Hacking Middle School Math with One Weird Test

This recent post over at the Women in Astronomy blog about spatial skills and success in STEM resonated with me. Here’s why.

One day in my 6th grade math class, we took an unusual test. Mrs. Bergstrom distributed the exam without fanfare or introduction, saying simply that this would not affect our grade but that we should still do our best. It was something new they were trying.

Which of the bottom shapes is a rotation of the top one? Much to my delight, a surprise sixth-grade math test was filled with questions like these. Image from sharpbrains.com
Which of these shapes is a rotation of the top one? Much to my delight, a 6th-grade math test was filled with questions like these. Image from sharpbrains.com

When the sheet landed in front of me, my eyes brightened. Instead of the usual numbers and word problems, this test had a series of unusual shapes. It was filled with multiple-choice questions about rotating blocks, pattern recognition, geometrical shapes, and cross-sections. There were no numbers! This was amazing! I breezed through the exam and wondered hopefully if these questions were the beginning of a new unit in math class.

Up to that day, I was always a good student, but my marks in English were always higher than math. I never really learned my times tables properly, much to my father’s chagrin. I harbored a passionate loathing of long division that remains to this day. I wasn’t bad at math, per se, but I was only borderline qualified for my school district’s gifted program in 3rd-5th grade. I loved science and space, but also reading and writing, and anything to do with numbers made me feel insecure and helpless in a hurry.

So you can imagine my parents’ surprise when they got a telephone message from my middle school, asking me to come in for further testing to see if I could skip 7th-grade-honors Pre-Algebra altogether and go straight to Algebra.

Apparently the mysterious math test I so enjoyed was actually a spatial reasoning test. They were using it to identify students who had potential to succeed in math. After a conversation with my parents, I went to my guidance counselor’s office with Mom. There, they gave me a more traditional “place out of Pre-Algebra” test that was much less fun. The questions got harder as I went, but I felt equipped to answer all of them and did my best.

My counselor graded the exam right in front of me. There were 20 questions; I was allowed to miss 4 and still pass. So of course I missed the first three “easy arithmetic” questions, and another toward the middle of the test, just barely squeaking through. But that was all I needed. Suddenly, I was Really Good at Math.

The next fall, on my first day of 7th grade, I entered the 8th-grade-honors Algebra classroom with trepidation. I was the only 7th grader. A few people assumed I was lost, but Mrs. Howes was kind and welcoming. The class, on the other hand, was extremely hard. I was not used to school being hard. But in the third week, shortly before the first test, another 7th-grader joined the class: my friend Jessica. (When her parents heard that I had skipped Pre-Algebra, they insisted she also take the placement exam. She aced it.)

At first, having Jessica in the class was intimidating. Everything seemed to come easily to her, even though she joined the class late. I got a D on the first test; she got an A. There were murmurings of whether or not I belonged in the class. But louder than those murmurings were voices of support: from Mrs. Howes, from my parents, and even from Jessica. I belonged in Algebra, and once I set aside time to study and learned how to work multi-step homework problems, I succeeded.

I have no idea why I was so good at spatial reasoning in 6th grade. It was certainly never formally taught to me. But by being in the right place at the right time, I was somehow able to graduate from “you’re a smart kid who’s OK at math” to “you’re a smart kid who’s a math whiz.” From that, it followed that my bad grade on an Algebra test must be a fluke, and I got the support I needed to succeed. Today I’m a PhD candidate in Astronomy, and calculus-based math is one of my most important tools. I can only imagine the difference we could make if all middle-schoolers knew math was fun, and that they were good at it.


2 thoughts on “Hacking Middle School Math with One Weird Test

  1. What a great story. I was skeptical after that inital “D” and was waiting for the punch line — is she doing math now? — astronomy PhD candidate! Wonderful.

    I have heard this story before, mind you. There seems to be a certain “effort” barrier (“once I set aside time to study”) making a big difference. So often students are able to coast through subjects on their native wit alone. Algebra is not too bad, but it does require enough practice to internalize.

    Btw, how are your times tables now? Did you hunker down and memorize those or do you just work them out as you go?

    The other part of your formula was “learned how to work multi-step homework problems”. I would love to hear more about that. Was it just a matter of patience? Or was it just the very idea that the answer may not be obvious at the start but if one identified and followed the stepping stones it would appear? Other?

    My Algebra pep talk is that it is not that hard and it is not that easy. I might strengthen that: Algebra is easy unless you think it is. i think survivor stories like yours need careful attention from those of us trying to help those who struggle with it, especially since most of us had no problem with it. Our perspective is no help!

    Final question, when you say “math is fun”, do you mean in and of itself (I myself thought it was just a bunch of neato puzzles and did not care if it had any application) or something else?

    Thanks for a valuable post.

    1. Thanks. There was definitely an effort barrier for me, because the problems were longer and more complex than before. Add that to my tendency to make stupid arithmetic mistakes and you have a recipe for frustration.

      I am still uncomfortable with mental math, and I rely on calculators for arithmetic and algebraic manipulation more than I’d like. I know probably half my 1–9 times tables without thinking. As for the multi-step problems… I think it was a combination of patience and realizing that there are stepping stones to be found in the first place. Part of it was also mindset: “this is the math class I belong in, so if I am lost then I need to seek out help” vs. “they must have put me in the wrong math class, so if I am lost then I am stupid and should be in a different class.” It is a hard leap from being smart to working hard (see, e.g., http://womeninastronomy.blogspot.com/2014/01/why-so-few-growth-mindset.html).

      I tend to see math as a means to an end—a tool I use to do science. Real-life science is never going to say “you aren’t allowed to use a calculator.” It’s a lot more about defining the problem at hand and figuring out which tools (mathematic and otherwise) you need to pull out to tackle that problem. I try to stress this when I teach intro astronomy labs: understand why we are doing a mathematical operation, learn how your calculator or computer can help, and then use it smartly.

      Even if math is “just” a tool, it’s a gorgeous one that underlies so much of the physical world. It is incredibly fun to wrap your mind around that and experience the occasional “aha!” moment. For example, the first time I learned about pi: the number of times a circle’s diameter wraps around its circumference. That’s just CRAZY, and has more physical applications than you can imagine. That said, just like other academic subjects, not all math has to be applied to be worth learning. I enjoyed the “neato puzzle” aspect as well, but when I got to Geometry after Algebra I absolutely loved applying all these weird letter variables to physical shapes. Trigonometry still blows my mind if I think too hard about it, and the surprisingly tricky geometry needed to define the orbit of a binary star is one of my favorite things.

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