Every other week, our founder and CEO, John Tapper, shares what's on his mind, from his thoughts on math education to what's been inspiring him lately. This is your chance to hear directly from the person who started it all. We believe great ideas are worth sharing, and Founder's Corner is our way of bringing you closer to the heart of All Learners Network (ALN) and the passion that drives everything we do.
In my last Founder’s Corner piece I began the story of Maria, an adult woman who had difficulty with math because of childhood trauma. Our work together lasted a little under two years. It was one of the most inspiring and influential relationships of my teaching career.
Maria’s math understanding consisted almost entirely of things she’d memorized. She knew all her math facts (addition facts, times tables) and she had foggy ideas about some of the algorithm rules. She could perform all four arithmetic operations with whole numbers but fractions were another story. Fractions (and decimals, to some degree) were a major source of stress. I found this out when I asked her why she had suggested that ½ + 1/3 was 2/5. In response she simply stared at me, without speaking, for an uncomfortable period of time. I thought she might be having an aphasic event until she said, “top plus top, bottom plus bottom. That isn’t right, is it?”
I asked her whether a half and some more was more or less than 2/5. She got a disgusted look and said, “I have no friggin idea.” This is where we began. Rather than explain how fractions work, we made a 3x4 rectangle. I asked her to show me a half of the rectangle (six of the spaces), a quarter (three of the spaces), a third (four of the spaces).
When I asked her how many spaces in the rectangle were 2/3, she shrugged. I rephrased the question, “Show me two 1/3s.” Maria pointed to two rows of four. Then she paused and her face lit up, “You mean, 2/3 is just two one thirds?!”
“Yep. It’s no harder than that.” Rather than tell her what she should know, I let her find her way to it. This is how we worked together for the next year and a half. I gave her the chance to figure out new concepts for herself and supported her thinking with questions and “pointers” that referred to what she already knew.
Maria’s response to working with fractions was easy to connect to her inability to focus (due to her complicated personal circumstances). Her trauma experience happened right at the time when math programs are diving deeply into fractions. She was taught in a way that demanded she listen and repeat the teacher’s understanding, something she was just incapable of doing. Unfortunately, understanding fractions, rational numbers, is a “gatekeeper” understanding for much of what comes later (Booth & Newton, 2012). Not understanding fractions kept Maria behind in math throughout middle and high school, as it does for so many students (Booth, Newton, & Twiss-Garrity, 2014).
When students become anxious, they lose working memory resources (Finell, 2022). I could actually watch this happen with Maria. Trauma left Maria with less working memory resources and the anxiety she experienced stripped what was left. When she was confronted by a perplexing fraction question or problem, she would stare into space. To test what was happening, I asked her what her favorite song was (she liked a country song called Broken). At these times, she couldn’t remember the name of this song. Once, when I asked her this, she became noticeably irritated.
When Maria couldn’t remember the name of her favorite song, we would practice focus and relaxation. I’d have her close her eyes and breathe in for a count of four and out for a count of seven. When she could remember the song (this never took very long) I’d ask her what the first line of the song was. When she could remember the first line, we’d continue with learning.
Over the course of two years Maria and I worked through rational numbers, proportions and into topics in Algebra 1. She was better at algebra, which doesn’t happen often. When we worked on algebra, whether it was figuring out functions or solving for unknowns in equations, she was a quick study. She offered frequent insights and seemed to genuinely enjoy doing algebra. She especially enjoyed working with technology that allowed her to play with linear functions; showing the graph of a function and asking for a prediction of the equation.
“OK so it crosses the y axis at -3, so we know there’s a -3 on the end… the line is a little steeper than 1… I’m not going to count the steps, I’m going to guess… Let’s say 4? So the equation is y=4x-3.” Then we’d look to see how close she was. She really liked doing this and she became quite skilled at using her reasoning to make sense of each graph.
The other part of the app would present her with an equation and she had to create the graph. She resisted my suggestions to use a graphing app. She liked to draw the graph on graph paper with a ruler. My patience would be tested from the amount of time she would spend making the graph perfectly. But Maria got good at rendering linear function graphs from equations. It even helped her a little with fractions when she had to figure out what a slope of 2 ½ would look like.
Maria eventually took the placement test and her score put her into the advanced math class. She got a B in that class. After her placement test, I got a phone call from the proctor of the test (Maria gave her my number). The proctor said she wanted to know why Maria would periodically close her eyes and sit still. She asked Maria but Maria said she couldn’t really explain so I got this call. I explained the technique and the proctor was intrigued. The other reason she had called me was that Maria had a very unusual scoring profile. Her score on the algebra items was almost perfect but she didn’t do as well on the arithmetic. I explained that, for Maria, algebra wasn’t carrying any emotional baggage.
I lost track of Maria over the years. She and her wife, Amy, moved away from Hartford out to New Mexico. I don’t know if she ever became a nurse. I took away a great deal from our learning together. Maria showed me how difficult personal circumstances can make learning. She showed me how valuable perseverance can be if a student can find a reason to persevere. Most importantly, she showed me that everyone can learn math.
Booth, J. L., & Newton, K. J. (2012). Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37(4), 247–253. https://doi.org/10.1016/j.cedpsych.2012.07.001
Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110–118. https://doi.org/10.1016/j.jecp.2013.09.001
Finell, J. (2022). Working memory and its mediating role on the relation between math anxiety and math performance: A meta-analysis. Frontiers in Psychology, 12, 798090. https://doi.org/10.3389/fpsyg.2021.798090
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