# Was Mrs. Thomas Right?

*December 3, 2017*

### Thirty-four years have passed, but a memory from my first grade math class remains vivid. My classmates and I were working on a subtraction worksheet, as Mrs. Thomas probed the room, checking for incorrect answers and making sure that everyone was on task. I was using my fingers to solve **7 – 3** when I sensed her hovering behind me. Sheepishly, I looked up and met my teacher’s stare. It wasn’t a look of anger, but instead deep disappointment. I was one of her best math students and she let it be known:

*You shouldn’t be using your fingers.*

### I’m frequently asked if Math teachers should allow students to use their fingers, and I always find it difficult to respond. The questions tend to be phrased with semantics and tone that expects a quick, obvious answer, but the benefits and pitfalls of the practice can’t be answered concisely.

### Fingers provide children with security and concrete proof that they answered problems correctly, helping them make better sense of abstractions. Most children are born with ten fingers and – barring a catastrophe – will always have them. Because of their familiarity and accessibility, those ten-digits are the most valuable math manipulative students will ever use.

### Still, teachers have – and continue to - discourage or disallow using fingers to make calculations. I don’t recall Mrs. Thomas explaining why she wanted us to automatize facts, but it’s not hard to guess her reasons. Spending brain energy on computation can distract from conceptualizing word problems. Making simple arithmetic mistakes while performing algorithms leads to inaccurate answers and less procedural practice. Mrs. Thomas understood:

*Not knowing basic math facts impedes critical thinking and accessing more challenging content.*

### Years before entering first grade, I often played sports with my father, sisters, and friends. I kept score by adding on and calculated lead or deficit margins by counting up and down. I had a conceptual understanding of addition and subtraction and performed simple calculations hundreds of times. Mrs. Thomas knew this, so when she noticed me using my fingers, she correctly judged that I was being lazy. She wasn’t necessarily implying that I should have memorized the answer to **7 – 3**, but that I should be stretching myself to solve it mentally. In doing so, she was pulling me towards automaticity.

### In that moment, Mrs. Thomas demonstrated outstanding pedagogy. Teachers and students often assume that struggling to answer questions is inefficient, unproductive learning, while answering effortlessly is the opposite. Although it might be counterintuitive, memory strengthens and mental stamina improves when we stretch our skill sets and struggle to arrive at answers.

### At the time, Mrs. Thomas’ no finger policy was right for me. The same rule, however, could’ve been debilitating for students who weren’t as comfortable with number concepts. For some first graders, numerals seem abstract in isolation. Thus, solving equations mentally creates complexity layers that overtax their brains. Left crutching on memory to guess/answer correctly, their practice lacks connective meaning. Not having helpful solving strategies, such as using fingers, stunts their progress, unnecessarily magnifying fluency weaknesses.

### The best math educators wean their students off finger dependency through efficient solving strategies combined with intensive, deliberate practice. For a child who hasn’t yet memorized **7 – 3**, consider the following approaches to solve, listed in order from most *concrete* and least efficient, to most* abstract* and most efficient:

### Aware of their students’ conceptual understanding and fluency comfort, the thoughtful practitioner guides each learner to use strategies that help them confidently solve problems. After students master a strategy, the teacher encourages them to practice using a more efficient method.

### ***

### In 1983, Mrs. Thomas was a veteran teacher, and her traditional style was reflective of the educational era in which she taught. Mathematical theory and practice has changed a lot since then. No longer regarded as a set of rules and procedures to memorize, mathematics has undergone a pedagogical transformation. This progressive philosophy emphasizes conceptual understanding and flexible thinking, while deemphasizing memorization and procedures.

### Today, most elementary school math teachers embrace a *Traditional* or *Progressive* approach. Traditional instructors typically emphasize memorization at the expense of effective, meaningful strategies. Progressive educators often encourage children to use their fingers, but don’t always guide them to use more efficient strategies. Pitting one philosophy against the other is a false choice. Both need to be cultivated.

### Memorizing basic skills is still advantageous to succeeding in the subject and teachers who ignore this simple truth have allowed the pendulum of mathematical pedagogy to swing too far from its origin. Great elementary educators teach the subject through thinking and reasoning, guiding their students to understand concepts before internalizing them through appropriate strategies and deliberate practice. In doing so, they implement best practices, while honoring the subject’s orthodoxy, centering the swinging pendulum.