like a cyclist riding a bicycle – the landscape may have easy level sections but also hills that require more energy input to get over them, and so we wanted to test whether they actually avoided these hills.

After seven years of working in partnership with the biologists, my colleagues and I proved our hypothesis was correct. Vesicles do follow lower energy “valleys” in the landscape, avoiding molecules which create the high energy hills in the energy landscape – taking the easiest path. The overall result is just the same as the biologists had found – the vesicles end up in the same end location and they reuse similar routes over and over again. But the difference lies in the way in which they do it, and it was not by following the cell skeleton as biologists had first believed – they take an easier route. It really shows the power of maths and how it can change the way we see things.

Mathematical models allow you to capture many gigabytes of raw data in a compact form in a way that is impossible for a biologist with a microscope. You can make modifications to the model easily and show how vesicle behaviour may change during disease, when they are disrupted or mutated. It could then reveal which molecules to target in future treatment studies – and lay the groundwork for larger and more thorough modelling of complex biological processes.

A modelled energy landscape. Shutterstock

The integration of cutting-edge microscopy with cell biology and mathematical modelling could be applied to many other problems in bio-medicine and will accelerate discovery in the years to come. The movement of molecules and other cell components is just one example of where the power of mathematics is unrivalled, but it is by no means its limit.

Useful is an understatement

Maths is often criticised by the public for lacking in “real-world” applications, but it is being applied to many real-world problems all the time. Groundwater contamination, financial and economic forecasting, plume heights in volcanic eruptions, the modelling of biological processes and drug delivery are just a few places where maths is making a huge difference.

I’m proud to say that I co-authored a paper with my biology colleagues, and I hope to see more mathematicians coming to the fore for science research in the future. Mathematics plays a central role in so many of the world’s scientific breakthroughs and deserves a headline role in more academic publications. Power to the mathematician – they’re behind more discoveries than you think.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Gabriel Lord

A small dot on an old piece of birch bark marks one of the biggest events in the history of mathematics. The bark is actually part of an ancient Indian mathematical document known as the Bakhshali manuscript. And the dot is the first known recorded use of the number zero. What’s more, researchers from the University of Oxford recently discovered the document is 500 years older than was previously estimated, dating to the third or fourth century – a breakthrough discovery.

Today, it’s difficult to imagine how you could have mathematics without zero. In a positional number system, such as the decimal system we use now, the location of a digit is really important. Indeed, the real difference between 100 and 1,000,000 is where the digit 1 is located, with the symbol 0 serving as a punctuation mark.

Yet for thousands of years we did without it. The Sumerians of 5,000BC employed a positional system but without a 0. In some rudimentary form, a symbol or a space was used to distinguish between, for example, 204 and 20000004. But that symbol was never used at the end of a number, so the difference between 5 and 500 had to be determined by context.

What’s more, 0 at the end of a number makes multiplying and dividing by 10 easy, as it does with adding numbers like 9 and 1 together. The invention of zero immensely simplified computations, freeing mathematicians to develop vital mathematical disciplines such as algebra and calculus, and eventually the basis for computers.

Zero’s late arrival was partly a reflection of the negative views some cultures held for the concept of nothing. Western philosophy is plagued with grave misconceptions about nothingness and the mystical powers of language. The fifth century BC Greek thinker Parmenides proclaimed that nothing cannot exist, since to speak of something is to speak of something that exists. This Parmenidean approach kept prominent historical figures busy for a long while.

After the advent of Christianity, religious leaders in Europe argued that since God is in everything that exists, anything that represents nothing must be satanic. In an attempt to save humanity from the devil, they promptly banished zero from existence, though merchants continued secretly to use it.

By contrast, in Buddhism the concept of nothingness is not only devoid of any demonic possessions but is actually a central idea worthy of much study en route to nirvana. With such a mindset, having a mathematical representation for nothing was, well, nothing to fret over. In fact, the English word “zero” is originally derived from the Hindi “sunyata”, which means nothingness and is a central concept in Buddhism.

The Bakhshali manuscript. Bodleian Libraries

So after zero finally emerged in ancient India, it took almost 1,000 years to set root in Europe, much longer than in China or the Middle East. In 1200 AD, the Italian mathematician Fibonacci, who brought the decimal system to Europe, wrote that:

The method of the Indians surpasses any known method to compute. It’s a marvellous method. They do their computations using nine figures and the symbol zero.

This superior method of computation, clearly reminiscent of our modern one, freed mathematicians from tediously simple calculations, and enabled them to tackle more complicated problems and study the general properties of numbers. For example, it led to the work of the seventh century Indian mathematician and astronomer Brahmagupta, considered to be the beginning of modern algebra.

Algorithms and calculus

The Indian method is so powerful because it means you can draw up simple rules for doing calculations. Just imagine trying to explain long addition without a symbol for zero. There would be too many exceptions to any rule. The ninth century Persian mathematician Al-Khwarizmi was the first to meticulously note and exploit these arithmetic instructions, which would eventually make the abacus obsolete.

Such mechanical sets of instructions illustrated that portions of mathematics could be automated. And this would eventually lead to the development of modern computers. In fact, the word “algorithm” to describe a set of simple instructions is derived from the name “Al-Khwarizmi”.

The invention of zero also created a new, more accurate way to describe fractions. Adding zeros at the end of a number increases its magnitude, with the help of a decimal point, adding zeros at the beginning decreases its magnitude. Placing infinitely many digits to the right of the decimal point corresponds to infinite precision. That kind of precision was exactly what 17th century thinkers Isaac Newton and Gottfried Leibniz needed to develop calculus, the study of continuous change.

And so algebra, algorithms, and calculus, three pillars of modern mathematics, are all the result of a notation for nothing. Mathematics is a science of invisible entities that we can only understand by writing them down. India, by adding zero to the positional number system, unleashed the true power of numbers, advancing mathematics from infancy to adolescence, and from rudimentary toward its current sophistication.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Ittay Weiss

Credit: Monty Rakusen Getty Images

Scientists have created a “map” of odor molecules, which could ultimately be used to predict new scent combinations 

The human nose finds it simple to distinguish the aroma of fresh coffee from the stink of rotten eggs, but the underlying biochemistry is complicated. Researchers have now created an olfactory “map”—a geometric model of how molecules combine to produce various scents. This map could inspire a way to predict how people might perceive certain odor combinations and help to drive the development of new fragrances, scientists say.

Researchers have been trying for years to tame the elaborate landscape of odor molecules. Neuroscientists want to better understand how we process scents; perfume and food manufacturers want better ways to synthesize familiar aromas for their products. The new approach may appeal to both camps.

One earlier strategy for mapping the olfactory system involves grouping odor molecules that have similar molecular structures and using those similarities to predict the scents of novel combinations. But that avenue often leads to a dead end. “It’s not necessary that chemicals with the same chemical structures will be perceived similarly,” says Tatyana Sharpee, a neurobiologist at the Salk Institute for Biological Studies in La Jolla, Calif., and lead author of the study, which appeared in August in Science Advances.

Sharpee and her colleagues analyzed odor molecules found in four familiar and unmistakable scents: strawberries, tomatoes, blueberries and mouse urine. The researchers calculated how often and in what concentrations certain molecules turned up together in these scents. They then created a mathematical model in which molecules that occurred together frequently were represented as closer in space and molecules that rarely did so were farther apart. The result was a “saddle”-shaped surface—a hallmark of a field called hyperbolic geometry, which obeys different rules from the geometry most people learn in school.

The researchers envision an algorithm, trained on this hyperbolic geometry model, that can predict the scents of new odor combinations—or even help to synthesize them. One of Sharpee’s collaborators, behavioral neuroscientist Brian Smith of Arizona State University, wants to use this method to create olfactory environments in places devoid of natural scents.

Such a tool would be useful to scientists and odor manufacturers alike, says olfactory neuroscientist Joel Mainland of the Monell Chemical Senses Center in Philadelphia, who was not involved in the study. The ultimate goal is to know enough about how odors work to replicate natural smells without the natural sources, Mainland says: “We want to identify a strawberry flavor without worrying about replicating the ingredients that are in a strawberry.”

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Credit of the article given to Stephen Ornes

It should come as no surprise that the first recorded use of the number zero, recently discovered to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history going back over 3,000 years and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.

As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of trigonometry, algebra, arithmetic and negative numbers among other areas. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.

The number system

As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as the Vedas. In these texts, numbers were commonly expressed as combinations of powers of ten. For example, 365 might be expressed as three hundreds (3×10²), six tens (6×10¹) and five units (5×10⁰), though each power of ten was represented with a name rather than a set of symbols. It is reasonable to believe that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.

Brahmi numerals. Wikimedia

From the third century BC, we also have written evidence of the Brahmi numerals, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics.

The concept of zero

Zero itself has a much longer history. The recently dated first recorded zeros, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the Babylonian and Mayan cultures in the early centuries AD and arguably in Sumerian mathematics as early as 3000-2000 BC.

But only in India did the placeholder symbol for nothing progress to become a number in its own right. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the democratisation of mathematics.

These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though Fibonnacci’s book liber abaci.

Solutions of quadratic equations

In the seventh century, the first written evidence of the rules for working with zero were formalised in the Brahmasputha Siddhanta. In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.

Rules for negative numbers

Brahmagupta also demonstrated rules for working with negative numbers. He referred to positive numbers as fortunes and negative numbers as debts. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.

This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative.

For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that negative numbers were absurd. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.

For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total.

Basis for calculus

This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his development of calculus in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.

Leibniz: Beaten to it by 500 years.

But Indian mathematician Bhāskara had already discovered many of Leibniz’s ideas over 500 years earlier. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that would not be rediscovered in Europe for centuries.

The Kerala school of astronomy and mathematics, founded by Madhava of Sangamagrama in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would later be repeated in Europe including Taylor series expansions, infinitessimals and differentiation.

The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation suffers from the Eurocentric bias, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by providing key players at the forefront of every branch of mathematics.

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Credit of the article given to Christian Yates

The probabilistic revolution first kicked off in the 1600s, when gamblers realized that estimating the likelihood of an event could give them an edge in games of chance.

Today, statistics has become the dominant way to communicate scientific findings. But courts can be hesitant to incorporate statistical evidence into decisions. Indeed, they have historically been antagonistic toward probabilities and are loath to be swindled by slippery statistics.

However, as an educator of statistics who has consulted in a variety of contexts and has served as expert witness to the U.S. District Court in Montana, I find that both my experience and my review of the evidence suggest that courts increasingly feature statistical thinking – whether or not it is identified as such.

Society needs to prioritize educating juries in the language of statistics. Otherwise, juries will be forever at the mercy of convincing, yet potentially invalid, testimony. Courtroom decisions should be based on facts and probabilities, not manipulation by a skilled prosecutor or defense attorney.

Thinking statistically

Probabilities changed the way human beings thought about outcomes. They are a useful tool for expressing our uncertainty about events in the world.

Will it rain today? It will or it will not, that much is certain. But probability allows us to express our ignorance about whether it will rain and quantify the degree to which we are uncertain. Stating “it will probably rain today” constituted a very innovative and different way of thinking.

Probabilities play a role in our daily lives, in decisions from whether to take an umbrella to work to whether to purchase flood insurance. We can consider “statistical thinking” to be any situation where probabilities are involved.

To some extent, humans are intuitive statisticians. For instance, research suggests we can revise a belief in the light of new evidence as prescribed by a statistical theorem, if the probabilities are given in a relatively intuitive rather than abstract fashion.

Statistical reasoning pervades many of the conclusions we draw regarding scientific phenomena. Even physics has had to acknowledge the reality of probabilities. So, if the courts use scientific findings as evidence, probabilities should naturally make their way into courtroom decisions.

Evaluating the evidence

If juries do not understand the nature of statistical conclusions, then they will be tempted to believe that scientific evidence is conclusive and deterministic, rather than probabilistic. For example, probabilities show us that cigarette smoking does not necessarily lead to cancer. Rather, extensive nicotine addiction likely leads to cancer.

Heads or tails? armydre2008/flickr, CC BY

Evidence can only fit a theory probabilistically. If we flip a coin 10 times and get 10 heads in a row, that suggests the coin may not be fair, but does not “prove” that it is biased.

Consider the analysis of DNA found at the crime scene. Is the DNA that of the accused? Maybe. Not definitively. A statistician might say, “The probability of this degree of DNA match occurring by chance is extremely small. The match may be due to chance, but since this probability is so small, we may conclude that it likely did not occur by chance, and use it as evidence against the accused.”

Of course, human judgment is fickle. Until jurors are trained to make rational decisions based on facts and probabilities, they will continue to be easily swayed by convincing litigators.

In the 1995 trial of OJ Simpson, for example, the bloody gloves found at the crime scene constituted powerful evidence against the accused. The samples obtained were extremely likely to belong to the defendant.

A statistically educated jury would not fall for Johnnie Cochran’s classic defense: “If it does not fit, you must acquit.” They would know in advance that no evidence, whatever the kind, fits a theory perfectly.

Cochran’s statement was, statistically speaking, utter nonsense. Of course no model fits perfectly, but which is the more probable model? That’s the task jurors ultimately face, even if they often perceive it as a “guilt” versus “no guilt” decision.

Whenever courts work with DNA matches, they must incorporate acceptable risk and error. But if such uncertainty can be quantified accurately, then it can serve as an aid in decision-making.

Statistical thinking indeed plays a role in the decision between guilt and innocence in a criminal trial. When a jury renders a “guilty” verdict, there is always the chance that the accused is not guilty, but that the many circumstances of the case simply lined up against him or her to lead the jury to a guilty verdict. In other words, the probability of the observed evidence under the assumption of innocence is so low that the evidence likely occurred under a more probable “narrative” – that of guilt.

But, when we make such a decision, we do so with a risk of error. This could be quite devastating to a defendant falsely put to death when all along he or she was innocent. For example, when researchers applied DNA testing to death row inmates in Illinois, they found that the results exonerated several inmates.

Errors in probability-based decisions can indeed be costly. Without a grasp of how virtually all decisions are based on probabilistic thinking, no jury can be expected to adequately assess any evidence in a rational way.

Base rates

Courts also struggle with whether and how to use base rates, another type of statistical tool.

A base rate is the probability of some characteristic being present in the population. For instance, say an individual takes a diagnostic test for a disease, such as HIV. The probability that she has the disease would be higher if she were sampled from a high-risk group – for example, if she shares needles to support a drug addiction, or engages in promiscuous sex with risky partners.

Courts often ignore base rate information. In Stephens v. State in 1989, the Wyoming Supreme Court heard testimony that “80 to 85 percent of child sexual abuse is committed by a close relative of the child.” They ultimately dismissed this, concluding that it was difficult to understand how statistical information would help reach a decision in an individual case.

In another case, a justice of the Minnesota Supreme Court proclaimed that she was “at a loss to understand” how base rates could help predict whether a particular person posed a danger to the public.

Part of the problem is that this information can appear biased against the accused. For instance, consider again the defendant accused of child sexual abuse. The probability that he is guilty might be evaluated in light of the fact that most perpetrators of abuse are relatives of or closely related to the family. This could be interpreted as biasing the evaluation against the accused. However, the courts have considered base rates in employment discrimination cases, an area where perhaps this information seems more naturally relevant (for example, Hazelwood School District v. United States).

If the courts are willing to use base rate information in discrimination cases, they should be encouraged to consider them in other cases as well, even if they seem less intuitive.

Learning to think statistically

Courts should make it a priority to instruct juries on how to interpret probabilistic evidence, so that they are not at the mercy of a convincing, yet potentially misleading, prosecutor or expert witness.

For example, juries might learn elementary statistics through coin-flipping lessons. This could help them, at minimum, find a way to think about the usual “beyond a reasonable doubt” instruction in a criminal trial.

When the assumption of innocence is rejected in favor of guilt, one does so with a risk of being wrong. How much risk is a jury willing to tolerate? Five percent? One percent? Surely such risk must also depend on the severity of the proposed punishment. Every decision is an exercise in risk and cost benefit analysis.

Until juries learn elements of statistical thinking, they are likely to continue making verdict decisions without the appropriate framework in mind. Probabilities have taken over the world, and this fact needs to be recognized by the courts.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Daniel J. Denis

Credit: Evelyn Lamb

Stacking oranges leads to computer-assisted mathematics. But does it feel like mathematics?

Earlier this month, I attended the Joint Mathematics Meetings in Seattle. One of the reasons I enjoy going to the JMM is that I can get a feel for what is going on in parts of mathematics that I’m not terribly familiar with. This year, I attended two talks in a session called “mathematical information in the digital age,” that got me thinking about what mathematicians do.

First, a confession: I went to the session because I like oranges. The first talk was by Thomas Hales, who is probably best known for his proof of the Kepler conjecture. In short, the conjecture says that the way grocers stack oranges is indeed the most efficient way to do it. The proof was a long case-by-case exhaustion, and Hales was not satisfied with a referee report that said the referee was 99% sure the proof was correct. So he did what any* mathematician would do: he took more than a decade to write and verify a formal computer proof of the result. I attended the talk because I figured there’s a small chance that any talk that mentions the Kepler conjecture might have oranges for the audience.

Hales’ talk was called simply “Formal Proofs.” These are not proofs that are written using stuffy language, with every single step written out, but proofs that can be input into a computer and verified all the way down to the foundations of mathematics, whichever foundations one chooses.

Hales began his talk with some examples of less-than-formal proofs, starting with a passage from William Thurston in which he used the phrase “subdivide and jiggle,” clearly not a rigorous way to describe mathematics. (Incidentally, Thurston also did mathematics with oranges. He would ask students to peel oranges to better understand 2- and 3-dimensional geometry.)

Although I never met Thurston, I am one of his many mathematical descendants. his approach to mathematics, particularly his emphasis on intuition and imagination, has permeated the culture in my extended mathematical family and has had a great deal of influence on how I think about mathematics. That is why it was so refreshing for me to go to a talk where intuition wasn’t a primary focus.

Hales was certainly not insinuating that Thurston was a bad mathematician. Thurston was only the first mathematician he used as an example of less-than-rigorously stated mathematics. A few slides later he mentioned the Bourbaki book on set theory. Yes, even that paragon of formal mathematics sucked dry of every drop of intuition, is not really full of formal proofs.

Hales’ talk was a nice overview of the formal proof programs out there, some mathematical results that have been proved formally (including some that were already known), and a nice introduction to where the field is going. I’m particularly interested in learning more about the QED manifesto and FABSTRACTS, a service that would formalize the abstracts of mathematical papers, a much more tractable goal than formalizing an entire paper.

The most amusing moment of the talk, at least to me, was a question from someone in the audience about the possibility of using a formal proof assistant to verify Mochizuki’s proof of the abc conjecture. Hales replied that with the current technology, you do need to understand the proof as you enter it, so there aren’t many people who can do it. The logical response: why doesn’t Mochizuki do it himself? Let’s just say I’m not holding my breath.

The second talk I attended in the session was Michael Shulman’s called “From the nLab to the HoTT book.” He talked about both the nLab, a wiki for category theory, and the writing of the Homotopy Type Theory “research textbook,” a 600-page tome put together during an IAS semester about homotopy type theory, an alternative to set theory as a foundational system for mathematics. The theme of Shulman’s talk was “one size does not fit all,” either in the way people collaborate (contrasting the wiki and the textbook) or even in the foundations of mathematics (type theory versus set theory).

I don’t know if it was intended, but I thought Shulman’s talk was an interesting counterpoint to Hales,’ most relevantly to me in the way it answered one of the questions Hales posed: why don’t more mathematicians use proof assistants? Beyond the fact that proof assistants are currently too unwieldy for many of us, Shulman’s answer was that we do mathematics for understanding, not just truth. He said what I was thinking during Hales’ talk, which was that to many mathematicians, using a formal proof assistant does not “feel like” mathematics. I am not claiming moral high ground here. It is actually something of a surprise to me that the prospect of being able to find and verify new truths more quickly is not more important to me.

You never know what you’re going to get when you wander into a talk that is well outside your mathematical comfort zone. In my case, I didn’t end up with any oranges, but I got some interesting new-ti-me perspectives about how and why we prove.

*almost no

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Credit of the article given to Evelyn Lamb