How learning to ride a bike is like/not-like learning math, and why it should be!

November 20, 2018February 27, 2021 by Jon Orr

What are the moments that truly matter?

For me they are moments where we learn or accomplish something we are exceptionally proud of. They are moments that make us stand back and say “Wow! I did it” They are moments that we say “wait, let me get a picture of this.” They are #Snapworthy moments.

For me they are, playing that first song on guitar, catching a fish for the first time, watching my daughters swim that full length of the pool, scoring a basket in basketball, scoring a goal, singing a song in front of an audience, or riding a bike for the first time.

Learning the skills needed to accomplish these feats takes a similar path. The process has a similar experience.

We learned these things through the process of productive struggle.

Take my daughter Lucie for example. She was the last to learn how to ride a bike (tough when you have a twin too). She finally learned in a similar way, most likely, to how you learned how to ride a bike; by getting on, trying to balance while coasting, then falling over! Then, trying again.

Every time she fell she learned something. She would adjust and try again. She was struggling productively.

The difference between just struggling and productive struggle is Feedback.

Going through the struggle, using feedback and then making small gains gives us a rewarding experience. It makes us want to keep going. We build perseverance. We want to do better. This is one of the key ingredients to make moments that matter.

You know, we learn to ride bikes this way but traditionally we don’t learn or teach math this way. Vice versa is also true. —> we don’t learn to ride a bike the same way we learn math.

For a moment Imagine that we did. Imagine we structured a course on riding bikes like we structure our traditional math classes.

Here’s what the syllabus of bike class might look like, especially if it was taught in our schools. (I’ve adapted an analogy here from Dr. William Rankin).

Day 1:

“Welcome to your first class on bicycle riding. It’s going to be a great semester! We’ll start off week 1 with learning all about the tires. Tires are super important they’re the life of the bike. Learning about tires is important because it will help us be ready when we ride a bike.

During week 2 and 3 we’ll go over how the pedals work. Pedals are vital, they help make the bike move. In those weeks we’ll learn how that happens so when we start riding bikes we’ll be ready.

In week 4 we’ll have a test on the tires and pedals and then we’ll move on to study the handle bars. We won’t revisit the tires and pedals again until the end of the year so make sure you study for this test!

Weeks 5-8 is for Brakes. Brakes are vital to controlling the bike. I know they are related to the handle bars but handle bars were last unit. We don’t want to mix the two.

Weeks 9-10 are for Gears! I know they’re part of the pedal, wheels, and handle bars, but we’ll just talk about gears those weeks. You’ll need to use them when you start riding your bike.

Well ……That’s all we have time for in this course….

If you take our next course we’ll learn all about balance, whoa, that’s a biggie when it comes to bike riding.

When do we actually ride bikes?

That’s when you graduate!“

Silly right!!

What did you do when you learned how to ride a bike? You just jumped on and rode! Just like Lucie did. You felt a purpose to what you were doing. You learned as you were riding. It was a memorable moment.

But that silly bike class is the way we traditional teach math class. We tell our students that a purpose of math learning is so they can solve problems in the real world! We hold it over their heads that real problem solving is only for when you’re in the real world — done all your schooling.

We’ve traditionally taught math concepts in siloed units as if one math strand isn’t connected at all to another.

We say now,

JUST RIDE BIKES

Teach through problem solving. Productive struggle teaches the resilience we are looking for in our students. Just get on the bike and ride it!

In many of my past “problem solving lessons” I wasn’t really teaching students how to become better problem solvers.

If we’re giving step by step guides to solving problems in our classes are we really teaching problem solving? How much genuine problem solving are we doing in our math classes?

Teach content through problem solving. It’s the productive struggle – feedback cycle that really teaches our students to build resilience and their problem solving skills. It’s the productive struggle – feedback cycle that will create moments that your students will feel pride in. Those are the moments that matter. Just get on the bike and ride it!

[UPDATE]

Carla, a participant from our Making Math Moments That Matter online workshop pointed me to a fascinating video –> “The Backwards Brain Bicycle” from SmarterEveryDay. The video illustrates the notion that we may have the knowledge of how something works but we don’t always have the understanding of making it work.

Watch:

How does the message of this video relate to math education? –> We may have the knowledge that we need to Just Ride Bikes so that our students can become better problem solvers while at the same time creating meaningful moments but we don’t understand exactly how to do that.

We ourselves need to Just Ride.

We have to unlearn what we understand about teaching math class so that we can build a new path towards Making Math Moments That Matter.

Resources to help “ride bikes”

DOWNLOAD THE BUILDING RESILIENT PROBLEMS SOLVERS GUIDE

ACCESS THE SPIRALLING MATH CLASS VIDEO SERIES & GUIDE

New to Using 3 Act Math Tasks?

Creating Math Moments: How we can transform typical textbook problems into moments that matter.

November 14, 2018February 27, 2021 by Jon Orr

In an ongoing effort to demonstrate that we can apply the 3 part framework of Spark Curiosity, Fuel Sense Making, Igniting Teacher Moves to any lesson in math class we’ll tackle the common problem of finding the equation of a line between two points. Like this basic problem:

Sometimes textbooks may even jazz it up a bit to give it some context like this one.

Let’s re make this lesson to fall under our 3-part framework.

Even though this is a grade 9 and 10 expectation here in Ontario you’ll find that this problem is quite accessible for many grades.

In particular you could use it to uncover:

Find the slope (rate of change) of a line between two points;
Find the equation (rate of change) of a line between two points;
Model real-life relationships involving constant rates;
Model linear relationships using tables of values, graphs, and equations.

Spark Curiosity

We’ve been arguing that instead of finding a context that will make students interested we should follow the curiosity path which we’ve described in the first two lessons in our 4 part video series instead.

Let’s consider the big idea here: We want students to build an algebraic representation of a linear relation using only two values.

To withhold information and build anticipation we will strip all the numbers and questions and ease into the lesson. To help create the classroom culture that values student voice, student thinking, and growth we’ll ask students to fill in the two blanks here:

Setting the floor low will help our students feel attached to the math problem that is coming. The more attached and invested they will feel the more internal motivation they will have to pursue the problem to the end.

In behavioural economics there is a theory known as The Sunk Cost Fallacy.

Or also known as Escalation of commitment.

From Wikipedia,

Escalation of commitment is a human behavior pattern in which an individual or group facing increasingly negative outcomes from some decision, action, or investment nevertheless continues the same behavior rather than alter course. The actor maintains behaviors that are irrational, but align with previous decisions and actions.

You see, we humans are inclined to avoid loss. We will continue with a project or line of thinking if we feel that if we abandoned it we would incur loss. Even if that abandonment was better for us.

For example, my first car, a 1993 Ford Escort – you know, this is the car that had the automatic seatbelts. When you sit down and turn the car on the seat belt came up and automatically moved over your shoulder. One winter the heater in the car stopped working and I paid over $1000 to have it fixed. Then not long after something else broke on the car and instead of saying enough with this car I said, “Well, I just paid $1000 to fix it if I don’t fix it now then it’s like my $1000 was wasted.” This bias I just exhibited is an example of the sunk cost fallacy. I wanted to throw bad money after good. The $1000 I previously spent was a sunk cost and there’s no way I could get that back so the $1000 shouldn’t play a roll in my new decision to fix the car. I should decide to fix the car or not fix the car without letting that $1000 affect this decision.

The sunk cost fallacy makes us feel that if we invest time, money, resources into a project or decision that we should keep going with that project or decision so we avoid loss. The escalation of our commitment keeps us in the game.

In math education we can use our students own tendencies of avoiding loss for their own good. By setting a low floor in activities, we are easing our students into those activities and lessons so that it will be harder for them to just quit and give up once they are deeply invested in the activity . They won’t want to feel that what they’ve done so far in the activity was a waste of time and resources. They’ve sunk a cost into the activity and will continue with it to avoid loss. You can read another application in education of the Sunk Cost Fallacy from Robert Kaplinsky.

Your students will fill in various items and values for this problem. In my class this was a fun moment as we shared out what they wanted to buy and for how much.

Fuel Sense Making by Revealing Information

So now we’ll move down the Curiosity Path and narrow the focus to give a little more information.

How much would 12 shirts cost?

Students can make quick predictions before revealing the information slowly…

We don’t want to waste all the work we’ve done on escalating our students commitment so we’ll move down the curiosity path a little bit more and avoid the rushing to the algorithm. Students will use the given information and their prior knowledge to build a strategy to solving this problem.

Fuelling Sense Making by Anticipating

We are strong believers and practitioners in the PDF or the book 5 Practices For Orchestrating Productive Mathematics Discussions. So in preparation for this lesson we used our Anticipation, Selection, and Sequencing template to brainstorm possible solutions and strategies our students will try.

You can grab a blank copy of this template here.

To maximize your mathematical discussions you may want to sequence the strategies from most common to least common.

For example;

You can expect many students to try to find a unit rate to solve this problem. This is quite natural! It makes sense to find the price per shirt. However, not all situations are directly proportional. We can ask our students: How do we know this is a direct proportional relationship?

When students find the unit rate for 12 shirts at $122 and then again for 24 shirts at $209 they will see that it doesn’t cost the same per shirt! WHAT!?

Something else is going on here. You may want to give a small hint here asking, “hmmm, If 12 shirts cost $122 does 24 shirts – which is double the amount of shirts cost $122 x 2? How much more does 24 shirts cost? What would 36 shirts cost?

Students who noticed this right away may draw a double number line to show the changing prices and eventually determine the cost per shirt.

Students who have found the cost per shirt will still notice that simply multiplying the cost per shirt by the number of shirts STILL doesn’t get the cost — there is some other value that consistently needs to be added – The initial value or fixed cost.

Have a discussion at what this fixed cost could be — shipping charges? Overhead costs? Printing rental fee? ect. With this new calculation rule students can move on to verify that it does indeed work with 200 shirts, and then finally find the cost of 1100 shirts. You may even want to steer your discussion towards finding an algebraic representation of this relation.

Some students may represent this pattern as a table instead of a double number line. Depending on your grade level you may also want to use the word slope to represent the cost per shirt. If you see this solution from your student you’ll want to push for an algebraic representation

You may see some students turning toward Desmos and graphing the points to find an algebraic representation. We definitely anticipated this having taught this lesson in a grade 10 applied class.

The order you present these strategies/solutions will depend on your lesson goal. If you are trying to achieve the goal from the top of this post (Finding an equation of line between two points) then you most likely will want to end with finding the algebraic representation and then showing how you can use Desmos to verify that representation.

Finally we can show students that if the relation is linear, we really only need two points.

We feel that if we can take this particular learning goal and modify the delivery and teacher moves to create a math moment that matters we can do this with any textbook problem. What lesson should we make over next?

If you haven’t checked out our 4-part video series yet, get over there now:

How to Create Resilient Problem Solvers

DOWNLOAD THE BUILDING RESILIENT PROBLEMS SOLVERS GUIDE

We also have a course inside our Math Educator PD Academy all about how to transform textbook problems into curiosity machines. Best of all, you can join the Academy free for 30 days and cancel anytime.

That should be enough time for you to dive in and learn a ton from that course.

Hour Glass Multiples

November 10, 2018November 11, 2018 by Jon Orr

Sparking Curiosity & Fuelling Sense-Making with the Least Common Multiple.

In this 3-Act Task students will be presented with a puzzling video of 3 “hour glass” sand timers. They’ll solve a brain-teaser like problem while ultimately learning about common multiples and the least common multiple (LCM).

In this particular, this task can be used for

estimation;
spacial sense;
volume;
counting in multiples;
least common multiple;

Act 1: Sparking Curiosity

Ask students to create a notice/wonder table or you can use one that Kyle Pearce and I built for our online workshop Making Math Moments That Matter.

Ask your students to write down anything they notice and anything they wonder while viewing this video:

<span data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;" class="mce_SELRES_start"></span>
Then have them share with elbow partners and then finally with the entire class.

Some possible notices and wonders:

I see three different colour timers.
Is that sand?
Whose house is that?
Are they timing the same amount?
What times will they time?
Will all three timers ever end at the same time? If so, when?
Is the timer in minutes?
I think the yellow timer times for 3 minutes.

After capturing all the notice and wonders on the front board steer the class to working on the problem

“Will all three timers ever run out of sand at the same time? If so, when? If never, why not?”

Assume that we will keep turning over a timer after the sand runs out.

Take a few minutes to have your students estimate when the timers will all run out at the same time –> “Predict with reasoning”.

Act 2: Reveal Information to Fuel Sense-Making.

To avoid rushing to the algorithm push down the curiosity path some more. Instead of just handing over all the necessary information to solve a problem ask the students what they want to know more about. For example student 1 might say “I’d like to know the times of all the timers”. As a teacher your next question should be: “I see, and if I gave you that information what would you do with it?” We can learn what our students understand and are thinking with their response to one prompt. By asking them to anticipate what they need forces them to develop a problem solving strategy.

After hearing a few students out, give them this information: But make them guess first. What time does each timer time?

Reveal the timers:

After this reveal send students to their vertical spaces to explore the strategies they began in the anticipation stage to determine when the timers will run out of sand at the exact same time.

Strategies you may see:

Drawings that show how much time is left every time one timer runs out.
lists of the multiples of 2, 3, and 5.
tables that track minute by minute.

Fuel Sense-Making to Consolidate Learning.

Depending on your grade range and student ability you’ll want to frame your consolidation so showcase your target learning goal.

I’m sure most learning goals will include a triple number line showing how multiples of 2,3, and 5 overlap.

Clearly show using the lines how the 2 and 3 minute timer will be turned over at the same time at the 6 minute mark. Then show them all the common multiples between 2 and 3.

Finally bring in the multiples of 5 to the mix.

As part of your consolidation show this video which overlays the common multiples as they occur in the reveal video. Students can clearly see that when the timers are turned over at the same time we have a common multiple.

Here is a reveal video without the number line overlay.

Try this lesson out in your class and report back here in the comments to tell us how it went.

Are you new to 3-Act Math problems? Grab our guide to running these problems in your classroom. Learn tips, suggestions, and avoid common mistakes of using these types of tasks.

New to Using 3 Act Math Tasks?

Acknowledgements.

I want to thank Michael Jacobs for turning my thinking towards thinking about the least common multiple. The creation story of the above task comes from this hour glass timer I bought from David’s Tea

I just bought this triple “hour” glass timer! I think we can pull some math from it. What do you wonder? What do you want to know more about? Let’s build some math lessons together. What can we do with this? #iteachmath #mtbos #mathchat pic.twitter.com/6FCz8ivpsO

— Jon Orr (@MrOrr_geek) November 6, 2018

Mike said,

if you flip it over every time the sand runs out from one timer, how many times will you flip it over before all the timers are empty at the same time?

— Michael Jacobs (@msbjacobs) November 6, 2018

Bryan also was thinking it was screaming LCM.

Doesn’t this scream LCM to you??🤩

— Bryan Penfound 🌐 (@BryanPenfound) November 7, 2018

Which made me start thinking about how that couldn’t work with all three timers attached. So I set off to buy some new timers. I found the ones you see in the problem above.

Promote Struggle – A Hero’s Journey in Math Class

October 31, 2018February 27, 2021 by Jon Orr

How many times have I seen a student give up before they even start an unfamiliar problem in my class? A lot! It happens way too much. How can we build resilience and determination in our students? One thing we can do is to let them experience unfamiliar problems regularly and help them struggle through the process of working on a solution.

Let me share with you how the Hero’s Journey story arc can help with learning productive struggle in math class.

While in Miami for the Apple Distinguished Educators Institute we saw a speaker from Pixar Randy Nelson discuss the aspects of Story. More specifically he spoke about the Hero’s Journey. That talk really hit home for me. Below is how I interpreted his message and how it relates to my classroom.

A Hero’s Journey

All of these characters take a hero’s journey….

Since I’m a math teacher describing the Hero’s Journey is best done with……a graph (English teachers will know it’s shown as a cycle).

On a time vs. Tension graph the Hero’s Journey looks like this: Time is the length of the journey….or story. The tension is felt by the audience.

In the beginning the hero is introduced, the main conflict is introduced, his/her world starts to change. As the story continues the hero must battle the forces of evil & go through struggle. They must experience conflict. It’s the conflict that the hero learns about themselves. They learn their strengths and weaknesses. It’s the struggle that makes the ending awesome. Its the struggle that make the hero see the solution. It’s the lessons they’ve learned in the struggle that let’s them go aha! I know what I need to do! The story would mean nothing to the hero and the audience if the climax was much earlier in the timeline. As the story ends the character returns to a NEW normal. They take their learning and come out stronger on the other side.

This curve we see above is nothing new to us. This curve is what learners go through. It’s a Learner’s Journey too.

Now, if we take a look at our traditional math classrooms we have a format much like this:

Photo credit: Kyle Pearce

Let’s look at that structure on the Time Tension graph.

After we take up homework, we introduce the new lesson or topic or problem to work on. It’s unfamiliar so tension in our students starts to increase. But what happens is that as the tension rises it immediately falls back down. And my good buddy Kyle Pearce mentioned to me that the tension doesn’t fall all the way back to the axis….a good number of our students feel that tension permanently.

Why does the tension fall immediately?

We make that happen. We relieve students of their pain by immediately telling them HOW to solve the problem.

It’s Our examples & solutions. Students don’t get a chance to struggle & discover, Therefore the math formula, strategy or algorithm means nothing to them! The memorizers will memorize and do ok, and the non-memorizers lose again. The ideas and strategies have no real value to them.

I think students should feel the need for the math they learn. They should experience struggle ….just like the hero.

Let’s take the old model of our lessons and transform it to match the Hero’s Journey. It’s the struggle that adds value to their learning. Let’s move the reveal of math rules etc farther in the timeline. Let’s let the students productively struggle through math problems. The reveal of the “math” will mean so much more after students see and/or feel the need for it.

DOWNLOAD THE BUILDING RESILIENT PROBLEMS SOLVERS GUIDE

An example in my class this week came when I wanted to teach students how to determine an equation of a quadratic function when given some key points.

I gave them this simple Desmos Activity Builder slide from Match My Parabola

Students already knew about vertex form of a quadratic function so I knew they could put in most of this equation. It’s the “a” value that they really didn’t know how to get efficiently. So I saw a lot of this…

Students used trial and error to find -1/4 as the right “a” value. But we then asked “How do we know that’s the right one?” We then discussed plugging in a point to check to see if the right side equals the left side. They had a few more slides just like this but with different points. By the end of the last slide you could see that they really wanted a more efficient way of determining the “a” value than guessing and checking. This is where I stepped in and we discussed the idea of using one of the points and the equation to solve for the “a” value. Everyone was on board! They all had struggled before we discovered an efficient strategy. They all wanted it. If I had started class by showing them the first slide and then just telling them how to do it, I would see lack of understanding of why and bored faces.

It’s the struggle that makes the math worth it! Let’s let our students be Heroes. How are you promoting struggle in your classroom? I would love to hear of your ways. Leave a comment below.

Click here to grab the Desmos Activity Builder Activity I showed above.

The Hero’s Journey & Pentomino Puzzles

To help you wrap your mind around the Hero’s Journey as a lesson model I’ve created a Hero’s Journey Lesson Template. The exercise is to choose a lesson you have coming up in your class. How can you modify that lesson so that the flow follows a hero’s journey? Use the template below to help plan your lesson out.

Exemplar: I used the template to model how I use the Pentomino Puzzles activity to teach solving linear equations.

You can see that we slowly build up the need for a helpful efficient strategy to solve the puzzles. When my students have struggled and persevered 3 or 4 times to solve a tough puzzle, the timing is now perfect for us to step in and help them develop that skill of solving equations.

Download your copy of the Hero’s Journey Lesson Template.

Want to dive deeper into learning how to teach through the Hero’s Journey? Dive into our self-paced online math educator pd course.