3 Stages of Math Acquisition

Back when I took Calculus in grad school I understood everything my professor wrote on the board. I was a freaking genius. "Yep, that's easy," I'd tell myself. "Really, they call this Calculus?" Then, as you may have expected from my arrogant opening sentence up there, it soon became mind-numbingly obvious that there were different levels of understanding when it came to Calculus. While it was true that I understood [most of] what was written on the board, doing the work by myself, at home, alone, without the tutor I ended up hiring to help me every single Sunday at $20/hour (cheap, right?? He was half my age), was impossibly difficult.  I asked my professor during one of the after-class help sessions why I understood the class material and not the homework, half expecting him to tell me that he made the class notes easier than the homework on purpose.

"It's a different skill," he told me. "It's one thing to connect a step I write on the board to the step before it; it's a much different thing to know what next step to take on your own."  


This was 5 or 6 years ago and this conversation will always stick with me. I've had a hard time since trying to explain my professor's words to my own students because it's a hard difference to explain. Why is it so different to understand a problem being solved on the board and knowing what to do on your own?  Isn't understanding understanding?  

Math, as it turns out, really is a foreign language and I've come to find out it's learned in a similar way. Math is acquired in much the same way as Spanish, Portuguese, French, Arabic, etc. I'll refer to Spanish here in this comparison.

1: Understanding what people are saying in Spanish = understanding the next solving step written on the board.

The first step to learning a foreign language is understanding what people are saying. Math is the same. Understanding what the teacher wrote on the board and being able to connect the step she just wrote to the step above it is step 1. For example:

A student may say, "Oh, I get it," when seeing this because he knows that -7 is the opposite of +7 and 7 needs to be subtracted from both sides. He may not have seen the step on his own but will say, "Oh, right!" when he sees the teacher write it. This is like a second language learner developing her receptive vocabulary where she is able to understand conversation but not yet able to speak it.

2: Speak conversational Spanish = telling the teacher what step to write next on the board.

The second step of Math acquisition is conversational Math, that is, being able to tell the teacher what to do next while she is solving the problem on the board with student input. While some students are comfortable answering teacher prompts for, "What's the next step?", other students in the room are still unable to see the next step on their own and are relying on the vocal students to let them "see" what comes next.

3: Write Spanish = completing classwork and homework independently

A lot of my students grew up speaking Spanish or Portuguese but have no idea how to write it. Have you seen this in the classroom where students who are able to answer, "What's the next step?" during the mini-lesson sit and stare at their blank independent classwork? Are they being lazy? No way! 

Whenever I introduce function notation I ask the class if anyone knows a foreign language. I then give my old example that "Feliz Cumpleanos" means the same as "Happy Birthday" just like f(x)= is the same as our comfortable y=.  If I'm feeling really scientific and channeling my old Biology teacher, I may give the comparison between 24 degrees Celsius and 75 degrees Fahrenheit describing the same temperature with 2 totally different numbers.

My students grew up fluent in 2 languages. But asking them to write in their parents' native language? No way! Being able to write Spanish is a very different skill from being able to understand or speak it. Completing independent classwork challenges a student to recall class examples, to know what comes next by being fluent in the equation-solving process, to use note examples, to allow what's in their little brains to travel down their arms through their pencils to form numbers on the page, and most importantly to have the confidence to take a chance at solving the scary problems all on their own. This is a huge jump from understanding why a step came next or answering teacher prompts during a mini-lesson. Learning Math is so much like learning a foreign language.

Now, where's that Spanish workbook I ordered from Amazon last summer...?

Thank you Surfer Kids Clip Art for the great kid in blue!