Month: March 2024

A mathematical historian at Trinity Wester University in Canada, has found use of a decimal point by a Venetian merchant 150 years before its first known use by German mathematician Christopher Clavius. In his paper published in the journal Historia Mathematica, Glen Van Brummelen describes how he found the evidence of decimal use in a volume called “Tabulae,” and its significance to the history of mathematics.

The invention of the decimal point led to the development of the decimal system, and that in turn made it easier for people working in multiple fields to calculate non-whole numbers (fractions) as easily as whole numbers. Prior to this new discovery, the earliest known use of the decimal point was by Christopher Clavius as he was creating astronomical tables—the resulting work was published in 1593.

The new discovery was made in a part of a manuscript written by Giovanni Bianchini in the 1440s—Van Brummelen was discussing a section of trigonometric tables with a colleague when he noticed some of the numbers included a dot in the middle. One example was 10.4, which Bianchini then multiplied by 8 in the same way as is done with modern mathematics. The finding shows that a decimal point to represent non-whole numbers occurred approximately 150 years earlier than previously thought by math historians.

Giovanni Bianchini worked as a merchant in Venice for many years before being appointed to an administrative role with a major estate owned by the powerful d’Este family. In this role, he also managed assets and investments, giving him a strong background in mathematics. He also published astronomy texts, demonstrating his ability to plot planetary motion and to predict when an eclipse would occur.

The finding suggests that Bianchini played a more important role in the development of math fundamentals than previously known. Van Brummelen notes that, as a merchant, Bianchini would have traveled extensively, including to places in the Islamic world, where math concepts were being developed, possibly influencing his use of non-whole numbers and ways to represent them more easily.

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Credit of the article given to Bob Yirka , Phys.org

 

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

 

The just-released report of the inquiry into price gouging and unfair pricing conducted by Allan Fels for the Australian Council of Trades Unions does more than identify the likely offenders.

It finds the biggest are supermarkets, banks, airlines and electricity companies.

It’s not enough to know their tricks. Fels wants to give the Australian Competition and Consumer Commission more power to investigate and more power to prohibit mergers.

But it helps to know how they try to trick us, and how technology has enabled them to get better at it. After reading the report, I’ve identified eight key maneuvers.

1. Asymmetric price movements

Otherwise known as Rocket and Feather, this is where businesses push up prices quickly when costs rise, but cut them slowly or late after costs fall.

It seems to happen for petrol and mortgage rates, and the Fels inquiry was presented with evidence suggesting it happens in supermarkets.

Brendan O’Keeffe from NSW Farmers told the inquiry wholesale lamb prices had been falling for six months before six Woolworths announced a cut in the prices of lamb it was selling as a “Christmas gift.”

2. Punishment for loyal customers

A loyalty tax is what happens when a business imposes higher charges on customers who have been with it for a long time, on the assumption that they won’t move.

The Australian Securities and Investments Commission has alleged a big insurer does it, setting premiums not only on the basis of risk, but also on the basis of what a computer model tells them about the likelihood of each customer tolerating a price hike. The insurer disputes the claim.

It’s often done by offering discounts or new products to new customers and leaving existing customers on old or discontinued products.

It happens a lot in the electricity industry. The plans look good at first, and then less good as providers bank on customers not making the effort to shop around.

Loyalty taxes appear to be less common among mobile phone providers. Australian laws make it easy to switch and keep your number.

3. Loyalty schemes that provide little value

Fels says loyalty schemes can be a “low-cost means of retaining and exploiting consumers by providing them with low-value rewards of dubious benefit.”

Their purpose is to lock in (or at least bias) customers to choices already made.

Examples include airline frequent flyer points, cafe cards that give you your tenth coffee free, and supermarket points programs. The purpose is to lock in (or at least bias) consumers to products already chosen.

The Australian Competition and Consumer Commission has found many require users to spend a lot of money or time to earn enough points for a reward.

Others allow points to expire or rules to change without notice or offer rewards that are not worth the effort to redeem.

They also enable businesses to collect data on spending habits, preferences, locations, and personal information that can be used to construct customer profiles that allow them to target advertising and offers and high prices to some customers and not others.

4. Drip pricing that hides true costs

The Competition and Consumer Commission describes drip pricing as “when a price is advertised at the beginning of an online purchase, but then extra fees and charges (such as booking and service fees) are gradually added during the purchase process.”

The extras can add up quickly and make final bills much higher than expected.

Airlines are among the best-known users of the strategy. They often offer initially attractive base fares, but then add charges for baggage, seat selection, in-flight meals and other extras.

5. Confusion pricing

Related to drip pricing is confusion pricing where a provider offers a range of plans, discounts and fees so complex they are overwhelming.

Financial products like insurance have convoluted fee structures, as do electricity providers. Supermarkets do it by bombarding shoppers with “specials” and “sales.”

When prices change frequently and without notice, it adds to the confusion.

6. Algorithmic pricing

Algorithmic pricing is the practice of using algorithms to set prices automatically taking into account competitor responses, which is something akin to computers talking to each other.

When computers get together in this way they can act as it they are colluding even if the humans involved in running the businesses never talk to each other.

It can act even more this way when multiple competitors use the same third-party pricing algorithm, effectively allowing a single company to influence prices.

7. Price discrimination

Price discrimination involves charging different customers different prices for the same product, setting each price in accordance with how much each customer is prepared to pay.

Banks do it when they offer better rates to customers likely to leave them, electricity companies do it when they offer better prices for business customers than households, and medical specialists do it when they offer vastly different prices for the same service to consumers with different incomes.

It is made easier by digital technology and data collection. While it can make prices lower for some customers, it can make prices much more expensive to customers in a hurry or in urgent need of something.

8. Excuse-flation

Excuse-flation is where general inflation provides “cover” for businesses to raise prices without justification, blaming nothing other than general inflation.

It means that in times of general high inflation businesses can increase their prices even if their costs haven’t increased by as much.

On Thursday Reserve Bank Governor Michele Bullock seemed to confirm that she though some firms were doing this saying that when inflation had been brought back to the Bank’s target, it would be “much more difficult, I think, for firms to use high inflation as cover for this sort of putting up their prices.”

A political solution is needed

Ultimately, our own vigilance won’t be enough. We will need political help. The government’s recently announced competition review might be a step in this direction.

The legislative changes should police business practices and prioritize fairness. Only then can we create a marketplace where ethics and competition align, ensuring both business prosperity and consumer well-being.

This isn’t just about economics, it’s about building a fairer, more sustainable Australia.

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

Credit of the article given to David Tuffley, The Conversation