I’m talking about the original, 1984 Karate Kid, not the remakes. Go back and watch it, if you’re not familiar.
In the Karate Kid, Mr. Miyagi teaches Karate to a young Danny LaRusso but it’s not in a way Danny expects. First, he asks Danny to wash and wax all the cars – there are at least four of them – and then wax them. Wax on (right hand). Wax off (left hand). Both in circular patterns. Then, he asks him to sand the floor using sanding disks (one for each hand), again in circular patterns. Next, paint the fence. Up and down, flex the wrist. Stand knees bent. Large boards, right hand. Small boards, left hand. Finally, paint the house. Side to side this time. Flex wrist.
Watching this for the first time as a teen with some friends, I wasn’t sure Mr. Miyagi wasn’t just trying to get some chores done around his house for free. Until, the structure was made visible. The structures: building strength, stamina, and muscle memory for specific karate moves were Mr. Miyagi’s goal all along.
When Danny began to complain that he hadn’t learned any Karate at all and that he needed to be ready for the upcoming tournament, Mr. Miyagi helped Danny see those structures: “Show me “Wax on, Wax off.”
Within minutes, the circular movements for waxing and sanding, along with the vertical and horizontal motions for painting were blocks for punches and kicks. The structures he had helped Danny build were made visible. Danny’s confidence grew and, well… re-watch the movie and see what happens.
That moment can happen in math classrooms, too. When it happens, it’s because, just like Mr. Miyagi, the teacher helping to make the structure(s) visible. Sometimes students say it out loud, too… “Wait, this is like that other thing we did.”
That moment is the whole game – a Miyagi Moment.
What they’re noticing isn’t a trick or a coincidence. They’re seeing structure — the relationships and properties that hold regardless of the numbers on the surface. And once a student starts seeing these structures and making sense of how they operate, mathematics stops being a collection of disconnected procedures and starts being a coherent system that actually makes sense – and is useful!
Structures like (properties of operations, equality, and place value) are everywhere in K–5 mathematics. A student who solves 6 × 7 by thinking “5 sevens is 35, plus one more seven is 42” is using the distributive property — they just don’t know it, yet. A student who finds a common denominator before adding fractions is using equivalence — same idea, different context. A student who notices that adding zero doesn’t change a number, ever, is bumping up against the additive identity — one of the most fundamental properties in all of mathematics.
The structure was always there. Most students just never got the chance to see it.
What changes when teachers make structure the focus — not as strategies to memorize, but as relationships to notice and use — is hard to overstate. Students stop asking “what am I supposed to do?” and start asking “what can I use to work with what I have?” That shift in question is a shift in mathematical identity. It’s the difference between a student who executes and a student who reasons.
The good news: you don’t need a new curriculum or a different set of standards. The structures are already in the mathematics you’re teaching. The move is to make them visible — to point at the relationship, name it simply, and ask students where they’ve seen it before.
They’ll surprise you with how much they’ve already noticed.
In the last two posts (Part 1 and Part 2), we looked at what fluency actually is and the three structures that sit underneath it. In this post, I want to make a connection that might surprise you.
Those same three structures — place value, properties of operations, and equivalence — are also at the heart of computational thinking.
I know. Stay with me.
What is Computational Thinking, Actually
Computational thinking isn’t about coding. It’s about a particular way of approaching problems — one that computer scientists and mathematicians have always shared. It has four moves:
Decomposition — breaking a complex problem into smaller, manageable parts
Pattern recognition — noticing what stays the same across different contexts
Abstraction — stripping away surface detail to see the underlying structure
Logical sequencing — ordering steps where each one depends on the one before
Sound familiar? It should. When a student solves 27 × 4 by breaking it into
and
they are decomposing. When they notice that the same move works with 27 × ¼, they are recognizing a pattern across abstractions. When they express that as a(b + c) = ab + ac, they have abstracted. These aren’t separate skills layered on top of mathematics. They are mathematics — the structural kind.
Where Numeracy Fits
Numeracy is what happens when a student can take those structures into the world and use them confidently. Not just in a math class, on a familiar problem type, with a procedure they’ve practiced. But in a grocery store, a spreadsheet, a situation they’ve never seen before.
That flexibility — the ability to deploy structure in unfamiliar contexts — is what separates a student who has memorized math from a student who understands it.
So fluency, numeracy, and computational thinking aren’t three different goals pulling in different directions. They’re three expressions of the same underlying capacity: the ability to reason from structure.
What this Means for Teaching
If the structures are the through-line, then the question worth asking about any lesson, any task, any resource is: does this make the structure visible?
Not just — is it a good activity? Not just — will students enjoy it? But: will students walk away having experienced something about how numbers behave that they can carry forward? Will they be able to use that understanding the next time they encounter something unfamiliar?
That’s the goal. Not fluency for its own sake. Not computational thinking as a trendy add-on. But students who see the structure — in numbers, in patterns, in problems they’ve never met before — and know what to do with it.
And when those students finish a challenging task and they tell you they want to do more — because somewhere along the way, they started asking “What if…?”, run with it. They’re posing rich questions and ideas to explore because they’re curious. Many of their ideas will likely involve some mathematical modeling. Another huge win.
So, the end of the lesson is not, necessarily, the end of a lesson. But it is the beginning of a mathematician.
Part 1 of this series was all about feeling the structure of mathematics. We looked at what fluency actually is — and what it compresses.
In this post, I’ll be discussing the three big structures behind all of the strategies for K-5, and how they connect to fluency and numeracy.
Across all of K–5 mathematics, three big structures do most of the heavy lifting. They don’t belong to any single grade level or content area. They run through multiplication, fractions, place value, and early algebraic thinking like a spine. Once you see them, you can’t unsee them.
Place Value
Our number system is built on a simple but powerful idea: position determines value, and each position is ten times the one to its right. That structure is what makes 27 × 4 manageable — we can split 27 into 20 and 7 precisely because of how our number system is organized. It’s also what makes the multi-digit addition and subtraction with whole numbers and decimals work. An algorithm isn’t the structure. It is the structure, compressed.
Properties of Operations
This is the one students use constantly without knowing it has a name. When a student solves 27 × 4 by doing 25 × 4 + 2 × 4, they’re using the distributive property. When they flip (3 × 8) x 5 to 3 x (8 x 5) because it’s easier to think about, they’re using associativity. These properties aren’t rules to memorize — they’re descriptions of how numbers actually behave. And they work the same way whether the numbers are whole numbers, fractions, or decimals.
(3 x 8) x 5
3 x (8 x 5)
Equivalence
This one tends to get underestimated. Equivalence isn’t just “same value, different form.” It’s the engine behind some of the most important mathematical moves students make. Finding a common denominator before adding fractions? That’s equivalence. Rewriting 28 as 20 + 8 before multiplying? That’s equivalence. Recognizing that
and
describe the same relationship? Also equivalence. Students who own this idea deeply have a tool that works across every content area they’ll encounter.
Here’s the thing about these three structures: they don’t stay in their lanes. Place value only does its full work because of equivalence. The distributive property only works with fractions because equivalence lets us rewrite them. Early algebraic thinking is what happens when students start to see all three as a unified system of structures rather than separate topics.
That’s not a coincidence. It’s the structure.
Fluency compresses the structures — place value, properties of operations, equivalence. Those structures are the shared foundation of both numeracy and computational thinking.
In my next post, I’ll connect these three structures to computational thinking — and why the connection has been there all along.
When we teach students about patterns, we’re really teaching them how to begin to think algebraically. This is especially true when we use Growing Patterns or Visual Patterns from Fawn Nguyen. When we use these patterns, we usually give the first three like you see in the image below.
We often ask students to build the next 2 or 3 stages which opens the door to describing how it grows which leads to a discussion about what is staying the same and what is changing (algebra).
But, what if we wanted to make this a more creative exploration to ignite curiosity and engage students more. What if we gave them this, instead:
What might your students do with this? How might their pattern grow? These are questions I asked myself before I did this exact thing with some fifth grade students. The results were amazing. A few students created the same pattern as above, but most created patterns that were much more interesting. And because they created themselves, they had a real “need” to know how to describe it. Some of their patterns can be seen in the images below:
Students shared how their patterns grew each had a chance to share how they saw what was changing and what was staying the same as their pattern grew.
The greatest thing about this is that instead of looking for another pattern to investigate more deeply, i.e., come up with a rule to find the number of red and yellow hexagons in any stage, we could just choose one of the students’ patterns. And we could do this for as many days as we had patterns.
This teacher wanted to use this as a springboard to plotting points on the coordinate grid, so I created this simple Amplify Patterns Task for them to complete the next day – and I added some their creative patterns into the task.
If you like, here is recording sheet that the students used. When I print these, I print two pages per sheet.
If you try this idea, I’d love to hear how it goes.
If you’re here because of you attended a session about What if…? at a conference, and would like to go straight to the conversation, click here to jump to the comments and type away! If you got here by some other means, please read on and feel free to join in the conversation as well.
Engaging students in rich tasks engages students, allows for multiple strategies, and brings everyone to the problem-solving table. But, often, at the end of the lesson, students stop thinking about math. Math class is over, so why should they keep thinking about it? What if there was a way to keep students curious? to keep students wondering? to give them permission to be creative? and keep them thinking about mathematics?
There is… It’s What if…?
Below are some resources that can be used to help incorporate What if…? into your mathematics teaching routine. You can start with one of these rich tasks. I’ve listed some other great resources for these types of problems at the end of this post.
After students have solved a rich task, we ask students to think about the problem they just solved and about what they’re still curious about. This might fall into one of three categories:
A change in the context of the original problem
A challenge of an assumption in the original problem
A new idea/alternate possibility added to the original
Students should only focus on one of these categories, initially. Trying to think in multiple directions will make this a frustrating experience, rather than a creative one. Most students tend to focus on adding a new idea/alternate possibility to the original problem or change one part of the context. Below you will find some structures to use to support students as they begin to think in terms of What if…?
Provide students with time to think privately about one of the questions in the template.
Provide students with time to talk with a partner/group about their What if…? ideas.
When it’s time to share as a class, be patient. It may take a minute to get sharing started – especially if this is the first experience with What if…?
In my previous post, I shared my personal experience with playing with mathematics as a child. That reflection prompted some digging about when we as humans begin to think mathematically, and I found something fascinating:
Experimental research shows that infants as young as 6 months have the ability to:
recognize the approximate difference between two numbers
keep precise track of small numbers, and
do simple subtraction and addition problems.
And when babies are mathematizing like this, they activate the same parts of the brain that are associated with mathematical thinking and reasoning in adults (I told you this was fascinating). So, before we can even speak, we have the ability to quantify. Our inherent curiosity and ability to think and even reason mathematically is on display here.
The idea that teaching and learning mathematics isn’t just computation, that it involves sense-making through reasoning, is research-based and builds on the natural curiosity, and the mathematical ideas and abilities we possess at a very young age.
Mathematics helps us make sense of and explain the world around us. It is the science that deals with the logic of shape, quantity, and patterns. Mathematics is a subject created based on the need to solve problems and, in my opinion, should be taught that way. It’s a beautiful, creative, and fascinating subject with applications in every field: teaching, economics, engineering, biology, chemistry, physics, entertainment, shipping, food service, geography, geology, technology, real-estate, and politics, to name just a few.
The common myth is that mathematics = computation. While computation is embedded within mathematics, it is really a very small part of a greater whole. The strong, flexible core of mathematics is all about reasoning and sense-making. The “computation part” of mathematics can be taught with this strong, flexible core in order to make sure that the computation students learn makes sense so that it can be applied to solve problems in the real world.
Ultimately, mathematics is about sense-making. The mathematics we use today to solve problems was developed by creative thinkers who asked questions like. “What if…?” “Maybe we could try…?” and “I wonder what would happen…?” This creative thinking is still happening today to solve problems like coastal erosion from tropical storms. You, your students, or your children can be one of these creative thinkers that uses mathematics and mathematical modeling to solve some of the world’s biggest problems. Let’s keep students thinking about mathematics as much as possible!
I’ve been very lax in posting for a while! In the past 5-6 years, I completed my specialist degree in K-8 Mathematics from UGA, began working as the Elementary Mathematics Program Specialist at the GaDOE, and had two kids graduate from high school.
This post, which has been sitting in my drafts for 3.5 years, is the beginning of the next phase. I have just recently retired from the GaDOE and am now working independently to deepen everyone’s (students, parents, teachers, schools, and districts) understanding of what it means to teach and learn mathematics. So, many more posts and resources are on the way!
Two Back-to-Back Events
I’ve been a teacher/coach/mathematics specialist for 30+ years and sometimes we never really know the impact we have. Sometimes we hear from current even former students and they thank us. Sometimes we get emails from parents thanking us for our work and dedication with their children. Sometimes we even get nominated and/or chosen for a teaching award. All of these are amazing and I think they keep us going. They help us get through some of the negativity that, unfortunately, can be a part of teaching.
I’ve received the thank-yous from students (current and former), which are always appreciated. I’ve received the emails from parents – often even more appreciated. I’ve even been nominated for teacher of the year a few times which was humbling in itself, but also appreciated, for sure. But that’s not why I feel humbled. Sometimes there’s more in the bigger picture of your teaching career that you didn’t know was there – something that is bigger than you could imagine. Sometimes you don’t realize the impact you have. And sometimes it hits you in the face all at once (or at least it seems like it’s all at once)!
As I mentioned, I finished my degree and graduated in December 2019. The night before graduation we had a dinner reception on campus. This was a time when all graduates from multiple programs could bring their families and show them where they’ve been spending the last two years of their lives working on the degree they’d receive the following day. After the meal, a professor from each program would say a few words about each graduate. Dr. Robyn Ovrick @RobynOvrick is our amazing professor at UGA Griffin and her words about me as a student and a teacher were enough to almost bring me to tears. I learned during her words that she had heard of me before I knew her, which I still find unbelievable. I don’t have her exact words to share but I will never forget their impact on me. When someone you respect and admire, as I do Robyn, reflects that back to you… out loud… in front of everyone, it’s very difficult to not be humbled! I barely held it together.
The next happened at graduation. But first a little back story. One of my classmates in our degree program was actually one of my former students. I taught Heather Kelley @heathermjkelley as a fifth grade student a long time ago. She has become an amazing teacher, friend, and colleague. Heather’s organizational skills helped me stay on track through all the research and weekly assignments in this program. It’s safe to say that, without her help, I might still be writing my literature review!
Prior to graduation, we were asked to nominate someone as a student speaker. Heather nominated me, but I threw it back to her. I’ve spoken a lot at conferences and workshops over the years and it was time for someone else’s voice to be heard. I had no idea that her speech would contain thoughts about her former fifth grade teacher. I was super humbled by her message! I still get choked up thinking about the wonderful things she said. Things that I had not even considered might have made a difference in her life.
My wife and father-in-law who were able to attend in person along with my parents and siblings who were able to tune in on-line, were surprised and overjoyed to hear this young woman include me in her speech. The video of her speech is below, if you’d like to hear it. And if you need someone to speak at an event, Heather would be a great choice.
At the dinner the night before, Heather joked that she was going to hand her speech to me to give. That would have been interesting, for sure!
It’s true that sometimes we never know the impact we may have, but sometimes – usually when we need to hear it – we have the pleasure to hear that we have made a positive impact in the lives of those around us.
The takeaway here, I think, is that no matter what you do in life, it all boils down to what Heather said in her speech (I’m paraphrasing here): “Find someone who nudges you out of your comfort zone, challenges you to be better, and encourages you to be the best person you can be in life. Better yet, be the one who encourages others to do their best and share it with the world!”
In this lesson, students are given 5 digits and their goal is to find the greatest product without actually doing the computation. The fifth grade students I used this with loved it. We took two days – one day to introduce the problem and a second day to try it again with different numbers, and find patterns. This is a fantastic problem because of the connections to so much more than place value!
Day 1
I started out with the same numbers Ms. Nguyen used in her example on her post. I did this because of time constraints on the first day. PTO performance dress rehearsals can really mess up a plan!
So the students were given the digits 8, 2, 4, 5, and 7. The task was to create two factors that would give the greatest product without actually doing the multiplication.
I asked students to take 90 seconds to think about it, then share their ideas with their groups. The math discussions were incredible. “582 x 47″ is less than hers because 582 x 74 has to be bigger. That one has only 47 groups of 582. This one has 74 groups of 582!” Similar comments/discussions happened at each table.
The students then shared their ideas for the two factors that would make the greatest product as I wrote them on the board:
582 x 74 = 782 x 54 = 872 x 45 =
825 x 74 = 752 x 48 = 752 x 84 =
754 x 82 = 572 x 84 =
I asked students to look carefully at their list and discuss with their tables which two they think should be removed and why. I did remind students that they should base their decisions on mathematical reasoning, not computation.
After about 90 seconds of discussion, I asked each table to identify the problem they think should go. After two tables shared, everyone agreed that these two (in red) should go.
582 x 74 =782 x 54 = 872 x 45 =
825 x 74 = 752 x 48 = 752 x 84 =
754 x 82 = 572 x 84 =
The students’ reasoning ranged from rounding to doubling and halving to just finding one more on the list that had to be greater. After that, students had to decide from the 6 left, which one would produce the greatest product. Most groups eliminated 2 or 3 more, but they struggled to find 1 because they thought it could go either way (see the green problems above).
Again, due to time constraints and PTO rehearsals, I asked them to choose one. The classroom teacher who was observing, had already found the products of all of the problems on the board. We asked for the products and wrote them on the board to some cheers of “Yes!” and some groans of “No!”
All agreed that it was a fun exercise. I loved it because the students were engaged in several of the mathematical practices, specifically constructing viable arguments and critiquing the reasoning of others This happens in other lessons, for sure, but it seemed more natural here because the disagreement was based on the reasoning used. Since not all students think the same way (and they shouldn’t), there were natural mathematical arguments discussions.
Before I left the classroom, I pulled out my deck of cards and had 5 students choose a number card to generate 5 new digits so that when they finished their PTO performance later in the evening, they could think some more about the math we did in class today. They were asked to come up with a 3-digit factor and a two digit factor that they think would give the greatest product.
Day 2
The next day, we went through the same process (the previous day’s work was on the board for them to refer to). The numbers the students drew were: 2, 9, 6, 7, 8
There were 12 ideas for the greatest product this time.
892 x 76 = 782 x 96 = 982 x 76 =
987 x 62 = 267 x 89 = 762 x 98 =
769 x 82 = 862 x 97 = 872 x 96 =
872 x 69 = 972 x 86 = 962 x 87 =
Again, I asked them to think for 90 seconds on their own, then share their thoughts with their tables about which problems could be eliminated based on mathematical reasoning. After sharing, I asked each table for their thoughts about which should go and why.
Again, the reasoning was amazing. The class, as a whole, came up with reasoning to eliminate 8 of the 12. They’re shown below in red.
892 x 76 = 782 x 96 = 982 x 76 =
987 x 62 = 267 x 89 = 762 x 98 =
769 x 82 = 862 x 97 = 872 x 96 =
872 x 69 = 972 x 86 = 962 x 87 =
The class got into a discussion about which of the remaining should go without prompting because they were so engaged in this problem! The class could not decide, but it was pretty well split between the green problems above.
Some thought it was 862 x 97 because:
“It’s almost 100 groups of 862 and 872 x 96 has one less group of a smaller number, but it isn’t enough.”
The other group countered with:
“We still have almost 100 groups of a larger number. We have one less group, but we have 10 more in each group!”
Again, the teacher was ready with the products and we checked all of the eliminated problems first to justify their earlier reasoning. We heard a few things that really made these two days worth it like: “See, I told you it was about 27,000” and “We were right get rid of that one!” Makes your heart swell up when kids say those things with mathematical confidence!
When we got down to the final two, they were on the edge of their seats! As the final products were revealed, there were no “I told you so’s” or mocking of others. The students really enjoyed the productive struggle of thinking and reasoning about greatest products. The students had a great time, but it wasn’t over yet.
As some in Queen Nguyen’s class, one student noticed a pattern from the work of both days. His explanation is described below:
“I noticed in both problems that the 2 was in the same place (red underline) and that it’s the smallest of the digits we used, so I thought about the largest numbers (digits) and checked to see if they’re in the same place and they are (Blue underline)!”
Another student chimed in with “There’s more. Look, the greatest digit is in the tens place for the second number. The next greatest digit for the first problem is 8 (green underline) and it’s in the hundreds place. For the second problem, the next greatest digit is 7 (green underline) and it’s in the hundreds place, too! And the third? greatest digit is right next to that in the tens place of the first number. And the digit before the smallest is in the ones place of the second number.”
The students were eager to check another set of numbers to see if this pattern they found could actually be a mathematical discovery. They wanted 5 more digits to use to check – they were asking to do more math! Before they left for the day, I found out that some students wondered if the pattern would change if it was a 4 digit times a 3 digit. Guess we’ll have to do another exploration!
All of this stemmed from asking students to reason about multiplication. In the process, all of their ideas were used to build a deeper understanding of multiplication and estimation. As a result, they made an interesting mathematical discovery based on the patterns they discovered and posed a new question to explore!
Thanks, again, to Fawn Nguyen for sharing this problem!
Let me just start with this. If you live in Georgia, say within a 2 hour drive to the UGA Griffin campus, seriously consider joining the Masters’ or EdS program. I’m in my first semester. It’s amazing! ‘Nuff said.
Maybe it’s just me… I thought I understood everything I needed to know about fraction equivalence… until this week. If you get to the end and think, “Oh, I already knew that!” I apologize. This is post is really for me to reflect a bit. If it helps anyone else make sense of fractions…well that’s just gravy!
It all started with an assignment for one of my graduate classes. The assignment was to read Chapter 3 from Number Talks Fractions, Decimals, and Percents by Sherry Parrish and reflect on one of the big ideas and the common misconceptions connected to those big ideas. I chose to reflect on fraction equivalence.
In the section on equivalence, Dr. Parrish talks about how students want to take fractions like 1/4 and multiply by two to get an equivalent fraction of 2/8. This misconception may be fostered by teachers who wish to make equivalent fractions easy for their students to remember. This is never a good idea! Because really… if you multiply 1/4 by two, that means you have 2 groups of 1/4. And 2 groups of 1/4 gives you 2/4 and 1/4 can’t be the same as 2/4.
What I learned next came from a phone conversation I had with Graham Fletcher about 15 seconds after I finished reading the chapter. Sometimes I just think he knows when I’m learning some math and gives me a call. He had a question about equivalent fractions. Over the course of about 45 minutes talking on the phone, I think we both deepened our understandings about what makes two fractions equivalent.
Take the rule of multiplying the numerator and the denominator both by the same number to make an equivalent fraction. If we look at 1/4 and multiply the numerator and denominator by two to get 2/8, we get an equivalent fraction, but this isn’t necessarily the whole story. To really understand fraction equivalence, I had to be asked to dive a little deeper. Graham asked me to dive deeper. As we talked, multiplying by one came up, then the multiplicative identity. These ideas definitely strengthened my understanding of fraction equivalence.
I thought I now had a deep understanding of fraction equivalence. But wait, there’s more. This is the best part. I went to class this past Saturday and Dr. Robyn Ovrick gave us this:
We were asked to fold the paper as many times as we wanted as long as all of the sections were the same size. Some of us folded once (guilty – I hate folding almost as much as I hate cutting). We shared our folds and Robyn recorded what several of us did on the smart board. Then she asked what we noticed. This is where everything came together for me. I tried to share my thoughts but I don’t think I was very successful. I was really excited about this. Here is my (1 fold) representation of an equivalent fraction for 1/4:
For my example, someone said the number of pieces doubled, and at this point (my eyes probably almost shot out of my head) I thought, but the size of the pieces are half as big. I’m usually pretty reserved and quiet, but this was too much. So, with a lot of help from colleagues in class who know me a bit better than the others it all came clear to me. We visually made equivalent fractions, but connected the visual to the multiplicative identity and even explained it in the context of paper folding.
Here it is.
The original paper shows 1/4. When we fold it in half horizontally, we get 2 times as many pieces and the pieces are half the size. This can be represented here:
The 1/4 represents the original fraction. The 2 shows that we got twice as many pieces, and the 1/2 shows that each of those pieces is half the size. With a little multiplication and the commutative property we can get something that looks like this:
Knowing that two halves is one whole is definitely part of this understanding, but seeing where it can come from in the context of paper folding allows an opportunity for a much deeper understanding. The numerator tells that there are twice as many sections as before and the denominator (really the fraction 1/2) says that the pieces are now half the size. We looked at another example of how someone folded 1/4 (someone who folded 8 times!) and noticed that it worked the similarly – we got 8 times as many pieces and the pieces were each 1/8 the size of the original. I don’t think anyone thought it wouldn’t work similarly, but it sure is nice to see your ideas validate something you thought you really understood before waking up that morning!
I’m still thinking about this and I keep making more connections. This morning, in a place where I think I do my best thinking (the shower!), I realized that this is connected to the strategy of doubling and halving for multiplication. I’ll leave you with that. Time for you to chew.
When I was growing up in (rural-ish) central New York, we had one TV. We received 5 local stations through the antenna on the roof (abc, nbc, occasionally cbs if the wind was blowing just right, then Fox came along, and a pbs station).
Growing up with five siblings meant that the first person in the living room got dibs on what show was on or there had to be a “discussion” to figure out what everyone would watch. Sometimes this ended in the TV being turned off by Mom or Dad with a “suggestion” that we go outside and get some fresh air. Other times, we would decide to figure it out on our own and end up on the local PBS station watching a man with a huge perm (this was the 1980s) paint beautiful scenes in about 25 minutes.
We (my 5 siblings and I) were all in awe while we watched Bob Ross paint wonderful paintings while talking to us (the viewers) about everything from his pet squirrels to painting techniques. And at the end of every episode I felt like I could paint just like Bob Ross! I never tried, but I felt like I could!
Recently, my kids have discovered the talent and wonder of Bob Ross through YouTube and Netflix. They love his words of wisdom:
“Just go out and talk to a tree. Make friends with it.”
“There’s nothing wrong with having a tree as a friend.”
“How do you make a round circle with a square knife? That’s your challenge for the day.”
“Any time ya learn, ya gain.”
“You can do anything you want to do. This is your world.”
And I love that they love these words of wisdom. You can find more here.
For Christmas this year, my son and I received Bob Ross T-shirts. Connor’s has just an image, while mine has a quote as well:
Bob Ross was referring to painting when he said these words; “In painting there are no mistakes, just happy accidents.” In other words, when you paint your mountain the wrong shape, treat it as a happy accident. It can still be a mountain, there may just end up being a happy tree or a happy cloud that takes care of your happy accident.
I think it works for math class, too. Recently, I modeled a Desmos lesson for a 7th grade teacher. The students had been working with expressions and equations but were struggling with the abstract ideas associated with expressions and equations. The teacher and I planned for me to model Desmos using Central Park to see how students reacted to the platform (this was their first time using Desmos) and how I managed the class using the teacher dashboard.
During the lesson, there was a lot of productive struggle. Students were working in pairs and making mistakes happy accidents. They were happy accidents! Because students kept going back for more. At times there was some frustration involved and I stepped in to ask questions like:
What are you trying to figure out?
Where did the numbers you used in your expression come from?
What do each of the numbers you used represent?
Before you click the “try it” button, how confident are you that the cars will all park?
The last question was incredibly informative. Many students who answered this question were not confident at all that their cars would all park, but as they moved through the lesson, their confidence grew.
One of the best take-aways the teacher mentioned during our post-conference was when she mentioned a certain boy and girl who she paired together so the (high performing) girl could help the (low performing) boy. The exact opposite happened. The girl was trying to crunch numbers on screen 5 with little success. The boy just needed a nudge to think about the image and to go back to some previous screens to settle some ideas in his mind before moving ahead with his idea that the answer is 8. Then, he got to expain how he knew it was 8 with the picture, conceptually, to his partner. The teacher’s mistake happy accident was in believing her students would always perform a certain way. When students are engaged in tasks that are meaningful, they tend to perform differently than when they’re given a worksheet with 30 meaningless problems on it (the norm for this class before Desmos). Ah-has all around and the “low student” shows that he knows more than the teacher thinks.
The icing on the cake? Several students walking out of the classroom could be heard saying, “That was cool.” or “That was fun.”
Let’s treat math mistakes as happy accidents, something to learn from and problem solve our way through. When students (all humans) make a mistake, synapses fire. The brain grows (More on this from Jo Boalerhere). What we do as teachers from this point, determines how much more the brain will grow. If we treat student mistakes as happy accidents, perhaps their brains will grow a bit more than if we continue to treat mistakes in the traditional manner.
Let’s hear it for Bob Ross. He probably never thought his words of wisdom about painting would be translated to the math classroom.
