math problem solving curriculum

IM Curriculum

What is a "problem-based" curriculum, 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

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

• to ensure students understand the context and what is being asked
• ask questions to advance students’ thinking in productive ways
• help students share their work and understand others’ work through orchestrating productive discussions
• 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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How To Choose the Best Math Curriculum

Tips, advice, and must-haves.

Choosing and implementing an effective math curriculum can be one of the most gratifying yet demanding tasks educators face. Math scores on standardized tests have always faced scrutiny, and the effects of the COVID-19 pandemic on math achievement make the development of a strong math program even more of a priority. Additionally, parents and community groups don’t always agree on what’s best for our children, and curriculum decisions can be controversial . 

Implementation of an effective math curriculum generally starts with the creation of a curriculum document. Then resources are found that best meet the math curriculum’s objectives. Some educators choose to adopt a publisher’s program in its entirety while others seek out a variety of materials. 

Math Curriculum Must-Haves

Whichever path you choose, a solid math curriculum should include the following:    

Alignment With State Standards

For public schools, a curriculum not in compliance with state standards is a nonstarter. State standards provide a logical prioritization of math topics and a clear progression of skills and knowledge development through the grade levels. They also provide an emphasis on mathematical process skills in addition to math content. Additionally, noncompliance can result in funding issues. (Public schools need to comply, private schools generally don’t, and rules vary by state for homeschoolers). Currently, 41 states and the District of Columbia have adopted the Common Core mathematics standards. 

An Emphasis on Problem-Solving Skills

Traditionally, problem-solving skills were an afterthought in math instruction, something to add at the end of lessons consisting of rote memorization and mechanical practice of a single procedure in isolation. Too often, problems were of the “a train leaves St. Louis at 60 mph …” variety, leading kids to hate math and wonder when they would ever use what they learned. Children should be presented with problems to solve that connect to their real-world lives, providing a meaningful context for learning abstract concepts. In addition, instruction should be explicit about the problem-solving strategies students have at their disposal and develop their ability to select a strategy in a thoughtful rather than automatic way.

Development of Mathematical Discourse

For any class in any subject, observe if the teacher is doing the talking/lecturing or if the students themselves are engaged in vibrant talk. This is particularly important in mathematics instruction, where student reasoning skills are developed when they are required to cite evidence to support their claims. Furthermore, reasoning skills are promoted when students are expected to respectfully critique their classmates’ arguments. Getting kids to respectfully engage in lively talk with their peers doesn’t happen automatically but is the result of high expectations and careful coaching.     

Meeting the Needs of Diverse Learners

Does the program provide for remediation and/or enrichment for students who need it? Can tasks be modified to meet the needs of the different skill levels in the class? Is the math curriculum equitable ?

Time for Metacognition and Reflection

When students have an awareness of their own thought processes ( metacognition ) and time to reflect on what they’ve learned, they can more easily connect new skills and knowledge to what they’ve already learned. They can select a problem-solving strategy they’re comfortable with that will help them be successful. Finding time to teach metacognition skills and give students time to reflect isn’t easy. The reflection part of any lesson is usually the first to go when time gets tight. 

Acceptance of Varied Representation of Math Ideas

Students best learn math when they can create models to represent concepts . At the elementary level, these models can include drawings, manipulatives, diagrams, and more that help them make abstract concepts concrete. Students can then move to more abstract ways of representing their math thinking by using numbers, symbols, and the like.

Use of Assessment To Guide Instruction

Traditionally, math programs have included summative assessments, given at the end of a unit of study to measure student achievement. It is also important to make frequent use of formative assessments, given during a unit to provide teachers with data on student understanding to help them make mid-course corrections during a lesson and to modify future lessons.   

Next Steps 

Once you’ve chosen your math curriculum, be sure to invest in the following areas to ensure you get the best results.

Supplemental Materials

To successfully implement the new math program, it’s necessary to invest in books , manipulatives, and other materials.

Instructional Time

Another investment that shouldn’t be overlooked is the investment of time. Teachers should be given sufficient time in their schedules for math instruction.

Professional Development

Finally, investment is needed to train the people who make the curriculum happen: the teachers! Teaching isn’t a matter of simply distributing materials and overseeing well-designed, engaging lessons. The skill of teachers helps students connect the activity they’ve engaged in during the lesson with a deeper understanding of the relevant math concept. In addition to training the faculty on the particulars of the new program, some teachers may need training in math content. It’s also a good idea to provide training in how the program flows from kindergarten to the upper grades. An elementary classroom teacher will be more effective when they understand what and how their students have learned in the lower grades and how their teaching prepares their students for the math they’ll learn in the upper grades. 

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We're All Math Beasts — But We Weren't Born This Way

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Common Core State Standards Initiative

Mathematics Standards

For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is “a mile wide and an inch deep.”

These new standards build on the best of high-quality math standards from states across the country. They also draw on the most important international models for mathematical practice, as well as research and input from numerous sources, including state departments of education, scholars, assessment developers, professional organizations, educators, parents and students, and members of the public.

The math standards provide clarity and specificity rather than broad general statements. They endeavor to follow the design envisioned by William Schmidt and Richard Houang (2002), by not only stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value and the laws of arithmetic to structure those ideas.

In addition, the “sequence of topics and performances” that is outlined in a body of math standards must respect what is already known about how students learn. As Confrey (2007) points out, developing “sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” Therefore, the development of the standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time. The knowledge and skills students need to be prepared for mathematics in college, career, and life are woven throughout the mathematics standards. They do not include separate Anchor Standards like those used in the ELA/literacy standards.

The Common Core concentrates on a clear set of math skills and concepts. Students will learn concepts in a more organized way both during the school year and across grades. The standards encourage students to solve real-world problems.

Understanding Mathematics

These standards define what students should understand and be able to do in their study of mathematics. But asking a student to understand something also means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

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A Teacher-Led K–8 Math Program

Ready Common Core Mathematics

Ready Common Core Mathematics helps teachers create a rich classroom environment in which students at all levels become active, real-world problem solvers. Through teacher-led instruction, students develop mathematical reasoning, engage in discourse, and build strong mathematical habits. This math program’s instructional framework supports educators as they strengthen their teaching practices and facilitate meaningful discourse that encourages all learners.

Ready Common Core Mathematics :

• Encourages students to develop a deeper understanding of mathematics concepts through the embedded Standards for Mathematical Practice
• Builds on students’ prior knowledge with lessons that make connections within and across grade levels and directly address the major focus of the grade
• Ready Mathematics 6–8, ©2020 Edition provides additional features for supporting English Learners, such as: 
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Ready Common Core Mathematics can be used as your core curriculum or to enhance your math instruction. Designed to develop strong mathematical thinkers, our math programs focus on conceptual understanding using real-world problem solving and help students become active participants in their own learning.

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Robust formative and summative assessment tools closely match the rigor and expectations of the state assessments and include lessons, mid- and end-of-unit assessments, and performance tasks at the end of each unit (Grades 2–5 only).

Teacher Toolbox for Mathematics

A perfect complement to Ready Common Core Mathematics , Teacher Toolbox for Mathematics is a digital collection of K–8 instructional resources that supports educators in differentiating instruction for students performing on, below, or above grade level. Regardless of the grade they teach, subscribers get access to the full range of Ready Common Core Mathematics  resources for all grade levels, in addition to multimedia content, assessment practice, discourse supports, and more.

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Robust formative and summative assessment tools closely match the rigor and expectations of the state assessments and include lessons, mid- and end-of-unit assessments, and performance tasks at the end of each unit (Grades 2–8 only).

A perfect complement to Ready Common Core Mathematics , Teacher Toolbox for Mathematics is a digital collection of K–8 instructional resources that supports educators in differentiating instruction for students performing on, below, or above grade level. Regardless of the grade they teach, subscribers get access to the full range of Ready Common Core Mathematics K–8 resources for all grade levels, in addition to multimedia content, assessment practice, discourse supports, and more.

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Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.