Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Rationale

Nowadays, shoppers don’t even need wallets to carry cash or credit cards because of mobile phones and online shopping. So why even bother using coins to teach math?

Firstly, most young learners like the idea of playing with money, and using coins to teach math is a great way to engage students. Coins (real or fake) are useful manipulatives and visual models that allow students opportunities to engage with meaningful real-life tasks, and talk about the concepts of addition, subtraction, multiplication, and division as they “do math” and “see math” with hand-on activities like sorting coins by sizes, colors, and denominations.

I’ve taught 5^th grade math in a self-contained classroom for over 12 years, and every year there have been students who didn’t know the value of quarters, or nickels, or cannot count by coins to make a dollar. The COVID-19 pandemic had disrupted coin production and circulation for almost 3 years, and also spurred an increased use of digital transactions instead of cash across the globe. When my school returned to in-person learning in 2021, many of my students were unfamiliar with the value of coins; some had never seen a nickel before. That year my school was selling pretzels for $0.75 each as a weekly fundraiser, a female student wanted to buy 4 pretzels. She gave me a five-dollar bill, and just walked away. She didn’t know that 4 pretzels at $0.75 each would only cost $3, not $5. This incident reminds me of the 1976 hit song “Money, Money, Money” by the Swedish Europop group ABBA. The song is sung from the perspective of a woman who works hard but still struggles with money, and the only way to survive is to look for a rich man to rescue her financially.⁵ In schools, math-gender stereotypes such as “girls are bad at math” and been shown to reduce girls’ confidence, motivation, interests and performance in math, and lead to women being less likely to pursue careers in the STEM fields.⁶

Later the following year, in 2022, during our weekly pretzel fundraiser, a male student thought a quarter (25 cents), a dime (10 cents), three nickels (15 cents), and five pennies (5 cents) were enough to buy a pretzel for 75 cents. When I told him that he needed 20 cents more, he didn’t know why. When I asked him to count and add all the values of his coins, he said with embarrassment that he didn’t know how. Yet, this male student had shown me on many occasions that he can articulate his math thinking with confidence, but at other times when he failed to calculate a math problem correctly, he would look defeated and sometimes expressed his frustrations with: “I’m too dumb to do this.” The harmful stereotype that “Only smart people are good at math” is so prevalent in our schools as well as in our society. Research studies have shown that students who were motivated, hardworking and skilled with good math strategies had gained significant math improvements in comparison to apathetic students with higher IQ.⁷

Throughout my teaching career, I’ve learned to be more empathetic to students who exhibit high math anxiety (an emotional reaction to math) and possible signs of dyscalculia (a math learning disorder akin to dyslexia). Dyscalculia often goes undiagnosed because it is not well-known and understood among educators; about 5-7% of elementary school aged children are estimated to be affected by dyscalculia.⁸ I’ve seen 5^th grade students who used their fingers and chins to add two numbers, unable to remember addition facts, and/or unable to understand simple equations.

Mathematics, more than any other subject, has the power to break students' spirits and crush their self-esteem at a young age. Many adults have never overcome their fear of math. In regard to coin values, most adults would have problems explaining mathematically how 4 quarters are equivalent to one dollar with a simple equation like 4 x ¼ = 1. The concept of adding quarters to make a dollar can be quite straight forward or a complete mystery to some children as well as some adults. The value of a quarter can be written as ¼ of a dollar, 25/100 of a dollar, $0.25, 25¢, or 25 cents. It can be confusing to understand how these digits: 1, 4, 25, 100 are related. Some adults also may not be able to explain it mathematically using multiplication or division: 4 quarters multiplied by $0.25 = $1.00, or $1.00 divided by $0.05 equals to 20 nickels. With coin awareness, students can explain why the decimal point of a number will move to the right when we multiply by 10, 100, and 1,000, and why the decimal point will move to the left when we divide by 10, 100, and 1,000.

Furthermore, money knowledge and healthy money habits are great starting points to introduce financial literacy. Money can help students to familiarize themselves with large numbers like millions and billions, and very small numbers like decimals. According to a 2019 study about math ability and math confidence, almost one-third of American adults (73 million people) cannot select the health plan with the lowest cost based on annual premiums and deductibles, nor can they figure out how to pay off their credit card debt based on minimum monthly payment and the annual percentage rate.⁹ In hospitals, overdosing and underdosing of medicines have happened so often that the mistake has been labelled as death by decimals.^10. Coins offer a real-life incentive for learning mathematics for elementary as well as high school students, and I hope this unit would also help teachers to sharpen their student’s math skills in money management.