NCTM 2016 Take-Aways Summary

My head is still swirling with everything that my NCTM experience made me think about math education. However, from all of my notes, I wanted to synthesize my bite-size takeaways from this conference. 

Really Important Reminders:

• Select tasks that combine different content areas
• Students should only be working with other people when it’s actually beneficial to work with other people.
• We should be engaging students not just in how to do math but in why and when.
• Having a problem solving protocol(s) is more important than what the particular protocol is
• Students often don’t distinguish between the tick-marks (addresses) and the intervals (distances) on a number line
• Formulas: throw most of them away

Things I can implement tomorrow in my classroom:

• “Pencils down, brains on” One minute to think about the problem
• For Estimation 180: “If you say a million [as your too high] you get this conceptually but be brave”
• Visual Patterns: Fold paper into 4 parts, and reveal and have students draw 1 step at a time. After each step ask students “How many in the next one?”
• Pull a kid from each table for a small group huddle and then send them back to be experts

Resources to Look In to:

• NCTM research brief on problem solving
• Math Visions Project
• Graham Fletcher’s Progression Videos (obviously in conjunction with the actual standards)
• Geoff Krall’s posts on PBL and revisiting his curriculum maps

And finally, I’m thinking a lot about engagement lately. Here are some things that I want to remember from the conference:

• “If I don’t ask you to guess, you don’t have a stake in it”
•  Using a high motivation Do Now as leverage for the rest of the class—if kids start excited and engaged, this sets them up for success in the rest of the lesson
• Students will be curious about (SWBCA) instead of students will able to (SWBAT).
• “People can’t understand solutions to problems they don’t have”
• Find a problem you are interested in and kids will probably be interested in it as well.

NCTM 2016 Days 2 & 3

Ignite:

Peg Cagle – promoting teaching as a profession

Michael Fenton – What we can learn about mathematical sequels from movie sequels

Good - best sequels take advantage of what we know about context and characters and hit the ground running - balance between familiarity and innovation - shed new light on original work

Bad - know when to stop - don’t wing it. You have to do the prep work - do not abandon what works

Marilyn Strutchens – equitable classroom strategies --> positive math identities: math autobiographies, low threshold/high ceiling, math teaching practices, formative assessment, social justice activities, ask students what they want to study, students can contribute to the development of math, engage families in doing math

Andrew Stadel – allow time constraints to improve teaching: 20% of most effective functions should take up 80% of the time

Annie Fetter – students will be curious about (SWBCA) instead of students will able to (SWBAT). “People can’t understand solutions to problems they don’t have”

Max Ray-Reik – borrowed lessons are good because they come with a story, but it’s hard to build coherence with a hodge-podge of borrowed lesson. The ideal curriculum is not free and open-source, but open-story and crowd-engaged.

Tracy Johnston Zager – fluid and powerful collaboration comes from thinking partnerships (generative, supportive working side-by-side), cross pollination (credit for ideas becomes interesting), math disputes (representing own ideas, listening to others, publically changing your mind), and peer feedback (how do I make this better). We should only be working with other people when it’s beneficial to work with other people.

Lee Stiff – All students deserve high quality education. We have created the achievement gap and we can change it.

Jennifer Wilson – The slow math movement.

Matt Larson – We should be engaging students not just in how to do math but in why and when.

 

Fumbling Toward Inquiry: Starting Strong in PBL

Geoff Krall

Take-aways:

• 5 design principles
• notch some early wins where students are talking to one another about math and thinking of themselves as mathematicians (start short and successful)
• provide an iterative framework and protocol (having a problem solving protocol(s) is more important than what the particular protocol is)
• choose tasks that support targeted instruction and group-mate experience (example pull a kid from each table for a small group huddle and then send them back to be experts)
• start slowly and strategically
• don’t go it alone
• Ira Glass: "Nobody tells this to people who are beginners, I wish someone told me. All of us who do creative work, we get into it because we have good taste. But there is this gap. For the first couple years you make stuff, it’s just not that good. It’s trying to be good, it has potential, but it’s not. But your taste, the thing that got you into the game, is still killer. And your taste is why your work disappoints you. A lot of people never get past this phase, they quit. Most people I know who do interesting, creative work went through years of this. We know our work doesn’t have this special thing that we want it to have. We all go through this. And if you are just starting out or you are still in this phase, you gotta know its normal and the most important thing you can do is do a lot of work. Put yourself on a deadline so that every week you will finish one story. It is only by going through a volume of work that you will close that gap, and your work will be as good as your ambitions. And I took longer to figure out how to do this than anyone I’ve ever met. It’s gonna take awhile. It’s normal to take awhile. You’ve just gotta fight your way through."
• Select tasks that combine different content areas
• Find a problem you are interested in and kids will probably be interested in it as well.

What I am wondering now: I used to primarily use tasks at the end of unit for students to solidify and apply their learning. Now I am often using tasks to introduce a topic and draw upon students’ prior knowledge and reasoning. I want to move towards integrating problem solving with mini-lessons and other instructions. What structures do I need to make this happen?

My Journey from Worksheets to Rich Tasks

Michael Fenton

Take-aways:

	“Before”	Now
What Math Teachers Do	- Answer questions - Present “the notes” - Assign Practice	Facilitate opportunities for students to engage meaningfully with math
Role of Math Teachers	-Explanation-Giver - Answer-Provider	- Question-poser - Thought-provoker - Discussion-starter - Math-instigator

• Estimation 180:
• Have them struggle with a specific estimation before giving the tools of too low/too high
•  Too low/too high a confidence builder for students who think that wrong answers are “their game”
• “If you say a million [as your too high] you get this conceptually but be brave”
• Visual Patterns
• fold paper into 4 parts, and reveal and have students draw 1 step at a time.
• After each step ask students “How many in the next one?” (Acknowledge that this is unfair after step 1)
• Make a table, sketch a graph, write an equation. Check with Desmos.

What I am wondering now: I am familiar and excited about all of the activities that Michael Fenton brings up as ways for students to engage meaningfully with math—3-act problems, estimation 180, visual patterns, would you rather, my favorite no, open middle, which one doesn’t belong. I have used them in isolation in my classroom. But I am really wondering how to fit them into the larger structure. How can really intentionally I use them to build the  math practices while working toward specific content goals.

 

ShadowCon

Robert Kaplinsky – give power, gain influence

Gail Burrill – Listen to the voices in your head:

            - Think deeply about simple things (Ross)

            - Never say anything a kid can say (Reinhard)

- I know that my students understand when I see them in a place they have never been (Cuoco)

            - Let the students do the work (Wiliam)

Kaneka Turner – “When were you first invited to the math party and how did that feel?” Who will you invite to the math party?

Graham Fletcher – Really get to know your standards: be a wise consumer

- draw the standard or use tools to support your understanding

- be dumb… surround yourself with brilliance

- be vulnerable

Brian Bushart – explore, play, and find joy in doing math

Rochelle Guitérrez – These populations need access to math vs. We need this population’s contributions to mathematics

The Power of the Number Line: Understanding Fractions as Numbers

Ryan Casey

Take-aways:

• students often don’t distinguish between the tick-marks (addresses) and the intervals (distances) on a number line
• in the common core, the fraction a/b is defined as “the quantity formed by a copies of size 1/b”
• “mathematicians are convinced that if something is on a # line then it is a #”
• prerequisite skills for putting fractions on a number line
  
• principles of linear measurement
• measurement is iterative
• measurement involves partitioning
• measurement involves processes that combine partitioning and iterating

What I am wondering now:  I really like the framework of using whole number understanding and leveraging that to extend into rational (or irrational) number understanding. I am wondering what would be good activities to do with both rational and irrational numbers in order to build my eighth graders’ understanding of irrationals.

The Life-Changing Magic of Tidying the Math Curriculum

Jason Zimba

Take-aways: How to tidy: teaching vs. turning in to a topic

• Never ask about manipulatives on tests
• Forget about calculators K-5
• Formulas: throw most of them away
• Headings: only for distinct topics
• Methods: don’t teach too many (having too many around means less time on the challenging one that gets us over the next hurdle. Strategies go stale)
• “Do mental calculation with multi-digit numbers, but save it for cases where a readily apparent mental strategy is both faster and more reliable than the standard algorithm”
• Procedures: they are for procedural tasks
• “A concept is worth 1000 procedures”
• Terms that never appear in the common core
  
  
• Store topics inside of one another

What I am wondering now: What am I teaching as a topic that isn’t really a topic? Exponent laws/Simplifying exponents. Solving systems using elimination (save that for high school). Anything else? And what are the most important methods that will get us over the next hurdle? Am I focusing on those? I think I need to spend some time tidying, but doing it with the text of the standards and the input of at least the other people in my department.

NCTM 2016 Day 1

Motivating the Unmotivated: Access to Learning

Barbara Dougherty and Lisa Bendall

Take-aways:

• Motivation through games that are high interest, require collaboration or cooperation, and incorporate significant mathematical thinking (ex: Bowl-a-Fact and Find-a-Place)
• Using a high motivation Do Now as leverage for the rest of the class—if kids start excited and engaged, this sets them up for success in the rest of the lesson
• Public accountability for engagement—homework through expert groups or collaborative groups

What I am wondering now: What is the “just right” fit for skill level in games that require some sort of prerequisite knowledge? How do students get feedback to improve if they are consistently demonstrating misconceptions in these types of games? What are the characteristics of a game that is both engaging and builds mathematics?

Cuisenaire Rods and Number Lines: Multiplication and Division of Fractions

Adam Harbaugh, Kurt Killion, and Gay Ragan (Powerpoint here)

 Take-aways:

• Sequence of building understanding: word problem--> concrete model with Cuisinaire rods--> # line model --> matching pencil and paper algorithm
• They are only doing “sharing” division (how many 1/3’s in 3/4?) rather than “measurement” division (3/4 is 1/3 of what whole?)  because “measurement” contexts with two fractions are too contrived. Therefore, this leads to the development of the common denominator algorithm.

What I am wondering now: They require their students to make the model so that it is easily partitioned by both denominators. This makes the numbers come out nice and the model easier to use, but it does not promote sense-making. Where is the line between sense-making/flexibility and streamlining for success?

Choosing Tasks for Productive Struggle, Not Frustration

Jackie Murawska (Powerpoint here)

Take-aways:

•  Characteristics of problems that are good for productive struggle:
• the problem solver must decide what math to bring in
• the task uses real life (often messy) data
• the task requires mathematical modeling
• “If I don’t ask you to guess, you don’t have a stake in it”
• “They’re not making mistakes because they aren’t thinking. They are making mistakes because they are thinking.”

What I am wondering now: These characteristics define a subset of problems that are engaging and good for productive struggle. But I think that the type of modeling problem described is only one type. How would I categorize the other types the are highly compelling?

Linear or Quadratic? Engaging in Two Effective Math Teaching Practices

Amy Hillen and Jennifer Outzs

Take-aways: 

• “In this figure as the step changes, the _____ also changes.”
•  For assessing questions stay and listen, for advancing questions walk away

What I am wondering now: I started my year building linear relationships with a lot of visual patterns work. I want to revisit that structure now and use this much more open prompt in order to review linear vs. non-linear.

Supporting Productive Struggle in Secondary Classes

Michael Steele

Take-aways: 

• “Pencils down, brains on” One minute to think about the problem
• “[Students] eavesdropping [on other students] is a great sign because it means there’s something worth listening to”
• When students are starting to say “What if…” that is a good sign

What I am wondering now: What should help look like in my classroom?

Math Games to Develop Experimental Thinking

In the past several months, I have had two “doing math” experiences that involve games—one at PCMI’s Boston teacher leadership weekend and the other at the Boston Math Teachers' Circle. Each game had a similar structure: there was a goal that you had to meet to win. There were the rules of the game, which gave the overall structure. Then there were the conditions in the game, which were particulars that could be changed without changing the structure of the game. And finally there was some sort of move that you could make.  Here are 3 of the games with each of their elements described:

 

Game	To Win	Rules	Conditions	Moves
1-person Stone Game	All of the stones end up in one pile	- Take one stone each from two of the piles and put the stones in the remaining pile	- There are 3 piles of stones - The piles have 6, 7, and 8 stones	Which piles do you take from?
2-Person Penny Game I	Take the last penny	- You can take some number of pennies in your turn - Then pennies are in a line	- Start with 11 pennies - Take one or two pennies - Who goes first	How many pennies do you take?
2-Person Penny Game II	Take the last penny	- You can take some number of touching pennies in your turn - The pennies are in a circle	- Start with 13 pennies - Take one penny or two touching pennies - Who goes first	How many pennies do you take and from where?

Mathematical Thinking Prompted by the Games

With all of these games, I felt like my experience went beyond playing (to win) a game and into deep mathematical thinking. I was doing math as I explored one or more of these questions:

• Is it possible to win? Why or why not?
• Under what conditions is it possible to win?
• What moves do you need to make to ensure that you win?
• These conditions are necessary to win, but are they sufficient?
• Do these conditions set up all of the ways win or only a subset?
• How does winning work if you modify the conditions of the game?
• How does winning work if you generalize the conditions of the game?

If I replace the word “win” with “find a solution” in each of the above questions, I often consider the same questions when I am doing math that is not prompted by a game. For example, at the math teachers' circle last month, I spent about two hours working, starting with the prompt “Find two different-looking sets of three numbers that both have a mean of 5 and same standard deviation.” This problem still has structural, unalterable rules: how we calculate mean and standard deviation, and that the sets need to look “different.” It also has conditions that can be modified and experimented with: 2 sets of 3 numbers, and that the mean is 5.  However, what sets this apart from a game is that there aren’t moves.  I was still experimenting with different options, but it felt really different than moving stones or taking turns removing pennies.

Whether with a game or not, this type of experimenting, asking “what if” questions, and generalizing is at the heart of a lot of the mathematical thinking that I do when I take time to “do math” for myself. But my students very rarely, if ever, do this type of thinking. The closest that we come is me directing them to try out various options, look for patterns, and then generalize from that. But they don’t get to experience the experimenting that prompts them to ask these questions and then the joy of choosing what question(s) they will further explore based on what they think will be most interesting or worthwhile. And I think that is a problem. If this is the type of thinking that I find most fun, interesting, and fulfilling, then my students should be having that experience as well.

Implications for my Classroom

So what would it look like if I fostered a culture of doing this type of mathematical thinking in my classroom?

There would be two main goals:

1. Developing students’ ability and desire to ask “what if” questions where they change and eventually generalize the conditions of a game/problem. From now on in this post, I will refer to this type of process as experimental thinking.
2. Using experimental thinking to develop understanding of grade-level content standards. The first goal is less useful in my purposes as a teacher unless it is leveraged into this second goal.

 In order to fully develop these two goals, I could imagine the following trajectory:

1. Play games that do not have a specific content focus, but would hook students and prompt experimental thinking. We would play the game for 10-15 minutes where students would get to know the game and figure out a winning strategy. We would then pause and generate questions about what the game made us wonder. Students would then choose one or more of those questions to investigate. Over several games, we would categorize our questions, and hopefully build a framework of questions similar to the ones that I mentioned above.
2. Play games (or as one of the other people at the math teacher’s circle suggested “gametivities”) where there are moves that you can make, but the game lies in specific content. Playing the game would then prompt experimental thinking about this content
3. Have content specific math challenges where there is still a goal, rules, and conditions, but there are not “moves” any more. Students ask the experimental thinking questions to deepen their own understanding and explore the concept.

Here’s what I am imagining for a “game” to introduce graphical solutions to systems of equations. I think as it is right now, this would lie in the third part of the trajectory because it is missing the experimentation through game moves.  On desmos, I would give students a graph of a line, say y = 3x – 5. The challenge would be to write equations for three more lines that do not intersect this one. Once they had time to experiment with this, I would imagine that the further exploratory questions that this could prompt could be:

• what if we wanted the lines to have one intersection?
• what  if we wanted the lines to have more than one intersection?
• what if the line was something besides y = 3x – 5?
• what if we started with any lines?

But I would really like to figure out how to gamify this more. I think the bridge of content-specific goals that are in the context of something that actually feels like a game is important. Having the game structure adds extra motivation to reach the goal and an easier framework in which to mess around and just try things.

This framework would need to happen over a longer period of time--at least a couple of months, if not the whole year. I haven't built the groundwork for this type of thinking in my classroom this year, but I could imagine trying out phase one in the last couple weeks of school. We wouldn't build content through the experimental thinking, but it would give me a chance not only to have a go-through at setting the groundwork for experimental thinking, but also decide if I think that this trajectory would be high-leverage enough in terms of development of both mathematical thinking and specific math content in order to dedicate significant time to it next year.

A Conversation with Jon Star

I am currently part of a cognitive science inquiry group with about ten other math teachers. The goals of the group are to learn more about what cognitive science research says about teaching and learning and also somehow apply this research to our classrooms through some sort of new or revised structure or routine. So far, we have read Make it Stick and Why Don’t Students Like School and  analyzed them through the framework of generating questions that are prompted by our reading and the relationship of the ideas to our own teaching experience.

I have all sorts of questions, but this week we posed three of our group’s bigger questions with Jon Star, who is an educational psychologist at Harvard who focuses on flexibility in problem-solving and acquisition of algebra. I came away with so much to think about because he was able to give specific suggestions for practice that were very obviously grounded in research.

Question 1: What can we do to support students, many of whom have identified disabilities, whose working memory has a smaller capacity or who have other challenges with working memory?

Answer: Here is a list of strategies, all of which are centered around the principle of modifications/accommodations that do not sacrifice the learning goals. These are strategies that can be considered “good teaching” because they can benefit all students. However, the negative consequences of not using these strategies hurt some students more than others.

Strategies:

• Reduce arithmetic complexity. Harder numbers make problems more complicated, but does not necessarily require a deeper understanding. He suggests making things conceptually harder, not computationally harder.
• Be aware of the amount and type of words in a problem.
• Be smart and organized about how you are using illustrations and board space. Processing everything verbally is extremely taxing on working memory. Board space and other visuals can be used a surrogate working memory for students.
• Reduce the need for dual processing (Ex: Expecting students to read and listen at the same time)

I really appreciated that he also clarified that awareness of when you are/are not doing any of these things is the more important than making sure to use all of these strategies all of the time. There are good reasons for not following each of these strategies when there is a specific purpose.

Question 2: How do we foster a culture where students are motivated to persevere through spaced, varied, and interleaved practice, even though it is harder and the results are not as immediately obvious?

Answer: It is important to complexify what cognitive scientists are saying about practice.

• Nuance #1: Practice should differ in different phases of learning—the first exposure to a concept/procedure vs. solidification after basic mastery. Some blocked practice may be necessary in the initial phase for motivation and initial formation of knowledge.
• Nuance #2: Elaborative recall is the main principle that drives recommendations about effective practice. The more opportunities to reconstruct/apply/elaborate/expand knowledge, the more solidified it becomes. With blocked practice, the concept/procedure gets too automatic and students don’t reap the benefits of the recall. Also, once students automaticity with a procedure, that knowledge is very stable and difficult to reexamine/extend/reconfigure. Thus if students gain automaticity before associating with concepts, it is extremely hard to do this later on. Ideally, we should be building automaticity and connections simultaneously.

Question 3: Ideally, we are building procedural fluency from conceptual understanding. Where does practice fit in this arc and should the practice look different depending on where in the arc you are?

Answer:

• Point #1: It’s a misconception that there is an order to how students should learn concepts and procedures. It is not true that conceptual understanding needs to precede procedural fluency. It is not true that procedural knowledge cannot develop conceptual knowledge. Order is arbitrary—instead it is more important that they are connected and developed iteratively.
• Point #2: There are the same best practices for developing procedural knowledge as conceptual knowledge. We can define conceptual practice as an opportunity recall and reconstruct concepts (just as we would define procedural practice as opportunity to recall and reconstruct procedures).
• Point #3: It’s hard to articulate what it looks like for conceptual understanding and procedural fluency to be intertwined. One possible way to “see” this is through flexibility: students should be able to solve a problem more than one way and identify which strategy is “better” (ex: more elegant) and what the criteria are that decides what makes one solution strategy different than others.

One final gem from Jon Star: 

When you are listening to a performance of a piano concerto, often the performance itself is evidence enough of mastery of the skills and concepts behind the music. You wouldn’t need to ask the soloist to explain why they made the decisions they made. The same should be true for math. A student does not always need to explain in order to demonstrate conceptual and procedural knowledge. Sometimes the work/thinking they do speaks for itself.

On my mind now:

My biggest questions coming out of this conversation are about Star’s points about the (as one of my coworkers put it) commutative nature of developing conceptual understanding and procedural fluency.  My whole teaching framework is built upon the idea of building procedural fluency from conceptual understanding. NCTM’s Principles to Actions articulates this as one of their math teaching practices: “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” 

However, based on the challenges that I have experienced when trying to teach this way (When are they ready to move from concepts to procedures? How do I help students actually connect the procedures to what they have built conceptually? How do I encourage students to go back to the conceptual understanding in order to re-build procedures that they have forgotten?), I am definitely in a position where I want to read and think more about Star’s position. I think that a lot of the current emphasis on conceptual understanding is a direct response to a history of teaching procedures with no conceptual understanding. It makes sense to me that the most important part is that we are doing both and connecting them, not that there is an extremely delicate, perfect sequence to have procedural fluency build out of conceptual understanding. Star also stated that understanding a procedure is different than having conceptual knowledge of a procedure, which is something I would like more specifics on. What does understanding of a procedure look like then? So I’m waiting on some recommendations from Star for some further reading in order to help me incorporate (or not) this into my ever-changing math education philosophy.

Adapting Scope and Sequence for Remediation

Like many schools, we have students in every grade who start the grade with unfinished learning/misconceptions/gaps in standards that fall under previous grades. We operate with a general policy of trying to remediate through grade level standards—as an eighth grade teacher I am teaching primarily eighth grade content and always using that as the starting point/end goal of any reteaching of prior-grade-level content. However, we have not done very much work to make sure that this happens in a systematic way. I was really excited when the Standards Institute addressed exactly how necessary and challenging this work was and provided a framework for thinking about how to do this type of Tier 1 intervention.

This slide from the Standards Institute (Day 3 powerpoint for all of the math groups) illustrates how students will fall further behind and never be able to catch up if we spend a significant amount of time filling gaps in isolation from the current grade-level content.

As an alternative, they offered these ideas about where and what to remediate. The idea is to make a strategic plan instead of trying to review everything or only reacting after the fact to what you have learned about students’ foundational knowledge.

While this generally matches how I have been trying to remediate, I really appreciate how clearly and intentionally this process is laid out. As someone who is in a department where we don’t use a curriculum and primarily write our own sequencing and pacing, I think that this work is really our next step in building vertical and horizontal coherence. I am extremely excited to take this framework to my team for when we are doing our big-picture planning.

The Standards Institute also introduced me to two amazing tools that will be essential to this work:

1) Zimba Wiring Diagram: One writer of the common core’s interpretation of how the standards connect to each other both between- and within- grades.

2) Coherence Map: This is an interactive map that allows you to “zoom in” on the wiring map. You can choose a standard to start from and then follow it to a connection of your choice.
3) Content Emphases by Cluster: This designates which clusters are major, supporting, or additional standards. Shockingly, I had never seen this before.

Rigor vs. Cognitive Demand

Between work on creating my school’s instructional vision and attendance at the Standards Institute this week I have been doing a lot of thinking around what makes good instruction at different grain size—lesson, unit, curriculum, sequencing, and overall educational experience. Two concepts, which I would once have used interchangeably, have come up frequently.

Cognitive Demand:

Stein, Smith, Henningsen, and Silver define cognitive demand as “the kind and level of thinking required of students in order to successfully engage with and solve the task.” I have analyzed tasks with their four levels of cognitive demand in many settings. The biggest question that comes up for me is what is the ideal placement and balance between low- and high-cognitive demand tasks? One of the professors in my teacher prep program always suggested  “a steady diet of high cognitive demand tasks” which I found both wise and frustratingly vague.

Rigor:

The idea of rigor as a concept separate from cognitive demand is a newer one for me. In the past nine months, though, it has come up for me in two places—as part of my district’s curriculum review tools and at the Standards Institute. At corestandards.org, they define rigor as “deep, authentic command of mathematical concepts” and clarify that it is “not making math harder or introducing topics at earlier grades.”

Here are the elements of rigor. I’ve also included the definitions from the five strands of mathematical proficiency, because I think they better describe the three elements.

	Corestandards.org	Adding it Up Chapt. 4
Conceptual Understanding	The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.	Comprehension of mathematical concepts, operations, and relations.
Procedural skills and fluency	The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others.	Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Application	The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.	Ability to formulate, represent, and solve mathematical problems. (This is called strategic competence, not application, here)

The language of the shift calls for students to pursue the three elements with “equal intensity.” However, this leaves me with questions about how the three elements interact.  One of the tools that my district used in evaluating curriculum starts to further illustrate exactly what balance of rigor looks like. 

The authors of Principles to Actions state “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” With this in mind, we were really looking to see whether and how procedural fluency was explicitly built off of conceptual understanding.

And then there’s the question of applications. In my experiences with both the Standards Institute and on the curriculum review team, applications were generally defined as any problem with a “real world” context. I find this interpretation problematic for three reasons.

1. Instead of applications acting as a third leg in the three-legged stool of rigor, all conceptual understanding and procedural fluency problems can be categorized as applications or not
2. There wasn’t a larger conversation around how well the real-world context was executed (See Dan Meyer here on one of my biggest pet peeves).
3. There’s a reason that the phrase “real-world and mathematical problems” occurs in the common core 31 times rather than just “real-world problems.” I think we need to better define what it means to have an application that is in a purely mathematical context, but I don't think we should just drop it.

I am not yet ready to fully develop my ideas around how to better define application, but I think that the strategic competence description in Adding it Up would be a good place to start.

Relationship

Based on these definitions, it is clear to me that I can’t use the terms “rigor” and “cognitive demand” interchangeably. But these are both words that I hear thrown around a lot when trying to evaluate teaching and curricular materials. So what relationship do they have to each other?

In looking at the text of a problem and the execution by both students and teacher, I think it makes sense to identify the foremost element of rigor and the level of cognitive demand separately. However, I would imagine that conceptual and application problems have a higher likelihood of being high cognitive demand, and that procedural fluency problems have a higher likelihood of being low cognitive demand.

I also think that it’s worth trying to push a little bit beyond the concept of “a steady diet”—which I think can be applied both to high cognitive demand tasks as well as tasks of each type of rigor.  In the standards institute we started to look at arcs of the elements of rigor over a particular topic or unit. In general, we seemed to see the sequence of conceptual understanding --> procedural fluency --> application, though of course it was not quite that clear cut. I would also question the placement of application primarily at the end because I think it underestimates students and impedes their learning to say that procedural fluency is required before being able to apply a concept.

This makes me wonder if there is a similar general flow that makes sense for cognitive demand. I would certainly argue against low cognitive demand being a prerequisite for high cognitive demand. I think that there’s a lot of value in starting with high cognitive demand tasks because they are an ideal place to surface student reasoning and build upon the foundation that students already have. But I wonder if it is oversimplified to suggest high cognitive demand --> low cognitive demand --> high cognitive demand as an arc for a unit. It might make more sense to have a unit that is composed of a series of shorter cognitive demand arcs of that nature.