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!
A partner game for developing student flexibility with number combinations to five and ten (or even 20) that has connections to the first Thanksgiving.
If you teach Kindergarten, this post (and game) is for you! Ok, 1st Grade teachers can use this too, in the beginning of the year, or with more ten frames.
I can’t believe I haven’t written a post about this. I learned about Hubbub about 17 years ago, when I was just starting as a math coach in my district. I heard in mentioned in a video that some Kindergarten students were watching to learn about the first Thanksgiving. Apparently, children played Hubbub at the first Thanksgiving. I jotted down the rules and added a double ten-frame as a score board.
Materials needed:
5 two-color counters per pair of students
Double ten-frames (one per pair) (one per student, if working on composing numbers to 20)
Cubes (10 per student) (20 per student if each child has a double ten-frame)
Rules:
The object of the game is to score 10 points (or 20 points if you use a double ten-frame for each student).
Players alternate shaking and spilling the 5 two-color counters. A player continues their turn (shaking and spilling counters to earn points) until they shake and spill 4 counters of one color and 1 counter of another color.
Points are scored according to how the two-color counters land (see scoring sheet linked below):
If the counters show 3 of one color and 2 of another color, the player earns 1 point.
If all five counters show the same color, the player earns 2 points.
If the counters show 4 of one color and 1 of another color, the counters get passed to your partner.
Players keep score by placing cubes on a ten-frame for each point earned. Players never lose the points they earn.
Teaching the Game
This is a partner game. I always model this game with the teacher on the carpet and we talk about the rules of the game, how to toss the counters, and good sportsmanship.
I sometimes use same-different images as an opener to this game. The slides for these can be found below, along with my lesson notes (standards included).
I introduce the two-color counters and then I shake them in my hands and gently drop them on the carpet. The counters will show one of six combinations:
As you can see, the possibilities are grouped by combinations to five (3 yellow and 2 red or 3 red and 2 yellow, etc.)
These combinations have scores/ consequences associated with them.
As we model the game, each time a new combination shows up, we discuss what happens. The scores and consequences for each pair of combinations is shown below.
A players turn only ends if they roll 4 of one color and 1 of another color. This is important, because most games we play with students, players each take one turn after each roll. We continue to model playing the game discussing good sportsmanship and asking questions, like “How many more do I need to get to 10?” “How many points do you have?” “Who has more points?”.
When someone wins – almost always not me – we ask the students if they are ready to play. The teachers pair the students with their partners, and we give them their materials. Students then begin to play and we monitor, asking students questions about combinations to five and 10 as they play. Students get very excited and, since we model it, we often hear students who do not “win” say, “That was a good game, would you like to play again?”
If you use this game, please share your experiences. I’d love to hear how you may have changed the game to suit your students’ needs.
Full disclosure here – I never won this game (17 years of playing) until last week when I went to my wife’s school to teach this to the Kindergarten classes before Thanksgiving break. They took a picture because it was the first time I ever won!
This is the first in a series of posts about learning to think like a mathematician. This is my first memory of playing with numbers, questioning my own thinking, and making sense of new ideas. As my brother Peter said, when I shot the video clip below, “and this is how it all began!”
When I was around 5 years, my older sister showed me an adding machine that was in my grandmother’s closet. It was large (to a fiver year old) and very heavy. It was completely mechanical and had 81 numbered hexagonal keys – 9 rows of 9 keys (see image, above). Each column of keys was numbered with the digits 1-9. Pressing a digit in a given column would display that digit in a corresponding window along the bottom row. There was also a lever on the right side that could be pulled to reset all of the column windows to zero.
My first experience with this incredible machine, thanks to my sister, Susie, was to press the “1” in the lower right corner (the ones place) continuously. The tenth press felt different and the display at the bottom returned to zero, but the place just to the left turned to display a “1.” Pressing the same “1” button felt the same again for 9 presses, then on the tenth press if felt different again, and again the window below turned to a “0.” And the window to the left turned to a 2. I could make the second window count to ten, too? By only pressing the one button? This was fascinating! After a long time, and lots of presses of the same key, when all of the bottom windows displayed nines, I would press that same “1” button and it would “feel a little different” and something amazing and extremely satisfying happened. Watch what happened.
All of the 9s would alternately flip to 0s, like a row of dominoes flipping over. It was a sensory explosion! I could hear the dials flipping. I could see them flipping, and I could feel when it should happen. It seemed magical at the time.
I (we) were told not to play with this because it was an antique and I (we) might break it. I was a pretty good rule follower, but the allure of this machine was too much. I would sneak into my grandmother’s closet and lug that heavy machine out to play with it – a lot – even when I was a little older, just to get to see, feel, and hear that domino-like effect of numbers flipping over. Even now, it makes me smile to think about it.
I would spend a lot of time pressing that “1” key until it got to 9, then press it one more time and a “1” would pop up in the place to the left and a zero was in the spot below the key I was pressing. It never got old. I kept pressing that key, not just to see what I had come to know would happen, but to figure it out. I began to make predictions and ask myself questions, like when the display read “249” one more press and the digit in the place to the left of my key changed to a 5. “I bet it changes to a six next time.” When it did, I was hooked. This continued and every time the next column got to a 9, I’d quickly press my “1” key one more time to get it to click over. What I noticed, though, is that it took a lot longer to get each row to 9, but that wasn’t enough. I wanted to know how much longer. I kept going because the more nines I had, the cooler it sounded when they all flipped over! Eventually, I figured out that it took ten presses of that “1” key to make the next column change, and that column had to change 10 times to make the next column change. I discovered a pattern of tens.
What I didn’t realize, initially, was that I didn’t have to keep adding ones to get the full row of nines (I was still only about 5). I could just press each of the nine “1” keys 9 times, then add 1 more by pressing the far right “1” key. Then, satisfaction and amazement came much sooner! Once I figured this out, it was a much quicker experience but, frankly, a little less satisfying. After a while, I really took note of the other keys and realized that I could use some of them to my advantage as well. For example, I could press the “9” key in each column once and then press the far right “1” key. Still satisfying and much more efficient to get to the end result, but not as enjoyable as seeing this one button do so much. Watch the video below to see and hear what I loved so much.
This was one of my first experiences with playing with mathematics and the effects it can have in the sense-making and building deep understandings of mathematical ideas. I believe it is one of my earliest mathematics learning experiences, and I believe it had a huge impact in how I think, mathematically.
My hope is that this and some future posts may cause you to reflect on some of your own, similar, experiences. If so, please share. I’d love to hear your experiences. Stories like this, I think, have the potential to bring to light just how beautiful mathematics can be and the connections that can be made by studying this amazing subject!
Side note: I really wanted to take this amazing contraption apart to see how it worked. My parents are thankful that I never did, but I still wonder what the inner workings of this adding machine look like. Unfortunately, I never got to find out, but I am still very curious. Perhaps there’s a video out there that I can watch so I don’t ruin this antique with my tinkering.
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.
My lessons never stay the same. They’re always evolving. Recently, I’ve taken a look at some 3-Act Tasks I created and I noticed:
Some of the tasks are lacking an act.
Others have resources that no students ask for (at least students that I’ve worked with).
The quality is low (shaky camera, point of changes, etc.)
So, I finally had a minute (read 2 days) and revisited each. Below, you’ll see the tasks I’ve chosen to revisit. An explanation of the original, what I changed, and why I changed it follows. If you’d like to skip this and get to the revisited tasks, click here.
My very first attempt at a 3-act task was the Candy Bowl task. I was working in an elementary school at the time and Graham Fletcher had created problem to get 2nd and 3rd grade students reasoning about subtraction by removing the numbers from the problem context. His context involved the lunchroom and numbers of students in three classes. We talked on the phone about this for a while and though I liked the problem, I wasn’t crazy about the context. I sat in my room trying to think of a context that would be a bit more engaging for students to think about. And the Candy Bowl was created.
It was a good problem, but it really lacked one of the most basic parts of a 3-Act Task… The third act. The reveal was weak, because it relied on the teacher to give students validation. The updated version, which had to be done from scratch (apparently whoppers candies are no where to be found anywhere near Valentine’s day), can be found here with all new updated resources for Act 2 and new video including two reveals, depending on which question students decide to tackle.
Another one of my early tasks was Sweet Tart Hearts. I really liked this one from the beginning. There is a huge focus on estimation which allows for students to obtain solutions that are close, but not exact in most cases. This also allows for the teacher to facilitate a discussion about why answers may not be exact for a variety of reasons. But again, it really lacked that third act. The task was good, but the closing of the lesson was weak due to the fact that the students were relying on the “all knowing” teacher to give them affirmation.
Apparently Sweet Tart Hearts are a hot commodity a few days before Valentine’s day. I went out the other day for a quick run to pick up a bag. I had to go to 4 stores and finally found a bag (the last one). I thought it would take about 10 minutes to do this revisit. Surely the numbers for the colors would be similar to the last time. Not only was that not true, but Sweet Tarts changed the orange hearts to yellow! But, the revisit is all done and I’m very pleased with the new reveal which allows the video to reveal the answer and the teacher to focus students on the reasonableness of their solutions.
My final revisit is the Penny Cube. It is probably my favorite task. I’ve certainly heard more from teachers about this task than any of the others. I think I got the reveal right on this one. The problem I found with this task was that I thought students would ask for things that I would want. The first time I did this task with students, I guided them to the information I had ready for them. They didn’t care anything about the dimensions of a penny. They just wanted some pennies and a ruler. It’s amazing what you learn when you listen to students, rather than try to tell them everything you think they need to know. So, to all of the students out there, Thank you for making your voices heard!
So, this was the quickest fix. I just updated the Penny Cube page (all of the coin specifications are still there – in case anyone wants them).
Note: In this post I share how I changed my approach to teaching the Penny Cube task.
So, it took a few days, but I’ve revisited some tasks that have been bugging me for a while and I hope it’s for the best. I know I’ll probably give these another look in the future. I’ll just need to start in early January to make sure I get the candy I need.
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). This was a time when TV programming on the major networks actually ended at about 1:00 a.m. with a video of the American flag waving in the wind and the national anthem playing. When that was over, there was nothing on TV but static. This is something my kids can’t imagine. Not that they watch regular TV that often anyway (YouTube, Vimeo, etc.), but every time they turn it on, there are at least 100 shows to choose from on 4 TVs.
This wasn’t the case for my siblings and me. Usually, 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.
For the second year in a row, I had the privilege and honor to give an ignite talk at the Georgia Math Conference (Last year’s talk can be found here.) What makes ignite talk sessions great is that you get a taste of what several speakers are passionate about and you get to walk away with at least one ember of at least one of those talks beginning to burn in you!
Special thanks to Graham Fletcher for putting this all together (in pre and post production!). Graham is top notch, “for sure” (Must be a little of my inner Canadian there).
The featured speakers this year in the order of their talk:
Under the Dome 