Month: April 2001

We are a product of evolution, and are not surprised that our bodies seem to be well-suited to the environment.

Our leg bones are strong enough to allow for Earth’s gravitational pull – not too weak to shatter, not so massively over-engineered as to be wasteful.

But it could also be claimed we are special and the environment was formed and shaped for us.

This, as we know, is the basis of many religious ideas.

In recent years, such ideas have been expanded beyond Earth to look at the entire universe and our place within it.

The so-called Fine-Tuning Argument – that the laws of physics have been specially-tuned, potentially by some Supreme Being, to allow human life to arise – is the focus of Victor J. Stenger’s book.

Stenger presents the mathematics underpinning cosmic evolution, the lifetime of stars, the quantum nature of atoms and so on. His central is that “fine-tuning” claims are fatally flawed.

He points out that some key areas of physics – such as the equality of the charges on the electron and proton – are set by conservation laws determined by symmetries in the universe, and so are not free to play with.

Some flaws in the theory, he argues, run deeper.

A key component of the fine-tuning argument is that there are many parameters governing our universe, and that changing any one of these would likely produce a sterile universe unlike our own.

But think of baking a cake. Arbitrarily doubling only the flour, or sugar or vanilla essence may end in a cooking disaster, but doubling all the ingredients results in a perfectly tasty cake.

The interrelationships between the laws of physics are somewhat more complicated, but the idea is the same.

A hypothetical universe in which gravity was stronger, the masses of the fundamental particles smaller and electomagnetic force weaker may well result in the following: a universe that appears a little different to our own, but is still capable of producing long-lived stars and heavy chemical elements, the basic requirements for complex life.

Stenger backs up such points with his own research, and provides access to a web-based program he wrote called MonkeyGod.

The program allows you to conjure up universes with differing underlying physics. And, as Stenger shows, randomly plucking universe parameters from thin air can still produce universes quite capable of harbouring life.

This book is a good read for those wanting to understand the fine-tuning issues in cosmology, and it’s clear Stenger really understands the science.

But while many of the discussions are robust, I felt that in places some elements of the fine-tuning argument were brushed aside with little real justification.

As a case in point, Stenger falls back on multiverse theory and the anthropic principle, whereby we occupy but one of an almost infinite sea of different universes, each with a different law of physics.

In multiverse theory, most universes would be sterile (though we should not be surprised to find ourselves in a habitable universe).

While such a multiverse – the staple of superstring and brane ideas of the cosmos – is often sold as science fact, it actually lies much closer to the world of science speculation (or, to many, fiction).

We are not out of the fine-tuning waters yet, but Stenger’s book is a good place to start getting to grips with the issues.

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

*Credit for article given to Geraint Lewis*

Relating mathematics to questions that are relevant to many students can help address its image problem, argues Eugenia Cheng.

Mathematics has an image problem: far too many people are put off it and conclude that the subject just isn’t for them. There are many issues, including the curriculum, standardised tests and constraints placed on teachers. But one of the biggest problems is how maths is presented, as cold and dry.

Attempts at “real-life” applications are often detached from our daily lives, such as arithmetic problems involving a ludicrous number of watermelons or using a differential equation to calculate how long a hypothetical cup of coffee will take to cool.

I have a different approach, which is to relate abstract maths to questions of politics and social justice. I have taught fairly maths-phobic art students in this way for the past seven years and have seen their attitudes transformed. They now believe maths is relevant to them and can genuinely help them in their everyday lives.

At a basic level, maths is founded on logic, so when I am teaching the principles of logic, I use examples from current events rather than the old-fashioned, detached type of problem. Instead of studying the logic of a statement like “all dogs have four legs”, I might discuss the (also erroneous) statement “all immigrants are illegal”.

But I do this with specific mathematical structures, too. For example, I teach a type of structure called an ordered set, which is a set of objects subject to an order relation such as “is less than”. We then study functions that map members of one ordered set to members of another, and ask which functions are “order-preserving”. A typical example might be the function that takes an ordinary number and maps it to the number obtained from multiplying by 2. We would then say that if x < y then also 2x < 2y, so the function is order-preserving. By contrast the function that squares numbers isn’t order-preserving because, for example, -2 < -1, but (-2)² > (-1)². If we work through those squaring operations, we get 4 and 1.

However, rather than sticking to this type of dry mathematical example, I introduce ones about issues like privilege and wealth. If we think of one ordered set with people ordered by privilege, we can make a function to another set where the people are now ordered by wealth instead. What does it mean for that to be order-preserving, and do we expect it to be so? Which is to say, if someone is more privileged than someone else, are they automatically more wealthy? We can also ask about hours worked and income: if someone works more hours, do they necessarily earn more? The answer there is clearly no, but then we go on to discuss whether we think this function should be order-preserving or not, and why.

My approach is contentious because, traditionally, maths is supposed to be neutral and apolitical. I have been criticised by people who think my approach will be off-putting to those who don’t care about social justice; however, the dry approach is off-putting to those who do care about social justice. In fact, I believe that all academic disciplines should address our most important issues in whatever way they can. Abstract maths is about making rigorous logical arguments, which is relevant to everything. I don’t demand that students agree with me about politics, but I do ask that they construct rigorous arguments to back up their thoughts and develop the crucial ability to analyse the logic of people they disagree with.

Maths isn’t just about numbers and equations, it is about studying different logical systems in which different arguments are valid. We can apply it to balls rolling down different hills, but we can also apply it to pressing social issues. I think we should do both, for the sake of society and to be more inclusive towards different types of student in maths education.

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

*Credit for article given to Eugenia Cheng*

One of the essential skills students need to master in primary school mathematics are “multiplication facts.”

What are they? What are they so important? And how can you help your child master them?

What are multiplication facts?

Multiplication facts typically describe the answers to multiplication sums up to 10×10. Sums up to 10×10 are called “facts” as it is expected they can be easily and quickly recalled. You may recall learning multiplication facts in school from a list of times tables.

The shift from “times tables” to “multiplication facts” is not just about language. It stems from teachers wanting children to see how multiplication facts can be used to solve a variety of problems beyond the finite times table format.

For example, if you learned your times tables in school (which typically went up to 12×12 and no further), you might be stumped by being asked to solve 15×8 off the top of your head. In contrast, we hope today’s students can use their multiplication facts knowledge to quickly see how 15×8 is equivalent to 10×8 plus 5×8.

The shift in terminology also means we are encouraging students to think about the connections between facts. For example, when presented only in separate tables, it is tricky to see how 4×3 and 3×4 are directly connected.

Math education has changed

In a previous piece, we talked about how mathematics education has changed over the past 30 years.

In today’s mathematics classrooms, teachers still focus on developing students’ mathematical accuracy and fast recall of essential facts, including multiplication facts.

But we also focus on developing essential problem-solving skills. This helps students form connections between concepts, and learn how to reason through a variety of real-world mathematical tasks.

Why are multiplication facts so important?

By the end of primary school, it is expected students will know multiplication facts up to 10×10 and can recall the related division fact (for example, 10×9=90, therefore 90÷10=9).

Learning multiplication facts is also essential for developing “multiplicative thinking.” This is an understanding of the relationships between quantities, and is something we need to know how to do on a daily basis.

When we are deciding whether it is better to purchase a 100g product for $3 or a 200g product for $4.50, we use multiplicative thinking to consider that 100g for $3 is equivalent to 200g for $6—not the best deal!

Multiplicative thinking is needed in nearly all math topics in high school and beyond. It is used in many topics across algebra, geometry, statistics and probability.

This kind of thinking is profoundly important. Research shows students who are more proficient in multiplicative thinking perform significantly better in mathematics overall.

In 2001, an extensive RMIT study found there can be as much as a seven-year difference in student ability within one mathematics class due to differences in students’ ability to access multiplicative thinking.

These findings have been confirmed in more recent studies, including a 2021 paper.

So, supporting your child to develop their confidence and proficiency with multiplication is key to their success in high school mathematics. How can you help?

Below are three research-based tips to help support children from Year 2 and beyond to learn their multiplication facts.

1. Discuss strategies

One way to help your child’s confidence is to discuss strategies for when they encounter new multiplication facts.

Prompt them to think of facts they already and how they can be used for the new fact.

For example, once your child has mastered the x2 multiplication facts, you can discuss how 3×6 (3 sixes) can be calculated by doubling 6 (2×6) and adding one more 6. We’ve now realized that x3 facts are just x2 facts “and one more”!

Strategies can be individual: students should be using the strategy that makes the most sense to them. So you could ask a questions such as “if you’ve forgotten 6×7, how could you work it out?” (we might personally think of 6×6=36 and add one more 6, but your child might do something different and equally valid).

This is a great activity for any quiet car trip. It can also be a great drawing activity where you both have a go at drawing your strategy and then compare. Identifying multiple strategies develops flexible thinking.

2. Help them practice

Practicing recalling facts under a friendly time crunch can be helpful in achieving what teachers call “fluency” (that is, answering quickly and easily).

A great game you could play with your children is “multiplication heads up” . Using a deck of cards, your child places a card to their forehead where you can see but they cannot. You then flip over the top card on the deck and reveal it to your child. Using the revealed card and the card on your child’s head you tell them the result of the multiplication (for example, if you flip a 2 and they have a 3 card, then you tell them “6!”).

Based on knowing the result, your child then guesses what their card was.

If it is challenging to organize time to pull out cards, you can make an easier game by simply quizzing your child. Try to mix it up and ask questions that include a range of things they know well with and ones they are learning.

Repetition and rehearsal will mean things become stored in long-term memory.

3. Find patterns

Another great activity to do at home is print some multiplication grids and explore patterns with your child.

A first start might be to give your child a blank or partially blank multiplication grid which they can practice completing.

Then, using colored pencils, they can color in patterns they notice. For example, the x6 column is always double the answer in the x3 column. Another pattern they might see is all the even answers are products of 2, 4, 6, 8, 10. They can also notice half of the grid is repeated along the diagonal.

This also helps your child become a mathematical thinker, not just a calculator.

The importance of multiplication for developing your child’s success and confidence in mathematics cannot be understated. We believe these ideas will give you the tools you need to help your child develop these essential skills.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Bronwyn Reid O’Connor and Benjamin Zunica, The Conversation