# data[].attributes.body
Clarify, share, and understand learning intentions and success criteria.

Engineer effective discussions, tasks, and activities,  that elicit evidence of learning.

Provide feedback that moves learning forward.

Activate students as learning resources for one another.

Activate students as owners of their own learning.

# data[].attributes.body
Using careful questioning to guide the students toward the solution, or even towards the "right" question.

# data[].attributes.attachment_link.readable_text
September 2015 | Volume 73 | Number 1
Questioning for Learning

Allison Hintz, Lynsey Gibbons and Melinda Knapp

"I'm confused. How does figuring out the perimeter help us?"

"We don't know how to count the shaded parts."

"We're stuck!"

When math teachers hear students make comments like these, they often wonder how to respond. They might try to craft questions that help students deepen their thinking rather than merely reach the "right" answers. Or they might wonder how to pose questions that encourage students to think like mathematical sense-makers.

Teachers who reply to such comments with questions are on the right track: Posing purposeful questions is an important component of facilitating meaningful discourse in mathematics. Purposeful questions also support another key element of any meaningful math lesson: productive struggle.

Why Is Struggle Important?

Effective teaching of mathematics provides students both support and opportunities to engage in productive struggle with the content (National Council of Teachers of Mathematics [NCTM], 2014). When students engage in productive struggle, they delve more deeply into understanding relationships among mathematical ideas instead of simply seeking correct solutions. Productive struggle often leads to long-term benefits; with the stronger understanding that arises from such struggle, students can apply their math learning to new problems (Kapur, 2010).

But generating questions that yield productive struggle isn't always intuitive. When replying to comments that clearly signal difficulty ("We're stuck!"), a teacher could well perceive students' frustration or lack of immediate success as indicators that instruction has failed. The teacher might jump in to rescue students by breaking down the task and guiding them step by step (Have you thought about starting by drawing a graph? First, create a table.).

Although well intentioned, this approach deprives students of opportunities to engage fully in making sense of the mathematics (Stein, Smith, Henningsen, & Silver, 2009) or to develop identities as mathematical thinkers capable of reasoning with difficult ideas. A better approach is to pose pointed questions that enable students to engage in struggle that leads to real understanding.

Questions That Get Students Started

To see such struggle in action, let's drop in on two students, Collin and Marina, as they approach a math problem. The problem (see fig. 1) invites students to look at a shaded figure in relation to a grid and "find the percent, decimal and fractional amount of the shaded area." After solving the task, students discuss their reasoning.

Figure 1. Math Task 1: Modeling Percent

Find the percentage, decimal, and fractional amount of the shaded area in relation to the whole grid.
Source: From Bridges in Mathematics Curriculum (1st ed.) by Math Learning Center. Salem, OR: Author. Copyright Math Learning Center. Used with permission.

As Collin and Marina are working, their teacher, Mr. Craig, comes over to listen:

Collin: We're stuck.

Marina: That we need to find out how many squares are shaded out of 100 because that will tell us the percent.

Collin: Yeah, but we don't know how to count the little shaded parts.

Marina: (tracing her finger around the grey shaded portion) We think the shaded area is a square.

Mr. Craig: What do you know about finding the area of a square that might help you?

Mr. Craig asks an intentional question (What do you understand about this task?) to help these learners uncover what they already understand about the problem. This question gives them time to make sense of and understand the task and scaffolds their thinking-without the teacher stepping in to do the thinking for them. It also allows Mr. Craig to hear Collin and Marina's current ideas, which helps him to ask a more specific question (What do you know about finding the area of squares?) that scaffolds the struggle. The careful questioning supports Collin and Marina as they think about how to use what they know and encourages them to reflect on and explain their thinking.

To help students engage with a task and support their perseverance in solving problems, pose questions that help students access broad math principles and get a feel for how to approach and think through a rigorous task. Keep questions like these in your "back pocket" to guide students stumped by any math problem:

What do you already know? What do you need to know?

What do you understand so far?

What parts of this problem make sense to you?

What in this problem doesn't yet make sense to you?

How might we decide which approach makes more sense?

What are some math ideas we've worked on before that could help you with this new idea?

Questions That Make the Mathematics Visible

Teachers can also support productive struggle by asking questions that make the mathematics more visible. For example, let's look at the last question Mr. Craig asked: What do you know about finding the area of a square that might help you? This purposeful question guided Collin and Marina to think specifically about using what they know about finding the area of a square to think about proportion in this problem. In doing so, it built on something they had already noticed (that the shaded area was a square) and sharpened their focus on a mathematically productive strategy without telling them how to solve the problem.

This question also served the mathematical goals for this lesson-to understand that a percentage is a ratio of the whole and that percentages, fractions, and decimals are related because they can all represent the same portion. The question nudged the students toward more sophisticated ways of reasoning-away from counting squares one by one and toward using what they know about decomposing areas of shapes.

These kinds of questions are challenging to pose, in part because they require a deep knowledge of mathematics and an understanding of how students' mathematical ideas build over time. As we have learned through our work with local elementary teachers, a powerful way to generate questions that reveal mathematics concepts is to thoughtfully plan with others who are teaching similar lessons.

When teachers plan collaboratively, they help one another identify the mathematical goal and the types of questions that will make mathematics visible at points of struggle and beyond. They consider how to advance students' mathematical understanding during the lesson and get ready to help students think like mathematical sense-makers (Smith, Bill, & Hughes, 2008).

Questions like these would make the math ideas behind the problem in Figure 1 visible:

How are percentages, decimals, and fractions related?

What do you know about finding the area of a square that might help you?

How does that array relate to multiplication and division?

It's important to notice the rich nature of the task these students are working on. Finding the proportion of the area of a shaded figure within a larger figure enables students to engage in active inquiry and exploration.

Tasks that promote reasoning and problem solving are especially fruitful for working on teacher and student questioning during moments of struggle. Such tasks help students engage in meaningful discourse.

Maintaining the high cognitive demand built into these math problems as students grapple with ideas, however, is often challenging (Smith & Stein, 1998). Comments that tell students what to do lower this cognitive demand; students lose opportunities to engage in mathematical reasoning and may pick up unintentional messages that they're not capable of tackling difficult mathematical ideas.

Encouraging Student-to-Student Questioning

Facilitating meaningful discourse also means supporting student-to-student questioning. Teachers play an essential role in fostering a classroom culture that supports such exchanges. Selecting tasks that warrant discussion, cultivating discussion that centers on making sense of students' ideas, modeling purposeful questions, and orienting students to one another and the mathematics are important steps we can take to give kids opportunities to ask questions of one another and build on ideas collectively.

Consider how Henry and Amaya tackle the same task that Collin and Marina did-figuring out what percentage, decimal, and fraction a shaded portion of a figure represents in relation to a grid-with a larger area shaded (see fig. 2). In presenting this task, Mr. Craig has encouraged students not to count the squares one by one but to use what they know about decomposing areas of regular and irregular polygons to determine the percentage of the grid that's shaded. He hopes they will connect this knowledge to what they know about fractions and decimals.

Figure 2. Math Task 2: Modeling Percent.

Find the percentage, decimal, and fractional amount of the shaded area in relation to the whole grid.
Source: From Bridges in Mathematics Curriculum (1st ed.) by Math Learning Center. Salem, OR: Author. Copyright Math Learning Center. Used with permission.

To help students appreciate that good thinking often emerges from confusion and struggle, he's also suggested to the class that if they get stuck, they can ask one another, "How should we get started?" After Henry and Amaya share brief observations on this tough problem, Henry uses this question to help them move forward.

Henry: I'm thinking it's 25 [small square units]

Amaya: I'm thinking it's 50.

Henry: Let's count it.

Amaya: Remember, we have to try not to count [the units] one by one.

Henry: How should we get started?

[Amaya writes "10" on her paper]

Henry: So the perimeter is 10? How does figuring out the perimeter help us?

Amaya: 10 is the number of square units on the edge of the grey part [tracing her finger around the edge of the shaded portion]. I counted 20 half squares, so it's 10 whole, but I don't think 10 is the perimeter-I think it's the total area of the half squares on the edge. What do you think?

Notice how Henry and Amaya found a way to get started on this rigorous task. They both guessed how many small square units were in the shaded portion and then moved into discussing how to calculate what percentage of the whole array the shaded portion takes up. Most important, they persevered in solving the problem. They realized that it's acceptable to not know how to get started, but not acceptable to give up (NCTM, 2014).

A Vibrant Pursuit

Posing purposeful questions is at the heart of teaching and learning. At the intersection of questioning, productive struggle, and engaging students in meaningful mathematical discourse, we find fascinating territory.

At these points of discontinuity, it's helpful for teachers-and students-to ask open-ended questions (What do you understand so far?) that encourage students to persist and that invite them to develop ideas about how to move forward. It's also fruitful to pose questions that sharpen the focus on the salient mathematics and give learners a way to discuss mathematical structures and make connections among math ideas and relationships (NCTM, 2014). When teachers ask both types of questions, they deepen student thinking and frame mathematics as a more vibrant pursuit than getting a right answer.

References

Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523-550.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.

Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3), 132-138.

Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3(5), 344-350.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (2nd ed.). New York: Teachers College Press.

Allison Hintz is an assistant professor in the School of Educational Studies at University of Washington-Bothell. Follow her on Twitter @allisonhintz124. Lynsey Gibbons is an assistant professor in the School of Education at Boston University. Follow her on Twitter @lynseymathed. Melinda Knapp is an instructor in the School of Teacher and Counselor Education at Oregon State University-Cascades. Follow her on Twitter @KnappMelinda.

# data[].attributes.attachment_link.description
September 2015 | Volume 73 | Number 1 Questioning for Learning Allison Hintz, Lynsey Gibbons and Melinda Knapp "I'm confused. How does figuring out the perimeter help us?" "We don't know how to count the shaded parts." "We're stuck!" When math teachers hear students make comments like these, they often wonder how to respond.

# data[].attributes.headline
 For example, if you were to ask your six-year-old to help you set the table for dinner, you could ask in a myriad of ways:Request 1: Please set the table with 7 forks, and 7 spoons.Request 2: Please set the table with forks and spoons for 7 people.Request 3: Please set the table with a fork and a spoon for yourself, your sibling, your parents (2 in this example), your grandparent, and your aunt and cousin, who are coming to dinner.Request 4: Please set the table for us and grandma, like you did last night, but your aunt and cousin are also coming for dinner.Request 1 is very explicit and is a simple procedural application of a mathematical skill: counting. Your child is instructed on how many of each item to get and can count 7 forks, and then return to the drawer to count 7 spoons.Request 2 is also a procedural application, though it can be achieved through multiple skills and thereby extends their application beyond simple execution of an explicit task. Your child could count and carry twice as in request 1. Or they could add 7 + 7, counting to 7 as they remove forks from the drawer, and then continuing to count to 14 while removing spoons from the drawer. Or they could pair each fork with a spoon as they take them from the drawer and count 7 sets.Request 3 begins to introduce conceptual thinking. In this request, your child is given the explicit instruction to get forks and spoons plus information about the people their task will serve, rather than an explicit number of people. They must first conceptualize those people around the dinner table, and then equate that to the quantity 7. From there, they could apply any of the procedural skills described above.Request 4 requires more conceptual thinking. It isn’t posed as a calculation or computation at all. Your child will first need to think about how many people were present at dinner the night before. Since their grandparent was present, the table will have 1 more than its usual base of 4. They’ll then have to consider that their aunt and cousin will join the table tonight and add those two additional diners. Then, they’ll have to think about which utensils might be needed. Lastly, they’ll have to go to the drawer and retrieve the appropriate quantity of different utensils.

# data[].attributes.subject
 For example, if you were to ask your six-year-old to help you set the table for dinner, you could ask in a myriad of ways:Request 1: Please set the table with 7 forks, and 7 spoons.Request 2: Please set the table with forks and spoons for 7 people.Request 3: Please set the table with a fork and a spoon for yourself, your sibling, your parents (2 in this example), your grandparent, and your aunt and cousin, who are coming to dinner.Request 4: Please set the table for us and grandma, like you did last night, but your aunt and cousin are also coming for dinner.Request 1 is very explicit and is a simple procedural application of a mathematical skill: counting. Your child is instructed on how many of each item to get and can count 7 forks, and then return to the drawer to count 7 spoons.Request 2 is also a procedural application, though it can be achieved through multiple skills and thereby extends their application beyond simple execution of an explicit task. Your child could count and carry twice as in request 1. Or they could add 7 + 7, counting to 7 as they remove forks from the drawer, and then continuing to count to 14 while removing spoons from the drawer. Or they could pair each fork with a spoon as they take them from the drawer and count 7 sets.Request 3 begins to introduce conceptual thinking. In this request, your child is given the explicit instruction to get forks and spoons plus information about the people their task will serve, rather than an explicit number of people. They must first conceptualize those people around the dinner table, and then equate that to the quantity 7. From there, they could apply any of the procedural skills described above.Request 4 requires more conceptual thinking. It isn’t posed as a calculation or computation at all. Your child will first need to think about how many people were present at dinner the night before. Since their grandparent was present, the table will have 1 more than its usual base of 4. They’ll then have to consider that their aunt and cousin will join the table tonight and add those two additional diners. Then, they’ll have to think about which utensils might be needed. Lastly, they’ll have to go to the drawer and retrieve the appropriate quantity of different utensils.

# data[].attributes.attachment_link.readable_text
During the breaks at a couple of recent professional development trainings, my colleagues and I asked teachers to share their definition of "deep conceptual learning." Common threads ran through the individual responses. Teachers agreed that deep conceptual learning involves:

Being able to transfer learning from one subject or lesson to another

Learning through trial and error

Thinking through a process to problem solve, as opposed to simply applying procedures

Being able to explain the thinking that goes into the solution

At MIND, deep conceptual learning is as common a topic of conversation as it is in schools. While basic procedural skills like counting and computation are critical, they are just the tip of the iceberg.

Procedural vs. Conceptual Learning

When I was a kid, we primarily learned math through computation and memorization. We did word problems, but these didn't require much thinking. "If Jane has three cookies and Spot takes one from her hand, how many cookies does Jane have left?"

I distinctly remember the moment when I needed to shift from procedural to conceptual thinking - not that I knew that this is what I was doing at the time.

It was 1970-something, and we didn't have the common core standards that encourage teachers to teach conceptually. Not to say that they didn't - Mrs. Evans, my first grade teacher, most certainly did. But my third grade teacher wasn't as strong in teaching conceptually. She could write the math on the chalkboard and correct us when we made an error in our rote math facts recall, but she couldn't explain the concept behind the numbers.

Addition was simple. We added things to other things. If Sally gave Jane a cookie, Jane had more cookies. Easy. Subtraction was simple too. Especially since we called it "take-away" - exactly what Spot did to Jane's cookie in the question above.

But multiplication? That was a new concept - a trickier concept. Confusingly to me at the time, when my teacher taught multiplication she didn't say "multiply" she said "times." I could tell time. I even knew that every number around the clock face meant that five minutes had passed. But "three times three equals nine" meant absolutely nothing to my young brain.

My teacher taught us to recite times tables. I had no idea how a table got into the mix, and why there was no clock involved in the times. I simply did not get it. I was frustrated, and it was the first time I ever felt lost in math. I tried to ask my teacher for help, but she told me to go back to my seat. I went home that day dejected and discouraged.

Fortunately, my mom was one of those moms who often asked "How was your day?" when I got home from school. On that particular day in third grade, I burst into tears. My Mom sat me down, listened, and then proceeded to explain that "times" meant "multiply" and "multiply is adding groups."

Huh? Adding groups?

She showed me with some walnuts from the trees in our yard. Three walnuts in this group, three walnuts in another group, and three more walnuts. How many walnuts? Nine! Oh, adding groups! I got it! My mom taught me the concept that my teacher was unable to explain.

Procedural vs. Conceptual Understanding at Home

My Mom was quick to grab a bunch of walnuts to teach me the concept of multiplication. We can do the same for our children at home. The way we pose questions allows our kids to apply procedural or conceptual thinking to everyday problems.

For example, if you were to ask your six-year-old to help you set the table for dinner, you could ask in a myriad of ways:

Request 1: Please set the table with 7 forks, and 7 spoons.

Request 2: Please set the table with forks and spoons for 7 people.

Request 3: Please set the table with a fork and a spoon for yourself, your sibling, your parents (2 in this example), your grandparent, and your aunt and cousin, who are coming to dinner.

Request 4: Please set the table for us and grandma, like you did last night, but your aunt and cousin are also coming for dinner.

Request 1 is very explicit and is a simple procedural application of a mathematical skill: counting. Your child is instructed on how many of each item to get and can count 7 forks, and then return to the drawer to count 7 spoons.

Request 2 is also a procedural application, though it can be achieved through multiple skills and thereby extends their application beyond simple execution of an explicit task. Your child could count and carry twice as in request 1. Or they could add 7 + 7, counting to 7 as they remove forks from the drawer, and then continuing to count to 14 while removing spoons from the drawer. Or they could pair each fork with a spoon as they take them from the drawer and count 7 sets.

Request 3 begins to introduce conceptual thinking. In this request, your child is given the explicit instruction to get forks and spoons plus information about the people their task will serve, rather than an explicit number of people. They must first conceptualize those people around the dinner table, and then equate that to the quantity 7. From there, they could apply any of the procedural skills described above.

Request 4 requires more conceptual thinking. It isn't posed as a calculation or computation at all. Your child will first need to think about how many people were present at dinner the night before. Since their grandparent was present, the table will have 1 more than its usual base of 4. They'll then have to consider that their aunt and cousin will join the table tonight and add those two additional diners. Then, they'll have to think about which utensils might be needed. Lastly, they'll have to go to the drawer and retrieve the appropriate quantity of different utensils.

By changing how the requests are phrased, you can move your child between procedural and conceptual understanding. You can do this in nearly any scenario. When I first started teaching math to my daughter, I didn't ask her to add 2 + 3. I asked "If you have three jelly beans and I give you two more, how many will you have?" She responded instantly, "Five mama! Can I have five jelly beans?" She learned the concept that basic adding means "more" before she ever saw a plus sign. And, she learned it visually, precisely how ST Math teaches.

Conceptual and Procedural the ST Math Way

ST Mathstarts out each objective with visuals, without written numbers or symbols, building the conceptual before the procedural or computational. In this way, students learn more deeply.

We see evidence of this every time we're in an ST Math classroom and a teacher introduces a new concept. The teacher frequently starts with the procedural and there's always at least one student who excitedly declares, "Oh, we did this with JiJi (the penguin mascot from ST Math)!" The students see the concept come about in the symbols and numbers their teacher is presenting, sometimes even before they've gotten to the symbols or numbers for that objective in ST Math.

Students learn via trial and error. They apply learning from previous lessons - problem solving by attending to immediate feedback and taking that information into their own perception-action cycle. They explain, or engage in academic discourse, when they share with peers, talk about their thinking with their teacher, or participate in a small group or whole class lesson. With ST Math games that straddle grade bands, students relate the concepts they learned in prior years to new concepts.

In the first grade level of the ST Math puzzle "Bouncing Shoes," students start by counting. Each ostrich has two legs and needs two shoes, each dog has four legs and needs four shoes. Two ostriches and three dogs need 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 shoes.
St Math's "Bouncing Shoes" puzzle
When Bouncing Shoes appears again in second grade, students move from counting to adding: 2 + 2 + 4 + 4 + 4 = 16. Again, remember kids see the visuals of legs and shoes way before they see 2s and 4s and +s. They may start this game by counting, but the way the legs are offered in pairs and then pairs of pairs gets them thinking more deeply. They begin to see the groups of 2 or 4.
Play ST Math puzzles from grades 3-4 that build on multiplication and division.
Students encounter Bouncing Shoes again in third grade when they begin to multiply. In the beginning, students often either group all the ostrich legs together and then multiply 4 x 4, or they split the dogs' legs into pairs and multiply 2 x 8.

Students can later use algebraic thinking to see this same problem as 2x + 4x = 16. They build upon their prior experience of seeing the legs and shoes as groups way back when they're counting. By the time they get to pre-algebra, their minds are already steeped in the concept.

Great teachers like Mrs. Evans and ST Math do this for our kids in the classroom. But we can start them on that path before they start school and continue to support their learning as they grow.

Want the latest from ST Math? Sign-up for our ST Math newsletter and be the first to hear about our offers and announcements.

# data[].attributes.attachment_link.title
Conceptual Understanding vs. Procedural Fluency

# data[].attributes.attachment_link.description
During the breaks at a couple of recent professional development trainings, my colleagues and I asked teachers to share their definition of "deep conceptual learning." Common threads ran through the individual responses.

# data[].attributes.body
Mathematical representations are visible or tangible productions – such as diagrams, number lines, graphs, arrangements of concrete objects or manipulatives, physical models, mathematical expressions, formulas and equations, or depictions on the screen of a computer or calculator – that encode, stand for, or embody mathematical ideas or relationships.

# data[].attributes.headline
Tasks that Provide Opportunities to Reason and Relate

# data[].attributes.subject
Tasks that Provide Opportunities to Reason and Relate

# data[].attributes.body
Problem-based tasks (PBTs) are math lessons built around a single, compelling problem.

# data[].attributes.attachment_link.readable_text
"Providing students with opportunities to grapple with math has led to amazing things happening in my class. Students are totally excited and are driven to figure out not just how to solve a problem but why it works ."

Jessica Proffitt, Fifth-Grade Teacher at Two Rivers

Download the above slides in the Additional Resources section.

Want to learn more about problem-based tasks in math? Read our deep dive.

Do you have a question about problem-based tasks or feedback about what we've shared? We'd love to hear from you. Send us an email at info@learnwithtworivers.org.

# data[].attributes.attachment_link.description
"Providing students with opportunities to grapple with math has led to amazing things happening in my class. Students are totally excited and are driven to figure out not just how to solve a problem but why it works ." Jessica Proffitt, Fifth-Grade Teacher at Two Rivers Download the above slides in the Additional Resources section.

# data[].attributes.body
The Standards for school mathematics describe the mathematical understanding, knowledge, and skills that students should acquire from prekindergarten through grade 12. Each Standard consists of two to four specific goals that apply across all the grades.

# data[].attributes.attachment_link.title
Principles, Standards, and Expectations - National Council of Teachers of Mathematics

# data[].attributes.attachment_link.description
Educational decisions made by teachers, school administrators, and other professionals have important consequences for students and for society. The Principles for school mathematics provide guidance in making these decisions. The six principles for school mathematics address overarching

# data[].attributes.body
Jo Boaler discussion.

# data[].attributes.attachment_link.readable_text
As parents and teachers, we do just about everything we can to make sure that children don't struggle. It turns out we are making a terrible mistake. Research shows that struggling is absolutely critical to mastery and that the highest achieving people in the world are those who have struggled the most. The more I communicate this message to parents and teachers the more stories I hear of complete personal transformation.

Neuroscientists have found that mistakes are helpful for brain growth and connectivity and if we are not struggling, we are not learning. Not only is struggle good for our brains but people who know about the value of struggle improve their learning potential. This knowledge would not be earth shattering if it was not for the fact that we in the Western world are trained to jump in and prevent learners from experiencing struggle.

An international study of mathematics teaching found that teachers in Japan put their students in places of struggle 44 percent of the time in classrooms-they saw this less than 1 percent of the time in U.S. classrooms. What do we parents and teachers do instead? We jump in and show the way, offering steps to a solution to help save our students from struggle. This is in large part because this new science is not widely available and we are culturally trained to feel bad, and to rush in and help, when this is probably the last thing we should do.

The research on the impact of struggle turns out to help adults too-in all sorts of jobs. I interviewed sixty-two people for my new book, "Limitless Mind." Many of them shared similar accounts of how they used to go into meetings afraid they would be found out for not knowing something. After learning about the importance of not knowing and of engaging in struggle they now proudly show up and say "I don't know, but I will find out." They display a mindset of discovery and curiosity, which has helped their lives in many ways.

Once we stop the charade of knowing everything, and embrace knowing less, with a willingness to sit with uncertainty, unexpected things happen.

When I was teaching middle schoolers in a research math camp a few years ago one girl stood out to me; she was nearly always wrong in her thinking, but she was always engaged, arguing her case, pushing to understand better. An observer of the class would have described her as a low achiever, but she improved more than any other of the 84 students we taught that summer. Her standardized test score in mathematics improved by 450 percent after 18 struggle filled lessons. Our messages to the students-that struggle would be valued and mistakes are productive-had helped her feel good about struggle and embrace it.

When I tell young learners that struggle and mistakes are the best times for our brains it is freeing. Students no longer give up on problems when they find them hard-they push through the struggle to the wonderful places on the other side. When students look at me with a puppy dog face and say: "This is hard," I say, "That is fantastic. That feeling of 'hard' is the feeling of your brain developing, strengthening and growing".

In 2016, two young computer scientists rocked the world of mathematics by solving a previously unsolved math problem, an event that many described as audacious. The two young men reflected that it was knowing less that allowed them to solve the difficult problem. It freed their mind to think in better ways.

I am not arguing that knowledge is bad or knowing answers is not helpful. What I am saying is that knowledge is less important than a mindset of discovery and curiosity. We cannot achieve anything creative without being comfortable with mistakes and struggle-and we should all embrace times of struggle, knowing they are helping our brains. When we adopt a limitless perspective, approaching different jobs and conversations with a comfort with uncertainty and struggle, with a willingness to learn from others and with a flexible approach to problems, outcomes improve-in learning and in life.

Millions of students start the school year each year excited for all they will learn, but as soon as they struggle or see someone solve a problem with ease, they start to doubt themselves and mentally shut down. This starts a less productive learning pathway for them. Instead they should value the time of struggle and know that they are on their way to being better, wiser and equipped with a stronger brain. Getting answers right is OK, being stuck and finding them hard is fantastic.

# data[].attributes.attachment_link.title
Why Struggle Is Essential for the Brain - and Our Lives - EdSurge News

# data[].attributes.attachment_link.description
As parents and teachers, we do just about everything we can to make sure that children don't struggle. It turns out we are making a terrible mistake. Research shows that struggling is absolutely critical to mastery and that the highest achieving people in the world are those who have struggled the most.

# data[].attributes.headline
How to Use Rubrics to assess evidence of student reasoning.

# data[].attributes.subject
How to Use Rubrics to assess evidence of student reasoning.

# data[].attributes.body
A rubric for assessing mathematical reasoning will help teachers be aware of the reasoning actions and formatively assess the reasoning articulated and displayed by students.

# data[].attributes.body
Asking questions encourages students to think through their answer and apply current knowledge to solve a new problem. Incorporating these six types of Socratic questions will enhance your student's math learning by developing critical thinking skills and math strategies.

# data[].attributes.attachment_link.title
Using Socratic Questions To Enhance Math Learning

# data[].attributes.body
Distinction between procedural and conceptual.

# data[].attributes.attachment_link.readable_text
ABSTRACT
We identify a recent trend in school mathematics as well as in some of the research literature in mathematics education: an emphasis on the practical uses of mathematics and an increased emphasis on verbalizations as opposed to numerical and computational skills. With tools provided by John Dewey, an early advocate of contextual and practical knowledge, we analyse the common research framework for discussing mathematical knowledge in terms of the procedural and the conceptual. We argue that procedural and conceptual knowledge should not be seen as opposites, and that the tendency to treat them as such might be avoided by emphasising the notion of operational skill. We argue that this is important in order for the students to gain both the contextual knowledge and the computational skill entailed in mathematical knowledge.

1. Introduction

During the 1990s, an international reform movement in mathematics education emerged in several countries (Boesen et al., ; Niss, Bruder, Planas, Turner, & Villa-Ochoa, ). Traditionally, curriculum documents have been based on what mathematical content that should be studied, that is, notions, concepts, theories, methods and results (Boesen et al., ; Österholm, ). The new trend instead aims to clarify the relationship between mathematical content and practice by introducing generic competencies such as conceptual understanding, problem solving, reasoning and communication skills, which can be noted in the mathematics curricula in several countries, for instance the Nordic countries (e.g. Kilpatrick, Swafford, & Findell, ; National Council of Teachers of Mathematics [NCTM], ; Niss, ; Niss & Höjgaard-Jensen, ). One expression of this trend is the increasing emphasis on verbal knowledge as well as applied mathematics, sometimes referred to as 'real-world mathematics', often motivated by a need to make sure that the students learn something that they can use outside the test situation, i.e. that the students understand how the mathematics they learn in school connects to life outside the classroom (Gainsburg, ; Grønmo, ; Pongsakdi, Laine, Veermans, Hannula-Sormunen, & Lehtinen, ). This is often explained in terms of making sure that the students learn mathematics creatively, based on conceptual knowledge, rather than just memorizing empty formulas (e.g. Boesen et al., ; Jonsson, Norqvist, Liljekvist, & Lithner, ).

The attempt to involve the students and contextualize the information is not new. In fact, it has been central for many school reformers, among them the American philosopher and progressive educationalist John Dewey. Even if Dewey came to be seen as the main progressive ideologue in many countries, he was not always understood and applied the way he would have preferred. Later in his career he came to criticize parts of the progressive movement quite severely (Dewey, ), and he in turn was criticized for being too academically oriented as well as for lacking a sense of direction (Karier, ). The latter was an effect of Dewey not wanting to blueprint a new social order; he thought that education should prepare the students to create their own future, to become active participants in a democratic society open and in the making (Dewey, ). This is why the method captured in the slogan Dewey has become famous for, 'learning by doing', was so central to his educational philosophy. After the Second World War and the rise of fascism, as strengthening democracy became widely recognized as an important aim for education in many Western countries (Raywid, ), Dewey had an impact on the school systems of e.g. Britain, Austria, Germany, and Sweden (Biesta & Miedema, ; Brehony, ; Hartman, Lundgren, & Hartman, ). In Sweden, Dewey has been very influential at least since 1946 (Hartman et al., ), to such an extent that some writers speak of a 'state progressivism' (Broady, ). Lately, the allegedly 'soft' educational ideals of progressivism have received increasing criticism in Sweden. Many popular debaters blame the perceived problems, such as lack of discipline and low achievement, on a progressive influence with its alleged relativization of knowledge and placing more value on student participation and activation than on imparting robust knowledge (Enkvist, ; Kornhall, ; Larsson, ). A similar criticism has been mounted by e.g. Michael Young () against the modern British educational system.

Despite the criticism against democratic classrooms and student participation, the concern to provide contextual knowledge that can be used in life outside the classroom remains, and has even been accentuated in the past decades. For instance, in many countries there has been increased focus on the results of the PISA-test (Programme for International Student Assessment), where, in mathematics, real world and narrative word problems take precedence over more traditional classroom mathematics that focus on numerical skills (Organization for Economic Cooperation and Development [OECD], ). A related tendency, visible in the research as well as in current mathematical curriculum documents, is an increased emphasis on verbal skills such as reasoning, discussing and communicating mathematics, coupled with less emphasis on the actual 'doing' of mathematics through symbolic manipulations, equation solving, and the use of algorithms (Bråting & Österman, ). The theoretical underpinnings for this development have been provided by, among other things, a certain interpretation of the distinction between conceptual and procedural knowledge. This distinction has become a platform for empirical studies as well as theoretical frameworks, where the distinction has often gained the status of a dichotomy (Kieran, ).

Our aim in this paper is to illuminate the development from numerical and computational skills toward increased emphasis on verbalizations in mathematics education, through contrasting it to the educational philosophy of John Dewey. More specifically, we will use Dewey's distinction between the psychological and logical aspects of a subject (Dewey, 1959b/) in our analysis of the theoretical framework in which mathematical knowledge is described as a product of conceptual and procedural knowledge. It can be helpful to spell out Dewey's philosophy of education in contrast to the more recent theoretical developments within educational research, in order to gain a better view of the latter. We will examine the philosophical assumptions underlying many scholars' view of what constitutes mathematical understanding, and argue that the resulting opposition that we can discern between conceptual and procedural knowledge has led to an increased emphasis on verbal as opposed to numerical and computational skills. Since we argue that this opposition is problematic for the aim of learning mathematics, we propose to emphasize operational skill, the 'doing' of mathematics. The operational aspect could be said to be embedded in the existing framework, however, we argue that it has been increasingly neglected over time and therefore stands in need of more attention. We believe that this would sharpen the conceptual tools needed to discuss mathematical knowledge, and allow researchers to discuss the importance of numerical and computational skills.

We will concretize our discussion with examples from the Nordic countries, which are interesting to study both from educational research and policy perspectives. As neighbouring countries, the educational systems in the Nordic countries share common values and ideologies for geographic, cultural and historical reasons, and during the last 50 years the Nordic welfare state has been established as a unique model with a strong emphasis on equity of access to high-quality education (Yang Hansen, Gustafsson, & Rosén, ). The implementation of generic competencies in the curriculum documents was developed relatively early in the Nordic countries. Already in the late 1990s, the Danish KOM-project (Kompetencer og matematiklæring) was initiated, with the aim to characterize mathematical knowledge in terms of generic competencies (Niss, ; Niss & Höjgaard-Jensen, ). Furthermore, in the Nordic countries as well as in the English-speaking countries, applied mathematics (or 'real world mathematics') has been a driving force in the development of school mathematics, in contrast to countries in Eastern Europe and East Asia (Grønmo, ). The strong emphasis on applied mathematics is revealed in the results of international evaluations such as TIMSS (Trends in International Mathematics and Science Study) and PISA, where the performance of students from the Nordic countries has been relatively weak on tasks in pure mathematics, especially in algebra, and better on tasks that have a daily life context (Bråting, Madej, & Hemmi, ; Grønmo, ). However, it is interesting that the overall results on these evaluations have differed between the Nordic countries over the years.

2. Education according to Dewey

The practical use of mathematics is one of the main concerns of school mathematics today (OECD ), just as it was for Dewey, who objected to a view of abstract knowledge as an end in itself and stressed the connection between education and real life, 'the organic relation of theory and practice' (Dewey, 1959a/, p. 84). Only if the knowledge the students gain in school is important to them, if it addresses their concerns and interests, will it truly be educative, he claimed. This could be misunderstood as advocating a school where the students set the agenda and consequently spend their time playing or socializing rather than learning, and where the teachers lack the authority to discipline them. However, this is not what Dewey had in mind. He specifically argued that 'student interest' was not to be identified with whatever the students happened to want at a specific time and place:

It will do harm if child-study leave in the popular mind the impression that a child of a given age has a positive equipment of purposes and interests to be cultivated just as they stand. Interests in reality are but attitudes toward possible experiences; they are not achievements; their worth is in the leverage they afford, not in the accomplishment they represent. To take the phenomena presented at a given age as in any way self-explanatory or self-contained is inevitably to result in indulgence and spoiling. Any power, whether of child or adult, is indulged when it is taken on its given and present level of consciousness. Its genuine meaning is in the propulsion it affords toward a higher level (Dewey, 1959b/, pp. 99-100).

Dewey sees children as naturally active, and it is the teacher's job to see to it that this energy is channelled toward realizing their full potential. In order to do so, the teacher needs to have thorough knowledge of the subject matter, and know each individual student well enough to be able to direct him or her in the right way. This requires psychological and social understanding as well as expertise in the subject matter, and is more demanding than applying ready-made schemata that ignore the individuality of the students, their different backgrounds, abilities and interests (Dewey, ). However, the difficulty reflects the importance of the task. For Dewey, the educational ideal is after all a societal ideal-to foster able young adults who can think critically and actively participate in, and form, the society they are a part of (Karier, ). This means that the freedom essential for educational purposes is not freedom from external control, but rather the freedom resulting in power of self control (Dewey, ). Education should foster a spirit of criticism and cultivate a habit of inquiry (Dewey, ), which cannot be conveyed as truths, but require an active undertaking on the part of the students. The Deweyan slogan 'learning by doing' is therefore not merely a heuristic device, but necessary for fostering the critical thinking that he considered essential for members of a modern democracy.

In order to address the students, to speak to them rather than at them, the teacher needs to have an understanding of their lives, of their thoughts and interests. She needs to listen to them and attend to them. And in order to help them develop, to reach higher and to continually learn something new, she needs to have thorough knowledge of the subject matter as well as the ability to direct the students' attention to ever more abstract and difficult matters. Dewey illustrates this as a movement from the psychological aspects of the subject to the logical. To emphasise the psychological aspect is to stress student interest and experiences and to relate the subject matter to what the student is familiar with; to emphasise the logical aspect is instead to focus on the subject as it appears to the expert, as an abstract body of knowledge driven by its internal laws and rules (Dewey, 1959b/). The teacher needs to be familiar with both of these aspects in order to teach the students; the teacher of mathematics must master mathematics as an abstract body of knowledge, but also know how to make it intelligible and interesting to the students. Dewey's point is that these aspects cannot be separated, they are both part of a well functioning education. The child's interest (the psychological aspect) and the subject matter (the logical aspect) are two limits that define a single process: 'Just as two points define a straight line, so the present standpoint of the child and the facts and truths of studies define instruction' (Dewey, 1959b/, p. 97). This means that the teacher ideally starts from what interests the students, but that which gives direction to the instruction, the goal, is the 'organized bodies of truth'-mathematics as abstract knowledge. Education, then, is what takes place in between these two defining points, the movement from student experience to abstract subject matter.

In mathematics, this movement could be exemplified by how children first learn to count concrete objects, and how in school small children are first presented with everyday examples involving familiar objects. During the years to come the instruction will become more and more demanding and move toward what Dewey calls the logical aspect of mathematics: toward mathematics as a goal in itself without a connection to practical interests. However, it is important to keep in mind that these two aspects are interdependent. They are two sides of the same coin, impossible to categorically separate. The logical aspect, the part of mathematics that is driven forward by the internal development of the field rather than by the need of practical applications, would be unthinkable unless mathematics also played a practical role in human life. And the practical applications of mathematics (in for example technology) are dependent on the progress made in the theoretical field of mathematics research. In school these two aspects need to go hand in hand as well, in order for the students to see the education as meaningful as well as be able to develop mathematically.

We propose that there are similarities between the didactical perspective on mathematics and Dewey's psychological aspect, where student interest, context and practical mathematics are central, whereas the logical perspective entails the context-independent, timeless aspects of mathematics such as symbolic manipulation (i.e. the 'pure mathematics' mentioned in footnote 2 at page 5 above). As we have seen, Dewey thought that these aspects go hand in hand in the educational process-it is imperative that the mathematics taught in the classroom connects to the students' experiences and to other subjects. This is similar to the aspirations of modern educational theorists, who investigate how mathematics could be made meaningful, how to connect the mathematics taught in the classroom with practical tasks in the real world (Gainsburg, ; Pongsakdi et al., ). Just like Dewey, they want to avoid an educational system where the students only learn empty formulas and rules by rote, without real understanding. However, the theoretical framework of Dewey and modern educational theorists differ in important respects. Whereas Dewey saw the practical aspects of mathematics as inseparable from the timeless aspects of mathematics, such as the operational aspects, and thought it important that both be included in the learning process, we will argue that many modern educational theorists tend to separate these aspects (e.g. Hiebert & Lefevre, ).

3. Mathematical knowledge in educational theory

How mathematical knowledge should be characterized is a recurring issue within the research field of mathematics education. Over the years this issue has, among other things, given rise to a distinction between procedural and conceptual knowledge which has served as a platform for empirical studies as well as the formation of theoretical frameworks (cf. Kieran, ). This research has also influenced the content of mathematics curricula in several countries, not least in the Nordic countries. Some of this research stems from the beginning of the 1970s. Here, 'instrumental understanding' denotes what happens when students learn a number of fixed and specific plans to solve a specific kind of task (Skemp, ). Skemp is critical of this type of understanding, since the students do not get an overall understanding of the relationship between the individual stages and the final goal of the exercise. Instead, he emphasizes 'relational understanding', where the students gradually form a conceptual structure that can produce an unlimited number of plans for getting from any starting point within a student's schema to any finishing point (Skemp, ). According to Skemp, the understanding will then become the goal in itself, and the plans are no longer fixed and immediately tied to a particular class of problems, as in the case of instrumental understanding.

Hiebert and Lefevre () make a distinction between 'conceptual knowledge' and 'procedural knowledge'. The latter is described by Hiebert and Lefevre as knowledge consisting of procedures used for solving mathematical problems, where the procedures are 'chains of prescriptions for manipulating symbols' (Hiebert & Lefevre, , pp. 7-8). This description is similar to Skemp's () explanation of instrumental understanding cited in the above paragraph. Skemp uses the phrase 'learning a number of fixed and specific plans' which is similar to the term 'procedure' and 'chains of prescriptions' used by Hiebert and Lefevre (). The term conceptual knowledge, on the other hand, is described as knowledge rich in relationships that 'can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information. Relationships pervade the individual facts and propositions, so that all pieces of information are linked to some network' (Hiebert & Lefevre, , pp. 3-4). This is similar to Skemp's description of relational understanding. Skemp speaks of a 'connected web' and a 'network' of knowledge, whereas Hiebert and Lefevre () use the expression 'conceptual structure' in order to characterize this kind of knowledge. Both describe a situation in which the student not only uses specific mathematical concepts, but also has an understanding of the whole system of concepts.

Building on the ideas above, Lithner () has developed a theoretical framework for mathematical reasoning. Lithner refers to, among others, Skemp's () and Hiebert and Lefevre () frameworks, but points out that none of these are actually 'aiming at characterizing the reasoning itself' (Lithner, ) and therefore Lithner intends to fill this gap. On the basis of his own empirical studies, Lithner argues that rote learning, understood as a process of learning something by repeating it until it is memorized, is a main factor behind learning and achievement difficulties (Lithner, , ). To specify and communicate the findings of his empirical studies Lithner distinguishes between two different types of mathematical reasoning: 'imitative' and 'creative' reasoning, where the former is connected to rote learning. By imitative reasoning, Lithner means recollection of a complete answer to a specific task, for instance when a student memorizes every step of a proof (but is unable to explain the meaning of each step). It can also be described as a strategy where one recalls a specific solution algorithm, which is similar to Hiebert and Lefevre () description of procedural knowledge in terms of learning 'chains of prescriptions' and Skemp's explanation of instrumental understanding as 'learning a number of fixed and specific plans'.

In contrast, Lithner () wants to emphasize the importance of creative reasoning. This takes place when a student creates a reasoning sequence novel to him or her (or recreates a forgotten one), where there are arguments supporting the strategy choice, arguments that are anchored in intrinsic mathematical properties of the components involved. This is related to Hiebert and Lefevre () explanation of conceptual knowledge as 'a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information' as well as Skemp's () description of relational understanding as an 'overall understanding of the relationship between the individual stages and the final goal of an exercise'. The main difference between imitative and creative reasoning is that in the latter case the solution is created, whereas in the former the solution follows a known path or is immediate through recollection (Lithner, ).

Even though Skemp () focuses on understanding, Hiebert and Lefevre () on knowledge, and Lithner () on reasoning, we believe that their common aim is to account for the difference between merely learning something by heart without any real understanding and without knowing how to apply what one has learned, on the one hand, and actually understanding a mathematical operation and being able to apply that knowledge in new situations as well as understanding the connections between the different concepts involved, on the other. Since both researchers and teachers agree that teaching students empty formulas without conveying any understanding of their meaning is not a desirable educational goal, theorists (e.g. Hiebert & Lefevre, ; Jonsson et al., ) try to develop methods to improve the students' conceptual knowledge and minimize procedural knowledge.

There are also studies that go against this trend and instead focus on the positive outcomes of instrumental understanding, procedural knowledge or imitative reasoning. For instance, Star () argues for a renewed focus on procedural knowledge in the research of mathematics education, based on the following three points: (1) The development of students' procedural knowledge has not received enough attention; (2) one possible explanation for this is that current characterizations of procedural knowledge reflect limiting assumptions about how procedures are known; and (3) reconceptualising procedural knowledge to remedy these assumptions would have important implications for both research and practice. Star () also points out that procedural knowledge was widely studied within cognitive psychology in the 1980s (e.g. Brown & VanLehn, ) and thereafter the relationship between procedural and conceptual knowledge has been a research topic in developmental psychology (Rittle-Johnson & Alibali, ; Rittle-Johnson, Siegler, & Alibali, ; Schneider, Rittle-Johnson, & Star, ).

Furthermore, Carolyn Kieran has written a critical paper where she claims that the distinction between conceptual understanding and procedural skills is a 'false dichotomy in mathematics education' (Kieran, , p. 153). She argues that conceptual understanding and procedural skills cannot be seen as separate entities since the elaboration and constitution of procedures are conceptual in nature and contribute to the understanding of the mathematical objects treated. Moreover, procedural skill contains a significant conceptual component in itself since procedures, even those that function as automatized, need to be regularly updated, revised and extended (Kieran, ). According to Kieran (), the dichotomy between procedural skills and conceptual understanding has been particularly damaging in algebra, since algebra has come to be viewed as a domain dominated by procedures of symbolic manipulations where the conceptual is generally absent. On the basis of her own project on algebra learning Kieran () illustrates algebraic examples of the co-emergence of procedural (or technical) and conceptual components as well as the interplay between them. However, we will not go into details on algebraic expressions, instead we will focus on the trend that Kieran is critical of, i.e. the view of the procedural as inferior to the conceptual. We will try to trace the reasons for the dichotomy and spell out what has been lost in the process, what we call the operational aspect of mathematics.

4. The goal of teaching mathematics

Dewey and educational theorists today (e.g. Hiebert & Lefevre, ; Pongsakdi et al., ) agree that students should not learn a separate classroom-mathematics that they only ever use in school, but a mathematics connected to their daily activities, something that can help them manage and make sense of other areas of their lives. School mathematics has been criticized for being boring and too focused on calculations without context and rote learning of algorithms, which leads to students only gaining surface knowledge, i.e. techniques that they can only use in the classroom:

Much time in mathematics classes is spent on learning and rehearsing algorithms, which are supposed to provide students with a quick and reliable way to cope with many of the tasks ahead (Boesen et al., ; Hiebert, ). There are, however, doubts as to whether these algorithms actually give rise to any deeper understanding of the principles of mathematics, or whether the extensive use of algorithms is counterproductive (Hiebert, ) (Jonsson et al., , p. 21).

The extensive use of algorithms, as described above, only leads to superficial, procedural knowledge. This is generally thought to be the case when the teacher walks the students through a solution with the help of an algorithm, and the students are subsequently asked to solve a series of problems using the same algorithm. This teaching method is strongly criticized by many researchers: 'There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning' (Jonsson et al., , p. 20).

Imitative reasoning is what Dewey tried to avoid by stressing the importance of 'psychologizing' the subject, making it interesting to the students through relating it to their everyday lives outside the classroom, and then gradually moving in a direction of more abstract knowledge. Abstract thinking is indeed a goal for the modern educational researchers we have reviewed above as well, but they do not share Dewey's understanding of 'abstract thinking'. For Dewey, it was an ability to solve problems mathematically, to engage in mathematics as does a researcher, who is interested in the proofs, refutations and calculations as ends in themselves, rather than as a means to achieve practical ends. Practical applications are of course important in people's everyday lives, as well as in the professional lives of economists, engineers and physicists, to name a few, but it is not the subject matter of higher mathematics. According to Dewey, not every student should become a professional mathematician, but it is important that the goal is always set high (in every subject), so that the students are pushed to ever more difficult tasks, in order to develop their thinking and reach further than they might have expected (Dewey, ).

Today, however, there is a growing tendency to understand abstract thinking, the deep mathematical understanding, as that which is expressed in arguments, modelling and contextualization (e.g. Pongsakdi et al., ). This tendency is also visible in the curricula of many countries today (NCTM, ), where mathematical knowledge is first and foremost tied to verbal abilities: to be able, in words, to explain why an operation is preferable to another, why a solution is correct etc. For instance, in the current Swedish national curriculum for mathematics ( Läroplan för grundskolan, förskoleklassen och fritidshemmet 2011) the knowledge demands are based on generic competencies, such as the ability to describe mathematical concepts and to communicate mathematics in various contexts, rather than the ability to calculate and reach the right results. The commentary material to the Swedish national curriculum emphasises that students should learn to use 'metacognitive reflections in order to think out loud, look for alternative solutions as well as discuss and evaluate solutions, methods, strategies and results' (National Agency for Education, 2011, p. 1). This stands in contrast to the Swedish curriculum of 1980 ( Läroplan för grundskolan ) where the importance of teaching the students numerical and computational skills was stressed by citing specific areas of mathematics that the students needed to master in order to manage important practical affairs and to become able citizens. Moreover, the Swedish 1980 curriculum prescribed that the students should learn algorithms, while in the 1994 and 2011 curriculums the term algorithm was removed. Simultaneously, there are signs that the students' operational skills have suffered. In a recent study, Norwegian students were shown to have weaker numeracy skills compared to earlier generations (Billington & Gabrielsen, ). In Sweden, the algebraic tasks in educational material for upper secondary school have become less complex compared to the 1960s, 1970s and 1980s (Jakobsson-Åhl, ). Furthermore, Filipsson and Thunberg () argue that a discrepancy in what is considered important mathematical knowledge has emerged between the Swedish secondary school and the universities. At the university level, routine skills in arithmetic and algebraic computations are considered absolutely necessary when learning mathematics, whereas in the secondary school mathematics curricula, focus lies on mathematics problem solving skills and how students understand concepts, rather than on practising computational skill (Brandell, Hemmi, & Thunberg, ; Filipsson & Thunberg, ). This becomes problematic in cases where the computational complexity of exercises at the university level is way above the level of complexity that the students are used to from secondary school (Filipsson & Thunberg, ). Consequently, many students encounter serious problems when they begin to study mathematics at university level, and initial problems may lead to prolonged or unsuccessful studies or even cause students to quit their studies in mathematics (Brandell et al., ). It seems that these problems are not unique for Sweden, there are similar problems in other countries, for instance in England (Kouvela, Hernandez-Martinez, & Craft, ) and Canada (Kajander & Lovric, ). Here we argue that an increased focus on operational skills in secondary school, not least in the mathematics curriculum documents, is necessary in order to facilitate the transition from secondary school to university level mathematics.

5. What is mathematical understanding?

To better understand the development from numerical and computational skills toward verbalizations, we have to look more closely at how algorithms are perceived. A common view is that the use of algorithms does not further mathematical understanding. The advantage of using algorithms is that they make it easier for students to reach the right conclusion, but the ability to use an algorithm is not an indication of any deeper understanding of the principles of mathematics (Hiebert, ). Brousseau (, p. 130) is often quoted in this context: 'The reason why an algorithm is regarded as efficient in solving a task (but not for learning) is that it is designed to avoid meaning'. The application of algorithms is often considered mechanical and it is sometimes argued that only carelessness can lead to a mistake (Lithner, ). In fact, imitative reasoning, the application of rules and algorithms, is at times considered the opposite of creative reasoning:

The aspect of creativity that is emphasized in this framework is not 'genius' or 'exceptional novelty,' but the creation of mathematical task solutions that can be modest but that are original to the individual who creates them. Thus, creative is the opposite of imitative (Bergqvist & Lithner, , p. 256).

If imitative reasoning-the application of rules and algorithms-is taken as the opposite of (desirable) creative reasoning, educators might become weary of numerical skill more generally. If the important thing is the thought process behind the calculations, rather than the calculations themselves, verbal explanations are needed to show whether the students really understand the tasks that they have completed. The calculations could, after all, be a result of merely mechanically applying rules or algorithms. On this approach, the efforts of a student who uses an algorithm to solve a problem and can check whether she, on the basis of the algorithm, has the right answer, is equated with the efforts of a student who copies the task from another student and has no idea what algorithm was used, nor how to make sure that the answer is correct.

What is the reason for the negative conception of using algorithms, and the idea that the mathematical knowledge should be expressed verbally? In the following, we will discuss 'understanding' rather than 'knowledge', since we want to focus on the process taken to underlie mathematical knowledge, rather than the exhibited competence. It seems to us that what is at stake here is the philosophical assumptions made about understanding. A common dualistic view is adopted, where understanding is taken to be a mental process, a process of which the calculation is the result. The result is seen to be contingent on the actual understanding, which is hidden-just like for the mind-body dualist, the bodily expression of a feeling is contingent: we can hide our feelings or force a smile even when we are sad. We cannot trust the facial expressions to reveal a person's true feelings, as we cannot trust the calculations on a paper to reveal the mental process that lies behind them. Thus, we cannot know how the student ended up with the numbers that s/he has written down. This view of the performed calculation as empty in itself, coupled with an idea of two distinct kinds of reasoning, one desirable (creative) and the other not (imitative), results in a 'suspicion' toward the mathematical operations the student has written down as potential expressions of merely memorizing empty rules or procedures. Therefore, we need something else, something in addition to a traditional mathematical task to make sure that the calculations originate in real understanding-and what we end up with is a verbalized account.

That understanding is primarily thought of as a mental phenomenon is shown by the importance attached to brain activity, in studies investigating how different teaching methods affect the brain. For instance, in a Swedish research study (Karlsson Wirebring, Lithner, Jonsson, & Liljekvist, ), a group of students got to solve mathematical problems using only given algorithms and rules, through imitative reasoning, another group solved the same problems using what was characterized as creative reasoning. After working with the tasks for a week, their brain activity was monitored and the results of the tasks were compared between the groups (Karlsson Wirebring et al., ).

One problem with sharply separating understanding as mental activity and the completion of a mathematical task as the result of this understanding, is that the gap between the test situation and real life problems widens. This becomes apparent in a Finnish study, concerned precisely with making mathematics applicable to real world situations. One way of reaching this aim, it is argued, is to create realistic, narrative word problems for the students to solve, like the following:

Paula is preparing some food and drinks for her birthday party. She buys two packets of chips (1 packet costs 2.50 euros), a big packet of mixed candies (1 packet costs 3.60 euros), and 4 bottles of lemonade (1 bottle costs 1.25 euros). Three friends come to the party. How much does the snacks and drinks cost for each participant? (correct answer: 3.4) (Pongsakdi et al., , p. 32)

A realistic feature of the problem is that the students have to figure out that they should divide the total cost with all four participants, not just the three guests. In this task, which was used to evaluate the students' reasoning ability, full points were awarded for students who modelled the task correctly, even if they made computation errors and thus arrived at the wrong answer (Pongsakdi et al., ). Since the students' test results improved after rehearsing with realistic, narrative word problems, the researchers concluded that word problems have a positive effect on students' problem solving skills. But it is notable that 'problem solving skill' here only refers to the ability to model the task, not the ability to actually complete it. Imagine a real life situation, where Paula was in charge of calculating the per person admission fee to an event. To the people depending on her it would not suffice that she thought correctly, unless she also got her sums right. In fact, in most real life situations, whether we are concerned with paying taxes or figuring out what medical dose to take, we need to arrive at the correct solution to avoid suffering negative consequences. In the most common sense of 'problem solving', the students who arrived at the wrong mathematical result failed to solve the problem. That the students' mastery of pure mathematics (in this case exemplified by calculations with decimal numbers) is the basis for all types of applied mathematics, is illustrated in the mathematical cycle of applied mathematics defined in the US Standards from 1989 (National Council of Teachers of Mathematics [NCTM], ), as well as in the PISA framework. Nevertheless, in English-speaking and Nordic countries, this has, to some extent, been misunderstood to mean that applied mathematics can be seen as an alternative to pure mathematics (Gardiner, ; Grønmo, ).

Again, it can be useful to remind ourselves of Dewey's advice to not separate the psychological and the logical aspects of mathematics: in order to solve mathematical problems successfully you need the reasoning ability, context-sensitivity and situational understanding as well as an ability to perform the calculations needed. In Pongsakdi et al. () example above it turned out that the students' reasoning ability did not suffice to solve the problem at hand, they also needed to be able to perform the calculations (calculating with decimal numbers). For Dewey, this insistence is connected to his rejection of dualism and the conception of intelligence as a purely mental phenomenon. The idea that we could know whether someone is a deep thinker through examining their brain is therefore ludicrous to him: thinking is internally connected to its expression, to how we manage in our practical activities. To suggest that 'problem solving skill' could be evaluated independently of whether the skill actually solves the problem it was set out to solve would not make sense to him.

6. Learning by doing: operational skill

We argue that the distinction between procedural and conceptual knowledge has led to a dichotomy where operational skill gradually over the years has fallen between chairs. In light of our hypothesis that, following Dewey, the different aspects of mathematics need to go hand in hand, we find it particularly important to emphasize operational skill, which combines procedural and conceptual elements. The operational aspect of mathematics is a methodological aspect that the students learn by engaging in mathematical activity, for instance performing calculations, solving equations, manipulating the symbols in a mathematical expression, proving a mathematical statement, etcetera. It is not a matter of mechanically applying formulae, and it need not be identified with a distinct mental process, but is part of a practice that the mathematician needs to master in order to develop his or her mathematical intuition and see relevant connections where others would not. Even if mastering the operational aspect requires practice, it entails a context-free, timeless dimension (and is therefore abstract in Dewey's sense). The key to operational skill lies in mastering the mathematical symbolism and the rules that have been worked out through centuries of mathematical progression. An example of this would be to understand the right way to use an algorithm.

One way of understanding the operational aspect of mathematics is to describe it in terms of the symbolic nature of mathematics, in the sense that mathematics has been perceived ever since the late 16th century. On this view, mathematics is not something we first learn and then do, but rather something we need to learn by doing. A symbolism is not merely a system of notation in the typographical or linguistic sense, but is connected to the mathematical activity itself. It is the operational aspect of a symbol, its function in a calculus, its role in the manipulation and transformation of expressions, which can be said to constitute it as a symbol (Stenlund, ). Dewey's logical aspect of mathematics can thus be identified as the symbolic conception of mathematics: to master the logical aspect is to master a symbolic calculus.

We argue that in the received conceptual framework for explaining mathematical knowledge, through procedural and conceptual knowledge, the different aspects of mathematical knowledge are often treated as clearly demarcated and separate sets of skills. This leads to a neglect of the operational aspect of mathematics, which could be said to fall into both categories. Particularly when the procedural and the conceptual are perceived of as opposites, there is very little room left for conceptualizing or evaluating the reflective use of algorithms and rules. Procedural knowledge need not only be a matter of rote learning; in learning a method for solving a problem the students learn to master a calculus and solve a problem mathematically. However, the ability to use received notations, symbols and rules requires a lot of practice, rather than the 'novelty' and 'originality' associated with conceptual knowledge. In this sense it combines aspects of both the conceptual and the procedural and resolves the apparent opposition between them.

In short, we propose to highlight the actual doing of mathematics, that we have termed operational skill. We find ourselves in agreement with Kieran, who argues that the dichotomy between procedural skill and conceptual understanding is misplaced since 'procedures are conceptual in nature' (Kieran, , p. 161). The risk of teaching the students mindless rules seems to be one of the reasons for why the distinction procedural/conceptual has turned into something of a dichotomy, and we realize the importance of avoiding this kind of rote learning. But it is important to point out that merely memorizing procedures or imitating solutions without understanding might not properly be called (procedural) knowledge at all, but simply failed education. However, practicing rules and procedures is often a necessary step in learning to use them operationally and with understanding, and should therefore not be equated with mere imitation and regarded as unimportant for learning mathematics, since gaining mathematical knowledge is dependent on the practical mastery of symbols and procedures. Mathematical knowledge gained by means of analogies, visualizations and verbal explanations are important means for creating a setting in which the student finds herself at home in the material, but verbal explanations, often identified as evidence of conceptual knowledge, cannot replace the symbolic calculations that the student needs to master in order to solve a mathematical problem. We therefore want to emphasise operational skill, in order to capture the importance of numerical and computational skills for mathematical knowledge, not as mindless rule-following, but as a thoughtful use and application of skills often necessary for understanding the mathematical solution.

7. Conclusion

We have argued that the existing framework for explaining mathematical knowledge in terms of procedural and conceptual knowledge is often interpreted as a dichotomy between the two aspects essential for education: practical modelling and verbal explanations (Dewey's psychological aspect), on the one hand, and the operational, symbolic aspects that are free of context (Dewey's logical aspect), on the other. Partly as a result of how procedural and conceptual knowledge are defined, numerical and computational skills have become less important in mathematics education, and meta-mathematical, communicative skills are seen as more important, as embodying what is thought of as 'real' mathematical knowledge.

As mentioned, there are signs that students' numerical and computational skills have suffered over the years: (Billington & Gabrielsen, ; Filipsson & Thunberg, ; Jakobsson-Åhl, ). There are many reasons for these developments. The emphasis on practical uses of mathematics and 'real-world mathematics' at the expense of numerical and computational skills could be one of them. Sutherland () shows that countries that emphasize real life (contextual) mathematical understanding tend to be less concerned about numerical and computational skills. Therefore, we suggest a new approach to the research framework, with more emphasis placed on operational skill. As the latter includes mathematics as a practical activity that the students need to engage in, at the same time as it is concerned with the symbolic, context-free nature of mathematics, it is indeed a call to revive the Deweyan slogan of 'learning by doing' in mathematics education.

Disclosure statement

No potential conflict of interest was reported by the authors.

1. From its start in the 1890's, progressivism was not a unitary movement but a multi-faceted protest against pedagogical narrowness and one-sidedness (Cremin, ).

2. 'Pure mathematics' should here be understood as the context-independent, timeless aspects of mathematics, or as a mathematical activity that does not explicitly take its application into consideration. A concrete example would be symbolic manipulations in algebra.

3. Cf. Michal Young's concept of 'powerful knowledge' as that which can help the students make sense of the world and take them beyond their own experiences (Young, ).

4. Dewey's psychological and logical aspects of a subject is in some respects similar to Basil Bernstein's horizontal and vertical discourse (Bernstein ).

5. 'Abstract' for Dewey refers to a subject as self-contained, driven by its own rules rather than by practical applications. 'The outcome, the abstract to which education is to proceed, is an interest in intellectual matters for their own sake, a delight in thinking for the sake of thinking.' (Dewey, , p. 298).

6. Here Kieran () refers to Hiebert and Lefevre () distinction between procedural and conceptual knowledge.

7. Mind-body dualism is intuitive up to a point, hence the attraction it casts on laymen as well as philosophers. However, when consistently applied it can lead to problems such as scepticism about other minds. Mind-body dualism was rejected, among others, by Dewey () and Wittgenstein ().

8. This could be one reason for the Nordic countries' weak TIMSS-results in algebra which was mentioned in the introduction section above (Bråting et al., ; Grønmo, ). Algebra is a topic where symbolic manipulations and context-independent problems (i.e. typical examples of pure mathematics) frequently appear.

9. 'Thinking is not a separate mental process; it is an affair of the way in which the vast multitude of objects that are observed and suggested are employed, [...] Consequently, any subject, topic, question, is intellectual not per se but because of the part it is made to play in directing thought in the life of any particular person' (Dewey, , pp. 156-157).

References

, & (2017). John Dewey and mathematics education in Sweden. In I. Bjarnadottir, F. Furunghetti, , , & G. Schubring (Eds.). 61-72), , Turin, Italy (Roma: Nuova Cultura). 'Dig where you stand' 4.Proceedings of the fourth international conference on the history of mathematics education, ICHME4 (pp.

, , & (2019). Development of algebraic thinking: Opportunities offered by the Swedish curriculum and elementary mathematics textbooks. Nordic Studies in Mathematics Education, 24(1), 27-49.

Filipsson, L. , & (2008). Paper presented at the Aims versus expectations - A Swedish study of problems related to the transition from secondary to tertiary education in mathematics. Monterrey, Mexico.International Congress on Mathematical Education, ICME 11, ,

(2004). Paper presented at the What is mathematical literacy? 10th International Congress on Mathematics Education, ICME-10, , Copenhagen, Denmark.

, & (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In (Ed.), Conceptual and procedural knowledge (pp. 1-27). Hillsdale, NJ: Erlbaum.

National Agency for Education. (2011). Läroplan för grundskolan, förskoleklassen och fritidshemmet, Lgr11 (National curriculum of Sweden). Stockholm: Fritzes.

(2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. In & (Eds.), 116-3rd Mediterranean Conference on Mathematical Education (pp. 124) Athens: Hellenic Mathematical Society.

Organization for Economic Cooperation and Development, OECD. (2009). Learning mathematics for life - A view perspective from PISA. OECD Publishing.

Österholm, M. (2018). The role of mathematical competencies in curriculum documents in different countries. In J. Häggström, Y. Liljekvist, J. B. Ärlebäck, , & (Eds.), 27-36). Gothenburg: NCM & SMDF.Perspectives on professional development of mathematics teachers. Proceedings of MADIF 11 (pp.

Pongsakdi, N. , , , Hannula-Sormunen, M. , & (2016). Improving word problem performance in elementary school students by enriching word problems used in mathematics teaching. Nordic Studies in Mathematics Education, 21(2), 23-44.

(2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404-411.

(2014). The curriculum and the entitlement to knowledge [Electronic resource]. Retrieved from http://www.cambridgeassessment.org.uk/Images/166279-the-curriculum-and-the-entitlement-to-knowledge-prof-michael-young.pdf

Additional information

Author information

Tove Österman is a researcher in theoretical philosophy at Uppsala University, Sweden, e-mail: tove.osterman@filosofi.uu.se. Her interests are educational philosophy and philosophy of mathematics. She is author of 'What Does Calculating Have to Do With Mathematics? Wittgenstein, Dewey, and Mathematics Education in Sweden', Peters Michael A., Stickney, Jeff (eds.) A Companion to Wittgenstein on Education (New York: Springer, 2017).

Kajsa Bråting is Associate professor (docent) in mathematics education and Director of postgraduate studies at Uppsala University, Sweden, e-mail: kajsa.brating@edu.uu.se. Her interests are mathematics education, epistemology and history of mathematics. She is author of 'On the relations between historical epistemology and students' conceptual developments in mathematics.' Educational Studies in Mathematics, 89(2), 251-265.

# data[].attributes.attachment_link.title
Dewey and mathematical practice: revisiting the distinction between procedural and conceptual knowledge

# data[].attributes.attachment_link.description
ABSTRACT We identify a recent trend in school mathematics as well as in some of the research literature in mathematics education: an emphasis on the practical uses of mathematics and an increased emphasis on verbalizations as opposed to numerical and computational skills.

# data[].attributes.body
Effective instruction begins with asking probing, intentional questions.

# data[].attributes.attachment_link.readable_text
Have you ever taught a math lesson that seemed to be going well-with students active, engaged and producing the right answers-but when you ask kids to explain the reasoning behind their work, they clam up or veer off on a tangent?It happens to the best of us, which is why it's so important to be intentional with our questioning and to think about not only the types of questions we are asking but also the pattern in which we're asking them.

Below we've rounded up eight tips to help you focus your questioning and, by doing so, deepen students' mathematical reasoning.

Don't let "information gathering" questions dominate your lesson.

These are answer-seeking questions such as "What is the formula for finding the area of a rectangle?" "When you write an equation, what does the equal sign tell you?" While these questions have their time and place (they help to establish what students know), they don't reveal higher-order thinking or reasoning.

Ask probing questions that require students to explain, elaborate or clarify their thinking.

This includes articulating the steps they took to solve a problem or complete a task. For example, "As you drew the number line, what decisions did you make so that you could represent seven-fourths on it?" "Can you show and explain more about how you used a table to find the answer?"

(Source: Minds in Bloom)

Give students adequate time to respond.

Research has shown that we often give students less than five seconds to answer a question in math class! When we move so quickly, we lose the chance to see how students are making sense of mathematics.

Ask students to make the mathematics visible.

By this we mean discussing the connections between mathematical ideas and relationships. For example, "How does that array relate to multiplication and division?" "In what ways might the normal distribution apply to this situation?"

Encourage reflection and justification.

These questions reveal deeper understanding of students' reasoning and actions, including making an argument for the validity of their work. For example, "How might you prove that 51 is the solution?" "How do you know that the sum of two odd numbers will always be even?"

(Source:@TEACHHEATH)
Avoid "funneling" the conversation.

This means landing on a desired conclusion in advance and giving little attention to student responses that veer from the desired path. You may be focused on students arriving at a particular answer, for example, or using a certain strategy to the exclusion of others. In a funneling question pattern, the focus is often on information gathering with one or two higher-order questions at the end.

Try "focusing" your questions instead.

While funneling question patterns are rigid and preset, focusing questions attend to students' needs and are open to being investigated in multiple ways. A focusing question pattern blends information gathering, probing, reflection and justification. It involves asking students what they notice and encouraging them to communicate their thoughts clearly.

This video case study shows the difference between funneling and focusing questions in action.

Encourage students to ask questions of one another.

It is much easier to facilitate a focusing question pattern when you are not the only one in the room directing conversation. Encourage students to support, challenge and question one another. For example, "Why do you think that?" "Could you have solved the problem in a different way?"

You may find that students' own questions take the conversation to a level you never imagined reaching!

(Source: Making Shift Happen)

This article is adapted from the NCTM publication Principles to Actions: Ensuring Mathematical Success for All.

# data[].attributes.attachment_link.title
8 Ways to Pose Better Questions in Math Class - We Are Teachers

# data[].attributes.attachment_link.description
It's so important to be intentional and to think about not only the types of math questions we are asking but also the pattern in which we're asking them.

# data[].attributes.headline
Strategies for Building Procedural Fluency

# data[].attributes.subject
Strategies for Building Procedural Fluency

# data[].attributes.body
This math fluency checklist is a great tool that can be used to encourage students to use different strategies to solve math problems efficiently and accurately.

# data[].attributes.attachment_link.title
Math Fluency Checklist | Worksheet | Education.com

# data[].attributes.attachment_link.description
This math fluency checklist is a great tool that can be used to encourage students to use different strategies to solve math problems efficiently and accurately.

# data[].attributes.headline
Start with Conceptual Understanding and Build Fluency

# data[].attributes.subject
Start with Conceptual Understanding and Build Fluency

# data[].attributes.body
"Essentially, conceptual understanding is knowing more than isolated facts, it is also knowing connections between those facts and having those facts well organized."

# data[].attributes.attachment_link.readable_text
According to Adding It Up (H/T Dan Meyer):

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

Essentially, conceptual understanding is knowing more than isolated facts, it is also knowing connections between those facts and having those facts well organized.

This image represents someone who understands some isolated facts.

And this image represents what conceptual understanding looks like:

I propose that those line segments between the points, representing the connections between isolated knowledge, are themselves a type of knowledge, consequently our second image is really more like the first image.

What is clearly missing in this analogy is that conceptual understanding is knowledge that is well-organized, but I claim that organization of knowledge is itself another type of knowledge. Also missing is knowing why the knowledge is important and the contexts in which an idea are useful, but again these also seem just like other types of knowledge.

Let's take a look at a problem I had to solve recently and see if we can use it to unpack what conceptual understanding is.

According to the solution key, the writers of this question are assessing students' ability to solve quadratic equations. However, that likely requires applying the quadratic equation to all four equations given, which would be time-consuming and potentially error-prone.

When I solved this problem, I saw four quadratic functions, each transformed from y = x ², and based on those transformations, I was able to eliminate (1) and (3) as options since they have real solutions. I also know that if 1 - i is a solution, then so is 1 + i, and that the sum of the roots of a quadratic function are related to the coefficient of x in the equation. I could not remember if the sum of the roots is equal to the coefficient of x or to -1 times the coefficient of x, so I multiplied out ( x - a)( x - b) to double-check and determined that if a and b are the roots of a quadratic equation, then -( a + b) is the coefficient of x in that equation (assuming a lead coefficient in the equation of 1). 1 - i + 1 + i = 2 so (4) must be the answer.

My actual solution strategy doesn't matter all that much, what matters is that in order to produce it, I needed to know some smaller things and that those smaller things could be seen as connected together to create a larger idea. Instead of seeing this as the difference between an expert and a novice, I prefer to think of knowing things as being a gradient between knowing a little bit and knowing more than a little bit.

Conceptual understanding is knowing ideas in an interconnected and organized way and knowing more ideas along the novice-expert spectrum.

The reason why conceptual understanding is an important goal is because otherwise we might be tempted to rely on teaching kids tricks instead of mathematics.

What mathematics does a child learn from learning this procedure for adding fractions?

Very little! It is extremely difficult for a child to connect this procedure to other mathematics that they know, consequently using a trick like the butterfly method is likely to lead to students knowing an isolated idea and not much else. Note that I am not opposed to memory aids, I just think they should be based on solid mathematical reasoning and they definitely should not be taught instead of the mathematics.

# data[].attributes.attachment_link.description
According to Adding It Up (H/T Dan Meyer): Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful.

# data[].attributes.body
Mathematics goals are essential for indication of what is expected of students to learn as a result of instruction.

"Formulating clear, explicit learning goals  sets the stage for everything else."

# data[].attributes.body
Not all tasks provide the same opportunities for student thinking and learning.
Student learning is greatest in classrooms where the tasks consistently encourage higher-level thinking.
Tasks with high cognitive demands are the most difficult to implement and are most often transformed during instruction.

# data[].attributes.body
It is important to connect teaching and learning with different representations such as, contextual, visual, verbal, physical, and symbolic.

Students, especially young students and students with learning disabilities, benefit from using physical objects, or acting out processes during problem solving.

# data[].attributes.body
Discourse that focuses on tasks that promote reasoning and problem solving is a primary strategy for developing understanding and meaningful learning in math.

Five Practices to support student responses:
1. Anticipating student responses before the lesson begins.
2. Monitoring students' progress and engagement.
3. Selecting students to present their information in class.
4. Sequencing students' responses in a specific order for discussion.
5.  Connecting students' responses to key ideas.

# data[].attributes.body
Some questions teachers can ask to lead discussion:
What do you notice about the graph?
Do you see a pattern in the data?
What are the measures of center for the pennies?

# data[].attributes.body
While types of questions are essential, it's the pattern of questions teachers use during discussion are very important.

The teacher decides the path to take to lead the discussion, which allows the students to engage based on that path, and come to conclusions using higher-level questioning and thinking.

# data[].attributes.body
Being fluent means students have flexibility and choice among the methods and strategies used to solve problems.

Fluency will build upon exploration that students do to solve problems.

Fluency should not be rushed, and should not only include memorization of facts.

# data[].attributes.attachment_link.title
Teaching Habits that Promote Productive Struggle in Math

# data[].attributes.attachment_link.title
Establish Mathematics Goals to Focus Learning

# data[].attributes.attachment_link.title
How To Use Open Ended Math Tasks In Your Classroom

# data[].attributes.attachment_link.title
Promoting Procedural Fluency in Math Class with Number Talks

# data[].attributes.headline
Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

# data[].attributes.subject
Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

# data[].attributes.headline
Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.

# data[].attributes.subject
Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.

# data[].attributes.headline
Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

# data[].attributes.subject
Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

# data[].attributes.headline
Use and connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

# data[].attributes.subject
Use and connect mathematical representations. Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

# data[].attributes.headline
Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.

# data[].attributes.subject
Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.

# data[].attributes.headline
Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.

# data[].attributes.subject
Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships.

# data[].attributes.headline
Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures l as they solve contextual and mathematical problems.

# data[].attributes.subject
Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures l as they solve contextual and mathematical problems.

# data[].attributes.headline
Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

# data[].attributes.subject
Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.