This post marks a couple of milestones for Musing Mathematically. First, this is the 50th post overall. For some reason that seems significant. Second, this post marks the blog’s first coined phrase–Atomic Skills.
I love the term atomic skills, but I can’t remember when I started using it. I believe it was the result of my limited vocabulary attempting to explain the current disconnectedness of math education. An atomic skill is a foundational skill. An atomic skill is a skill that holds no real ‘stand-alone’ significance, but can build toward a very significant solution. Atomic skills are usually practiced in isolation of each other in a very repetitive fashion. In school mathematics, atomic skills often make the difference between a good and bad student. Students classify errors with atomic skills as “stupid mistakes”.
The term “atomic” fits for two main reasons:
- Atomic skills are the building blocks of mathematical problem solving much like atoms build our physical universe. They are the pieces from which elegant mathematics arise.
- Atomic skills often cause the most explosive conflicts among mathematics educators. Traditionalists stress the importance of these skills, while reformers advocate for their incorporation into significant contexts.
(I use the terms “traditionalist” and “reformer” very loosly; I think all math educators fall somewhere on a continuum between the two)
The flagship for Atomic skills has always been, and will always be, the multiplication tables. For some reason, these basic facts have infatuated teachers, parents, policy makers, and flashcard manufacturers to no end. Successful students can perform the mulitplication tables automatically. By themselves, they represent a set of patterns; it is not until they are granted a context that they take on real significance.
To be clear, I agree that successful students should know their mulitplication tables. To further clarify, I believe that the incessant drilling of these facts is not the best way to attain them. In order for these atomic skills to be meaningful to students, they must be learned in a context.
Enter Project Based Learning.
I am not arguing for the elimination of atomic skills in education. I am calling for a re-invention of how they are presented. Students need to be presented skill-rich tasks that require them to work on atomic fluency to reach solutions. Too many textbook problems require a single atomic skill to solve–especially in the lower grades. Worse than that, when students begin to show deficits in mathematics we just heap on more repetitions of the same process that failed them originally. An expertly designed project or task provides a numeracy-rich setting embedded with large amounts of atomic skills. Students are required to use, and subsequently learn, basic skills to arrive at a larger and more significant solution.
My most recent stab at this is my “Classroom Remodeling” project. It is a real problem that I faced during my implementation of Project Based Learning in my class. The task is built with a degree of vaguity as well as some strict mathematical guidelines.
The task materials are available for download here. It was originally designed in GeoGebra, but students found that paper cutouts worked better than the digital model. This also included work with scale.
Students need to use atomic skills to arrive at a final project that balances:
- Spatial restraints
- Cost restraints
- Capacity restraints
These three factors require students to repeatedly interact with various atomic skills:
- 3 digit addition
- 1 digit by 3 digit multiplication
- Mulitples of 3 and 4
- Conversion between inches and feet
- Tax calculations (if included in the task)
The teacher’s job is to elicit the deeper thinking as the groups progress. For example, a group may be struggling to find 2 more seats for students in their lecture. Maybe replacing a 30″x60″ table with two 24″x60″ tables would provide the same group capacity (4) while increasing the lecture capacity by 2. (from 2 to 4). How will that effect the cost? Will it put you over budget? Will there be space? Are circular tables an option? The thoughts take on an exploratory nature, but students are still required to master the embedded atomic skills. One alteration leads to a question in another area, but every stage is fueled by calculations.
Many of the problems, tasks, prompts, and projects in this blog are designed in this vein. The atomic skills are rehearshed within a much grander scope. This way students are working on automaticity of low-level skills while increasing larege scale teamwork, problem solving, and numeracy skills. It may be analagous to shoving an antibiotic pill into a bowl of yogurt.
Cleverly designed projects don’t devalue the importance of atomic skills; they put them in their proper place–as subsidiary, yet explosive, skills.