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Illustrative Mathematics (IM) believes each student

has brilliant mathematical ideas

has experiences and ideas that are valuable

can make sense of and solve problems

can learn grade-level mathematics

How can educators act on these beliefs, in teaching and learning with IM 6-12 Math™, to craft learning communities and experiences in which each learner knows, uses, and enjoys mathematics?

Illustrative Mathematics (IM) believes that each student deserves access to grade-level mathematics. Teachers act on this belief when we teach with IM 6-12 Math™ because IM built the curriculum directly from the standards. IM also believes that each student can make sense of and solve problems and has experiences and ideas that are both valuable and brilliant. One way teachers can act on these beliefs is to use the instructional opportunities and strategies offered within each lesson.

This post is the first in a series of three in which we seek to help teachers to

1. Highlight instructional opportunities teachers have within IM 6-12 Math™ to enhance access students have to the math. The more we understand these opportunities, the more we notice and use them.

2. Be on the lookout for and identify additional opportunities to enhance access and challenge within the curriculum based on students' strengths. All students can learn grade-level mathematics with the right opportunities and access.

3. Prepare and plan for an upcoming lesson with access in mind. We teachers know our students' strengths and when we plan our lessons with them in mind, we help to remove unnecessary barriers to learning.

We look forward to hearing from you about what resonates with your own experience and how this series may inform your instructional practice moving forward.

Each IM lesson is designed to start by inviting students into the mathematics of the day. Then, activities take students on a deep dive into concepts. Finally, the teacher partners with students to consolidate and apply learning within the activity syntheses, lesson synthesis, and cool-down. Each learner can learn grade-level math with careful, intentional planning.

Within each lesson, how can we enhance students' access to the mathematical content? What instructional opportunities are already written into the teacher materials? Let's examine a lesson to find out.

Keeping the Equation Balanced (Grade 8 Unit 4: Linear Equations and Linear Systems] builds towards these learning goals:

Calculate the weight of an unknown object using a hanger diagram, and explain (orally) the solution method.

Comprehend that adding and removing equal items from each side of a hanger diagram or multiplying and dividing items on each side of the hanger by the same amount are moves that keep the hanger balanced.

These goals address and build towards : Analyze and solve linear equations and pairs of simultaneous linear equations.

What does it look like for students to meet these goals? One way to explore what it looks like for students to meet the learning goals of a lesson is to examine the cool-down.

Students might:

understand that if a hanger is balanced then there is equal weight on both sides, and if a hanger is unbalanced, there is unequal weight on both sides

know that to keep the hanger balanced, they must add or remove shapes so that the weight remains balanced (equal on both sides)

understand that identical shapes have the same weight

find equal-weight shape equivalencies

describe adding or removing shapes that maintain equal weight on both sides, either in writing or by drawing

So, how does the lesson set students up for success on the cool-down and guide them in meeting the learning goals? It starts with inviting students into the mathematics of the day with the Activity 2.1: Hanging Socks warm-up. This warm-up is designed to activate student thinking around balance and equality.

What do you notice? What do you wonder?

[from Grade 8, Unit 4, Lesson 2: Activity 2.1 ]

This visual framework for examining balance can enhance access to increasingly abstract representations (hanger diagrams, then equations) later in this lesson and unit.

In this warm-up, and throughout the lesson, students will draw on 3 cognitive functions-conceptual processing, visual-spatial processing, and language-to meet the learning goals. For example, they may rely on

conceptual processing to understand that if a hanger is balanced then there is equal weight on both sides, and if a hanger is unbalanced, there is unequal weight on both sides

visual-spatial processing to know that to keep the hanger balanced, they must add or remove shapes so that the weight remains balanced (equal on both sides)

language to describe, in writing or by drawing, adding or removing shapes that maintain equal weight on both sides

Why does this matter? There are eight ways of thinking, called cognitive functions, that have been found to be integral to learning mathematics. The warm-up calls on three of the eight.
Brodesky et al. (2002). Accessibility strategies toolkit for mathematics. Education Development Center.
When we give our students a mathematical task, we ask them to think in one or more of these ways. When the cognitive functions called upon in a task align to students' strengths, they are able to more readily and successfully engage with the task. On the other hand, when one or more of the cognitive functions represent a challenge for students, students may encounter barriers to learning and completing the task. In this case, teachers can use the instructional opportunities and strategies offered within each lesson in IM 6-12 Math™ to provide alternate means of accessing the task, without undermining its cognitive demand. (Enhancing access students have to the mathematics is one of the key commitments of the Universal Design for Learning Guidelines outlined on the website.)

Following the notice and wonder warm-up, students engage with Activity 2.2: Hanging Blocks. The purpose of this task is for students to understand and explain why they can add or subtract expressions from each side of an equation and still maintain the equality, even if the values of those expressions are not known.

[from Grade 8, Unit 4, Lesson 2: Activity 2.2 ]

In the next activity, Activity 2.3: More Hanging Blocks, students have the opportunity to solve two more hanger problems and write equations to represent each hanger.

[from Grade 8, Unit 4, Lesson 2: Activity 2.3 ]

Once again, we may notice that both the conceptual processing and visual-spatial processing cognitive functions are in play. What might this mean for students? If conceptual and visual-spatial processing happen to be cognitive strengths for students, they might experience this task joyfully as they tinker with important mathematical concepts. On the other hand, if these cognitive functions present students with a challenge, we teachers have an opportunity to enhance access students have to conceptual and visual-spatial processing so that they too may experience joy and confidence in tinkering with the mathematics of the task. Upon reading the embedded instructional support, we have some instructional decisions to make. Let's examine some of these options.

Let's focus on Hanger A. How might we simplify it in a way that students will engage with the same learning? Is there a way to simplify it so that students will also get the same answer as their classmates? Based on the suggested instructional strategy, we could simplify Hanger A in several ways as illustrated below:

Which of these options can enhance access students have to the mathematics, attend to the cognitive functions in play, and build to the learning goals of the lesson? Let's think back to the second learning goal of the lesson: Comprehend that adding and removing equal items from each side of a hanger diagram or multiplying and dividing items on each side of the hanger by the same amount are moves that keep the hanger balanced. If we choose Option 1, we remove an opportunity students have to tinker through adding or removing equal items from a hanger. So, what are the impacts to learning if we choose Options 2 or 3? We recognize that both Options 2 and 3 ensure students tinker with adding or removing equal items from a hanger while giving students the opportunity to strengthen their ability to internalize comprehension. Upon closer inspection, we can see that Option 3 will yield the same value for the blue square as peers working on the original hanger. Option 3 will serve to foster a sense of community across the classroom.

Why not use the simplified hanger for everyone? The original hanger foreshadows future lessons, where students add or subtract both constants (circles) and variables (squares) from both sides of the equation. Use of a simplified hanger might meet today's objectives, yet it could affect the overall progression of learning for the unit. Great care must be taken when making accommodations to enhance access. The use of a carefully selected, simplified hanger, along with other instructional moves, will enhance student access to grade-level work as they build the conceptual underpinnings of analyzing and solving linear equations.

Next Steps

As you prepare for an upcoming lesson, we invite you to

look and listen for cognitive functions that are strengths in each of your students

reflect on the mathematical focus of the lesson as evidenced in the learning goals and cool-down

identify the intended cognitive functions of each task

consider the impact of instructional decisions and task accommodations on your students' ability to meet the learning goals

Please be on the lookout for the next entry in this series, and share insights gained by going through this process. We'd love to learn from you.

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Leveraging IM 6-12 Math Teacher Materials to Enhance Access to Grade-Level Mathematics - IM CERTIFIED® BLOG

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At Illustrative Mathematics we are committed to creating a world where learners know, use, and enjoy mathematics. We believe that every student can learn grade-level mathematics with the right opportunities and support. Our approach is to remove unnecessary barriers and provide teachers with options for additional support so that every student can engage in rigorous mathematical content. We've been busy this year working on some exciting enhancements to the teacher tools and supports to empower teachers to deliver instruction that meets the specialized needs of English learners and students with disabilities.

By Liz Ramirez
In the newly released version of IM 6-8 Math, there are new and enhanced supports for students with disabilities and English learners. These supports are embedded within lessons, incorporated into the unit narratives, and featured in the teacher's guide. Highlights include:
Supports for Students with Disabilities
Within the Lessons
New in IM 6-8 Math: additional supports for students with disabilities are included in every lesson, and have been completely redesigned to give teachers more concrete guidance to meet the diverse needs of their students. The new supports follow the Universal Design for Learning Guidelines ( http://udlguidelines.cast.org), and each aligns to one of the three principles of UDL:

1. Engagement to help students self-motivate and become more independent.

2. Representation to leverage students' individual strengths by inviting them to engage with the same content in different ways.

3. Action and Expression to provide students with a variety of tools, and to encourage flexibility and choice in the ways students demonstrate their understanding and their reasoning about mathematical ideas with others.

Here is an example of how the principle of engagement is reflected in the new supports, from Grade 6 Unit 5 Lesson 14: Using Operations on Decimals to Solve Problems:

In this lesson, students apply their knowledge of operations on decimals to two sporting contexts. In this activity, students may be given the opportunity to create a visual display to showcase their solutions and prepare a short presentation in which they explain their reasoning and calculations. The support suggests providing a rubric or checklist that will help guide students during the creation of their display. This additional guidance supports students with the process, allowing them to focus on the mathematical content of their displays.

Here is an example of how the new supports reflect the principle of representation, from Grade 8 Unit 4 Lesson 2: Keeping the Equation Balanced.

In this lesson, students recall a representation that they have seen in prior grades: the balanced hanger. During this activity, they encounter a hanger with an unknown weight that cannot be determined. The support suggests providing students with access to a physical model of the hanger diagrams. Often, students with disabilities benefit from hands-on activities, which allow them to make sense of a problem and make connections between concrete ideas and abstract representations.

Lastly, here is an example of how we considered the principle action and expression in our development of the new supports for students with disabilities, from Grade 7 Unit 6 Lesson 5: Reasoning about Equations and Tape Diagrams (Part 2)

In this lesson, students are working up to represent situations that can be represented by an equation of the form p(x+q)=rp(x+q)=r. The work of this task is to draw a tape diagram to represent each situation. Tape diagrams are used to help students understand why these situations can be represented with equations of this form, and to help them reason about solving equations of this form. While tape diagrams may be familiar for many students, some students will continue to benefit from reviewing how to draw the diagrams, which will also help to solidify understanding of the connections between the situations, diagrams and equations.

In the Teacher Guide
The revised Supporting Students with Disabilities section of teacher guide now includes:

General instructional strategies that teachers can use to help make a lesson or activity more accessible, such as:

Providing students with access to physical manipulatives to help illustrate connections between concrete ideas and abstract representations.

Using graphic organizers such as word webs, Venn diagrams, tables, or other metacognitive visual supports to provide structures that illustrate concepts or support students with organizing their thoughts and ideas.

Building in time for brain breaks to refocus and re-energize the physical and mental state of students during a lesson.

Accessibility for Students with Visual Impairments: Additional guidance and information about features built into the materials that make them more accessible to students with visual impairments such as:

A color palette using colors that are distinguishable to people with the most common types of color blindness.

Tasks and problems designed such that success does not depend on the ability to distinguish between colors.

Mathematical diagrams presented in scalable vector graphic (SVG) format, which can be magnified without loss of resolution, and can be rendered in Braille.

Where possible, text associated with images is not part of the image file, but rather, included as an image caption that is accessible to screen readers.

Alt text on all images, to make the materials easier to interpret for users accessing the materials with a screen reader.

For additional information about the Universal Design for Learning framework, or to learn more about supporting students with disabilities, visit the Center for Applied Special Technology (CAST) at www.cast.org/udl.
Supports for English Language Learners
Within the Lessons
In the earlier version of the curriculum, only certain lessons featured additional supports for English learners. We're excited to announce that we now offer additional ELL supports for every lesson with suggested strategies for teachers to provide access to the language demands of each lesson.

All of the work we've done stems from the framework for supporting English learners designed for us by the Understanding Language team at the Stanford Center for Assessment, Learning and Equity (UL/SCALE). Each of the activity-level supports each is tied to one of the 8 Mathematical Language Routines (MLR). They are similar to instructional routines in that they structure student interactions but also provide an additional focus on language development.

Here is an example of the standard support we use to support English learners during number talks:

All of the supports indicate the language demands (reading, writing, speaking, listening, conversing, or representing) that are supported for the activity.

Here is another example of a support from Grade 7 Unit 2 Lesson 3: More about Constant of Proportionality. During this activity, students make connections between constant speed and proportional relationships, with special attention to the constant of proportionality. The support suggests MLR1 Stronger and Clearer Each Time as a structure to give students an opportunity to revise and refine both their ideas and their verbal and written output.

To make the MLRs more accessible to teachers, we updated the descriptions for each routine in the teacher guide, and included a step-by-step description of "how it happens." To help teachers learn the routines, an extended description of the support is included the first time each MLR appears in a course.

In the Teacher Guide and Unit Narratives
Here are a few highlights of enhanced supports for English learners, and language development for all students:

Sentence Frames: A new addition to the teacher guide, examples of open-ended sentence frames teachers can use to amplify and support the different ways students will communicate their understanding.

Enhanced Unit Narratives: Each unit begins with an overview that describes the story of the unit and the mathematical work that students will engage with. We've added two components to each overview to give teachers unit-specific language and vocabulary support.

A progression of disciplinary language that describes how students will use language as they learn the mathematical content of that unit.
Example: excerpt from Grade 7 Unit 2 Narrative

2. New terminology tables that show where new terms are introduced, and help teachers to know when students can be expected to understand a word or phrase receptively and when students should be producing these words or phrases in their own speaking or writing.

Student Glossary: The student glossary includes more images and student-friendly language.Example: Grade 6 Unit 2 - Ratio

For additional information about supporting English Language Learners, visit Stanford's Understanding Language site at https://ell.stanford.edu.
at www.cast.org/udl.

Next Steps

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Updates to Supports for Students with Disabilities and English Language Learners in IM 6-8 Math - IM CERTIFIED® BLOG

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At Illustrative Mathematics we are committed to creating a world where learners know, use, and enjoy mathematics. We believe that every student can learn grade-level mathematics with the right opportunities and support. Our approach is to remove unnecessary barriers and provide teachers with options for additional support so that every student can engage in rigorous [...]

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By Kate Nowak
How do we help our students build mathematical understandings that endure past the unit test? If we want students to construct strong, reliable bases of mathematical knowledge, our instruction needs to do more than present explicit procedures-even when that's done well. Providing lots of opportunities for students to reason can help. So can understanding and leveraging the progression of learning across grade levels. But what does that look like in practice? Let's examine a single topic in grade 7: solving inequalities.

Suppose that you teach grade 7 math, and by the end of the year you want students to reliably solve an inequality like this and graph its solutions:

$18>3(10-2x)$

So you try the problem yourself, and it looks something like:

$18>3(10-2x)$

$6>10-2x$

$2<x$

$x>2$

What do students need to be able to do to reliably duplicate what might have gone on in your brain while solving this inequality? Potentially:

1. Start solving it like you would solve an equation.

2. Remember that if you multiply or divide each side by a negative number, the inequality sign changes from > to <, or vice versa.

3. Once the inequality is in the form $x>a$, you can graph the arrow pointing in the same direction as the > symbol, but first you have to restate the inequality with the variable on the left, or it won't work.

Let's assume they can already solve the equation (the first thing) and now we're focused on how to get them to do the second two things.

One issue is that problematic conceptions result from this procedural approach to solving inequalities. From the AMS blog: "We define a conception as 'problematic' if it describes an understanding that obscures connections to related ideas, introduces mathematical inconsistencies, or is likely to hinder students from developing powerful meanings of future topics." I'm mainly referring to the third thing, choosing which part of the number line to shade based on which way the inequality symbol is "pointing." These ideas aren't meaningfully connected, which becomes obvious when you try to use the same trick to graph $2<x$.

Another issue is that the 3-step process used to solve the inequality is framed as a list of procedures, and if your plan (as mine once was) is that students will all remember those procedures indefinitely without connecting them to understanding, you are going to be disappointed. In the third step, the procedure (draw your arrow in the same direction the inequality symbol is pointing) has its own procedure (make sure the variable is on the left side of the inequality; rewrite if needed). When your procedure has a procedure, you should really rethink what you are doing.

Some people try to demonstrate a procedure and then retroactively make connections to conceptual understanding. These are the folks who think (as I once did) that good teaching is a matter of really clearly explaining. "Here's why you have to flip over that inequality sign! Chalk and talk and chalk and talk and chalk and talk. See?" Sorry, babe. Cat's out of the bag. Only half of them were listening and only two of them followed your explanation to the end.

So what should we do instead? Let's review the progression of these ideas starting back when the kids were still cute. Grade 1 is where they should first encounter the symbols >, <, and =. They should be doing things like "I know that $34$ is greater than $27$, and I can write that like this: $34>27$. If alligators are invoked, I don't have a problem with that. As mentioned in the progression document for Number and Operations in Base Ten, K-5, using these symbols can be a challenge for young learners, and it can help for them to think of putting the wide part of the symbol next to the larger number. But no worries, there are more chances to use these symbols to compare three-digit numbers in grade 2, fractions with the same denominator or the same numerator in grade 3, multi-digit numbers in grade 4, fractions with different numerators and different denominators in grade 4, decimals to the hundredths in grade 4, and decimals to the thousandths in grade 5.

If students never internalized the meaning of these symbols after all that, you might have to frontload some instruction on it. Hopefully, though, a quick reminder of what the symbols mean will do.

In the pivot to algebra in grade 6, we get variables involved as well as the number line, with these two standards:

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example $-3>-7$ means $-3$ is located to the right of $-7$.

Write an inequality of the form $x>c$ or $x<c$ to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form $x>c$ or $x<c$ have infinitely many solutions. Represent solutions of such inequalities on number line diagrams.

In grade 7, there may be unfinished learning related to these ideas from grade 6. If the kids learned that the "direction" of the inequality symbol corresponds to the direction of shading on the number line (but only if $x$ is on the left!), it would help to build some more meaningful schemas. The understanding we want to develop is that on the number line, greater numbers are to the right and lesser numbers are to the left. The symbol $>$ means "greater than" and the symbol $<$ means "less than" (which they have known since first grade). There are no tricks needed (and tricks don't last in the long run), rather practice at interpreting, comprehending, and writing statements with variables that use these symbols. For inspiration, see IM Grade 6 Unit 7 Lessons 8-10.

Then in grade 7, that understanding is applied to more sophisticated problems. We get the $\le$ and $\ge$ symbols, we have operations on signed numbers so we can solve stuff like $7+y \lt \text-2.5$ and $\text-10 \gt 3x$ and we are solving equations like $3x+7=15$ so we can solve even more complicated inequalities. But let's focus on how we're going to tackle that whole "flip the inequality around when you divide by a negative" thing. The answer is, we're not. This falls into the category of "shortcuts that are perfectly fine if students notice the generalization on their own and reliably use it on their own, but it's counterproductive to make it the trick we teach everybody." Instead we're going to:

Develop the understanding that the solution to an associated equation gives you the edge or boundary of the solution to the inequality.

Explore lots of inequalities that represent a context, and reason about what the solution would have to be to make sense in the context. For example, "The inequality $-12x+100?25$ represents an account with $\$100$ in it where someone withdraws $\$12$ each month for $x$ months. They want to maintain at least $\$25$ in the account." If $x$ is a relatively small number like 1 or 2, they meet their constraint. If $x$ is a relatively large number like 10, they don't. So the solution should end up saying $x$ is less than some positive number.

Recognize that the involvement of negative numbers sometimes makes it hard to predict what the solutions will be, and reason about the solutions to relatively simple inequalities using tables and number lines.

Then put it all together to solve an inequality like $18<3(10-2x)$. Solve the associated equation to determine where the boundary is, and then use any of a variety of strategies to decide whether values less than or greater than the boundary value make the inequality true.

The procedures outlined at the beginning of this post might get some of the students successfully through the exam at the end of the year, even most of them if you are really repetitive about it. But the understanding students develop from an approach that doesn't introduce problematic conceptions will endure and serve students well throughout their later math coursework and beyond.

Next Steps

Achieving all this understanding doesn't have to take too long. We offer an approach that takes five lessons in IM Grade 7, Unit 6, Lessons 13-17. Check it out!

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Rethinking Instruction for Lasting Understanding: An Example - IM CERTIFIED® BLOG

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By Kate Nowak How do we help our students build mathematical understandings that endure past the unit test? If we want students to construct strong, reliable bases of mathematical knowledge, our instruction needs to do more than present explicit procedures-even when that's done well. Providing lots of opportunities for students to reason can help.

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Developing Conceptual Understanding and Procedural Fluency - IM CERTIFIED® BLOG

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By Melissa Schumacher, Curriculum Writer Which is more important for students to have: conceptual understanding or procedural fluency? Does one have to be taught before the other can emerge? Some argue that procedure has to come first, because without procedural fluency, students won’t be able to do…

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This blog post is the fourth in a series of four blog posts exploring the student experience of problem-based learning. The first three posts are available here: (1) "How Do Students Perceive Problem-Based Learning?" (2) "Inviting Students to the Mathematics" (3) "Concrete Representations that Give Students a Way to Get Started."

Okay, so the kids can get started and represent their thinking. But are they really learning? Am I really teaching? What are we doing here?

In this series of blog posts, we're exploring how students have to shift their thinking about what being a good math student looks and sounds like in order to be successful in a problem-based classroom. What did you notice about the representations and tools offered to students? Are your students using those tools to support their reasoning and problem-solving? What are you curious about now, as you think about what it takes for students to be successful in a problem-based classroom?

Here is a reminder of one model for problem-based instruction that we use at IM:

Think back to the beginning of this series when we categorized how students often learn to behave in math class. When materials present students with all the tools and strategies they need before posing any problems, it's less likely that students will experience learning from a classmate's unique approach. It's more likely that students won't value their own mis-starts and incomplete ideas, because a complete model is always available. It's more likely that conversations will end after the complete, correct answer is shared and checked against the authority. If only one approach is demonstrated, there's little reason to compare and build connections among approaches that lead to the same answer.

In a problem-based curriculum, students do much of their learning during whole-group synthesis of different solution approaches. In other words, a lot of the learning happens when the teacher helps students compare and reflect after they've worked on the problem for a while. In the previous blog post, we described that the teacher might call the whole class back together when the students are at various problem-solving stages: one group might have finished the original problem a while ago and is now trying to solve it an additional way, while another group has a wrong answer, and a third group is only halfway through the problem. Why might the teacher do that? Why wouldn't the teacher work with each group to make sure they are getting through this problem correctly?

Most of the learning in a problem-based classroom happens when students compare their ideas to one another. Armed with this knowledge, we can be more comfortable facilitating this sort of robust, synthesizing conversation. The teacher does not have to correct the group with a wrong answer because they know the whole class will benefit from comparing that group's thinking to their own, and will understand better why the right answer is right. The teacher doesn't have to wait for each group to finish because they know that the students had opportunities during the launch and work time to develop a better understanding of the problem. The teacher can set a goal of everyone being ready to make connections between their ideas and their classmates' ideas, even if they struggled with the problem.

But how does this learning from others happen? It happens when students have shifted from thinking that the learning happens when the teacher explains (at the beginning of class) or corrects (at the end of class). It happens when students believe that learning happens from actively comparing their ideas to others' ideas. It happens when students try hard to learn new, more efficient or effective ways of thinking.

In order for students to learn this way, they need to:

Be willing to share their ideas, right or wrong.

Be willing to listen to and learn from others' ideas, right or wrong.

Be curious about the work of other students, even if they have already solved the problem one way.

Actively work to make sense of what other people are saying, even if they are confused.

Be willing to try someone else's method and work until they feel confident in their reasoning to solve similar problems in the future.

Listen for structure, patterns, representations, and methods, not just answers and steps.

These norms are very different from the implicit or explicit norms they may have picked up in other math classes. How do we help students make the necessary shifts?

Culture change often happens when we bring active attention to it. Some methods teachers have used include:

Using talk moves that support students to listen to one another and share their thinking such as:

Giving independent think time both when starting to work on a task and after students hear another student's ideas or methods.

Using think-pair-share or turn-and-talk opportunities. Students can practice what they might say to the whole group, or tell someone what they understand and don't understand about something they just heard, or tell a partner what connections they see between the strategy on the board and their own thinking.

Asking multiple students to revoice what they heard a classmate say. Note that student revoicing tends to help students listen to each other and their peers. Teacher revoicing has the opposite effect, and decreases students' willingness to listen to peers.

Giving kids explicit feedback on their participation in discussions, before they focus at all on the content of discussions ("Tyler, I notice that when Jada shared her thinking, you asked her a question to understand it better, and that helped you both improve your ideas.")

Writing down examples of students actively using the classroom norms, and displaying them during the discussion as another way to give feedback (sometimes called a "Participation Quiz"). For example, you might display a seating chart, and then when you overhear a student say, "Can you explain how you thought to...?" you would write that quote down next to their group as a positive example of participation.

Having kids create a running and growing list of class norms, and then helping students to actively enforce them (such as "don't make fun of people being wrong," or "try drawing a picture first before you ask a teacher for help," or "give people quiet think time before jumping in.") One way to grow this list is when you catch students engaged in productive behaviors, naming them and asking, "Should we add this to our list?"

Problem-based curricula like Illustrative Mathematics' work best when kids believe they can tackle problems on their own, when they have tools to share their reasoning (not just their steps), and when they are willing and able to learn by comparing their ideas to their classmates' in a discussion focused and guided by the teacher. Those skills are not always honed in their previous math class, if teachers used a different kind of curriculum. Therefore, explicit culture change for students is as important as picking a great curriculum.

Next Steps

What do you notice as big shifts for your students when you use a problem-based curriculum? What are you still curious about when you think about how students learn from solving problems? How will you explore your questions in your reading, teaching, and conversations with colleagues?

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Explicit Classroom Norms to Teach Kids How to Learn From Solving Problems

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This blog post is the fourth in a series of four blog posts exploring the student experience of problem-based learning. The first three posts are available here: (1) "How Do Students Perceive Problem-Based Learning?" (2) "Inviting Students to the Mathematics" (3) "Concrete Representations that Give Students a Way to Get Started."

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"Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first."

By Kristin Gray

I've come to think that approaching a lesson plan is like approaching a 500-piece puzzle. It's hard to know where to start. It takes time. It involves endless trial and error. But when it's finished - when all the pieces have been put in place - there's a sense of pride and accomplishment.

I'm a math teacher, and when I shared this reasoning with other teachers for the first time, I included a lot of beautiful words around mathematical connections, but in the middle somewhere, I used the phrase "making connections explicit." The room seemed to recoil. I paused to reflect on why teachers were withdrawing. I asked them, "Was it the word 'explicit?'" There were nods throughout the room.

Trust me, I get it. The word "explicit" has developed a cringe-worthy connotation among peers in the math community. In the past, if I were asked to paint a picture of a math teacher engaging in "explicit" teaching, I would have painted either a teacher standing up at the front of the room, telling students how to solve a problem, or a teacher crouched down next to a struggling student, "correcting" the way the student approached the problem because they were taking the "long way there."

These scenarios make us cringe, because we know they don't lead students to true understanding. I think there are parallels in teaching other subject areas as well. Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first.

In a math classroom, an "explicit" teacher tells students how to approach math, typically by way of a procedure. There's no discovery. But, when we purposefully plan using a problem-based curriculum in our classroom - setting up investigations for our students, designing and implementing routines and fostering intentional talk - we can change what it means to be "explicit."

When we plan and teach lessons - in any subject area - we make hundreds of decisions around a specific learning goal. We make decisions about the questions we'll ask to help students see connections. We make decisions about which particular pieces of student work to share at the end. In each of these choices, we are being explicit about the idea we want to bring to light.

We are explicitly planning, not explicitly teaching.

I've had many moments of frustration when planning and teaching a problem-based curriculum. Ideas can go unnoticed. Connections can go unmade. I sometimes used to spend the end of class standing at the board, frantically trying to explain something, because I was worried students were about to leave without getting "the point" of the lesson. Somehow, the kids didn't read my mind! Moments like this came about because my planning wasn't explicit enough.

I've found Mary Kay Stein and Peg Smith's "5 Practices Framework" a useful tool for explicit planning. The framework forces me to think continuously about the goal, choose an activity that supports that goal, plan questions to move students toward the goal and sequence student work in a way that creates a productive, purposeful discussion toward an explicit idea. I believe this could work in any subject area, as a way of supporting our students in their development of true understanding.
Mary Kay Stein and Peg Smith's "5 Practices Framework:"

1. Anticipate what students will do - what strategies they will use - in solving a challenge.

2. Monitor student work as they approach the challenge in class.

3. Select students whose strategies are worth discussing in class.

4. Sequence those students' presentations to maximize their potential to increase students' learning.

5. Connect the strategies and ideas in a way that helps students understand the concepts learned.

Mary Kay and Peg also describe a Practice 0 of Choosing a Mathematical Goal and Appropriate Task in their book - like selecting the right puzzle to put together on a particular day.

This framework can apply across various contexts and subject areas, but I think it's embodied in the open education resource (OER) curriculum by Illustrative Mathematics. Each lesson and unit tells a mathematical story in which students arrive at a specific mathematical landing point. While students may not all arrive in the same way or at the same time, the problems and discussions are structured to ensure students don't leave the day's work without understanding the goal.

Explicit planning requires us to understand the final picture - the goal - before the puzzles pieces can come together. It requires us to carefully connect each piece, because one missing piece will leave the final picture unfinished. While planning lessons for our unique groups of students, we may not at first see exactly how each puzzle piece will connect, and sometimes we can feel stuck. But our colleagues and students can help until all the connections are clear - and we're proud of the final picture we see.
Next Steps

Check out the video to see teachers learning about these 5 Practices in a recent IM professional development.*

Find a sample lesson plan written using the 5 Practices here: Grade 6, Unit 3, Lesson 4, Activity 1: Road Trip in Illustrative Mathematics 6-8 Math Curriculum.

If you are reading the 5 Practices book , I would love to hear your personal connections and reflections as you read.

If you are using the 5 Practices to plan for activities in your classroom, I would love to hear about your experiences.

to learn more about professional development of the 5 Practices for your classroom or school.

*Professional Development based on NCTM's 5 Practices for Orchestrating Productive Mathematics Discussions by Smith and Stein.

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The 5 Practices Framework: Explicit Planning vs Explicit Teaching - IM CERTIFIED® BLOG

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"Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first." By Kristin Gray I've come to think that approaching a lesson plan is like approaching a 500-piece puzzle.

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Math teachers can talk all day about math! We get super excited when we encounter someone else who enjoys these conversations as well.

But too many of our students don't get to experience that joyful discourse, which limits their understanding of and positive experiences with math.

How would our students' math learning and perception about math improve if they also had regular opportunities to talk about math? How might these opportunities help us create a world where learners know, use, and enjoy mathematics?

Let's explore the Math Language Routines, which are embedded in IM K-12 Math.

What are math language routines?

Stanford Graduate School of Education's Understanding Language and the Stanford Center for Assessment, Learning, and Equity (SCALE) designed eight Math Language Routines (MLRs) to help teachers "amplify, assess, and develop students' language" in math class. These routines are structured, but adaptable to the content and classroom context. The routines "emphasize the use of language that is meaningful and purposeful, not inauthentic or simply answer-based . . . [and] can and should be used to support all students learning mathematics." (from Understanding Language/SCALE's Principles for the Design of Mathematics Curricula: Promoting Language and Content Development ) There are four specific design principles used to entice students to discuss the concepts and ensure that the discussions provide opportunities for students to learn something new. The Math Language Routines also provide opportunities for students to engage in the Common Core's Standards for Mathematical Practice .

As students engage in math language routines, MLRs help students to advance their thinking in the following ways:

OPTIMIZING OUTPUT as students describe their own math thinking to others orally and visually or writing as physical, graphical, pictorial, or symbolic representations.

SUPPORT SENSE-MAKING as students scaffold their thinking and amplify language so students can make their own meaning between concepts and representations.

CULTIVATE CONVERSATION by providing opportunities to simultaneously make meaning, communicate that meaning, and refine the way content understandings are communicated.

MAXIMIZE META-AWARENESS as students organize their own experiences, ideas, and learning for themselves through reflection of the experience of discussing their math ideas, reasoning, and language.

This doesn't just mean that teachers tell students to talk about math and they do. We know that if we simply open the space and tell students to talk, much of the discussion will veer off-topic. Math Language Routines help teachers create a structure in which students can bring those four design principles to life.

Let's see what opportunities each MLR provides for students to practice exploring their math thinking and the MLRs' connections to the Standards for Mathematical Practice .

MLR 1: Stronger and Clearer Each Time - Students write a response to a math problem, then verbally share their response with a partner to get feedback from the listener to improve the response, and revise their original written response based on the feedback they received.

Students learn to both listen to their own thoughts and the thoughts of another with the purpose of strengthening their "first draft" thinking. This MLR strengthens student communication of their thinking. They make their explanations clearer through repeated "reviews" and revisions, which allow them to practice giving, receiving, and applying constructive feedback, and develop listening skills. This MLR helps students communicate their mathematical reasoning with greater clarity and focus.
(This connects to Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them, 3: Construct viable arguments and critique the reasoning of others, and 6: Attend to precision. As students verbalize and review in writing the strategies they used and justification of those strategies, students have the opportunity to get feedback which can be used to critique the precision of their solution strategy and answer.)
MLR 2: Collect and Display - Students access their own and others mathematical ideas as the teacher scribes the language, strategies, and concepts students use during partner, small group, or whole-class discussions using written words, diagrams, and pictures.

By creating stable representations of student thoughts, the teacher extends the involvement of students in the math classroom. Student contributions are displayed for everyone to see. During that discussion as well as future conversations, students refer to these written ideas to improve the clarity and focus of their own communication. They can use the vocabulary, reference previous relationships between concepts, and identify new connections. This MLR helps make student thinking visible and promotes student ownership over mathematical ideas.
(Standards for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others, and 6: Attend to precision. As students view the thinking of their classmates from the representation, they can address any differences they see between their representation and their classmates' representation in type and kind, further pushing students to find the most efficient or precise representation to solve the problem.)
MLR 3: Critique, Correct, and Clarify - Students rewrite a math response from an example that is incorrect, incomplete, or otherwise ambiguous.

In addition to providing students with a space to further engage in math, students employ a "growth mindset" and consider revisions to their work in order to strengthen their communication. This MLR helps students "try on" others' math thinking in order to improve it.
(Standards for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. Similar to doing error analysis, students have the opportunity to engage with a strategy for solving that might be different from theirs in order to find ways to improve the situation representation, address and clarify misconceptions about the math, and "borrow" parts of the strategy that were relevant to add to their toolbox.)
MLR 4: Information Gap - In a group, each student has different parts of the math context and they work together to piece together that information orally and/or visually to bridge the gap between the parameters of a math situation and a question to solve a math problem.

As students work to make sense of individual pieces of information about a math context, they also identify what information they need in order to answer the math question. This creates an environment where students work together as they consider which strategies are the most efficient and what information they need to know to solve the problem. This MLR helps students explore math contexts with a purpose.
(Standards for Mathematical Practice 2: Reason abstractly and quantitatively, and 4: Model with mathematics. As students imagine the kinds of information they need to know and explain why they need to know that information, they are building an understanding of the problem and possible solutions related to the parameters of the problem and the kinds of data they need to solve for an answer.)
MLR 5: Co-Craft Questions and Problems - Students take a provided math "answer" to create a problem that could result in that answer or use a given context to create a problem that can be answered using that context.

With this MLR, students use their creativity to explore a math context as they iterate their understanding in order to identify questions that can be asked and answered by the context. As students share their understandings with others, they also uncover the importance of slowing down to explore and think before they try to answer a question, which results in students understanding the math more deeply. This MLR helps students define math contexts.
(Standards for Mathematical Practice 4: Model with mathematics. By creating their own questions and problems, students learn to visualize the parameters of the problem as they consider what kind of answers are possible with the information the problem provides.)
MLR 6: Three Reads - Students are guided to read the problem three separate times with three separate purposes with quick discussions between each read.

When students slow down to consider how they can use the information in a problem, they notice things about their understanding of the problem context and their own math understanding that they wouldn't have noticed before. Giving students a purpose to read helps to focus students to identify (1) the context of the mathematical situation, (2) relationships that exist within that context, and (3) strategies that can be used to answer the question using the problem context. This MLR helps students think metacognitively about their math problem solving.
(Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them. As students set a purpose for reading each of the three times, students practice and see the importance of understanding the problem context and using the information they know from the problem in order to identify a strategy for solving that relates directly to the problem context and available information.)
MLR 7: Compare and Connect - Students identify, compare, and contrast their own understandings with other students' mathematical approaches, representations, concepts, examples, and language.

As students encounter other's math thinking, this MLR opens the floor for students to consider how they see and talk about math. This MLR specifically helps students interact with each other's thinking.
(Standards for Mathematical Practice 4: Model with mathematics, 6: Attend to precision, and 7: Look for and make use of structure. As students compare (find differences) and connect (find similarities) between their strategy for solving and another classmate's strategy for solving, students mine their understanding of the math concept in relation to the structure and steps of another student's understanding of the math. This allows both students to attend to precision because to be understood by another, they have to be precise with their explanation.)
MLR 8: Discussion Supports - Students access strategies in multi-modal ways, like visuals, aural, animation/movement, and symbols as modes, to help students better make sense of complex language, ideas, and classroom communication.

Enhancing access to the mathematical context through these multi-modal supports helps students develop understanding while also continuing to work on their mathematical communication. This MLR helps students deepen their mathematical understandings, and find ways to communicate their understandings in different ways.
(Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them, 3: Construct viable arguments and critique the reasoning of others, 5: Use appropriate tools strategically, and 6: Attend to precision. Using discussion supports helps students to see the reasoning of others as well as identify which tools to solve are most appropriate. In using these supports, students can be successful with in-depth or multistep problems with support from teachers and classmates.)
Next Steps

Consider what your students need to encounter math more deeply and in collaboration with others. More heads are better than one!

Looking for more support for distance learning ? Check out IM's collection of resources.

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Math Language Routines: Discourse with a Purpose - IM CERTIFIED® BLOG

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Consider this:
A child whose primary language isn't English does not participate in the classroom to the degree that their English-speaking peers do. Parent-teacher conferences are held with family members who are eager to hear what the child is learning in class, the skills that they have in math class, and where the teacher wants them to grow more across the year. Everyone at the table cares deeply about the child's success, yet language is hindering communication for the student, the teacher, and the student's family.
If it doesn't already sound familiar to you yet, it soon will. According to the National Education Association , by 2025, 25 percent of students in classrooms across the country will be English learners, or ELs.

Often, perceptions of ELs in the classroom are built on the assumption that language barriers limit the students' access to grade-level content. Many teachers frontload vocabulary work for ELs, asking students to memorize lists of mathematical vocabulary devoid of context, believing that this is a necessary prerequisite for ELs to construct meaning or communicate their ideas. However, according to Judit N. Moschkovich in Language and Mathematics: Multiple Perspectives and Directions for Research, ELs "can participate in discussions where they grapple with important mathematical content," even as they are acquiring English.

A Third Grade Classroom Meets Fractions

In February 2021, a grade 3 classroom returned to school from remote learning. Due to the COVID-19 pandemic, the students hadn't been in school with peers for eleven months, and they were in the middle of their first lesson from on fractions: Unit 5, Lesson 1, Activity 2: Fold and Name .

The teacher launched the activity by giving each student 4 paper rectangles.Then she invited the students into the activity. "We are going to fold these rectangles to partition them. Fold each rectangle into 3, 6, 4, or 8 equal parts. Draw lines where you folded to partition the rectangles. Be prepared to share how you folded your rectangles."

Students began working independently. After 5 minutes, the teacher asked students to work in pairs to share how they folded their rectangles. As students worked, the teacher used the Math Language Routine Collect and Display (MLR2), described below, to record mathematical words and ideas the students expressed. Then she invited students to share their approaches to the task.

Student A, an EL in the class, raised his hand and described what he did:

"I took the paper and I folded it down and put the corners like this" (showing how he aligned the corners and edges of the rectangle). "Then I did this" (lining the corners up again, and folding it in half again).

Student B, a native English-speaker, then raised her hand and said, "Yes. I folded it in half like he did. Then I folded it in half again to make four equal parts."

More students shared, refining their language as they heard from their peers. The teacher, aware that Student A had begun the conversation by modeling his understanding using the paper rectangle, encouraged more. During the synthesis, she asked him, "How did you use the partitions of four equal parts to partition eight equal parts?"

This time, Student A responded with more excitement. "I folded the rectangle in half like this to make two parts, and then in half again to get four parts. Then I folded the four parts in half again to get eight equal parts."

How did the teacher position Student A?
The teacher saw that Student A mathematically understood the idea of partitioning and that his approach to folding yielded equal parts. She noted that he was using halves to partition fourths. Being an EL did not mean that Student A lacked understanding that, if shared, would benefit the class. The teacher recognized this and took the time to amplify his voice. By using the Math Language Routine Collect and Display (MLR2), she allowed him to become more comfortable with the language needed to describe the math that he was doing. His peers became a resource for him as they shared their strategies using mathematical terms, and his mathematical thinking was reinforced. In the end, she positioned him so that he could demonstrate his command over his mathematical understanding and the language that he was developing. It was exciting to see how much he contributed and the opportunities helped him grow more confident.

How can we all recreate opportunities like these in our classrooms?
Moskovitch suggests three guiding instructional principles for teachers that view language as a resource, rather than an obstacle, and focus on student achievement, rather than just on learning English.

Principle 1: Focus on listening to students' mathematical reasoning, not on their accuracy in using language.

Principle 2: Provide opportunities for students to use language as part of the mathematical processes, not as single words or definitions.

Principle 3: Provide opportunities for students to engage in the complexity of mathematical language, including multiple representations (objects, pictures, words, symbols, tables, graphs), modes (oral, written, receptive, productive), kinds of talk (exploratory, expository), and audiences (teachers and peers).

The eight Mathematics Language Routines, or MLRs, that are built into the math curriculum provide a way for teachers to put these principles into action. The MLRs support all students as they move from using informal language to using more formal mathematical vocabulary, but, as the Course Guide states, they are "particularly well-suited to meet the needs of linguistically and culturally diverse students who are simultaneously learning mathematics while acquiring English". The following examples highlight this.

Collect and Display
The purpose of Collect and Display (MLR2) is to preserve the diverse, fleeting, and often informal language students use as they engage in mathematical tasks. The teacher listens for and charts the words students use during partner, small-group, or whole-class discussions to describe an image or solve a problem using written words, diagrams, and pictures and posts them for later reference.

In doing the work of this MLR, teachers and students also engage in all three of Moskovitch's principles. In this Grade 5 activity, Unit 3, Lesson 13, Activity 1: Paper Strips , students divide whole numbers by three different unit fractions using the context of strips of paper. This helpful diagram leads to students recognizing important relationships between the divisor, dividend, and quotient even without explicitly using these words. As students compare and contrast the three problems, the teacher listens for the authentic language they use to describe division.

Compare and Connect
The purpose of Compare and Connect (MLR7) is to give students the opportunity to explore mathematical ideas by comparing the approaches and representations of others to their own. The teacher and students work together to create a visual display that they can use to find connections between the strategies, with the goal of deepening their understanding of the mathematical concept. They work to develop their understanding of the mathematical concept as well as their mathematical language.

In doing the work of this MLR, teachers and students engage in all three of Moskovitch's principles. In this Grade 1 activity, Unit 3, Lesson 24, Activity 2: Find the Number That Makes Each Equation True , students share different methods for solving 15 - 12. As they share, teachers listen to their reasoning, students explain their approaches, and they engage in the complexity of the mathematical language of subtraction.

Discussion Supports
The purpose of Discussion Supports (MLR8) is to support rich and inclusive discussions in math classrooms. Rather than the same routine over and over, the Discussion Supports are a collection of pedagogical strategies and teacher moves that can be combined to support discourse during almost any activity. One of the strategies is the use of sentence frames.

Sentence frames support all three of Moskovitch's principles by providing students with a structure to communicate. The sentence frames included in the curriculum are open-ended so that they amplify language rather than constrain it. Some are generic, such as, "First, I _____ because..." and "I noticed _____ so I..." Others support the specific content and language functions of particular lessons such as, "This card shows an even number because..." and "This card shows an odd number because..." from Unit 8, Lesson 3, Activity 2: Card Sort: Even or Odd in Grade 2.

If we focus on how students are able to use informal language to express their understanding rather than the use of accurate and technical vocabulary, we are more likely to recognize the contributions that they bring to the classroom community. 2025 is less than 4 years away. In order to make sure all of our students are able to know, use, and enjoy mathematics, we need to open the door for students to use whatever language helps them to make sense of the mathematics.

Next Steps

We shared a glimpse into one of our classrooms. Now it's your turn! What opportunities will you create for your students to use informal language? How will you honor their language and position them as competent contributors in your math classroom?

Share your experiences and success stories with us on Twitter using #LearnwithIM.

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Unlocking Learners' Thinking Using the Mathematical Language Routines - IM CERTIFIED® BLOG

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As teachers, we ask a lot of questions. It's part of the job. It's how we learn about our students.

Questions serve a variety of purposes during instruction. We can use questions at the beginning of an activity to launch a task , during a lesson to informally assess, and while providing feedback to advance student thinking. Let's look at how we can use intentional and purposeful questions to set students up for success, assess their understanding, and push their learning forward.

Questions help set up a math task and engage all students in the task.

At the beginning of a lesson or task, we ask questions to orient students and ensure they understand what they need to do. These questions also build engagement and buy-in from students. You might launch learning by asking questions like the following:

What do you notice about the problem?

What do we mean by _____?

What are we trying to figure out?

How is this task similar to one we've completed before? What's new or different?

These questions open entry points into the activity for all students. They stimulate student thinking without lowering expectations by prompting or suggesting a specific solution path.
Questions allow teachers to individualize assessment and instruction.
As students work on math problems, you likely circulate throughout the room and monitor their progress. This is a great opportunity to ask questions to formatively assess and extend student thinking.

Assessing questions provide information about students' knowledge and skills and help clarify what students understand about the work.

Advancing questions push a student to think in new ways to build their knowledge.

The following are general examples of assessing questions:

How did you figure this out?

Why did you draw this the way you did?

Why did you cross that out?

How do you know your answer is correct?

The following are general examples of advancing questions :

What would happen if we changed the number(s) to _____?

What is another method you can use to solve this problem?

What is a rule or formula that would work for all problems like this one?

Assessing questions can also help you find patterns in student work and inform how you conduct the Student Debrief or Lesson Closing.

Advancing questions naturally create differentiation because they allow you to meet students at their current point of success and move their thinking forward.

Let's look at a specific example to clarify the roles of assessing and advancing questions. As you review the following work of three different students on a Grade 5 fraction problem, think about what each student might know and what would further extend their learning.

A newspaper's cover page is text, and photographs fill the rest. If ³⁄ ²⁄ ₅ ₈ of the text is an article about endangered species, what fraction of the cover page is the article about endangered species ?

Student A

Student B

Student C

Students A and C arrived at a correct answer, but Student C identified a common factor in both the numerator and denominator, which resulted in a more simplified answer. Student A seems to have labeled the drawing incorrectly but still performed the correct multiplication. Student B is struggling to get started. Here are some questions you can ask to assess or advance each student.

Assessing Questions

A dvancing Ques tions

Student A

How did you show ³⁄₈ on your drawing? How did you show ²⁄₅ ?

Did you start with the drawing or the arithmetic next to the drawing? Why?

What you labeled in your drawing as shows ³⁄⁸⁄₈. Why is that? How would you label your drawing now? ₈ actually

John solved this problem differently and said the article is ³⁄₂₀ of the cover page. Is he correct? How do you know?

What if the problem started with different fractions, and of the page was text and ¹³⁄60 ⁷⁄50 of the text was an article? Would that be easy to draw? How would you solve that problem?

Student B

How does your drawing represent the information given in the problem?

How many vertical lines did you draw inside the big rectangle? What fraction of your big rectangle is each smaller rectangle?

How many vertical lines would you need to split the big rectangle into eighths?

What's the other important fraction in the problem? How many lines would you need to split the big rectangle into fifths?

If you start with a new rectangle, how can you use what you learned to redraw the problem?

Student C

Why did you decide to multiply ³⁄₈ by ²⁄5 ?

What does ³⁄₈ represent in the problem? ²⁄₅ ?

If ³⁄₂₀ of the cover page is the article about endangered species, what fraction of the page is something other than the article? What fraction of the page is other text?

Carlos solved this problem differently and said the article is ⁶⁄₄₀ of the cover page. Is he correct? How do you know?

Marie solved the problem and said the article about endangered species is ²³⁄₄₀ of the cover page. Without doing calculations, how could you convince her she made a mistake?

Although each student's response was unique, we found ways to advance each student's thinking through questioning. Posing questions promotes reflection and thinking that will lead to deeper conceptual understanding and more meaningful connections between mathematical ideas. While every student is different, every student is capable of greatness. With well-placed, purposeful questions, we can advance each student's learning and help them achieve their greatness.

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Advancing Student Learning with Questions

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As teachers, we ask a lot of questions. It's part of the job. It's how we learn about our students. Questions serve a variety of purposes during instruction. We can use questions at the beginning of an activity to launch a task , during a lesson to informally assess, and while providing feedback to advance student thinking.

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By Elham Kazemi, Co-Author of Intentional Talk and Professor at University of Washington

"Gifts from the earth or from each other establish a particular relationship, an obligation of sorts to give, to receive, and to reciprocate."

Robin Wall Kimmerer, Braiding Sweetgrass, p. 25

Robin Wall Kimmerer's book, Braiding Sweetgrass, has given me a lot to think about. Since I picked it up this summer, I find myself going back to it again and again. Kimmerer writes about Indigenous wisdom and our natural world. She is an enrolled member of the Citizen Potawatomi Nation, Distinguished Teaching Professor of Environmental Biology at SUNY, and the founder and director of the Center for Native Peoples and the Environment. Braiding Sweetgrass is a book to be savored and read and listened to many times over. Each chapter in the book is organized by the teachings of sweetgrass first and foremost and then by many organisms and plants in our natural world-strawberries, maple trees, water lilies, and so on. In the chapter called "The Gift of Strawberries", she writes about the wonder and generosity of spirit she feels whenever she comes upon a field of wild strawberries. She explains, "Strawberries first shaped my worldview of a world full of gifts simply scattered at your feet. A gift comes to you through no action of your own, free, having moved toward you without your beckoning. It is not a reward, you cannot earn it, or call it to you, or even deserve it. And yet it appears. Your only role is to be open-eyed and present. Gifts exist in a realm of humility and mystery-and as with random acts of kindness, we do not know their source. "(p. 24-25).

I think about the way I found joy in learning about children's mathematical thinking and experience and was ever-changed by a chance meeting with Megan Franke. In my first graduate school class, I was assigned to meet and introduce Megan in a proseminar about educational research. As Megan shared the tenets of Cognitively Guided Instruction (CGI) in that graduate seminar, I remember being in awe that teaching could be all about listening and learning from children's ideas and experiences.

Since reading Kimmerer, I have been thinking about children's willingness to share their ideas as gift-giving. What do we, as teachers, do with the gifts children freely give us every day? I have been thinking more deeply about what it means to receive these gifts. How can we be "open-eyed and present" as children share their gifts with us?
"It's funny how the nature of an object-let's say a strawberry or a pair of socks-is so changed by the way it has come into your hands, as a gift or as a commodity. The pair of wool socks that I buy at the store, red and gray striped, are warm and cozy. I might feel grateful for the sheep that made the wool and the worker who ran the knitting machine. I hope so. But I have no inherent obligation to those socks as a commodity, as private property. There is no bond beyond the politely exchanged, 'thank yous' with the clerk. I have paid for them and the reciprocity ended the minute I handed her the money. The exchange ends once parity has been established, an equal exchange. They become my property. I don't write a thank-you note to JC Penney.
Kimmerer describes how the field of wild strawberries was her childhood playground, retreat, and classroom all wrapped up together. She talks about experiencing all that the fields gifted her and that she tried to give back to the strawberries by weeding the ground so they would find more fertile soil for their runners. She then writes, "Gifts from the earth or from each other establish a particular relationship, an obligation of sorts to give, to receive, and to reciprocate" (p. 25). In the classroom, when children gift us their ideas, what kind of relationship do we establish with them? How do we reciprocate?

Let me share another passage from her chapter on strawberries where she made me think further about reciprocity:

But what if those very same socks, red and gray striped, were knitted by my grandmother and given to me as a gift? That changes everything. A gift creates ongoing relationship. I will write a thank-you note. I will take good care of them and if I am a very gracious grandchild, I will wear them when she visits even if I don't like them. When it's her birthday, I will surely make her a gift in return. As the scholar and writer Lewis Hyde notes, 'It is the cardinal difference between gift and commodity exchange that a gift establishes a feeling-bond between two people.'" (p.26).

I have come back to this passage and reflected on the way I have come to understand young people's thinking and experiences in classrooms. If children are willing to freely share their ideas with us, then we enter into a particular kind of ongoing relationship with them, and one that I would hope is worthy of the gifts they give us. Children would expect us to attend to their ideas, to appreciate them, to help them listen and take in each other's ideas. I would want the "feeling-bonds" that would be created as a result to be one of respect and curiosity for their brilliance. I would hope that we would be judicious in our use of instructional materials and our engagement with learning standards to continually reciprocate children's ideas and experiences, to follow their lead, to encourage them to keep asking questions, to keep seeking.

What I most want for our children is to find school a place where they are known, seen, and heard and where they come to know themselves and others through ongoing, reciprocated relationships.

Next Steps

Listen to Elham Kazemi and other math education experts as they discuss the importance of student thinking in math instruction in the webinar: Experience IM K-5 Math™: A Focus on Student Thinking. In it you will also learn how the curriculum supports teachers to understand and listen to student thinking.

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Gifts: A Reflection on Student Thinking - IM CERTIFIED® BLOG

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By Elham Kazemi, Co-Author of Intentional Talk and Professor at University of Washington "Gifts from the earth or from each other establish a particular relationship, an obligation of sorts to give, to receive, and to reciprocate." Robin Wall Kimmerer, Braiding Sweetgrass, p. 25 Robin Wall Kimmerer's book, Braiding Sweetgrass, has given me a lot to think about.

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Teacher Moves that Shift Power Dynamics in the Classroom

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Practical steps for elevating students’ status, centering students in the learning process, and positioning students in positive ways.

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Redefining What it Means to Be "Good at Math"

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How to broaden our definition of mathematical competence to more accurately reflect the discipline and increase equity.

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A world where all learners know, use, and enjoy mathematics.

Perhaps the most mysterious verb in the IM vision-a world where all learners know, use, and enjoy mathematics-is the first one: .

Knowing mathematics means being able to "do the math"

While learning mathematics as an undergraduate math major, I often found myself in conversations with people about what it was exactly that I was doing. Often they seemed to see it as doing the same thing they understood from school, but just much more difficult, like adding up nine twelve-digit numbers or calculating the integral of a very complicated function. This view of mathematics as complicated procedures is out of fashion with educators these days, partly because it reduces mathematics to something that computers can do, but I suspect it is still a fairly common understanding among the general public. Someone who can add two-digit numbers fluently in their head or using a pencil and paper algorithm knows math.

Although this point of view misses a lot, it captures something important. Some of the most beautiful products of mathematical thought are procedures that achieve powerful results seemingly without any need for conscious thought. The base-ten number system is one of the pinnacles of human ingenuity, enabling the reduction of complex calculations to simple iterative procedures that require only the knowledge of single digit sums and products. And although the utilitarian value of these procedures has been superseded by calculators, their value as ingenious tools remains. Humans love tools and enjoy using them. To this day I add up columns of numbers in my head, just because I can. Of course, calculators are fun tools to use as well and expand the repertoire of explorations that you can make with numbers.

Knowing math means seeing patterns

Another popular view of mathematics is that it is about seeing and understanding patterns. As an approximation, this isn't bad; it captures a lot of what students learn in mathematics in K-12. An algorithm can be thought of as a sort of pattern. Understanding the steps in the standard algorithm for addition goes well beyond being able to "do the math." Geometric patterns have a place in developing geometric understanding.

Still, mathematics as patterns doesn't quite capture everything. Is a geometric proof a pattern? How do patterns fit into my understanding of how to read a graph? If seeing patterns is understood more generally as seeing structure, then this view of mathematics becomes more accurate. To think of a proof, you often need to see some structure in a geometric figure; to read a graph you need to understand the structure of axes, scales, and coordinates . Two of my favorite practice standards -MP7, look for and make use of structure, and MP8, look for and express regularity in repeated reasoning-add a gloss to "seeing patterns" that is useful in understanding what it means to know mathematics.

Knowing mathematics means interpreting and expressing mathematics

For me, this gets to the heart of the matter. Interpreting means, for example, reading a mathematical expression and being able to describe the calculation it represents. Expressing means being able to write an equation representing a word problem. Implicit in the words interpreting and expressing is the assumption that whatever you are interpreting or expressing has meaning. After all, we all understand in language arts that reading and writing does not mean reading and writing gobbledygook. And yet, for many students, that's exactly how they see patterns of symbols they are required to read and write in the course of solving math problems. Faced with solving the equation (x - 2)(x - 9) = 0, the "do the math" approach might lead you to expand the expression on the left to put the equation in standard form and then factor it again or use the quadratic formula (I have seen this!). Whereas interpreting the equation as telling you that the product of two numbers is zero leads to a much quicker way to find the solutions.

Implications for educational technology

The COVID-19 pandemic has increased interest in technology assisted instruction. Our understanding of what it means to know mathematics guides our desired outcomes from such instruction. There are plenty of technological solutions that can train students to do the math and graphical interfaces can often help students see patterns. But if the goal is to have students attach meaning to their work, then they need to be interacting with another sentient being, a teacher or a student. (Granted, there are some students who can learn math by interacting with their own brains, but such students are rare.) Until we have AIs capable of consciousness, educational technology will have to focus on facilitating discussions among students and between real-life teachers and students. I think this is an underexplored territory in educational technology; whereas there are quite advanced technologies for adaptive testing, handling mathematical input, and making sophisticated mathematical displays, technologies for handling the orchestration of productive mathematical discussions and the implementation of sophisticated instructional routines are underdeveloped.

There's always a danger with technology that the piece of technology itself is at the center of the table, and everyone is trying to figure out what to do with it. Putting meaningful learning at the center of the table, and then asking what technology can do to facilitate it, is more likely to result in solutions that will truly help teachers and students adapt to the new world we find ourselves in.

Next Steps

Think about the technology you use in the classroom. How does it help students know the math?

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What does it mean to know mathematics? - IM CERTIFIED® BLOG

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Perhaps the most mysterious verb in the IM vision-a world where all learners know, use, and enjoy mathematics-is the first one: . While learning mathematics as an undergraduate math major, I often found myself in conversations with people about what it was exactly that I was doing.

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What Does It Mean to Use Mathematics? - IM CERTIFIED® BLOG

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Our vision at Illustrative Mathematics is a world where all learners know, use, and enjoy mathematics. In my last post I picked up that first verb and talked about what it means to know mathematics. In this post I'd like to talk about what it means to use mathematics.

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By William McCallum
When I started this series of blog posts on what it means to , , and enjoy mathematics, I thought this one would be the easiest. Math is fun, right? How could you not enjoy mathematics? I certainly do-I love the complexity in underlying simplicity, the pleasure of seeing a pattern and then finding an explanation for it, the intense joy when you have been working on a problem for a long time and finally solve it. But so what? Our vision at IM is not a world where mathematicians know, use, and enjoy mathematics. We want to reach all learners. And many learners have had their capacity for enjoying mathematics damaged by their experiences of school mathematics. So I thought it would be interesting to take a piece of school mathematics that is as ubiquitous as it is awful-PEMDAS-and see what pleasure there is to be obtained from it.

In my first post in this series, on what it means to know mathematics, I identified three levels: doing the math, seeing patterns, and interpreting and expressing mathematics. From the point of view of knowledge these form a hierarchy, each more sophisticated than the last. But from the point of view of enjoyment, they are all on the same level. It's a bit like enjoying food or wine: some people enjoy it without thinking too much about it, others really get into it. Enjoyment is good at whatever level. So it is with mathematics. So let's start with doing the math.

A good place to look for people enjoying things (and not) is Twitter. Every now and then a viral math problem erupts on Twitter. Someone asks "What is 6/2(1+2)?" and people get into arguments about whether the answer is 1, as in

These people are enjoying mathematics! They learned how to do the math, in this case PEMDAS, in school, and they are enjoying using that knowledge. Of course the disagreement results from the fact that PEMDAS is a flawed way of talking about order of operations (see my blog post about order of operations ). But these people had fun arguing about it.

M oving on to the next level of knowledge, seeing patterns, there is a pattern to be found in these viral math problems.

What is the pattern? The use of the division symbol and the indication of multiplication by concatenation are from two different stages of a student's journey, elementary school and middle school. It is this mixing up of notation that is the key to making these problems work. PEMDAS doesn't make sense in middle school, once you start writing algebraic expressions. If you replace the numbers inside the parentheses with letters, you get:

You can't "do the parentheses" first in this expression. There is nothing to do in there. Furthermore, in middle school you'd be more likely to use a backslash instead of a division symbol, as in:

This pushes you further in the direction of the interpretation of the original expression as 3(1+2) = 9. The people who make up these problems are having fun with the pattern, exploiting the tension between the two systems of notation.

Finally, at the level of interpreting and expressing mathematics, we have this tweet:

Following the suggestion of @standupmaths to use more brackets, we see that the two answers, 1 and 9, come from different ways of clarifying the expression:

That feeling of finally getting what is going on with this problem is one of the great pleasures of mathematics.

In the grand scheme of things, this viral math problem is not of great consequence. All the more does it illustrate that, even in its most inconsequential aspects, there is enjoyment to be had in mathematics, at all three levels of knowledge.

Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009-2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.

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What does it mean to enjoy mathematics? - IM CERTIFIED® BLOG

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By William McCallum When I started this series of blog posts on what it means to know, use, and enjoy mathematics, I thought this one would be the easiest. Math is fun, right? How could you not enjoy mathematics? I certainly do-I love the complexity in underlying simplicity, the pleasure of seeing a pattern and [...]

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What Students Should Know and Be Able to Do

Our ultimate purpose is to impact student learning and achievement. First, we define the attitudes and beliefs about mathematics and mathematics learning we want to cultivate in students, and what mathematics students should know and be able to do.

Attitudes and Beliefs We Want to Cultivate

Many people think that mathematical knowledge and skills exclusively belong to "math people." Yet research shows that students who believe that hard work is more important than innate talent learn more mathematics. ¹ We want students to believe anyone can do mathematics and that persevering at mathematics will result in understanding and success. In the words of the NRC report Adding It Up, we want students to develop a "productive disposition-[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy." ²

¹ Uttal, D.H. (1997). Beliefs about genetic influences on mathematics achievement: a cross-cultural comparison. Genetica, 99(2-3), 165-172. doi.org/10.1023/A:1018318822120

² National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. doi.org/10.17226/9822

Knowledge

Conceptual understanding: Students need to understand the why behind the how in mathematics. Concepts build on experience with concrete contexts. Students should access these concepts from a number of perspectives in order to see math as more than a set of disconnected procedures.

Procedural fluency: We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.

Application: Application means applying mathematical or statistical concepts and skills to a novel mathematical or real-world context.

These three aspects of mathematical proficiency are interconnected: procedural fluency is supported by understanding, and deep understanding often requires procedural fluency. In order to be successful in applying mathematics, students must both understand and be able to do the mathematics.

Mathematical Practices

In a mathematics class, students should not just learn about mathematics, they should do mathematics. This can be defined as engaging in the mathematical practices: making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning.

What Teaching and Learning Should Look Like

How teachers should teach depends on what we want students to learn. To understand what teachers need to know and be able to do, we need to understand how students develop the different (but intertwined) strands of mathematical proficiency, and what kind of instructional moves support that development.

Principles for Mathematics Teaching and Learning

Active learning is best: Students learn best and retain what they learn better by solving problems. Often, mathematics instruction is shaped by the belief that if teachers tell students how to solve problems and then students practice, students will learn how to do mathematics.

Decades of research tells us that the traditional model of instruction is flawed. Traditional instructional methods may get short-term results with procedural skills, but students tend to forget the procedural skills and do not develop problem solving skills, deep conceptual understanding, or a mental framework for how ideas fit together. They also don't develop strategies for tackling non-routine problems, including a propensity for engaging in productive struggle to make sense of problems and persevere in solving them.

In order to learn mathematics, students should spend time in math class doing mathematics.

"Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving." ³

Students should take an active role, both individually and in groups, to see what they can figure out before having things explained to them or being told what to do. Teachers play a critical role in mediating student learning, but that role looks different than simply showing, telling, and correcting. The teacher's role is

1. to ensure students understand the context and what is being asked

2. ask questions to advance students' thinking in productive ways

3. help students share their work and understand others' work through orchestrating productive discussions

4. synthesize the learning with students at the end of activities and lessons

Teachers should build on what students know: New mathematical ideas are built on what students already know about mathematics and the world, and as they learn new ideas, students need to make connections between them. ⁴ In order to do this, teachers need to understand what knowledge students bring to the classroom and monitor what they do and do not understand as they are learning. Teachers must themselves know how the mathematical ideas connect in order to mediate students' learning.

Good instruction starts with explicit learning goals: Learning goals must be clear not only to teachers, but also to students, and they must influence the activities in which students participate. Without a clear understanding of what students should be learning, activities in the classroom, implemented haphazardly, have little impact on advancing students' understanding. Strategic negotiation of whole-class discussion on the part of the teacher during an activity synthesis is crucial to making the intended learning goals explicit. Teachers need to have a clear idea of the destination for the day, week, month, and year, and select and sequence instructional activities (or use well-sequenced materials) that will get the class to their destinations. If you are going to a party, you need to know the address and also plan a route to get there; driving around aimlessly will not get you where you need to go.

Different learning goals require different instructional moves: The kind of instruction that is appropriate at any given time depends on the learning goals of a particular lesson. Lessons and activities can:

Introduce students to a new topic of study and invite them to the mathematics.

Study new concepts and procedures deeply.

Integrate and connect representations, concepts, and procedures.

Work towards mastery.

Apply mathematics.

Lessons should be designed based on what the intended learning outcomes are. This means that teachers should have a toolbox of instructional moves that they can use as appropriate.

Each and every student should have access to the mathematical work: With proper structures, accommodations, and supports, all students can learn mathematics. Teachers' instructional toolboxes should include knowledge of and skill in implementing supports for different learners.

³ Hiebert, J., et. al. (1996). Problem solving as a basis for reform in curriculum and instruction: the case of mathematics. Educational Researcher 25(4), 12-21. doi.org/10.3102/0013189X025004012

⁴ National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. doi.org/10.17226/9822

Critical Practices

Intentional planning: Because different learning goals require different instructional moves, teachers need to be able to plan their instruction appropriately. While a high-quality curriculum does reduce the burden for teachers to create or curate lessons and tasks, it does not reduce the need to spend deliberate time planning lessons and tasks. Instead, teachers' planning time can shift to high-leverage practices (practices that teachers without a high-quality curriculum often report wishing they had more time for): reading and understanding the high-quality curriculum materials; identifying connections to prior and upcoming work; diagnosing students' readiness to do the work; leveraging instructional routines to address different student needs and differentiate instruction; anticipating student responses that will be important to move the learning forward; planning questions and prompts that will help students attend to, make sense of, and learn from each other's work; planning supports and extensions to give as many students as possible access to the main mathematical goals; figuring out timing, pacing, and opportunities for practice; preparing necessary supplies; and the never-ending task of giving feedback on student work.

Establishing norms: Norms around doing math together and sharing understandings play an important role in the success of a problem-based curriculum. For example, students must feel safe taking risks, listen to each other, disagree respectfully, and honor equal air time when working together in groups. Establishing norms helps teachers cultivate a community of learners where making thinking visible is both expected and valued.

Building a shared understanding of a small set of instructional routines: Instructional routines allow the students and teacher to become familiar with the classroom choreography and what they are expected to do. This means that they can pay less attention to what they are supposed to do and more attention to the mathematics to be learned. Routines can provide a structure that helps strengthen students' skills in communicating their mathematical ideas.

Using high quality curriculum: A growing body of evidence suggests that using a high-quality, coherent curriculum can have a significant impact on student learning. ⁵ Creating a coherent, effective instructional sequence from the ground up takes significant time, effort, and expertise. Teaching is already a full-time job, and adding curriculum development on top of that means teachers are overloaded or shortchanging their students.

Ongoing formative assessment: Teachers should know what mathematics their students come into the classroom already understanding, and use that information to plan their lessons. As students work on problems, teachers should ask questions to better understand students' thinking, and use expected student responses and potential misconceptions to build on students' mathematical understanding during the lesson. Teachers should monitor what their students have learned at the end of the lesson and use this information to provide feedback and plan further instruction.

⁵ Steiner, D. (2017). Curriculum research: What we know and where we need to go. Standards Work. Retrieved from https://standardswork.org/wp-content/uploads/2017/03/sw-curriculum-research-report-fnl.pdf

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Illustrative Mathematics What is a "Problem-Based" Curriculum? - Teachers | Kendall Hunt

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Our ultimate purpose is to impact student learning and achievement. First, we define the attitudes and beliefs about mathematics and mathematics learning we want to cultivate in students, and what mathematics students should know and be able to do. Many people think that mathematical knowledge and skills exclusively belong to "math people."

# data[].attributes.attachment_link.readable_text
What Students Should Know and Be Able to Do

Our ultimate purpose is to impact student learning and achievement. First, we define the attitudes and beliefs about mathematics and mathematics learning we want to cultivate in students, and what mathematics students should know and be able to do.

Attitudes and Beliefs We Want to Cultivate

Many people think that mathematical knowledge and skills exclusively belong to "math people." Yet research shows that students who believe that hard work is more important than innate talent learn more mathematics. ¹ We want students to believe anyone can do mathematics and that persevering at mathematics will result in understanding and success. In the words of the NRC report Adding It Up, we want students to develop a "productive disposition-[the] habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy." ²

¹ Uttal, D.H. (1997). Beliefs about genetic influences on mathematics achievement: a cross-cultural comparison. Genetica, 99(2-3), 165-172. doi.org/10.1023/A:1018318822120

² National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. doi.org/10.17226/9822

Knowledge

Conceptual understanding: Students need to understand the why behind the how in mathematics. Concepts build on experience with concrete contexts. Students should access these concepts from a number of perspectives in order to see math as more than a set of disconnected procedures.

Procedural fluency: We view procedural fluency as solving problems expected by the standards with speed, accuracy, and flexibility.

Application: Application means applying mathematical or statistical concepts and skills to a novel mathematical or real-world context.

These three aspects of mathematical proficiency are interconnected: procedural fluency is supported by understanding, and deep understanding often requires procedural fluency. In order to be successful in applying mathematics, students must both understand and be able to do the mathematics.

Mathematical Practices

In a mathematics class, students should not just learn about mathematics, they should do mathematics. This can be defined as engaging in the mathematical practices: making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning.

What Teaching and Learning Should Look Like

How teachers should teach depends on what we want students to learn. To understand what teachers need to know and be able to do, we need to understand how students develop the different (but intertwined) strands of mathematical proficiency, and what kind of instructional moves support that development.

Principles for Mathematics Teaching and Learning

Active learning is best: Students learn best and retain what they learn better by solving problems. Often, mathematics instruction is shaped by the belief that if teachers tell students how to solve problems and then students practice, students will learn how to do mathematics.

Decades of research tells us that the traditional model of instruction is flawed. Traditional instructional methods may get short-term results with procedural skills, but students tend to forget the procedural skills and do not develop problem solving skills, deep conceptual understanding, or a mental framework for how ideas fit together. They also don't develop strategies for tackling non-routine problems, including a propensity for engaging in productive struggle to make sense of problems and persevere in solving them.

In order to learn mathematics, students should spend time in math class doing mathematics.

"Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving." ³

Students should take an active role, both individually and in groups, to see what they can figure out before having things explained to them or being told what to do. Teachers play a critical role in mediating student learning, but that role looks different than simply showing, telling, and correcting. The teacher's role is

1. to ensure students understand the context and what is being asked

2. ask questions to advance students' thinking in productive ways

3. help students share their work and understand others' work through orchestrating productive discussions

4. synthesize the learning with students at the end of activities and lessons

Teachers should build on what students know: New mathematical ideas are built on what students already know about mathematics and the world, and as they learn new ideas, students need to make connections between them. ⁴ In order to do this, teachers need to understand what knowledge students bring to the classroom and monitor what they do and do not understand as they are learning. Teachers must themselves know how the mathematical ideas connect in order to mediate students' learning.

Good instruction starts with explicit learning goals: Learning goals must be clear not only to teachers, but also to students, and they must influence the activities in which students participate. Without a clear understanding of what students should be learning, activities in the classroom, implemented haphazardly, have little impact on advancing students' understanding. Strategic negotiation of whole-class discussion on the part of the teacher during an activity synthesis is crucial to making the intended learning goals explicit. Teachers need to have a clear idea of the destination for the day, week, month, and year, and select and sequence instructional activities (or use well-sequenced materials) that will get the class to their destinations. If you are going to a party, you need to know the address and also plan a route to get there; driving around aimlessly will not get you where you need to go.

Different learning goals require different instructional moves: The kind of instruction that is appropriate at any given time depends on the learning goals of a particular lesson. Lessons and activities can:

Introduce students to a new topic of study and invite them to the mathematics.

Study new concepts and procedures deeply.

Integrate and connect representations, concepts, and procedures.

Work towards mastery.

Apply mathematics.

Lessons should be designed based on what the intended learning outcomes are. This means that teachers should have a toolbox of instructional moves that they can use as appropriate.

Each and every student should have access to the mathematical work: With proper structures, accommodations, and supports, all students can learn mathematics. Teachers' instructional toolboxes should include knowledge of and skill in implementing supports for different learners.

³ Hiebert, J., et. al. (1996). Problem solving as a basis for reform in curriculum and instruction: the case of mathematics. Educational Researcher 25(4), 12-21. doi.org/10.3102/0013189X025004012

⁴ National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. doi.org/10.17226/9822

Critical Practices

Intentional planning: Because different learning goals require different instructional moves, teachers need to be able to plan their instruction appropriately. While a high-quality curriculum does reduce the burden for teachers to create or curate lessons and tasks, it does not reduce the need to spend deliberate time planning lessons and tasks. Instead, teachers' planning time can shift to high-leverage practices (practices that teachers without a high-quality curriculum often report wishing they had more time for): reading and understanding the high-quality curriculum materials; identifying connections to prior and upcoming work; diagnosing students' readiness to do the work; leveraging instructional routines to address different student needs and differentiate instruction; anticipating student responses that will be important to move the learning forward; planning questions and prompts that will help students attend to, make sense of, and learn from each other's work; planning supports and extensions to give as many students as possible access to the main mathematical goals; figuring out timing, pacing, and opportunities for practice; preparing necessary supplies; and the never-ending task of giving feedback on student work.

Establishing norms: Norms around doing math together and sharing understandings play an important role in the success of a problem-based curriculum. For example, students must feel safe taking risks, listen to each other, disagree respectfully, and honor equal air time when working together in groups. Establishing norms helps teachers cultivate a community of learners where making thinking visible is both expected and valued.

Building a shared understanding of a small set of instructional routines: Instructional routines allow the students and teacher to become familiar with the classroom choreography and what they are expected to do. This means that they can pay less attention to what they are supposed to do and more attention to the mathematics to be learned. Routines can provide a structure that helps strengthen students' skills in communicating their mathematical ideas.

Using high quality curriculum: A growing body of evidence suggests that using a high-quality, coherent curriculum can have a significant impact on student learning. ⁵ Creating a coherent, effective instructional sequence from the ground up takes significant time, effort, and expertise. Teaching is already a full-time job, and adding curriculum development on top of that means teachers are overloaded or shortchanging their students.

Ongoing formative assessment: Teachers should know what mathematics their students come into the classroom already understanding, and use that information to plan their lessons. As students work on problems, teachers should ask questions to better understand students' thinking, and use expected student responses and potential misconceptions to build on students' mathematical understanding during the lesson. Teachers should monitor what their students have learned at the end of the lesson and use this information to provide feedback and plan further instruction.

⁵ Steiner, D. (2017). Curriculum research: What we know and where we need to go. Standards Work. Retrieved from https://standardswork.org/wp-content/uploads/2017/03/sw-curriculum-research-report-fnl.pdf

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Illustrative Mathematics What is a "Problem-Based" Curriculum? - Teachers | Kendall Hunt

# data[].attributes.attachment_link.description
Our ultimate purpose is to impact student learning and achievement. First, we define the attitudes and beliefs about mathematics and mathematics learning we want to cultivate in students, and what mathematics students should know and be able to do. Many people think that mathematical knowledge and skills exclusively belong to "math people."

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mathgeekmama.com-Math is a Foreign Language So Treat it Like One