Rethinking fractions and calculus
If you had a hankering for a hamburger, and you were trying to get the best bang for your buck, would you pick the 1/3 pound burger for $4.99 or the 1/4 pound burger for $4.50? If you chose the latter, you have just given credence to what educational experts believe to be a growing area of concern.
“We have a problem with fractions,” said Ken Wagner, Commissioner of the Rhode Island Department of Education (RIDE), after bringing up the edible example of a survey that showed Americans were alarmingly unable to ascertain which burger was actually bigger.
Wagner states that the problem with fractions starts from when kids begin to learn about them around the third grade. Rather than teach kids that fractions are simply representations of actual numbers on the number line – with a physical place between other, whole numbers – they are taught procedurally how to multiply or divide fractions as though they are some kind of separate mathematical entity.
This creates a rigidness in mathematical thinking that, Wagner warns, could persist all the way throughout the child’s educational life.
“If kids don’t have a deep understanding of fractions, not only will they, like me, have problems with baking and cooking years later because it’s applied fractions, they’re going to have a problem with algebra in eighth, ninth grade,” he said. “And when they have a problem with algebra, they’re going to have a problem with college entrance and college placement. They’re much more likely to get placed in non-credit bearing remediation. And the single thing you can do to help ensure a kid doesn’t complete college is to put them in non-credit bearing remediation when they get to college.”
Wagner said that if a child does not have a solid understanding of fractions by the time they finish fifth grade, they are as behind as a child who doesn’t have a good grasp on reading in third grade – an issue that garners far more attention and resources than the more complex issue of fractions.
“Fifth grade fractions is as big of an equity barrier as third grade reading,” Wagner said, “We know we have huge gender gaps – girls are not as comfortable with math as boys claim that they are – and it all starts with fractions. The disproportionate access to early algebra or advanced coursework in high school for girls versus boys; kids of color versus not; poverty versus not – all starts with fractions.”
Wagner believes that in order to get students to think differently and more productively about fractions, we need to be able to train teachers to teach fractions differently. Working towards that solution is a part of RIDE’s venturing into creating new teacher pathways that encourage prospective educators to go into fields such as mathematics, in addition to reworking how teachers experience professional development.
Wagner, and RIDE, is advocating for a “professional learning circle” model of professional development where teachers sit together and share best practices on what instruction methods work best to fulfill standards set by the state.
“Teachers helping teachers is the way to plug these gaps,” Wagner said. “We don’t have to raise the stakes and threaten people that, ‘If you don’t know this we’re going to discipline you or fire you,’ we just have to be honest with where our knowledge gaps are and let teachers work with teachers.”
Expanding upon the conversation of fractions, Wagner went on to think critically about how the nation’s higher education institutes should be willing to reconsider a long-held belief that calculus is the end-all, be-all of advanced mathematics; a topic that anybody who winds up in a math-related field needs to have deep knowledge of.
“The closely kept secret is very few people actually use pure calculus – it’s calculus teachers and calculus professors and calculus tutors, they’re the ones using calculus day-to-day,” Wagner said. “Calculus was actually an applied math problem. Newton was trying to solve a problem for how to calculate an area so he invented calculus. Calculus was invented for a particular purpose but it has become a thing unto itself, as if you’re not in the lofty spheres of math unless you’re using calculus.”
Wagner gave the example that both he – the commissioner of a state’s public education system who excelled in calculus and even enjoyed it – and the chairman of the Rhode Island Council on Elementary and Secondary Education, an engineer, both have never needed their knowledge of calculus for their careers.
He lamented that the fact that colleges and universities still maintain one dominant mathematics pathway – one that takes kids from algebra in eighth grade all the way through calculus in high school and into college – turns kids off to math who may otherwise be interested in mathematics were it actually taught in an applied way that caught their attention.
“We’re boxing kids out, because kids don’t see it as relevant,” he said. “I’m not advocating that we replace that [pathway], I’m just advocating that we add to that. We need an algebra to applied math pathway that’s tied to things like computer science, quantitative analysis, statistics – and that pathway will not only be very relevant from an economical perspective, it’s also going to engage kids in a way that we’re not currently engaging them.”
Wagner said that the ultimate determinant of this recommendation will be the universities and math departments themselves, as middle schools and high schools will simply have to continue preparing kids for a calculus pathway as long as that pathway is what they will inevitably face when getting to college.
Still, Wagner is holding out hope that this will change with more time and attention paid to the matter.
“Math is beautiful, math is useful, we just haven’t yet shown the kids and the teachers all the different ways in which math can be beautiful and useful,” Wagner said.