Fresh Air: NCTM Reflections

I’ve been reflecting on the NCTM Annual conference in San Francisco all over the place…in my head, with my spouse, with mathy friends etc and so I almost didn’t write a blog post.  San Francisco and the conference were fabulous:  Inspiring, motivating and fullfilling in so many ways, but I didn’t think I needed to process it in writing.   Then Sunday morning I got up and decided to attend the Sunday Celebration at Glide Memorial Church.   I haven’t felt the need to step foot in a church since my daughter was baptized, but the positive energy and love radiating from this place drew me in Sunday at 9 am.

As it turns out, the hour and a half at Glide was one of the most inspirational of the a weekend jam-packed with pockets of profound moments and inspirational experiences.  The guest poet was Chinaka Hodge .  She completely blew my mind in a way that makes me want to scour the internet and devour everything she’s ever created.  Another woman read a poem titled “It Began with Verbal Abuse” where she detailed an abusive relationship that left her with a missing tooth and an addiction to drugs.  Glide became her salvation.  The pastor read a poem about her mother’s experience as an Japanese-American in the 1940’s.  She founded Glide to help others like her.


View from my hotel room of Glide Memorial Church

So what is it about Glide or NCTM or the Math-Twitter-Blog-o-Sphere (MTBoS) that makes spending time together so special?  Echoing Lisa Henry’s closing remarks from TMC 15:  It’s the community (stupid).  But there’s more.  It’s our opportunity to tell people in 3-dimensions how much we have appreciate their work in 2-dimensions. In my world, this was captured quite perfectly while meeting Henri Picciotto.  After Anna (Blinstein) was gracious enough to introduce me, I asked him if he would mind getting a picture with me.  I think his exact words were “that’s kind of weird but ok”  (which made me adore him even more).


I’m not a regular part of Glide Memorial Church like I am the MTBoS, but the invitation is the same:  Come join our community.  We accept you as you are and we want to help you become a better version of yourself.  

Thank you everyone who helped fill my heart with such joy this weekend in San Francisco.



Something to Talk About

It’s April.  Have you heard of April?  It’s that month of the school year where all students can see is the finish line, and all teachers can do is expend a never-ending amount of energy trying to motivate students in these final weeks before summer.  I mean, come on kids, it’s not even May yet!  There aren’t even leaves or goslings or temperatures consistently above 40 for goodness sake! There is so much learning to be done before we send you off into the summer sun!

Anyway, I haven’t done a Number Talk in a couple of weeks because of spring break and MCA testing.  I decided to begin anew today, and this was a treat.  We reviewed the norms, fired up our brains and got to work on 250 x 28.  When I asked for students to volunteer their answer, I got 4 incorrect answers along with the correct answer.  Second period, much to my delight, they contributed four different incorrect answers as well as the correct one.  Students who are hesitant to share were willing to explain their strategies to the class.  With their permission, I’ve included their names with their strategies:

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These 9th graders are doing a spectacular job of working together in cooperative groups as well.  When I give them something to discuss with respect to probability and statistics, they really challenge one another respectfully and support one another appropriately.  This doesn’t necessarily happen naturally but is instead developed through consistent expectations of shared group responsibilities.  Overall, it’s a tough time of year to be a high school student with summer looming.  I’m very proud of the progress my students have made thus far this trimester in making sense of all the data around them.


No Worries. We got this.

If there is an idea that I repeat over and over and over more than any other, it would be that procedural fluency must build from a conceptual foundation.   National Council of Teacher’s of Mathematics explicitly state this in Principles to Actions as one of eight mathematics teaching practices:


Then begins third trimester with a class called Algebra 2 Concepts.  These students don’t all hate math.  But most of their experiences with school mathematics have made them feel defeated by a subject that has too much power in their lives.  And I’m faced with conceptually putting my money where my mouth is as we begin quadratics and factoring.

Factoring is tough for a lot of students, and most of the struggle I see boils down to a couple of things:

  1.  Factoring is introduced using rote procedure with no connection to multiple representations.
  2. Students never remember the procedures.

And speaking of procedures, I’m as guilty as anyone.  I’ve tried them all:  Slide and Divide, Guess and Check, The Pull-Out Method (don’t ask) and every other poorly-named, easily-forgotten factoring freak show on the planet.

And then I come across the Box Method.  I’ve seen (and taught) a modified version of this method before with some immediate success but no long-term staying power.   However, the progression I’m using with this class helps kids make the connection of a factored quadratic to an area model using length times width.  Every time I help a student, I am asking about the length and width with respect to area.  Procedurally, they are dividing out common factors.  But what they are representing is the length and width of a rectangle, which they solidly understand, when multiplied, is the area of that rectangle.

We took a short quiz today, and we had some tremendous results:

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Interesting story about that 2nd one with the sign error: this student originally learned factoring in algebra 1 using guess and check. She fought me a little on trying the area model. You can kind of see here that she originally tried the guess and check again. Then realized she couldn’t quite get it and fell back on the area model. I feel like she made a great connection today with it. So proud of her.

I’m so proud of these students. They challenge my resolve every single day, and I’m lucky to get an opportunity to help them learn.

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:



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.

Watching Solitaire in Silence

Remember Windows Solitaire? I have fond memories playing this fantastic digital distracter with my high school beau on his brand new Gateway computer.  We would take turns striving for success in this card-clicking frenzy, the other watching and waiting patiently for the deck to empty.

But have you ever watched someone play solitaire on the computer?  It is so…what word comes to mind?  Frustrating?  Infuriating?  Aggravating, perhaps?  And why is that?

Check out this screenshot:


What if the player was about to click on that blue, flowery deck of cards…would you be fighting the urge to save them from their potentially game-ending error of failing to move the sequence beginning with the six of spades to its rightful place atop the seven of hearts?  Or would you idly sit by and let them to figure out that solitaire is won by carefully searching for card moves before drawing from the deck?  Would you make any suggestions for improving their game once failure was inevitable?

I think this solitaire analogy is a lot like teaching.   I realized fully today why the “productive struggle” is so hard to sustain and perhaps why teachers so often fall back on traditional methods of delivering information to students:  Watching people struggle without intervening is difficult. Just as it’s natural to want to smooth out the path for our children, it’s also tempting to do the same for our students.  It’s just easier (and so much faster) to zip Maria’s (my daughter) coat or buckle her seat belt or pick up her toys.

As a simple, mathematical example, imagine one of your students is attempting to solve a quadratic equation. They start off like this:0126161730-1.jpg

Being the savvy algebra teacher you are, you can anticipate the error that the student is most likely going to make.  You’ve seen it hundreds, if not thousands of times.  Your inner teacher voice might be thinking, “For the love of humanity, Herbie (not your real name), set the dang thing equal to zero!  Quadratic formula!  IT’S GOT A SONG, FOR GOODNESS SAKE!”

Instead, you do not impede their solving and let them continue on their merry, algebraic way.  0126161732-2.jpg

Re-enter teacher voice in your head, “Now look what you did, Herbie.  You’ve gone and…wait…one of those answers is right.  Great.  Now we’ve really got issues.”

So what do we do about this?  Clearly the student needs some redirection and the teacher’s role is to guide the learning.  But had we intervened during earlier steps, we rob this student of a golden opportunity for brain growth.  Plus, we deprive the rest of the class the chance to learn from the misconception.  Even more, what a fantastic extension we have here:  why did the student get part of the problem correct and part incorrect?

In summary, we deny students the opportunity to learn from mistakes if we  prevent them from making mistakes in the first place.

Related Side Note:  I’m currently reading The Gift of Failure by Jessica Lahey.  Her introduction about her son’s shoelace-tying trials seems strikingly familiar.  And I can use this antedote as a reminder when encountering the zippers, and the seat belts, in addition to quadratic equations.



This is Our Theorem – College Algebra

“We came up with a theorem once at my old school.  The teacher has it in a frame behind his desk.”

This statement from one of my college algebra students made me both elated and sad at the same time.  Thrilled because this is the type of mathematics I believe all students should have the chance to engage in on a regular basis.  Disappointed because this type of discovery happens so infrequently in American mathematics classrooms that the incident warranted a sacred place on the wall of this teacher’s room.

In College Algebra, part of today’s learning objective was to define a polynomial function and determine some key features.  I have the awesome types of students that if I were to write down the surly definition and features of a polynomial function onto the whiteboard, each would follow in lock-step and write it in their notebooks solidifying it’s place among mathematical obscurity.

Today, we were going to break that cycle with something different.

But I needed to know where they were at, so I had them write down what they knew about a polynomial function.


After some discussion and leading questions, we were sure that linear, quadratic, cubic, quartic, x^5, x^6, and so on were all polynomial functions.  Awesome. We weren’t, however, as sure about functions including negative exponents, roots, sin/cos, or algebraic fractions.

What makes this group we are sure about special?  Last week, we spent a considerable amount of time on features of functions including domains, end behavior, intercepts, intervals, symmetry, and turning points.  In their groups, I had them examine the graphs of these alleged “polynomials” through the lens of the features of functions.

Two similarities emerged as significant:


Questions:  Was this true of all polynomial functions?  And if both conditions were not met, could we exclude it from our known polynomial functions?  Hiding my initial excitement, I then had them look at our list of “questionable” functions. For example, did “y = 9 + 1/x” meet each of these two criteria?

Christopher Danielson suggested that my class give this new theorem a name, so we could refer back to it with ease:

“Class. We have found that all polynomials blah blah blah…” [while writing the statement of the theorem on the board.]  In mathematics, when we have an important finding like this, and when all mathematicians have agreed the finding is true, it gets a name.  Sometimes it is named for a person, such as ‘Fermat’s Last Theorem’; sometimes it is named for what it says, as in ‘The Triangle Inequality’.  But that name makes it possible to refer to it going forward. It helps us to remember and to use the thing we figured out. So we need to name our theorem. Who has a name they’d like to suggest?”

Alas, the excitement of naming the theorem will have to wait until tomorrow.

Conceptual Function Foundation Follow-up

One of my favorite parts about teaching is having the opportunity to see growth in myself and my students.  I love when a lesson I have used with success previously gets even better the next time around, especially when it is a lesson that exemplifies my teaching philosophy.

Last trimester, I used the New Visions for Public Schools’ Algebra 2 functions unit to help build a conceptual foundation.  I wrote about that experience here.

Day 1:  To start off this unit on functions, they were in pairs:  one partner facing the projector screen and one facing the back of the room.  I then drew an arbitrary function on an unlabeled set of axes.  The person facing the screen needed to use words only (no pointing, no gestures) to help their partner draw the graph.  The person drawing was not allowed to ask questions, just draw what they hear.  For example:

0113160857.jpg 0113160857a-1.jpg

Follow-up question (before they turn and look): What could you tell your partner to help them improve their graph?

Afterward, we talked about what descriptions were helpful.  My goal was to turn those descriptors such as “hills” and “curvy lines” into more specific function features.  Classwork (with their group of 4) involved looking at specific graphical examples to define end behavior, turning points, positive and negative intervals, etc.

Day 2:  Gave students a set of 36 graphs.  They needed to sort the graphs into exactly four groups based on their function feature.  I then followed up by having them choose one graph that best represented each group.  Some examples of student work:

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Next week, we will look at their groupings and decide which ones highlight important features of the graphs.  Then, we will see if we can add some specificity and some real-world.  Seriously, I love the way this progression helps my students make sense of the function features.  It sure beats standing at the front and going through examples.