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!
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!
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.
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!
OK, so my timing on this is not great. This was actually written back in December (still a little tardy) and then the holidays ran over me. Blah, blah, blah. Nevertheless, everything below is still relevant.
Having attended and presented at conferences before, I have to say some conferences are good, and some not so good. NCTM Nashville 2015 was, in my opinion, the best I have attended – hands down!
Here’s why:
On Wednesday evening, in the opening session, Graham Fletcher, Robert Kaplinsky, Laila Nur, Andrew Stadel and Cathy Yenca set the tone for the conference. They spoke about their personal experiences of improving mathematics teaching and what they use to continuously improve their practice, they all spoke about how accessible and personalized PL for math teachers’ needs can be with a Math Blogs, Twitter, the #MTBoS (Math Twitter Blog-o-Sphere) that links them all together, and Web 2.0 tools that are not only changing the ways we think about teaching mathematics, but also the ways students engage in mathematics in their classrooms. One word: Powerful. And as I said before, it set the tone for the rest of the conference.
The rest of the sessions, at least the sessions I attended, all connected to the opening session. In the Desmos sessions I attended with Michael Fenton and Christopher Danielson, the presenters were able to take novices through the simplicity and beauty of this free graphing calculator (which is really much more – see my post on this here) and those of us who are just above the novices had plenty to learn as well. I even had a Desmos special tutoring session from Cathy Yenca and Julie Reulbach in the back of one of these sessions.
The twitter sessions I attended were always full and the session facilitators, as well as many attendees, lent a hand to those who wanted to get on board “this Twitter math train.” In addition, LOTS of people stopped by the MTBoS booth and were given some “small group” lessons on how to use Twitter, who to follow, and were given some general tips to make the whole experience low stress! Michael Fenton and John Mahlstedt were the facilitators of the Twitter sessions I attended. In each of these sessions, attendees were eager to learn more about Twitter and how it could help them become better math teachers. Even some not so eager people were asking questions near the end of these sessions!
The rest of the sessions I attended (I even co-presented one) had to do with modeling with mathematics – SMP 4. These sessions were probably the most valuable to me for two reasons:
We got to really dig in to some math and have some great mathematical discussions!
I got to experience more modeling in secondary mathematics which is great since I have just rejoined the secondary math world.
Ashli Black‘s session: Selecting and Using Tasks to Develop MP.4: Model with Mathematics was all about investigating characteristics of modeling tasks and working with pitfalls. I recommend following Ashli on twitter: @Mythagon. She really knows what modeling with mathematics should look like in the secondary math world, she’s a great presenter, and I’m thankful that she took the time to fill out the speaker form last year.
Michael Fenton‘s session on modeling provided a one-two punch – Modeling WITH Desmos! This was an incredible session. Michael’s presentation combining Desmos with mathetmatical modeling was. I was making sense of mathematics through the models created. I wish I had learned math this way, initially! While I can’t go back in time to learn this way for the first time, I can make sure that the students in my district have the opportunity. And it’s one of my goals for this year.
Andrew Stadel’s session: Model with Mathematics using Problem Solving Tasks. I have to admit, I’ve been using Andrew’s resources from his blog for a few years, but it was a real treat attending his session. He engaged us in a three-act task: Swing Wraps. This problem solving task engaged us in mathematical arguments, modeling, and sense making and a few other SMP’s. Mr. Stadel also did some modeling of his own through the types of questions he asked to the whole group and small groups, through his guiding of the discussion, and through his commentary about the importance of doing these types of problems.
So, in conclusion, here’s what this all boils down to:
Allow students to model the problems they solve with mathematics.
Take a look at Desmos – a long hard look – one that allows you to see it for more than just a free online graphing calculator that students can use to model with mathematics (that should be enough-but there’s oh-so much more to it!)
This post actually started as a rant as I was sitting through meeting after meeting with really nice people trying to sell products to “Fill the Gaps.” So, if it has a rant-y feeling, just know where I’m coming from. If no one really likes this, that’s ok. At least it’s out of my system for now. You see, when you’re “invited” to attend meetings to raise student achievement, you really need to show up, or who knows what will happen. So, in the effort to stand up for teachers and students, I attended all of them.
These were really nice people presenting to us, and they were very passionate about their products. I even largely agree with several of them on their basic philosophy.
At least one of the people listening with us in the room was sold on many of the ideas before we even started these meetings. Every slide or picture shown was met with a “That’s good!” or a “That’s really good!” I think if they showed us a shiny, new penny, this person would have said, “This is what our students need!” with the same reaction! The pictures of bulletin boards showing concept maps and vocabulary word walls and even students working may be good – or may not. Really, there’s no way to tell – especially with the picture of the students working. What were the students saying? Were they discussing mathematics? Were they using the vocabulary on the bulletin board? Were they making connections to the concept maps? Did they give and receive feedback about their work? Let’s see some video, so I can see how this is really working.
Again, philosophically I agree with their framework of instruction. However, the product is not really necessary if the PL these companies are willing to provide is effective.
Now, on to the PL. Lots of good strategies offered here. And more pictures of students “engaged.” My question: what are the students engaged in?” Are they engaged in the mathematics or the product? My initial response to this self-posed question was: Does it really matter? The students are working. After thinking about this for just a few seconds, though, I can say without a doubt that it does matter!
Engaging students can be tricky. A passerby, seeing students working silently in their seats, might conclude student engagement in a task. A passerby, seeing and hearing students discussing a task, may conclude non-engagement in a task as well as lack of classroom management. Really it’s hard to tell, in either case, whether there was any engagement or what kind of engagement there was.
So, what does engagement mean? It depends on what you want. One of my goals year after year is to engage students in the mathematics they’re studying. When I first started teaching, I wanted students to just be engaged, no matter what. As I think back, they were engaged – probably in my educational “performance.” I was the “fun” teacher that did crazy math lessons. As I grew professionally, my lesson focus evolved to take the students’ engagement away from me and toward the mathematical content. So, why is it so important? If students are engaged in creating the product (creating a poster, making a presentation, etc.) they may be learning mathematics, but how do we know. I’ve seen students engaged in creating beautiful products and walk away with little mathematical understanding. I’ve also seen students engaged in mathematics and creating not so beautiful products, but beautiful understandings and mathematical connections.
So, for all of the professionals in the room thinking this (or any of the other presentations we’ve seen) is the silver bullet. . . It’s not. The only silver bullet out there that’s going to raise student achievement is teacher PL grounded in understanding mathematics conceptually and building teachers’ pedagogical understandings and strategies. If we want high achieving students, we have to help teachers achieve their greatest potential. No program out there will do that, but if you really want to become a better math teacher, Twitter and the #MTBoS are a great place to start!
Georgia’s new state school superintendent, Richard Woods, recently wrote a column about teaching mathematics. “Funny math methods” was the catch-phrase taken from the article and sent out through the media. This was not unexpected. Frankly, I’m surprised it took this long. This was part of his campaign platform.
Though his column has prompted some emotional responses from math educators, it is imperative that this significant dialogue he has opened, continue. The best thing we can do for the students of Georgia is to keep this discussion going in order to come to a common understanding about the mathematical terms, strategies and ideas presented by Mr. Woods in his column. We can truly help the students of Georgia by making sure we are all speaking the same language.
Since I am unable to respond directly to Mr. Woods’ column, I would like to continue the dialogue here. I welcome any and all comments that keep this discussion moving forward in a positive light. I encourage all viewpoints, since one-sided dialogues don’t tend to be very productive.
Mr. Woods talks about hearing from parents unable to help their children with their math homework. I, too, have heard this from parents. My response to this is: If students are not able to do their homework independently, perhaps it should not have been assigned. This is difficult for many to hear. If you think about it, though, it really makes sense. If we want students to build their understanding of mathematics based on what they have learned, we have to make sure they have learned it before they can build on it. That said, I look at homework as falling into one of three categories:
Practice – students use understandings learned in class to practice and build a more solid understanding at home.
Preview – students are given a few problems to get them thinking about a new concept that is related to what they already know.
Extension – students take a problem or task they worked on in class and are asked to extend their understanding. For example, in middle school, students may discover a growth pattern and as an extension, they may be asked to create a growth pattern that grows twice as fast.
Notice that each of these types requires students to have an understanding before they begin. Understanding in mathematics, as in reading, is crucial for student success.
Mr. Woods also mentions the need for students to have a firm understanding of the fundamentals of mathematics. He goes on to say that basic algorithms, fact fluency, and standard processes for addition, subtraction, multiplication, and division contribute to building that strong foundation for student achievement.
This is interesting. First, algorithms have gotten a bad rap. But, there is a place for algorithms in the big picture of how students learn mathematics. An algorithm is just a mathematical term for a series of steps that can be followed to determine a solution to a mathematical computation. Problems occur when algorithms are taught just as a series of steps to memorize, rather than facilitating an understanding of the computation(s) first. Without understanding, the steps often don’t make sense and one or more of three things may happen next:
Students may complete algorithmic steps out of order.
Students may skip one or more steps of the attempted algorithm.
Students may confuse the steps of one algorithm with another.
These may seem like easy fixes -“just tell the students again”. Telling them where their errors are and having them practice more problems does not work. Without a conceptual understanding of what the computation means, students will continue to make these errors. Though students may be able to show some success in the short term, over the long term they will revert back to one or a combination of the error patterns above.
Completing algorithms incorrectly doesn’t even compare to one of the worst side effects of this procedural teaching: students who don’t realize their answers are unreasonable. For example, a teacher recently sent me the email below:
This student has some major misconceptions. With a conceptual understanding, this student could have reasoned that 1/3 of a pound is less than a whole pound, so the answer should be less than $5.25. Without conceptual understanding, the student is attempting to recall and use procedures they do not understand, is confusing procedures, and is unable to determine whether or not the solution they have found is reasonable. This is only one piece of numeracy that is lost in the procedural mathematics instruction that Mr. Woods seeks.
Fact fluency and the standard procedures for the four basic operations is next. I don’t think there is a math teacher anywhere in the world that doesn’t think fluency is important. In order to be clear though, memorization and fluency are not the same thing. Not even close. To keep this short and sweet, with the focus on students, I have copied the excerpt below from the GA DOE frameworks for mathematics. I think this sums everything up nicely (no pun intended). However, if you would like to learn more about fluency, click the links below.
Fluent students:
flexibly use a combination of deep understanding, number sense, and memorization.
are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.
are able to articulate their reasoning.
find solutions through a number of different paths.
The fundamentals that Mr. Woods should have mentioned are actually reasoning and sense making. This is what it takes to learn and do mathematics. So, again, understanding must take place. However, understanding cannot take place through the memorization of algorithmic steps alone. This is not just what I think. It’s what I know from years of teaching students mathematics. This is also backed by research, papers, & videos. The building of understanding is also fostered through a passionate, grassroots movement of mathematics teachers #MTBoS (Math Twitter Blog-o-Sphere). This is our place to collaborate, share, and work to improve our teaching of mathematics.
Next, Mr. Woods discusses teaching “funny math methods.” He specifically mentions “the lattice method” and he correctly states that this method is not state mandated and not required for students to achieve on state tests. Mr. Woods is absolutely right! This is not state mandated because it is a ridiculous strategy for multiplying multi-digit whole numbers. to be fair, it works – every time. If you follow the steps of this algorithm by making the grid correctly, and placing the digits of the numbers correctly in the grid, and placing the products of the digits in the right places, and drawing the diagonals correctly, and adding the digits along the diagonals correctly, and copying the product correctly from the grid written in standard form. You think that’s ridiculous? Here’s something even worse – it’s often used for those students who have trouble remembering the steps of the standard algorithm. This method is the definition of “funny math” and math teachers should not use this since it is does not make sense to students (or teachers) and it does not align with any of the standards for multiplication.
Mr. Woods says that mathematics has become over complicated. It hasn’t. It is only as complicated as it has been for centuries and that complication is exacerbated through teaching without sense making. We can teach students to think mathematically on their own. We can support and help them grow through their own understandings of mathematics. We can help students make sense of mathematics and learn to use this to make informed decisions, rather than listening to others make these decisions for them. We can do this because what we know now is how students learn mathematics. It is not through memorization. It is through sense making and reasoning. What we know now is that teaching students to think mathematically, through problem solving by building conceptual understanding provides students experience and allows them to make connections to algorithms they create and those created by others. What we know now is that this works best for all students. Not just average students, or above average students, or below average students. All students.
Finally, just to be clear:
We (mathematics teachers) are most likely more current than most on research-based, best practices in the mathematics classroom.
There is a place for the algorithms you wish to see in the classroom, and they are found in the appropriate grade level standards. However, using an algorithm is not the end-all, be-all for learning mathematics. There is always a need for students to be flexible, efficient, and accurate in their computations. Multiple strategies, based on student understanding, must be explored.
At the end of the column, Mr. Woods states that “Offering choices and clarification are some of the steps we are taking to address the concerns surrounding mathematics in our state.” I applaud these steps. Choices are always a good idea. Clarification is even better! Let’s work together to clarify the misconceptions about best practices in mathematics instruction. Armed with these common understandings, Georgia can lead the charge as a state united to raise student achievement in mathematics.
I look forward to all comments and continuing this dialogue to help build these common understandings.
I recently got back from Santa Fe. I was attending a conference there for a few days last week and afterward, I drove to El Paso to visit my brother’s family (he’s currently stationed in the middle east so I didn’t get to see him – unless you count face time) for a day before flying home.
Let me preface this story by saying that we all probably have a story similar this, but how we handle it can be a possible game changer.
Somewhere on my long drive, I stopped at a fast food restaurant to grab a quick bite. So, I went inside and got in line. The following outlines the beginning of our interaction as I stepped to the counter:
Cashier: May I take your order?
Me: Yes, please. I’d like a number 2.
Cashier: Large or medium.
Me: Medium, please.
Cashier: (after pushing more buttons than is conceivably necessary to enter my choice of “medium”): Your total will be $6.05.
I dug through my wallet (receipts from the trip and everything) and found that all I had was a $10 bill, so I handed it to her. She entered $10.00 correctly and the correct change of $3.95 showed up on the little screen. At just about that point, I remembered that I had a bunch of change in my pocket and said quite enthusiastically, “Oh, wait, I think I have a nickel.” Who wants to carry around $0.95 in change in their pocket.
The cashier didn’t miss a beat, and said, “So, your change will be $4.00 even.” I kind of smiled as I continued to look through my change, proud that she had a mental strategy to adjust to the situation and that she seemed quite confident and comfortable using it in this situation.
Unfortunately, I didn’t have a nickel, but I still didn’t want change falling out of my pocket into the depths of the rental car, never to be seen by me again. So, I told her, “Oh, I’m sorry, I don’t have a nickel, but I do have a dime.”
As I handed her the dime, I saw her face morph from a confident smirk to a confused, almost terrified look of despair . I had just taken her from a mathematical point of “Yeah, I can do this math stuff. I may not use the computer for the rest of my shift” to “Holy $#!+, what the #=|| just happened!”
I went into math teacher mode and waited patiently for her to begin breathing again. And then I waited for her begin thinking. She adjusted my change with my introduction of the idea of a nickel, why not a dime? After what seemed like 5 minutes (probably more to her), it was painfully obvious to all around that her anxiety in this situation was taking over her ability to tackle this problem. So, I tried to think of a “least helpful question” to ask. Now I put myself on the spot. If she only knew that we were both now feeling some of this pressure.
So, I finally asked her my question and she gave me the correct change within a few seconds. She smiled as she gave me my change and my new “unknown” student and I parted ways. I know I felt good about helping someone develop a strategy outside of the classroom. I hope she had a similar feeling about learning to make sense (no pun here) of making change.
Being a math teacher is a 24-7 job sometimes and we can find our students anywhere – even in a fast food restaurant in New Mexico!
What you would have asked the cashier in this situation. I’d love to hear what your “least helpful” question would have been. No pressure, take as long as you like. No one is waiting in line behind you!
Feed the hungry!
Oh, here’s my question: “If you could change the dime into some other coins, what would you change it for?”
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 