The toxic butterfly effect of school mathematics
The case for adaptivity
Tutoring is a joy and a privilege. My interaction with students anchors my understanding of where school mathematics falls short.
Take Inny (her name only partly disguised so that she may revel in the use of her case study), an eighth grade student who possesses an amazing vitality for mathematics. Much of our time together is spent traversing the rough terrains of extended problem solving.
But not this day, for Inny had come unstuck on a homework assignment. I ran my eyes over the question at hand. Given a linear expression involving x and y, Inny was required to sketch the function y. So, for example:
Sketch the function y, where 5x-3y+12=0 (*)
A tutor’s job is to pinpoint their student’s source of confusion. My first thought was that Inny had yet to grasp the link between a function’s algebraic form and its graphical representation.
But as Inny explained her difficulty, her voice marked with frustration, it became evident that my assumption was misplaced. Inny was adamant that she “had the whole y = mx +c thing down”. As we probed each dimension of the problem, I realised that Inny’s problem was more fundamental: she did not know how to get the above expression (*) in the standard y=mx+c form. She could not rearrange terms.
A head-scratching moment. How can any student be expected to sketch such functions by relating them to the y=mx+c form when they have not understood the basics of rearranging algebraic equations?
Inny vaguely recalled studying how to rearrange equations (‘oh yeah, that thing with scales’) but scarcely had time to understand the concept or practice the methods. Since the topic was only briefly covered in class, and had not been reinforced since, this single gap in her mathematical knowledge was blocking her entry into more advanced concepts.
After our brief sojourn to algebra, Inny got her mojo back. She did not need my help with sketching the graph itself — the conceptual basis of the question was always clear in her mind. Now that she had acquired the requisite prior knowledge of rearranging equations, the homework assignment was within her reach.
The essence of mathematics is found in rich problem solving, but this depends on having a solid foundation of core skills and concepts. The standard curriculum model pushes students through topics at a fixed pace. Since learning is rarely linear, this inevitably leaves students with core gaps in their knowledge.
This problem is especially serious in subjects like mathematics, where knowledge across the curriculum is highly interdependent. For example, students have precious little hope of commanding an understanding of fractions until they have mastered the basics of whole numbers and operations. Going deeper, they will need to get on top of fractions before playing with the concepts of probability.
A small gap in one topic will often amplify over time and spread in several directions as students encounter more advanced material.
This toxic butterfly effect has a crippling effect on students’ understanding and, worse, their confidence. It leaves educators with underwhelming impressions of students’ true potential.
Remember that Inny grasped the idea behind her homework assignment all along — she just lacked the procedural tools to manipulate the expressions. Inny was fortunate on this occasion to fill the missing piece in time. How unconscionable a thought that she may never have realised her aptitude for advanced algebraic concepts, simply because she was never given the time to address her gaps in basic skills.
It is vital that students are given the time and support to master foundational skills and concepts before advancing in the curriculum.
The breadth of students’ learning needs makes tracking their individual knowledge gaps a tall order. It can seem a near mathematical impossibility in a class of thirty students and a curriculum containing hundreds of standards.
Private tutors provide one way of targeting students’ individual learning needs, but human tutors do not come cheap. So-called intelligent tutoring software seeks to scale the benefits of private tutors by automating adaptive 1:1 instruction, and is growing in popularity among parents and educators. However these models evolve, they must be driven by a commitment to grounding every student in core mathematical skills and concepts.
Inny is a reminder of how delicately interwoven mathematical knowledge can be. We must make the case for adaptivity and bring it to the forefront of our curriculum models.
I am a research mathematician turned educator working at the nexus of education, innovation and technology.
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If you liked this article you might want to check out two of my other pieces: My nephew brought home this menacing maths problem and Why every 10-year-old can emulate this iconic gameshow moment.
