We Read. And then We Math.

Maria read tonight. And not that “man, can, fan” stuff.  She read from a legitimate book.  Frozen:  Big Snowman, Little Snowman.  The plot grabs you right from the get-go.  I don’t want to ruin it for those who haven’t read it yet.

We, obviously, have a long way to go.  But listening to her sound out and struggle with the words was fascinating.  And although I know the English language has its quirks, those rule-breaking words are blaring when you are listening to a 5-year old attempting to sound them out.

Here is an example page (Warning – spoilers):


Maria: I know that word is “on.” And I know the word “horse” has to be on this page.

Me: Good, now what about the first word?

Maria: Some nonsensical mispronunciation of the word “Anna”

Me:  Who is the character on this page?

Maria: Oh. Anna.  Anna…Mommy what is that funny letter (points to the “g”)?

Megan:  That’s guh.

Maria:  guh guh guh eh eh eh ta ta ta ssssssss.  Gets…On…the horse and chases after Elsa in the snow.

Megan:  Try this word after “on”

Meanwhile, I’m thinking there are five e’s on this page representing 3 different sounds.  How the heck does anyone ever learn to read!?  What a nightmare.  Then I realized:  context.  Every word in this book connects to multiple other words to form sentences and tell a story.  A mind-numbing Disney Princess story, but a story nonetheless.

And that’s what we are missing in mathematics.  The story.   Solving quadratic equations are taught separate from the graph of the quadratic itself and is that process connected to any other representation.  Students are taught steps to solving an equation but rarely is there any connection to how that equation was built and what it represents in the first place. Kids are taught to read by using contexts and eventually get to choose their own books to enjoy.  Do we ever let kids have “choice math” time?

And so then reading practice looks something like this:


While math practice looks something like this:

from worksheetfun.com

from worksheetfun.com

Steve Leinwand has spoken at length about this, and he sums it up nicely in this slide from one of his presentations:

In mathematics, we just keep asking the same things:  Where did Jane go?  Who went to  the store?  And so we miss opportunities for kids to make connections between ideas by justifying their paths to the solution.  And as long as all we are asking is “where did Jane go?”  our students won’t be afforded the opportunity to consider the motivation behind Jane’s adventure.

Mathematics at 5 Miles per Hour


Remember learning how to drive? Remember those early teenage years when you thought, “Why do we need permits and all that behind the wheel time? I’ve watched people drive my whole life! I’ve totally got this!”


Then you sit behind the wheel. Adjust the seat and the mirrors, secure your seatbelt…and then ossilate between the gas and the brakes until your dad yells at you to keep your eyes straight ahead of you and stop hovering over the brake pedal and good lord, girl, I’m gonna throw up if you don’t stop jerking the wheel. My God, you drive like your mother, and don’t go that fast on this road, I don’t care if the speed limit is 55, the car behind you can go around you. I don’t remember a whole lot of what my parents told me as I learned to drive. My mom probably said something like “changing lanes isn’t a right turn and a left turn. It’s a fluid movement.”
But I couldn’t learn to drive until I got out on the road and navigated it for myself. I needed to make sense of how the steering wheel worked together with the gas and the brakes. I needed to experience stopping distance and highway merging, white knuckled and fearlessly. I needed to drive on ice and in the rain. 30 miles per hour city streets and 65 miles per hour freeways. In short, observing my parents for 15+ years didn’t have nearly as much effect on my ability to drive as a few months behind the wheel myself. My parent now a guide rather than a presenter of information.


I hope the parallels to teaching math are fairly obvious here but I think too often we feel that because a student can follow our examples and imitate in practice, they’ve learned mathematics. In the short term, that might be true. But in the long run, do they know how soon they need to apply the brakes when approaching a stop sign at 45 mph? Or just that the brakes make the car stop?


But don’t kids need to practice their skills? Surely sense-making mathematics can’t completely replace routine practice, right? Here’s another antecdote: my high school had something called a Driving Range. Only a few high schools had them at the time, which was impressive until I realized that this Driving Range is not related to golf and wasn’t really that cool. During driver’s education class, we would go out to the driving range and practice our “skills.” At 5 miles per hour. Left turns, right turns, 3 point turns, yielding, merging, all done slower than an average runner. Sure we “practiced” all kinds of “skills.”


Could we apply them to any problem-solving situation on, for instance, a real road? Not a chance. Most math practice is just like the driving range. We explain to our students how the left turn works versus the right turn and then send them off to practice at 5 miles per hour on the driving range. And then wonder why they have no idea what to do when they they venture out onto the road when it’s snowing.
Many who have ridden with me might
disagree that I’ve mastered driving, but they keep letting me renew my license and operate a motor vehicle anyway. And I’m thankful my parents let me venture out past 5 miles per hour.
Kids don’t need more mathematical driving range practice. They need more behind the wheel. With an adult sitting next to them, encouraging them and guiding them. Because I believe that everyone with a driver’s license can drive at the highest levels.


*Quotes from Principles and Standards for School Mathematics, 1998.

Gratitude for a Graphing Calculator

I ended Friday here:

I got a lot of great responses from a ton of great people (thank you) bu then there was this:


To be honest, if Christopher Danielson wanted to come and teach a guest lesson on the history of writing instruments for crying out loud, I’d be bouncing-off-the-walls excited.  I think I did an ok job staying calm while he was there.  But goodness was this a treat for me and my students.

Professor Danielson taught one of my Master’s courses at the University of Minnesota about 11 years ago, and my first-year teaching self didn’t then appreciate the brilliance and talent of this man.  I saw observed him in my classroom today, effortlessly and masterfully, assess my students  their current level of understanding, guide them through an example and connect the new learning to their prior knowledge.  Thank you, Professor.  You helped me to develop better questions for my students.


And since I’m taking the time to thank Christopher, I want to show some gratitude to the other creators of the tool that has been a game changer in my and so many other classrooms:  Desmos.  Here are what rational functions used to look like in my classroom a few years ago.

I’d like to say this is better than nothing, but with misleading end behavior, I’m not sure it did us any good.  screenshot_2016-02-08-20-41-26-1.png

Here’s the same graph in Desmos:


There’s so much more to this than just calculator display.  Have you ever had a question about Desmos go unanswered?  Ever?  Have you ever used a technology tool in your classroom that was so intuitive that all students could feel successful using it? And finally, have you ever engaged with a staff of people who work more tirelessly and joyfully than the crew at Desmos, relentlessly improving their product to secure the positive future of mathematics classroom?  As a small part of the online mathematics teacher community, I want to thank everyone at Desmos from the bottom of my heart for the shift they have helped create in so many classrooms across the country and for the difference they continue to inspire in mine.

Rational Function Fan Fair

Sometimes when planning a unit, I browse through the Desmos Activity Builder.  When searching for Rational Functions, I came across Dylan Kane’s Building Rational Functions Activity.  Excellent.  I now had a muse.  Here is what I came up with for college algebra:


I like to gush over my students when they do awesome stuff, and this was no exception.  I love it when my classroom is abuzz with sense-making conversations.  I feel like this activity helped students become more comfortable with the structure of rational functions and how that equation structure is reflected in the graphs. Thanks, Dylan, for inspiring some awesome thinking in my class today.