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). 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.
Engaging students in math has always been a goal for me. No… more than a goal…. a passion! And it’s not always easy to do. For example, I used to hate teaching students how to find the sum of an arithmetic series. I didn’t hate it because it was difficult to teach or because students had an overwhelming difficulty learning it. I hated it because I was the only one that saw the beauty in it. I was the only one who was passionate about it.
This lesson was “fun” (I use the quotes to denote that this was a fun lesson for me – not so much for my students). But this all changed when I allowed my students the opportunity to think for themselves.
The task was very simple in concept: Find the sum of the series of the numbers 1-20.
Before going any further, it may be useful to know about the
Class norms:
Estimate first,
the answer is never enough,
reasoning, explaining and looking for patterns are all expectations,
if you found one way, look again, you may find a more efficient way,
get out of your own head and talk about the math with your partner/group while you work
Several started adding 1 + 2 + 3 + 4 + . . .+ 19 + 20. I noticed this and asked those groups for one word to describe their strategy. Sample responses: boring, lame, tedious (actually proud of that one), calculator worthy…
My reply to each of their descriptions: If your strategy is [insert one: boring, lame, tedious, or just plain calculator worthy] why do you feel the need to use it?
Sometimes students get stuck in their own thinking and just need to be made aware of it. To help nudge students to think in other ways, I had bowls of tiles with the numbers 1-20 written on them available for groups to use.
It took several minutes before students began to grab tiles and began to notice things like:
“Hey, Mr. W., we can make a bunch of 20s.”
“We got a bunch of 21s. 10 of them. It can’t be that easy, right?”
“We made 10s and 30s. How did you make 21s?”
“We did the 20s too. That’s the easiest way for us.”
It was a bit chaotic, and I didn’t know it then, but there was a passion building. This wasn’t just engaging, these students were ALL IN. They were more than engaged and wanted to learn more about the strategies they came up with. They wanted to share. Needed to know. And the answer was almost irrelevant. The connections between all of their strategies became the focus.
From here, getting to the algebra made sense. How would you find the sum of the numbers 1-50? 1-90? 1-100? What about 5-50? Some saw their ideas with the tiles transfer easily to an algebraic expression and equation. Others not so much. So, more time to talk and share. More time to find a strategy that is more convenient to generalize for a series of numbers of any range. The success of the students’ mathematical ideas gave them power to reach further – to take another chance.
Teaching the lesson this way was a definite improvement on the original. In this version, the students’ ideas matter, so students matter. In this version, students think for themselves and collaborate with others, and in turn get validation of their thinking, so students matter. In this version, students built some passion. They fed off of each other. And the content mattered because of the students’ interaction with it.
Is this lesson the best it can be? I’m not sure. So, I’ll continue to try to improve on it.
Last year I had the honor of being asked to write four posts for NCTM’s Math Teaching in the Middle School Blog: Blogarithm (one of the coolest math blog names out there). They were posted every two weeks from November through the end of December (which just shows that I can post more frequently if someone is reminding me every other week that my next post is due (thanks Clayton).
The four posts are a reflection of a lesson I taught with a 6th grade teacher, in September of last year, who was worried (and rightfully so) that her students didn’t know their multiplication facts. After a long conference, we decided to teach a lesson together. I modeled some pedagogical ideas and she supported students by asking questions (certain restrictions may have applied).
So there’s this thing going around about algorithms being a bad.
They’re not. What’s bad is when students learn an algorithm – any algorithm from anyone – without making sense of it on their own.
Enter (what is considered by some) the buzz word: “Strategy” (Guess what, the strategies being taught now are all algorithms).
I often hear teachers talking about teaching students several different strategies for (insert operation here). Good, right?
Not so much. Here’s the thing. If teachers teach all of these different strategies, without student understanding at the forefront, they may as well teach the standard algorithm. The worst part here is that students can actually be worse off being taught these multiple strategies without understanding than one algorithm without understanding.
Essentially, students are being force-fed strategies (aka algorithms) that they don’t understand and they feel like they need to memorize all of these steps for all of these strategies. We’re going down the wrong path here. Our destination was right, but we took a wrong turn somewhere.
It’s time to stop the madness!
How? you ask.
Let me tell you a story…
Back in early fall 2007, when I was still a toddler of a math coach, my beautiful wife’s grandmother passed away and the whole family went to her school on the weekend to help her get some lessons together for the few days she would be out. Truthfully, I was the only one helping since the kids were 7 and 4 at the time. Kim gave me jobs to do and I did them with precision and efficiency. One of the tasks she gave me was to make a 18 copies of a few tasks for her students to complete during her absence.
To help her out, I took my son, Connor (the second grader), with me to the copy room so she’d only have 1 child to keep track of while she was trying to work. When we got to the teacher work room, Connor watched as I placed the small stack of papers on the copy machine tray, typed in the number of copies (18) needed and then hit the copy button. Within seconds he asked me (in the most exasperated voice he could muster) “How many copies is that going to make?”
I swear, when things like this happen, mathematicians in heaven play harmonious chords on harps using ratios. I hear them and respond accordingly. This time, I brought Connor over to the copy machine screen and showed him the numbers.
Me: “Do you see that 5 right there? That’s how many papers, the copy machine counted, and that 18 right there? That’s how many copies of each piece of paper I asked the copy machine to make.”
Connor: “Oh…”
Commercial break: I didn’t really expect much more than an estimate. This was September and Connor was a second grader. He may have heard the word multiplication, but likely didn’t know what it meant.
And we’re back! His eyes looked up as he thought about this briefly and within seconds of his utterance of “Oh,” he said in a thinking kind of voice, “50…..”
Now, I’m not one to interrupt a student’s thought process – I work with teachers to keep them from doing it. I actually remember having a mental argument with myself about whether I should ask him a question. I was so excited in this moment, I couldn’t help myself. I asked (with as much calm as I could), “Where did you get 50?”
I kid you not, he replied by pulling me over to the screen on the copier and said, “You see that 1 right there (in the 18), that’s a ten. And 5 tens is 50.”
I could hardly contain myself. Naturally, since I had already interrupted him, I asked what he was going to do next. I was floored when he said that he didn’t know how to do five eights. I was floored because he knew how to multiply a 2 digit number, he just lacked the tools to do so. In the context of this copy machine excursion, Connor made sense of the problem, reasoned quantitatively, showed a good degree of precision, and I’m sure if he had some tools, he would’ve come to a solution within minutes.
As we left the teacher work room, with copies in hand, I asked him to think about it for a bit and see what he could come up with. When we got back to my wife’s room, I told her all about it. When I got to the part where he didn’t know how to do five eights, I called across the room to him and asked him if he figured out what five eights was. As he said, “No.” he paused and thought for a few seconds and said, “Can I do 8 fives? ‘Cause that’s 40.” Before I could ask him (thank God), “What about the other 50?” He said, “40…50…90!”
This second grade boy (My Son!) who had never been taught multiplication, what it means, or any algorithm for it, created a strategy for finding a solution to a contextual problem that most of us would solve using multiplication. He came up with the strategy. It was based on his understanding of number and place value and he created it. These are the strategies students need to use — the ones they develop.
I’ve told this story at least 50 times (I’ve even told it to myself while on the road). Afterward, I often challenge teachers to take their students to the copy machine and watch this play out for themselves. Some pushback does come out occasionally with comments like these (my responses follow each):
That’s because he’s probably gifted. He is, but that’s not a reason to not do this with any group of kids. Every student can and will do this when presented with contextual problems and access to familiar tools and where teaching through problem solving is the norm.
You probably worked with him on multiplication tables. Yes, and no. When Kim was pregnant with Connor and on the sonogram table with a full bladder, I leaned close to her stomach and started reciting multiplication facts to make her laugh (I’m cruel for a laugh sometimes) Other than the 4 or 5 facts I quickly rattled off that afternoon, I’ve never recited them since. I doubt that did much, if anything, for his math achievement.
You must work with him a lot with math. Not really!Other than natural math wonders that have piqued my kids’ interests and sparked some discussion, no. Questions they’ve had, like – “Dad, how many tickets do you think I have in this Dave & Busters cup?” are all we’ve spent any amount of quality time on. That and puzzles.
So, when it comes to strategy building, it all has to begin at the student level of understanding. The best way to do that is to let students develop their own strategies, share them with each other, and build more powerful understanding from there. Then, if they do get “taught” a standard algorithm somewhere down the road, it has a better chance of making sense.
The problem I’m seeing with personalized learning (overall and especially as it pertains to math instruction) is the common understandings about what it is, what it can look like, what it shouldn’t look like, and how it works as related to our own learning experiences are fragile at best.
Many school systems, including my own, are looking at personalized learning as a means to improve math instruction, raise math test scores, and increase student engagement. These goals are great and many systems have them in some form or another. However, when personalized learning forces teachers into using sweeping generalized practices that often trump solid content pedagogy, something is drastically wrong.
I don’t think this is necessarily the fault of personalized learning as a concept, but I do think it is problematic when common understandings become compromised. These compromised understandings lead to sweeping generalized practices like:
No whole group instruction – ever
Students should be on a self-paced computer program for personalized learning
Teachers have to create new groups of students every day/week to make sure learning is personalized
Teachers should do project based learning several times per unit to engage learners
Teachers need to use choice boards for every standard they teach.
This is not a definitive list – just what I’ve heard from within my own district over the last few years.
I may not have a response to each of these, but I can point out a few sources in addition to my thoughts:
No whole group instruction – ever – Dan Meyer’s post: http://blog.mrmeyer.com/2014/dont-personalize-learning/ my favorite idea from this is from Mike Caufield: “if there is one thing that almost all disciplines benefit from, it’s structured discussion. It gets us out of our own head, pushes us to understand ideas better. It teaches us to talk like geologists, or mathematicians, or philosophers; over time that leads to us *thinking* like geologists, mathematicians, and philosophers. Structured discussion is how we externalize thought so that we can tinker with it, refactor it, and re-absorb it better than it was before.”
2. Students should be on a self-paced computer program for personalized learning –Personalized learning is not something you get get from the App Store or Google Play or from any ed tech vendor.
Some other comments from Dan Meyer: Personalized Learning Software: Fun Like Choosing Your Own Ad Experience and from Benjamin Riley: “Effective instruction requires understanding the varying cognitive abilities of students and finding ways to impart knowledge in light of that variation. If you want to call that “personalization,” fine, but we might just also call it “good teaching.” And good teaching can be done in classroom with students sitting in desks in rows, holding pencil and paper, or it can also be done in a classroom with students sitting in beanbags holding iPads and Chromebooks. Whatever the learning environment, the teacher should be responsible for the core delivery of instruction.”
3. Teachers have to create new groups of students every day/week to make sure learning is personalized– I’m not sure this is the case. If teachers really know where their students are in their mathematical progressions (lots of ways to do this – portfolios, math journals, student interviews (GloSS and IKAN from New Zealand, etc.) These types of data are much more effective that computer testing programs because teachers are able to see and hear students’ thinking as well as their answers. In my opinion, you can’t get more personalized than that!
4. Teachers should do project based learning several times per unit to engage learners– anyone who has had PBL training knows that 1 per year is a good start! PBL takes time – to plan, and plan some more (most often with other content areas). If anyone expects more than one per year or semester initially, it’s time to have some Crucial Conversations!
5. Teachers need to use choice boards for every standard they teach– student voice and choice does not have to be a choice board. And really, how much of a choice do students have if we’re giving them all possible choices with no input from them?
To sum up: In order to really improve those goals of improving math instruction, increasing student engagement, and raising math test scores one thing is certain – an investment to increase teacher content and pedagogy knowledge must be at the forefront. There is no other initiative or math program that will help districts reach these goals more effectively than this!
Not understanding mathematics can be extremely frustrating for students. As a teacher, figuring out how to help students understand mathematics can be just as frustrating. My primary go-to resource for these situations is Teaching Student-Centered Mathematics, by John Van de Walle et. al. because it’s all about focusing on big ideas and helping students make sense of the math they’re learning in a conceptual way.
Recently, I was asked to model a lesson for a 6th grade class who was having difficulty working with percents. So, I turned to my go-to resource, and during planning, I realized that I didn’t know anything about these students other than that they were struggling with percents. So, I couldn’t assume anything. I ended up creating three separate lessons and combined them into 1.
First, I handed groups of students a set of Percent Cards and Circle Graph Cards. Their task was to match the percent with the corresponding circle graph. As students were working on this, I heard groups reasoning about how they were matching the cards. Many started with benchmarks of 25%, 50%, and 75%, while others started with the smallest (10%) and matching it to the graph with the smallest wedge. As groups finished, they were asked to find pairs, using the matches they made, that totaled 100%. Once finished, a discussion about their process for completing these tasks revealed a solid understanding of percent as representing a part of a whole.
Now to shake their world up a bit. I asked them to leave their cards because they would be using them again shortly. I introduced these Percent circles and asked them what they were. A brief discussion revealed some misconceptions. Some students said they were fractions, others said they were wholes because nothing was shaded. I altered my planned line of questioning to questions that eventually led to a common understanding of what fractions were and how the pictures of the fraction (percent) circles really showed wholes and parts (fractions).
Their next task was to match their cards with the equivalent fraction circle. This was incredibly eye-opening. Groups began to notice that some percent card matches could fit with multiple fraction circles (50% could be matched with the halves, quarters, eighths, and tenths). Thirds and eighths were the last to be matched. But their reasoning didn’t disappoint. One group noticed that the percents ending in .5 all belonged with the eighths because they were too small to be thirds (the other percents with decimals).
Students were eager to share their thoughts about what they learned about fraction circles and percents:
Fractions and percents are the same because the pieces look the same.
1/4 is the same as 25% and 2/8
I don’t get why the eighths end in .5.
The percents all can be fractions.
1/8 is 12.5% because it’s half of 25%
Finally, I asked students to solve a percent problem (now that they’ve all realized that fractions and percents can be used interchangeably). I gave them the m & m problem from this set of percent problem cards. The only direction I gave was that they had to solve the problem using some representation of the percent in the problem before they wrote any numbers.
My bag of M&M’s had 30 candies inside. 40% of the candies were brown. How many brown candies is that?
While this was problematic at first, students looked at their fraction circles and percent cards and realized they could use four of the tenths since each tenth was the same as 10%. Most students needed just one “least helpful” question to get on the right track: Where do the 30 m & m’s belong in your representation?
Most groups were able to make sense and persevere to solve the problem correctly, and explain why they “shared the 30 m & m’s equally among the ten tenths in the fraction circle” and why they “only looked at four of the tenths because that’s the same as 40%.”
My beliefs that were reinforced with this lesson:
We can’t assume understanding from correct answers alone. We need to listen to students reason through problematic situations.
Students really want to share their thinking when they realize that someone is really interested in hearing it.
Students crave understanding. They really want to make sense.
Procedures are important, but not at the expense of understanding.
Empowering students by allowing them to build their own understanding and allowing them to make connections allows students to feel comfortable taking risks in problem solving.
Please take a look at Jenise Sexton’srecent blog about percents with 7th grade students for some fantastic ideas about students using number lines and double number lines to solve percent problems. It’s SWEET!
Under the Dome 