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!
This is the first in a series of posts about learning to think like a mathematician. This is my first memory of playing with numbers, questioning my own thinking, and making sense of new ideas. As my brother Peter said, when I shot the video clip below, “and this is how it all began!”
When I was around 5 years, my older sister showed me an adding machine that was in my grandmother’s closet. It was large (to a fiver year old) and very heavy. It was completely mechanical and had 81 numbered hexagonal keys – 9 rows of 9 keys (see image, above). Each column of keys was numbered with the digits 1-9. Pressing a digit in a given column would display that digit in a corresponding window along the bottom row. There was also a lever on the right side that could be pulled to reset all of the column windows to zero.
My first experience with this incredible machine, thanks to my sister, Susie, was to press the “1” in the lower right corner (the ones place) continuously. The tenth press felt different and the display at the bottom returned to zero, but the place just to the left turned to display a “1.” Pressing the same “1” button felt the same again for 9 presses, then on the tenth press if felt different again, and again the window below turned to a “0.” And the window to the left turned to a 2. I could make the second window count to ten, too? By only pressing the one button? This was fascinating! After a long time, and lots of presses of the same key, when all of the bottom windows displayed nines, I would press that same “1” button and it would “feel a little different” and something amazing and extremely satisfying happened. Watch what happened.
All of the 9s would alternately flip to 0s, like a row of dominoes flipping over. It was a sensory explosion! I could hear the dials flipping. I could see them flipping, and I could feel when it should happen. It seemed magical at the time.
I (we) were told not to play with this because it was an antique and I (we) might break it. I was a pretty good rule follower, but the allure of this machine was too much. I would sneak into my grandmother’s closet and lug that heavy machine out to play with it – a lot – even when I was a little older, just to get to see, feel, and hear that domino-like effect of numbers flipping over. Even now, it makes me smile to think about it.
I would spend a lot of time pressing that “1” key until it got to 9, then press it one more time and a “1” would pop up in the place to the left and a zero was in the spot below the key I was pressing. It never got old. I kept pressing that key, not just to see what I had come to know would happen, but to figure it out. I began to make predictions and ask myself questions, like when the display read “249” one more press and the digit in the place to the left of my key changed to a 5. “I bet it changes to a six next time.” When it did, I was hooked. This continued and every time the next column got to a 9, I’d quickly press my “1” key one more time to get it to click over. What I noticed, though, is that it took a lot longer to get each row to 9, but that wasn’t enough. I wanted to know how much longer. I kept going because the more nines I had, the cooler it sounded when they all flipped over! Eventually, I figured out that it took ten presses of that “1” key to make the next column change, and that column had to change 10 times to make the next column change. I discovered a pattern of tens.
What I didn’t realize, initially, was that I didn’t have to keep adding ones to get the full row of nines (I was still only about 5). I could just press each of the nine “1” keys 9 times, then add 1 more by pressing the far right “1” key. Then, satisfaction and amazement came much sooner! Once I figured this out, it was a much quicker experience but, frankly, a little less satisfying. After a while, I really took note of the other keys and realized that I could use some of them to my advantage as well. For example, I could press the “9” key in each column once and then press the far right “1” key. Still satisfying and much more efficient to get to the end result, but not as enjoyable as seeing this one button do so much. Watch the video below to see and hear what I loved so much.
This was one of my first experiences with playing with mathematics and the effects it can have in the sense-making and building deep understandings of mathematical ideas. I believe it is one of my earliest mathematics learning experiences, and I believe it had a huge impact in how I think, mathematically.
My hope is that this and some future posts may cause you to reflect on some of your own, similar, experiences. If so, please share. I’d love to hear your experiences. Stories like this, I think, have the potential to bring to light just how beautiful mathematics can be and the connections that can be made by studying this amazing subject!
Side note: I really wanted to take this amazing contraption apart to see how it worked. My parents are thankful that I never did, but I still wonder what the inner workings of this adding machine look like. Unfortunately, I never got to find out, but I am still very curious. Perhaps there’s a video out there that I can watch so I don’t ruin this antique with my tinkering.
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!
For the second year in a row, I had the privilege and honor to give an ignite talk at the Georgia Math Conference (Last year’s talk can be found here.) What makes ignite talk sessions great is that you get a taste of what several speakers are passionate about and you get to walk away with at least one ember of at least one of those talks beginning to burn in you!
Special thanks to Graham Fletcher for putting this all together (in pre and post production!). Graham is top notch, “for sure” (Must be a little of my inner Canadian there).
The featured speakers this year in the order of their talk:
Hi, my name is Mike… and I love using Desmos with students.
This is not a bad thing at all. I’m not giving up time with my family to spend on Desmos. It’s just that whenever I think I’ve exhausted all of the ways to use this fantastic tool with students, the Desmos team adds a new activity or game that I can and want to use right away! These people know how to keep us wanting more!
Now, for all of you teachers out there that haven’t engaged your students in this amazing math tool, let me move from a user to a pusher. 4 reasons why you should use this amazing tool with your students:
It’s completely free! (not just this first time – all the time)
It’s a graphing calculator that works beautifully online or as an app for students to Model with Mathematics – SMP 4.
This is a screenshot of how my son, Connor, used the Desmos Calculator to make sense of transforming quadratic functions.
3. When you sign up as a teacher (again, for free) you can assign activities and games (yep, they’re all free to use, too) to your students and you can check their progress from your teacher page.
So, beyond the graphing calculator – which is amazing on its own – as a teacher you can assign an activity to your students based on the content they are investigating. Try Central Park – it’s my favorite activity. (If you like, you can go to the student page and type in the code qqbm. I set this up for anyone reading this post. Feel free to use an alias if you like).
And as far as games go, check out Polygraphs. It’s like the Guess Who? game for math class. Trust me, your students will love it and there are polygraphs for elementary as well as secondary. The polygraphs are all partner games, so students will need to work in pairs. I’ve even made a few:
4. As you get sucked in to this tool, you may begin to think to yourself, “Boy, I really wish there was an activity for ______. If only knew how to create an activity for my students to use on Desmos.” That’s taken care of, too, with Activity Builder and Custom Polygraph (and, yep, you guessed it – they’re free to use, too)
And before you begin to doubt whether you can create an online activity or polygraph, the Desmos team has already taken steps to make this extremely teacher friendly. Before you know it, you’ll have your own Desmos activity published!
Finally, as a great end of year gift, Dan Meyer blogged about the latest from Desmos – Marbleslides. If this doesn’t get you to use Desmos with your students. . . well, I’m sure they will think of something else, soon. But seriously, try this out. I have re-learned and deepened my own understandings of mathematics by trying and reflecting on many of these activities and games, and then having my own kids do them (and then they ask me why their teachers aren’t using them – “Can you talk to them, Dad?”). The conversations will be happening this semester for sure!
But the best part about all of this is that students get to use the calculator to investigate graphs and compare graphs and equations/functions. They get to notice and wonder about what matters and what changes a graph’s slope, and y-intercept for linear functions and what changes the vertex and roots of parabolas. They get to investigate periodics and exponentials and rationals and so much more. They get to engage in activities and games that have components that ask them to reflect on what they’ve learned in the games and activities themselves. The students are doing the mathematics.
Then, in class, we get engage students in talking about the math they’ve investigated! How sweet is that?
You see, as great as Desmos is, it can’t take the place of great teaching. It’s a tool that can help us become better at our craft and help our students gain a deeper understanding of mathematics! Sounds like a win-win!
So, I guess I don’t have a Desmos math addiction. Addictions have adverse consequences and I see none of that here! I just have – as we all do thanks to Desmos – access to a powerful mathematical learning tool! Thank you Desmos. I can’t wait to see what’s coming next!
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.
The following is a reflection on a 3-Act task I modeled for an 8th grade teacher last week. The 3-Act is In-N-Out Burger from Robert Kaplinsky and the plan I followed I completely stole from the amazing @approx_normal ‘s blog post on her work with the same 3-Act with administrators last year.
This past Thursday was the day we agreed on to model the lesson. So, this group of 8th grade students, who have never even seen me before, are wondering who this guy is that’s about to teach their class. And, just as planned, they were giving me weird looks when I showed them the first cheeseburger picture and asked them what they noticed. I believe one of them even asked, “Are you a teacher?”
Fast forward through to the “What do you wonder?” piece and the questions were amazingly well thought:
“How much weight would you gain if you at that whole thing (100×100 burger?)”
“How much do the ingredients cost for it (100×100 burger)?”
How much does it (100×100 burger) cost?
“Why would someone order that (100×100 burger)?”
“Did someone really order that (100×100 burger)?”
“How long did it take to make the (100×100 burger?)”
There were just a couple more, and they all came up very quickly. The students were curious from the moment we started the lesson. They are still working on precision of language. The parentheses in their questions above denote that this phrase was not used in the question, but was implied by the students. We had to ask what “it” or “that” was periodically throughout the lesson as they worked and as time went on, they did become more consistent.
The focus question chosen was:
How much does it (the 100×100 burger) cost?
Students made estimates that ranged from $20 to $150. We discussed this briefly and decided that the cost of the 100 x 100 burger would be somewhere between $20 and $150, and many said it would be closer to $150 because “Cheeseburgers cost like $1.00, and double cheeseburgers cost like $1.50, so it’s got to be close to $150.” That’s some pretty sound reasoning for an estimate by a “low” student.
As students began Act 2, they struggled a bit. They weren’t used to seeking out information needed, but they persevered and decided that they needed to know how much a regular In-N-Out cheeseburger would cost, so I showed them the menu and they got to work.
I sat down with one group consisting of 2 boys (who were tossing ideas back and forth) and 1 girl (Angel) who was staring at the menu projected at the front of the room. She wasn’t lost. She had that look that says “I think I’ve got something.” So, I opened the door for her and asked her to share whatever idea she had that was in her head. She said, “Well, I think we need to find out how much just one beef patty and one slice of cheese costs, because when we buy a double double we aren’t paying for all of that other stuff, like lettuce and tomato and everything.” The boys chimed in: “Yeah.” I asked them how they would figure it out. Angel: “I think we could subtract the double-double and the regular cheeseburger. The boys, chimed in again: “Yeah, because all you get extra for the double double is 1 cheese and 1 beef.” “Well done, Angel!” You helped yourself and your group make sense of the problem and you helped create a strategy to solve this problem! Angel: (Proud Smile)!
We had to stop, since class time was over. Other groups were also just making sense of the idea that they couldn’t just multiply the cost of a cheeseburger by 100, since they didn’t think they should have to pay for all of the lettuce, tomato, onion, etc.
They came back on Friday ready to go. They picked up their white boards and markers and after a quick review of the previous day’s events and ah-ha moments, they got to work. Here is a sample after about 15 minutes:
Many groups had a similar answer, but followed different solution pathways. I wanted them to share, but I also wanted them to see the value in looking at other students’ work to learn from it. So I showed this group’s work (below-it didn’t have the post-its on it then. That’s next.). I asked them to discuss what they like about the group’s work and what might make it clearer to understand for anyone who just walked in the classroom.
Here’s what they said:
I like how they have everything one way (top to bottom).
I like how they have some labels.
I’m not sure where the 99 came from. Maybe they could label that.
Where’s the answer…
During this discussion, many groups did just what @approx_normal saw her administrators do when she did this lesson with them. They began to make the improvements they were suggesting for the work at the front of the room. It was beautiful. Students began to recognize that they could make their work better. After about 5 minutes, I asked the class to please take some post-its on the table and do a gallery walk to take a close look at other groups’ work. They were to look at the work and give the groups feedback on their final drafts of the work using these sentence starters (again, from @approx_normal – I’m a relentless thief!):
I like how you. . .
It would help me if you. . .
Can you explain how you. . .
Some of the feedback (because the picture clarity doesn’t show the student feedback well):
I like how you showed your work and labeled everything.
I like how you broke it down into broke it down into separate parts.
It would help me if you spaced it out better.
I like how you explained your answer.
It would help me if it was neater.
I like how you explain your prices.
I like how you wrote your plan.
I like how you explain your plan.
I like how you told what you were going to do.
Can you explain how you got your numbers.
I like how you wrote it in different colors.
It would help me if you wrote a little larger.
Some samples with student feedback:
Not only was the feedback helpful to groups as they returned to their seats, it was positive. Students were excited to see what their peers wrote about their work.
Now for the best part! Remember Angel? As she was packing up to leave, I asked her if her brain hurt. She said, “No.” After a short pause she added, “I actually feel smart!” As she turned the corner to head to class, there was a faint, proud smile on her face. Score one for meaningful math lessons that empower students.
Please check out the websites I mentioned in this post. These are smart people sharing smart teaching practices that are best for students. We can all learn from them.
Let me explain. There’s a math epidemic (remember Ebola 2014+). Students are bleeding out from the gashes of their misconceptions of mathematics. The lack of teaching conceptual understanding along with sacrificed opportunities to make mathematical connections is the double edged sword. This is an epidemic, and some teachers, school systems and educational leaders are treating it like it’s a tiny scratch, instead of the pervasive threat to mathematical achievement that it is.
Here’s a familiar scenario: A school’s test scores come back after the spring testing season (or mid-terms). The scores show little growth from the previous year in the area of mathematics, and any change is not in a positive direction. The knee-jerk reaction to the valid question, “What can we do to fix this?” is to look for programs and technology that will fix the problem. These are the same individuals who, way back in August, looked us all in the eye and, with the greatest of sincerity, reminded us that the single most important factor determining student success is the quality of the teacher. Not new programs. Instead of growing the quality of teachers, we get programs that:
push speed over comprehension (imagine if we taught reading this way).
define fluency based on digits rather than efficiency, flexibility, and accuracy.
use technology to separate us from our students when we know that what we really need is to spend more time listening to them and creating an interactive classroom with technology as a support for this human interaction
are essentially Band-Aids
I often hear the phrase “back to the basics” in times like these. I’ve heard parents, administrators, and even a few teachers say this. I think everyone would agree that “back to the basics” should mean that students become computationally fluent. The idea of going back to this implies that we were doing something right before. And we all know that’s not true. After all we have generations of adults who are not computationally fluent and/or have extreme math anxiety. And how did that happen?
Answer 1: Timed tests. My sixth grade teacher called them speed tests. We did them every day, right after lunch. (I was never in the top 10).
Answer 2: Algorithms memorized by students with no understanding, presented by teachers with little understanding other than from a teachers’ edition.
Answer 3: Little or no real problem solving. Naked computation all around. No wonder students were turned off by mathematics!
Answer 4: No interaction. Math is a social activity. If you talk to any engineer, designer, architect, mathematician, statistician, etc. They aren’t doing their work in an office silently sitting in rows. They’re constantly talking to one another about the mathematics they’re using. The idea that all of us are smarter than one of us makes so much sense in the real world and it should make sense in the classroom as well.
If we went back to teaching math like we did 20-30 years ago (I think that’s what some of these folks were implying when they said “back to the basics.” We’d still be in the same boat. Anyone ever watch How old is the shepherd?. That was popularized over 20 years ago and the results haven’t changed. Going back is not an option. Building fluency is.
So what do we need to do in math classrooms? I have a few ideas to stop the bleeding and these are certainly not original to me.
Apply pressure to the wound. Give up on the ineffective treatment, not the patient. Apply pressure to stop the bleeding. Focus on tasks and activities that build number sense. Number Talks, Math Talks, Estimation 180, Visual Patterns any or all of these can be put in place at any level. And the best part is, students can easily be trained to begin to apply the pressure themselves. They have the power to stop the bleeding!
Close the wound. This can only happen with stitches. And it takes time to get the hang of it. The wound has to be closed with the thread of understanding. We can’t understand for them, so the wound has to be closed with the help of the students. The students create this thread as we stitch and we can’t do it without them. How do they create this thread of understanding? We have to stop telling so much and instead “be less helpful.” If we tell students too much, the thread breaks.
Treat any symptoms that may show up after the initial treatments above:
Symptoms
Name
Treatment
Students may begin to rely on rote procedures with no foundational understanding
Sometimes unintentionally caused by parents & other adults trying to help.
Misconceptionitis
Identify the misconception(s) and re-build understandings using the CRA (Concrete Representational Abstract) model
Students are finding unreasonable solutions to tasks & problems and they often seem unaware; clueless
Unreasonableness
This is often attributed to students just not thinking enough. Treatment should include a DAILY diet rich in estimation – prescribe www.estimation180.com
Students count (often on fingers when computing or rely on a calculator for the simplest of calculations and even then, they can get incorrect answers.
Influencia
This is often diagnosed along with unreasonableness (see above). Its roots lie in naked computation and memorization of facts rather than allowing students to build strategies and practice those strategies until they become fluent. First, counting strategies are the lowest level strategies. Students need to build more efficient strategies by exercising with investigations of number relationships through number talks, math talks, and strategy building. Stop giving speed tests.
Students have strategies for computation, but are not applying them in problem solving situations
No Solvia
Students need a heavy dose of problem solving every day. This must involve students engaging in the Big 8 Standards for Mathematical Practice. Problem solving tasks every day. Hydrate often with student reasoning. Adopt the classroom mantra: “The answer isn’t good enough.”
Begin new concepts with a problem before any formal instruction on the topic. See what students can do before assuming what they can’t do.
I’m a teacher and I know many of you reading this are the choir that need no preaching to. If you’re interested in saving the patient, stopping the bleeding, and raising math achievement, click on some of the links in this post. There’s so much to learn from those smarter than me. Also check out #MTBoS on Twitter. Lots of math goodness from the best out there.
A good friend and colleague, Krystal Shaw, tweeted this article about Krispy Kreme Donuts in the UK a while back and it immediately got me thinking. . . so I really liked it and wanted to use it with kids. To plan for the lesson, I started to take myself through this problem as if it were a 3-act task (I wasn’t sure it would become one, but I wanted to see where this would lead). I looked at the picture:
and jotted down what I noticed. Then I began wondering:
How many donuts are in that big box?
What are the dimensions of the box?
Is there more than one layer of donuts in the box?
How many rows of donuts are there?
How big is (What is the diameter of) a Krispy Kreme donut?
When I was finished (or thought I was finished) wondering, I began to seek the information needed to answer my questions.
I found some nice strategies for determining the number of donuts in the box. Strategies accessible for 4th grade students. I was happy, so I moved on to the next question: What are the dimensions of the box?
This is when it happened.
I was stuck.
Perfect.
Challenge accepted.
I looked at the pictures, found the information in the article, then began to question that information (and myself) as well as some critical friends. This problem was getting better and better as I walked myself through it. Fantastic! SMP 3: Construct viable arguments and critique the reasoning of others, such as a Krispy Kreme representative from the UK or a USA Today reporter. Maybe this question won’t have a third act, but the estimation and reasoning used to solve this could be extremely empowering for kids.
I challenge you to solve this problem with your class as well and share your results. Challenge yourself and your students to construct a viable argument and critique the reasoning of others. Does your math challenge the information in the article or support it. Either way, integrate writing into math class in a meaningful way:
Time for me to give this a try! More in about a week.
By the way:Krystal Shaw gave her amazing Mathletes after school club the task of writing a 3-act math lesson for their teachers to teach. I think she should post it on her blog to share with the MTBoS!
Yeah, you read that right! I know many of you are now probably thinking about at least one, or likely, a combination of these questions:
What could math teachers possibly have to learn from magicians?
How could there be a connection between these two very different careers?
How would Mike know?
Beginning with the last question probably makes the most sense. At an early age I developed a fascination with magic, sleight of hand to be specific. Any magician I saw perform – either on TV or live – filled me with wonder. Certainly, some of that wonder was directed toward how the trick or illusion worked, but even beyond that I wondered how I could learn to create this wonder in others. Since I was about 10, I have studied magic and about 7 years later I began performing magic shows at schools, for church groups, and even for a few holiday parties. Once I began my career as a teacher, my role as a magician changed and I focused most of my energy on teaching. I’ve lived the life of a magician and a teacher and over the last few years, and I’ve begun to notice the similarities between the two.
A magician’s goal is to entertain his or her audience while bringing about a sense of wonder. The means for accomplishing this goal involves the use of any combination of a number of tools including misdirection, psychology, sleight of hand, and story telling. If a magician does his or her job well, the feeling of being tricked doesn’t really enter into a spectator’s mind. The big idea here is the creation of wonder.
That’s the first thing teachers can learn! It doesn’t take a sleight of hand artist to build a sense of wonder in students. It takes some creativity and some work and dedication to the idea that all students deserve the chance to wonder and be curious. All students need that sense of wonder that builds inside them and creates an intellectual need to know and learn.
This is a great time to be a teacher of mathematics. Evoking this wonder in students in math classes is extremely accessible because of technology and the online math community know as MTBoS. There are hundreds of math teachers out there at all grade levels and in all areas who have realized the power of making students wonder. We’ve all been creating 3-Act Tasks and sharing ideas on blogs and webpages, twitter, and youtube or vimeo. All for free. They’re there for everyone to use – because we’ve all learned, through using these tasks, that it helps us build student curiosity, engages them in the mathematics and in their own learning, and it helps us build independent, creative mathematical thinkers. Here is more about why you should use 3-Act Tasks.
This brings me to the second thing we can learn from magicians: we can’t do this alone! If we work together, we all benefit! Most people probably think that magicians are private wizards who lock themselves in a room to practice and never share their secrets. That’s a bunch of crap! Magicians realized a long time ago that if they work together, they can work more efficiently and become more productive. Sometimes magicians work on a trick for a while, get stuck and then bring it to some friends they have in the magic community. These other magicians share their ideas, they brainstorm, and try possible solutions. Then they test the best solution on an audience. This can be very scary! Think about it. This is a trick they’ve never tried – they’ve practiced (A LOT), and maybe even performed in front of small audiences. They must be nervous! But they go out on stage or wherever their venue is and perform it. They have to! It’s how they pay their bills. Often, some of their friends who helped them are there to provide feedback. After several performances, and feedback, the script has been adjusted and the magic has been perfected, and it becomes a part of the magician’s repertoire.
Now think about how many math teachers still work. . . alone, in their room, not sharing their ideas. Magicians realized this was not very productive a long time ago. Other professions did the same. It’s time math teachers realize this too!
Take a look at the MTBoS, and see what you think. Look at some of the sites below and see if you find something you like. Try some ideas/lessons with your students. It’ll be a bit scary in the beginning, but soon it’ll become part of your repertoire! We’re all here to learn from one another because “All of us are smarter than one of us!” ~ Turtle Toms
What I’ve learned through this whole process is that I get the same feeling of success when I create the sense of wonder in students as I did as a magician creating wonder in an audience. . . but it’s even better with students!
Under the Dome 