I just attended an AMTE session on the Common Core State Standards and math teacher education. The panelists and audience listed many immediate needs as we plan how to implement the standards; the one that caught my ear was the need to align the Mathematical Practice Standards to the work that’s already been done, like the NCTM practices and the “Adding it Up” strands.
I figured that I could at least align the work I’ve already been doing about developing activity structures for teaching problemsolving and problemsolving strategies to the process standards. Here’s my first attempt at that.
The Practice Standards section describes certain behaviors that students who understand the concept in question might engage in:
consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut
The authors hypothesize that without understanding the concept, students will revert to procedures and not do any of above behaviors. I wonder if students who have mastered the process standards might not be able to apply those behaviors to new concepts, and seek to understand and learn the concept through good problemsolving behaviors. To me that suggests that the process standards can become an explicit strand of the curriculum, related to a learning to learn strand. The process standards don’t necessarily need to be applied after learning, as part of application and extension. Instead, mathematical practice standards can be taught with and prior to content standards, through explicit instruction in problemsolving.
So, how do the practice standards align with problemsolving strategy instruction? I’ll try to go practice by practice and think about strategy connections.
1. Make sense of problems and persevere in solving them.
The obvious connection here is the strategy we call Understand the Problem. Included in Understand the Problem is the core activity I Notice/I Wonder, which gives an entry point into the problem to students of all levels, but which can be perfected and improved even by expert problemsolvers (see, for example, the world’s hardest easy Geometry problem). I Notice/I Wonder begins to orient students towards recognizing givens and constraints, identifying mathematical quantities or objects in the problem, and describing relationships among them.
Other Understand the Problem activities help students to sketch or act out the problem, a “lite” version of the Change the Representation strategy that helps students further understand the problem scenario and bring out hidden relationships. Paraphrasing the problem is another key activity here. We also engage students in thinking as much as they can about what the answer can and can’t be. Activities support students to describe as much as they can about the final answer (units, estimates, ranges, etc.) and to work through the problem with an answer they know or suspect is wrong, just to get a further sense of the relationships in the problem.
The Practice Standards also include Change the Representation and Solve a Simpler Problem as strategies that students might employ as they make sense of and persevere in solving problems. Our Change the Representation activities support students to identify “what this problem is about” and use their understanding of mathematical ideas in the problem to brainstorm or select from a provided list of alternative representations. They work in groups to compare representations and try to connect their representations to the constraints and relationships they noticed as they first explored the problem. The Solve a Simpler Problem activities support students to articulate “what makes this problem hard.” We’ve found that students often identify the very things mathematicians notice, but then struggle to understand how they can then simplify the problem or even that they are allowed to simplify the problem. We offer students (and their teacher facilitators) several concrete suggestions for how problems can be simplified or analogous problems can be posed.
In terms of persevering to solve problems, we also offer activities to help students learn questions and techniques to Plan and Reflect and Get Unstuck. To help students plan we offer tips on moving from what you noticed and wondered to possible strategies to try, as well as a structure with group roles and tasks to help students think metacognitively as they select and carry out a plan. For getting unstuck we offer a selfdiagnosis based on the guess and check strategy (could I even think of a quantity to guess for? What would be a good guess? Do I know any calculations I could do if I had a guess? How would I know if my guess were correct?) and a series of questions students can ask themselves, a partner, or be asked by a facilitator to help them “unstick” themselves.
Finally, since students who understand problems check their work, we offer the activity “Testing 1, 2, 3, 4″ in the part 2 of the Guess and Check activity sequence.
2. Reason abstractly and quantitatively.
The ProblemSolving and Communication Activity Series takes students through a sequence of activities specifically designed to help students “make sense of quantities and their relationships in problem situations” and introduces the skills of “creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.”
Beginning with the noticing and wondering stages, we introduce a “level 2″ of noticing and wondering that helps students focus on quantities and their values, and relationships or calculations they could do. We stress naming quantities and providing units over identifying values (the numbers in the problem).
Students then use those quantities and relationships to get good at guess and check. Being able to identify the appropriate quantity to guess for, set up appropriate calculations, check results against appropriate constraints, and repeat the process in an organized way are vital to being able to set up a mathematical model of a situation. If the student can’t use guess and check successfully, it’s unlikely that they could set up and solve the problem algebraically.
We then encourage students to employ the Make a Table strategy. Partly this allows them to “take advantage of regularity in repeated reasoning” and notice the patterns in their calculations that could be represented in a mathematical model. It also focuses them again on the naming and organization of quantities. Answering the focal question, “what needs to be organized” further develops the habits of mind in this practice.
Lastly, we move students to the Make a Mathematical Model strategy. It again begins by listing quantities and relationships. Then students are supported to “build up” or “break down” the relationships to develop a complete model for the problem. An alternative strategy is to begin the problem by guessing and checking a few times, organizing work into a table, and then rewriting the calculations with a variable for the unknown quantity in the problem.
3. Construct viable arguments and critique the reasoning of others.
First let’s take a look at the “viable arguments” part. The Look at Cases and Use Logical Reasoning strategies both help students to develop their skills at constructing logical, complete arguments. Look at Cases provides students many examples of the kinds of cases they can explore and break situations into, as casebased reasoning is new to many novice problem solvers. Use Logical Reasoning focuses on the questions, “What must be true? What can’t be true? What might be true?” and then supports students to use ifthen reasoning to move items from the “might” column to the “must” or “can’t” column.
Broadening the focus to communication overall, throughout the series we introduce different communication formats and structures that get students writing and talking to learn, to get help, to share, and to present their ideas formally. The activities run the gamut from brainstorming to organizing thoughts to formally presenting. In addition, several activities include structures that have students comparing their work to the work of others and using suggested questioning techniques to help them learn how to give effective feedback.
4. Model with mathematics.
Mathematical modeling is a practice I’d love to do more with. The Make a Mathematical Model strategy has a stronger focus on actually building up the equation, table, flowchart, etc. that students will use to abstract and represent the relationships. We still need some activities that support problemposing and problemdefining. One way we’re moving in that direction is the posing of mathematical scenarios, without a specific question, about which we ask students to notice and wonder. Every PoW written in the last couple years includes a “scenario only” version, and we give a way a free scenario a week here: http://www.mathforum.org/pow/scenarios/.
5. Use appropriate tools strategically.
We haven’t gone as far as we could to teach students how to select and use appropriate mathematical tools in the activity series, because we didn’t want the reality of technologypoor classrooms to prevent students from using the activities. But in the Activity Series Examples documents, available for all 20092010 PoWs, we do model uses of tools like manipulatives, dynamic geometry software, spreadsheets, and graphing utilities. In addition, we’ve done some work to align the activity series with the capabilities of the TINspire here.
6. Attend to precision.
The activity “Testing 1, 2, 3, 4″ in Guess and Check is my favorite for helping students attend to issues of precision, accuracy, and reasonableness. In addition, some of the partner questioning structures in the Get Unstuck, Plan and Reflect, and Wonder activities help students develop habits of mind around asking for and reflecting on precision in language and calculation.
7. Look for and make use of structure.
Looking at the activity series as a tool to help students solve word problems makes it seem disconnected from this sort of looking for structure and connections in specific mathematical content areas. But stepping back and thinking about applying the activities in the context of teaching content might help illuminate how it is that we get students to become people who “look for and make use of structure.” Three activities really stand out for me.
The first is “I Notice/I Wonder.” Proficient noticers see the structure that’s there; proficient wonders seek out even hidden structures. As students practice noticing and wondering in the context of solving problems and learning content, they become better able to see and more disposed to look for structure.
The second is Solve a Simpler Problem. The activities there, while focused on standards methods of simplifying problems or looking for analogous problems, exploit the structure of problems or of the objects within problems. When students look to see how they could make a problem simpler and whether they’ve changed the problem, they are grappling with its underlying structure. When they change a problem to an analogous problem or relate it to a problem they’ve solved before, they’re again using the ability to see structure. And when they rerepresent the numbers in the problem, they are realizing that numbers are flexible and they have the power to decompose and recompose them as needed.
Finally, the Play activities help students dig deeper into noticing and wondering by carefully analyzing the “clues” or structure given in a problem or context and allowing themselves to tweak, change, restate, or rerepresent them. Here is where students learn to add auxiliary lines (or experiment with taking away given lines) in geometry problems, to recognize the significance of the constraints in the problem (i.e. to see that a problem with two unknowns and one constraint has many possible solutions), to try multiple strategies and representations before getting bogged down in one, and to generally approach problems and math content with an openness to exploring them to find deeper and deeper structures.
8. Look for and express regularity in repeated reasoning.
In the Make a Table strategy (which should really be called Make a Table and Look for Patterns) we offer several ways for students to explore and talk through patterns they see in repeated calculations. We encourage students to look for and describe patterns both horizontally and vertically, as well as to describe what’s happening “over and over again.” Even the simple activity provided in an extension of the Guess and Check activities (courtesy of Dr. Math), in which students do calculations without ever simplifying any steps, help them see and exploit repeated reasoning.
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While looking over my 5th graders math pages I noticed the “Make a problem simpler” strategy being used. It increased a SIMPLE division problem into several more steps. What, if any, disadvantages to you see to employing this and the other above mentioned strategies. Especially as it pertains to everyday use of math as we go through our everyday lives. Say for instance grocery shopping etc… While I see this would be useful to understand the structure of a problem and to see that there are more ways than one to solve the problem. I can see that it could be useful as they go on to college math, but does this make sense for “everyday living”? I Would appreciate your thoughts.
Thank you,
Common Core Newbie :)
P.S. Glad I found your blog. It gave me a better understanding of the goals of the CC Standards and their strategies.
Hi Donna, great question! I’m going to offer my thoughts but then I’m going to crowdsource this to some brilliant elementary math educators on Twitter. My thinking is heavily influenced by CGI (Cognitively Guided Instruction) — are you familiar with that framework for thinking about how kids learn arithmetic? The really, really short version of it is, kids usually need to go through a more labor intensive, step by step process before they can make sense of the more efficient algorithms we want them to learn. Think about a kindergarten kid solving 5 + 3. In September they might have to count out five blocks, then count out three blocks, then push the piles together and count out all 8 blocks. But as they solve more problems and explain their thinking and hear their classmates’ strategies, they might start to do things like count on from 5: “let’s see, 5 and three more, that’s 5… 6, 7, 8. 5 + 3 = 8″. And then as they memorize some math facts they might say, “I know 4+4 = 8 and 5+3 is just up one and down one from 4+4, so 5+3 has to equal 8 also.” And soon they just know it. 5+3=8 is stuck permanently in their memory, without them having to memorize a list of “facts”.
So it may be that your 5th graders are still making sense of what division means, how it relates to multiplication, why place value is secretly super helpful in division… and while they go through that process they use brilliant problem solving strategies like “Make a problem simpler.” To an expert, it looks like they are making it harder, but right now for them an algorithm is HARD because they don’t understand what this certain “recipe” of steps has to do with what they know about division, multiplication, and place value. The more they get to tell you about their thinking, and hear other students and your own strategies for solving division problems, and the more they get to do what makes sense to them the more they’ll be able to do those efficient processes when they are ready for them.
Speaking of every day living — it would be a fun project to take an everyday, grocery store math problem, and poll as many adults as you can find about how they would deal with that situation. I wonder if you will find lots of different approaches or one common approach. Fascinating!
Glad you found the blog helpful!
Hello Donna,
I am a K5 special education teacher, and I do use a strategy called “7 division” that sounds very much like what you are describing.
Problem: 244 ÷12
The strategy is to decompose 244 using multiplication and what you know about 12. This means that there are many entry levels and the math is accessible by all. Some people think that this will not promote the development of multiplication/fact fluency, but what I find is that the more students use this method, the more their understanding improves. If a student only knows 1×12=12 it doesn’t take long before they realize the inefficiency and wonder, hmmm could I use 2×12 and they they work on their doubling strategies. When these get better, they soon are pushing themselves to create the biggest multiplication problem possible to decompose. In reality this is probably the strategy we all use, we just do so in chicken scratch on the side of our work, or hold the numbers in our mind for the mental math. The beauty of this strategy is that you can see all of the student’s work and identify the misconceptions much more easily. Unfortunately this format will not allow me to underline the numbers, but I have my students highlight what they multiply by so they can see what then goes outside of the “7″ I was also not able to draw a vertical line so you can’t see my seven as clearly.
244/12
________________________
12×2=24 24424= 220  2
12×10= 120 220120= 100  10
12×5= 60 10060= 40  5
12×3= 36 4036= 4  3
no whole # x 12 =4 so my  +
remainder is 4  ____________
 20 Remainder 4





I asked my daughter how she would do the problem 244 divided by 12 if she had to do it in her head and she said. “Well, I would think, 10×12 is 120 so 20 x 12 is 240 and that is as close to 244 that 12 can get so now I have to subtract 240 from 244 and that gives me 4 so my answer is 20 remainder 4.” I think this kind of decomposing leads to multiplication and division fluency that will be taken into everyday living.
Good grief! The text did not line up at all like I typed it! I am sure that is very confusing! I am going to try to copy and paste it again, and try to imagine the top part of the seven underlining the math problem 244 divided by 12, and the vertical line of the 7 running between the difference of the subtraction problems and the multiplier used with 12 to decompose 244
244 divided by 12
12×2=24 24424= 220 2
12×10= 120 220120=100 10
12×5= 60 10060=40 5
12×3= 36 4036= 4 3
no whole # x 12 =4 my +
remainder is 4 20 Remainder 4
Hi Donna,
I love what Max said, and have two other thoughts to add:
1. I’d love specifics on the division problem. What was the problem, and how did kids make it simpler to solve it? For example, did they change one of the numbers to a landmark number, solve, and then adjust their answer? It’s hard to comment on their work without having a sense of what they did. Looking at student work is one of my FAVORITE things to do, so if you have a chance, please share an example! :)
2. I hear about number of steps a lot. One thing I wonder is if you’ve counted the steps to long division? It’s mindblowing! There are so many! So a solution that might seem like more steps to us, might actually be fewer. Or it might be more. It depends. I’m actually not concerned about number of steps, in general. What I want for my kids is for them to be efficient, accurate, flexible thinkers when computing. So, if they solve 149 divided by 2 by subtracting off 2, then 2, then 2, then 2…they are not being efficient. And they are likely not being accurate, because it’s an errorprone approach. In that case, I would see if I could get the student thinking about subtracting groups of 2. Repeated subtraction of 10s or 20s is getting better. And so on. Eventually, I want students to be able to solve that problem using a more strategic approach. For example:
a) Saying 2 x 7 = 14, so 2 x 70 = 140. (And I want real understanding of multiply by 10 there, not “adding a zero” as a trick.) OK. I dealt with 140, that leaves 9 left to divide by 2. 2 x 4 = 8. 70 + 4 = 74. So, 149 divided by 2 is 74 with one left over.
b) Or, a student might say 150 divided by 2 is 75; 148 divided by 2 is 74, so 149 divided by 2 is 74 1/2.
c) Or, a student might break 149 into 100 + 49. 100 divided by 2 is 50. 49/2 = (40/2 + 9/2) = 20 + 4.5. 50 + 20 + 4.5 = 74.5
Which of those has more steps? Well, b, which might be called solving a simpler problem and was clean for these numbers, but wouldn’t work so well for 169 divided by 7. Part of what I want my kids to be able to do is choose a strategy that’s a good fit for the problem at hand. For this problem, a, b, and c are all reasonable strategies, and could be done mentally in the grocery store, for example.
So, I’m not sure if that’s helpful, but I like to think about what kinds of problemsolvers I want my kids to be eventually. The road between here and there might be a circuitous, and bumpy, but that’s because they’re creating math anew, each time they figure something out. Learning isn’t linear. I think our (really challenging!) job is to create conditions where they’ll keep moving across that landscape, experimenting with different ideas, building new concepts, making new connections, trying out different, more powerful, more efficient techniques in different situations. That’s the fun of it!
I second Max’s recommendation of the three CGI books. They’re great for getting a feel for how children’s thinking develops.
Please keep asking!
Best,
Tracy
Hi Donna,
Great question. As a 5th grade teacher as well, I understand your concern for students’ efficiencies (and application) around the operations as I hear that concern often from both parents and teachers. Max and Tracey made excellent points about how students learn math, the use of multiple strategies, problem solving and the need for students to “muck” around in the math before finding an efficient strategy that makes sense to them. I don’t have much to add in that regard because they did SUCH a beautiful job, however I can speak to what I believe has improved student learning in my classroom and helped me overcome the obstacles like those in your comment.
I see your comment in two parts, doing the actual division process itself and then the application part of “when they need to use this in life, will they have an efficiency to their strategy?” As to the division process itself, I believe sometimes, as adults (me included), we take for granted the place value understanding necessary for students to make sense of operations. When students have gotten to 5th grade without those foundational understandings, it is then an overwhelming task (for the teacher AND student) to get to the root of their understandings and misconceptions and try to build from there. The best way I have found to get to these understandings and misconceptions is letting the students talk, talk and talk some more! I do Number Talks two to three times a week, centered around operations. I use it as a formative assessment to see what my students are thinking and how they can estimate, decompose/compose numbers, and use strategies described in Max’s original post. Not only am I listening, but the STUDENTS are listening. They are revoicing, restating, adding on, agreeing/disagreeing with each others’ strategies while I facilitate the discussion. It does not happen overnight, but students develop a need/want for a more efficient strategy. You would have a hard time finding a student who uses repeated subtraction for a division problem not listening for a less time consuming way to solve the problem. Through this talking and listening, students can find what makes sense to them and build on what they know. We sometimes do journal entries after our number talk in which I ask them to solve a similar problem, but this time to try a strategy they think they would like to understand better. I tell them when they get stuck to stop, write “I am stuck here because…” and the journal goes away. I am the only person who sees the journal and I respond back to each one.
As far as the application, I tried to think back to when I have really last used long division in my everyday life and honestly, I grab a calculator. I think that is the reality of most. Do I round, estimate, and come close to information I need to know? Absolutely. So that makes me rethink what is truly important about the division process….is it really so students can long divide in their everyday life when they leave me? Or is it more about building the foundation for relational understandings, looking for repeating reasonings, decomposing numbers, estimating, and problem solving? Honestly, this is just something I grapple with on a daily basis. Don’t get me wrong, my students will leave my class fluently dividing, however I, over the years, have taken a different perspective on the “Why” part of teaching it.
Like Tracey, I would love to hear more about the student work and the problem itself. I am addicted to student work! Maybe we could connect and share our 5th grade experiences? That would be great!
Thank you for involving me in the convo! Love it!
Kristin
@MathMinds
Hi, Donna
Everything Max, Tracy, and Kristin have said is all excellent. I almost considered not replying at all, but then thought, why not? And so here I am. :)
There are important questions to consider regarding the “Make a problem simpler” strategy. First, who is the problem being made simpler for, the teacher or the student? Second, what does “simpler” mean in the context of a given problem? Finally, what changes need to be made for that simplification to happen?
When students are applying this strategy, they are making the problems simpler for themselves, not anyone else. And, yes, the changes they are making may actually increase the number of steps. I know that this sounds counterintuitive to making the problem simpler, however there are a couple of things that could be going on.
First, the student may be sacrificing efficiency for understanding. When students are new to a concept, this makes sense, and we should applaud their efforts. Efficiency has to be built over time and with practice. I love Kristin’s strategy of using Number Talks to accomplish this goals.
On the other hand, it may very well be the case that the number of steps does not equate at all with the time or effort involved. Just because it takes the child a lot of writing and/or time to explain what they did step by step, that does not necessarily mean it required that much effort when they actually did all the steps. Doing the steps one by one can go faster than describing the steps one by one.
And of course there is also the chance that a student thought he/she was making a problem simpler, but in fact through a lack of understanding, he/she could have made it more complicated. You very well might need to probe some students more to find out what was going on in some cases.
As far as everyday life goes, being taught consistent methods for solving computation problems can be helpful to fall back on. However, isn’t it even more powerful to be taught how to be flexible in our thinking to analyze a situation and select a strategy for ourselves?
For example, my teachers taught me long division years ago, but as an adult I’ve learned that I can use the calculator in my phone to solve most division problems much more quickly.
For a nondivision example, I know that I can solve 100 – 98 with the standard algorithm using pencil and paper (it happens to require borrowing across 2 zeroes which is cumbersome), but I also know I can analyze the numbers to see almost immediately that they are 2 away from each other.
Or for an even more reallife example, if gas is $3.25 per gallon and my tank is 14 gallons, I know that 14 × 3 is 42. I also know that 14 quarters is $3.50, so the cost to fill my tank should be $45.50.
What ends up happening quite often is that adults who were taught rigid methods tend to use them in school when they are required, but outside of school, they fall back on invented strategies that made more sense or were easier to do mentally, for example while out shopping without paper and pencil. I’ve had numerous conversations with adults who sheepishly admit to the nontraditional strategies they’ve been using all these years. They often felt that they were wrong for using them and had to hide them because they didn’t match what they learned in school. They feel vindicated when I tell them they have been using sound mathematical reasoning all along. If only these people could have experienced this while they were still in school. Their strategies are just as effective and sometimes even more efficient in some circumstances.
So to get back to what you said in your comment, I would see your students’ work on this one division problem as a place to gather information about where your students are at regarding understanding division. Probe them to see why they did what they did to make the problem simpler, and ask them why they think their methods resulted in a simpler problem. It could help you immensely as you plan future activities for your students to help push their understanding, efficiency, accuracy, and fluency.