Common Core State Standards Initiative

  • Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up : adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Standards in this domain:

Ccss.math.practice.mp1 make sense of problems and persevere in solving them..

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize —to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize , to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

CCSS.Math.Practice.MP4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

CCSS.Math.Practice.MP6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

CCSS.Math.Practice.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x 2 + 9 x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3( x - y ) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y .

CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y - 2)/( x - 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x - 1)( x + 1), ( x - 1)( x 2 + x + 1), and ( x - 1)( x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

  • How to read the grade level standards
  • Introduction
  • Counting & Cardinality
  • Operations & Algebraic Thinking
  • Number & Operations in Base Ten
  • Measurement & Data
  • Number & Operations—Fractions¹
  • Number & Operations in Base Ten¹
  • Number & Operations—Fractions
  • Ratios & Proportional Relationships
  • The Number System
  • Expressions & Equations
  • Statistics & Probability
  • The Real Number System
  • Quantities*
  • The Complex Number System
  • Vector & Matrix Quantities
  • Seeing Structure in Expressions
  • Arithmetic with Polynomials & Rational Expressions
  • Creating Equations*
  • Reasoning with Equations & Inequalities
  • Interpreting Functions
  • Building Functions
  • Linear, Quadratic, & Exponential Models*
  • Trigonometric Functions
  • High School: Modeling
  • Similarity, Right Triangles, & Trigonometry
  • Expressing Geometric Properties with Equations
  • Geometric Measurement & Dimension
  • Modeling with Geometry
  • Interpreting Categorical & Quantitative Data
  • Making Inferences & Justifying Conclusions
  • Conditional Probability & the Rules of Probability
  • Using Probability to Make Decisions
  • Courses & Transitions
  • Mathematics Glossary
  • Mathematics Appendix A

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Practice standards.

Click on the number to find illustrations for each of the eight Standards for Mathematical Practice.

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council's report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy).

Check out our elaborations of the practice standards: Grades K-5, Grades 6-8

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Testing the NCTM 2020 Standards Using Rigorous Mathematics and Multiple Solutions to a Single Geometric Problem

  • Published: 23 June 2022
  • Volume 27 , pages 1061–1077, ( 2022 )

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  • Jay Jahangiri 1 ,
  • Victor Oxman 2 &
  • Moshe Stupel 3  

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In this section of Resonance , we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. “Classroom” is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science.

The Principles and Standards for School Mathematics introduced by the National Council of Teachers of Mathematics (NCTM) have been in operation for over two decades. Since then, the mathematical community has learned that standards alone will not realize the goal of higher levels of mathematical understanding by all students, and more is needed. An obvious focus was given to mathematics teachers at the forefront of delivering these standards. Hence, Standards for the Preparation of Secondary Mathematics Teachers was developed and approved by NCTM (2020) as the official position of NCTM on teaching and learning mathematics. To test these standards, we designed a ‘selected topic in geometry’ course to engage pre-service mathematics teachers in professional development, focused on multiple approaches to a specific geometry problem using rigorous mathematics. Our experiment is a testimony that the implementation of rigorous mathematics, coupled with many solutions to a single problem, allows teachers to actively engage their students constructively and productively without potential harmful effects on students’ performance, reducing the stress levels for both students and teachers.

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L M Gojak, Preface to NCTM: Principles to Actions , 2014. Retrieved from: https://www.nctm.org/About/President,-Board-and-Committees/Past-Presidents/Linda-M¯Gojak,-President-2012%E2%80%932014/ .

National Council of Teachers of Mathematics (NCTM 2020), Standards for Mathematics Teacher Preparation Retrieved from: https://www.nctm.org/Standards-and-Positions/CAEP-Standards/ .

G Polya, How to Solve It: A New Aspect of Mathematical Method , Princeton University Press, Princeton, NJ, 1973.

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J J Lo and R McCrory, Proof and Proving in a Mathematics Course for Prospective Elementary Teachers, ICME Study 19 , Vol.2, pp.41–46, 2009.

R Liekin, Multiple Proof Tasks: Teacher Practice and Teacher Education, ICME Study 19 , Vol.2, pp.31–36, 2009.

P Ording, 99 Variations on a Proof , Princeton University Press, 2019. ISBN 978-0-691-15883-9.

T Brown, A Contemporary Theory of Mathematics Education Research , Springer, ISBN: 978-3-030-55099-8, 5th October 2020.

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Jahangiri, J., Oxman, V. & Stupel, M. Testing the NCTM 2020 Standards Using Rigorous Mathematics and Multiple Solutions to a Single Geometric Problem. Reson 27 , 1061–1077 (2022). https://doi.org/10.1007/s12045-022-1397-z

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Problem solving is generally recognized as one of the most important components of mathematics. In Principles and Standards for School Mathematics , the National Council of Teachers of Mathematics emphasized that instructional programs should enable all students in all grades to “build new mathematical knowledge through problem solving, solve problems that arise in mathematics and in other contexts, apply and adapt a variety of appropriate strategies to solve problems, and monitor and reflect on the process of mathematical problem solving” (NCTM 2000, p. 52). But how do students become competent and confident mathematical problem solvers?

Kenneth Williams is interested in problem solving and writing in the mathematics classroom.

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NCTM Process Standards vs CCSS Mathematical Practices

The  NCTM  process standards, Adding it Up  mathematical proficiency strands, and Common Core State Standards  for mathematical practices are all saying the same thing but why do I get the feeling that the Mathematical Practices Standards is out to get the math teachers.

The NCTM’s process standards of problem solving, reasoning and proof, communication, representation, and connections describe for me the nature of mathematics . They are not easy to understand especially when you think that school mathematics is about stuffing students with knowledge of content of mathematics. But, over time you find yourselves slowly shifting towards structuring your teaching in a way that students will understand and appreciate the nature of mathematics.

The five strands of proficiency were also a great help to me as a teacher/ teacher-trainer because it gave me the vocabulary to describe what is important to focus on in teaching mathematics.

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4 thoughts on “ NCTM Process Standards vs CCSS Mathematical Practices ”

Does your “exposure” mean you personally watched Japanese classrooms or were you told/did you read how classes are taught?

Yes, including development of materials with Japanese math educators. We had a project with them. AlSo, the APEC (Asia Pacific…) project about Lesson Study for math which concluded last year devoted each year of the project for each of the process standards. I think the last one was about representation and communication. If you are familiar with Teaching Gap, the TIMSS video study by Stigler an Hiebert, you’ll get a picture of the Japanese math class. Of course, it doesn’t mean it’s happening they all teach that way.

We teachers are under tremendous pressure to be accountable for what students attain, which means insuring students do well on tests. It’s all about ‘data’ and Grade Level Content Expectations now. I fear both teachers and students will blow a fuse.

Really a nice addition in Mathematics Education

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How Teachers Are Implementing The NCTM Standards

Standards-oriented classrooms, obstacles to change, sites that support change, policy recommendations.

A 16-year veteran of high school mathematics teaching, Lamont Stewart describes a typical unit in his geometry class: I begin with a concrete example. When we were ready to start parallelism, I raised the question of the parking lot at the high school, whether it could be restriped to hold more cars. The kids measured lines, turning radii, angles. Then they proposed ideas. They were doing a heck of a lot of math. But I have to give a departmental exam, and sometimes it's a real challenge to get to the right point at the right time. Also, it's hard to give a grade to projects submitted by groups. Is it fair to give a high grade to someone who is a —al member of a productive group?
Another long-term teacher, Olivia Green, describes her geometry class: I start a unit by asking the students what they know and then posing a problem for them to investigate in the computer labs next to my classroom. They create certain figures, usually working in pairs, take measurements, and make and test conjectures. Then, we talk about their findings.

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics . Reston, Va.: NCTM.

Smith, M., and J. O'Day. (1991). “Systemic School Reform.” In The Politics of Curriculum and Testing , edited by S. Fuhrman and B. Malen. London and New York: The Falmer Press.

Wiske, M.S., and C.Y. Levinson. (1992). Coordinated Support for Improving Mathematics Education . (Technical Report TR92-2). Cambridge, Mass.: Educational Technology Center, Harvard Graduate School of Education.

Wiske, M.S., C.Y. Levinson, P. Schlichtman, and W. Stroup. (1992). Implementing the Standards of the NCTM in Geometry . (Technical Report TR92-1). Cambridge, Mass.: Educational Technology Center, Harvard Graduate School of Education.

• 1 With the exception of Kimberly Girard, I have used pseudonyms for the teachers discussed in this article.
• 2 With the exception of Kimberly Girard, I have used pseudonyms for the teachers discussed in this article.

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The Biggest Policy Challenges Schools Are Facing Right Now

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There are many education policy challenges facing schools at the moment.

Today, two educators share which ones they think are the most important ones.

‘Legislative Attacks’

Keisha Rembert is a lifelong learner, equity advocate, and award-winning educator. She is the author of The Antiracist English Language Arts Classroom , a doctoral student and an assistant professor/DEI coordinator for teacher preparation at National Louis University. Prior to entering teacher education, Keisha spent more than 15 years teaching middle school English and U.S. history.

George Orwell’s words in his book 1984 resonate deeply today: “Who controls the past controls the future. Who controls the present controls the past.” These words hold immense relevance as we traverse the landmine of educational bills that have enacted book bans; restricted the exploration of race, sexual orientation, and gender identity topics; and prohibited the teaching of historical truths or any discourse that may result in “ discomfort, guilt, or anguish .”

In the past year, education-focused legislative attacks have become palpable and personal. We have seen an influx of anti-LGBTQIA+ bills , totaling a whopping 283, nationwide. In Florida, the value of AP African American Studies has been questioned, undermined, and dismissed as “ lacking educational value. ”

And critical race theory has become persona non grata, a scapegoat to thwart discussions and actions toward racial justice in our polarized American political landscape. These examples highlight the trend of states’ attempts to not only control curricula, learning, and discourse but also to stifle justice and constrict bodies and intellectual progress, negatively impacting the whole of society.

According to a 2022 Rand Corp survey, one-fourth of the teachers reported being influenced by legislative actions, pending and imposed, to change their lessons. It is scary to think that state legislatures, without any educational expertise, wield the power to manipulate knowledge and rewrite history. In the words of Paulo Freire, “Leaders who do not act dialogically, but insist on imposing their decisions, do not organize the people—they manipulate them. They do not liberate, nor are they liberated: they oppress.” And thus, the barrage of these oppressive educational policies are not only unconscionable but also fundamentally untenable for student and societal success.

We find ourselves at a critical juncture, where the exclusion of diverse perspectives and the suppression of uncomfortable truths have the potential to distort our collective consciousness. It is in recognizing and embracing the history of the most marginalized among us that we truly learn about ourselves, our growth as a society, and the ideals to which we aspire.

These dehumanizing legislative impositions hinder our students’ understanding of our shared history and also represent a dangerous path that encroaches on our personal and academic freedoms. They undermine our capacity to nurture students’ critical-thinking skills and hamper our ability to cultivate a citizenry that values democratic ideals and engages thoughtfully in meaningful change.

As educators, we must continue to fight and offer our support to those living under oppressive state regimes. In our classrooms and beyond, we should:

  • Advocate academic freedom: We cannot be passive bystanders while the rights of our students, selves, and colleagues are at stake. We must actively engage in discussions and initiatives that protect and promote freedom of all kinds within our schools, communities, and nation. We must reject the notion that any student should be denied the invaluable opportunity to be exposed to truth, diverse and inclusive perspectives, ideas, and experiences. Our championing of freedom creates an environment that fosters critical thinking, humility, and a deeper understanding of our world.
  • Foster critical thinking and humility: The Rev. Martin Luther King Jr. said, “The only way to deal with unjust laws is to render them powerless by ignoring them.” It is time to lean into what we know is right and teach our students to do the same. To navigate this time of distortions and mistruths, our students need to be analytical thinkers who are discerning, open-minded, and equipped to challenge rhetoric and resist the manipulative forces that are restricting knowledge and controlling narratives.
  • Uphold the ideals of democracy and global humanity: In the face of state-led oligarchies, it is our duty to instill in our students civic literacy, agency, collective responsibility, and the need to dismantle oppressive systems. Our students must be justice seekers who build bridges as compassionate citizens.

If we are not vigilant, we risk facing a fate reminiscent of the residents of Oceania depicted in 1984 , where “every record has been destroyed or falsified, every book rewritten, every picture repainted, every statue and street building renamed, every date altered. And the process is continuing day by day and minute by minute. History has stopped. Nothing exists except an endless present in which the Party is always right.”

Censorship is antithetical to freedom; it begets spirit-murdering curricula violence, posing a direct threat to the mental and emotional well-being of students whose histories, identities, and personhood are silenced and deemed inconsequential and without value. By perpetuating harm, these laws also establish a dangerous precedent for future educational policies. The brevity of this moment demands action. If education is the ultimate pursuit of liberation, then the freedom it promises hangs in the balance.

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STEM Access

Kit Golan ( @MrKitMath ) is the secondary mathematics consultant for the Center for Mathematics Achievement at Lesley University in Cambridge, Mass.:

Despite the demand for mathematical thinkers, our country continues to push data-illiterate and math-phobic graduates into the workforce. As such, a vital issue facing public schools today is inequitable access to high-level math courses, which acts as a gatekeeper for many who might enter science, technology, engineering, and math (STEM) careers.

Most course sequences prevent students from reaching rigorous math classes, especially students of color. Often, students who do have access to these courses come from privileged backgrounds whose families have invested time and money outside of the school day to “race to the top.” Regardless, many colleges use AP Calculus as a determining factor for entrance and class placement even though most students don’t reach this or other high-level math courses that better align with their career aspirations due to systemic barriers.

Few districts have created flexible course sequences that allow students to reach high-level math classes by senior year, meaning many students who do not accelerate in middle school may never be able to reach higher math classes without taking multiple math classes simultaneously or attending summer school.

Many middle school students do not know their career trajectory; having the option to delay acceleration until junior year and take a compressed Algebra 2/precalculus course would allow more students to access rigorous courses without being barred in middle school. Additionally, because current Algebra 2 courses focus heavily on symbolic manipulation that modern graphing technology renders obsolete, a compacted course could focus more on developing the conceptual understandings needed by eliminating this content. Yet, few schools have made this transition despite the obvious benefits.

Truly, this is a larger issue of tracking and acceleration for some students. Despite the consensus that sorting practices have a disproportionately negative impact on outcomes for marginalized students (NCTM, 2018), many parents still advocate for their children to be accelerated. Because teachers frequently struggle to differentiate for mixed-ability math classes, students who are ready for additional challenges may slip through the cracks as their teachers attempt to support struggling students’ access to grade-level content.

I’m not advocating separating these students into different streams, as the reality is that no matter how well you think you’ve grouped students by ability, there is no such thing as a truly homogeneous class; student variation is one of the only constants in education! Instead, teachers need additional professional development, time, and support (and reduced class sizes!) to better be able to differentiate their classes to ensure that all students have both access and challenge.

This is a systemic issue that requires structural changes beyond individual teachers. Sadly, most middle and high schools rarely have schedules allowing students to gain additional experience with math unless they are pulled from arts or other elective courses. Meanwhile, community colleges have recently begun to replace “developmental math” (their “low track”) courses with co-requisite models where students would enroll in both a credit-bearing course and an additional support class designed to help them gain access to the math content of the former. How might K-12 schools replicate that idea to provide additional support to students who need it?

Ultimately, the issue facing public schools is whether AP courses should be considered a privilege for the few who have access to outside resources or if it should be accessible to any who are interested in pursuing that pathway. Under the current paradigm, only students who take additional math courses outside of their standard school day or who are able to double up on math courses early in high school are able to reach AP Calculus by senior year. It’s outrageous that students who take Algebra 1 “on time” in 9th grade are considered remedial math students when measured along the path to AP Calculus. It’s past time we updated high school math options to reflect the 21st-century needs rather than settle for the status quo of the past century.

NCTM (National Council of Teachers of Mathematics). (2018). Catalyzing change in high school mathematics: Initiating critical conversations . Reston, VA: Author.

avitalissue

Thanks to Keisha and Kit for contributing their thoughts.

They answered this question of the week:

What do you think is the most important education policy issue facing public schools today, why do you think it is so important, and what is your position on it?

Consider contributing a question to be answered in a future post. You can send one to me at [email protected] . When you send it in, let me know if I can use your real name if it’s selected or if you’d prefer remaining anonymous and have a pseudonym in mind.

You can also contact me on Twitter at @Larryferlazzo .

Just a reminder; you can subscribe and receive updates from this blog via email . And if you missed any of the highlights from the first 12 years of this blog, you can see a categorized list here .

The opinions expressed in Classroom Q&A With Larry Ferlazzo are strictly those of the author(s) and do not reflect the opinions or endorsement of Editorial Projects in Education, or any of its publications.

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  5. NCTM Process Standards: Problem Solving by Sara Falkin

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  1. Principles and Standards

    Principles and Standards for School Mathematics outlines the essential components of a high-quality school mathematics program. It emphasizes the need for well-prepared and well-supported teachers and administrators, and it acknowledges the importance of a carefully organized system for assessing students' learning and a program's ...

  2. PDF NCTM Standards (2020)

    NCTM Standards (2020) Secondary . Standard 2: Knowing and Using Mathematical Processes Candidates demonstrate, within or across mathematical domains, their knowledge of and ability to apply the mathematical processes of problem solving; reason and communicate mathematically; and engage in mathematical modeling.

  3. Algebra

    Use mathematical models to represent and understand quantitative relationships. Pre-K-2 Expectations: In pre-K through grade 2 each and every student should-. model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols. Grades 3-5 Expectations: In grades 3-5 each and every student ...

  4. Standards for Mathematical Practice

    The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving ...

  5. Practice Standards

    The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving ...

  6. Implementing The Standards: The Vision of Problem Solving in the

    Problem solving has been espoused as a goal in mathematics education since the late 1970s, with focused attention ansmg from NCTM's An Agenda for Action (1980). But problem solving should be more than a slogan offered for its appeal and widespread acceptance. It should be a cornerstone of mathematics curriculum and instruction, fostering the development of mathematical knowledge and a chance ...

  7. Developing a Problem-Solving Lesson

    Problem solving is one of the most important mathematical abilities that teachers can foster in students, as evidenced by its prominent role in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and An Agenda for Action (1980). Because of the increased availability of calculators and computers, knowledge of which computations are necessary is more important than ...

  8. Principles and Standards: Learning through Problems: A Powerful ...

    Problem solving has been a major focus of mathematics education over the past two decades. Much has been learned about the problem-solving capabilities of elementary school children and their use of strategies in solving problems. However, problem solving has typically existed apart from core curriculum objectives. In many situations, it is a focus only after specific concepts and skills are ...

  9. NCTM Releases New Position Statement on Linking Mathematics Education

    About NCTM Position Statements NCTM position statements define a particular problem, issue, or need and describe its relevance to mathematics education. These statements address important and timely policy issues relevant to mathematics education. ... They rest on the foundation provided by Principles and Standards for School Mathematics ...

  10. E-Examples

    This two-part e-example encourages students to use visualization, spatial reasoning, and geometric modeling to build their own patterns or recreate one from given square arrays. Algebra. 4.1.1 Making Patterns (Java) Includes an interactive figure for creating, comparing, and viewing multiple repetitions of pattern units. Algebra.

  11. Browse in Problem Solving

    Chris Harrow, Justin Johns, and Hassan Lakiss. Problems to Ponder provides 28 varying, classroom-ready mathematics problems that collectively span PK-12, arranged in the order of the grade level. Answers to the problems are available online. Individuals are encouraged to submit a problem or a collection of problems directly to [email protected].

  12. Testing the NCTM 2020 Standards Using Rigorous Mathematics ...

    The Principles and Standards for School Mathematics introduced by the National Council of Teachers of Mathematics (NCTM) have been in operation for over two decades. ... On understanding, Learning and Teaching Problem Solving, Wiley ... J., Oxman, V. & Stupel, M. Testing the NCTM 2020 Standards Using Rigorous Mathematics and Multiple Solutions ...

  13. NCTM Curriculum Standards for Grades 9-12

    Standard 1: Mathematics as Problem Solving. In grades 9-12, the mathematics curriculum should include the refinement and extension of methods of mathematical problem solving so that all students can--. use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content;

  14. Writing About the Problem Solving Process to Improve Problem ...

    Problem solving is generally recognized as one of the most important components of mathematics. In Principles and Standards for School Mathematics, the National Council of Teachers of Mathematics emphasized that instructional programs should enable all students in all grades to "build new mathematical knowledge through problem solving, solve problems that arise in mathematics and in other ...

  15. NCTM Process Standards vs CCSS Mathematical Practices

    The NCTM process standards, Adding it Up mathematical proficiency strands, and Common Core State Standards for mathematical practices are all saying the same thing but why do I get the feeling that the Mathematical Practices Standards is out to get the math teachers. The NCTM's process standards of problem solving, reasoning and proof, communication, representation, and connections describe ...

  16. PDF NCTM: Principles & Standards for School Mathematics ( PSSM ) Est: 2000

    NCTM: Standards for Mathematics: Pre-K to 12 Established: 2000 Page 2 Content (Process) Standards: Instructional programs for Pre-K to 12 should enable students to: Problem Solving Standard: • Build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts;

  17. How Teachers Are Implementing The NCTM Standards

    Lamont Stewart and Olivia Green 1 were among 50 innovative geometry teachers interviewed by researchers at the Harvard Graduate School of Education (Wiske et al. 1992). 2 All were recommended by school administrators or mathematics education specialists for their success in using practices consistent with the NCTM Standards. The sample was drawn from urban, rural, and suburban schools in all ...

  18. PDF NCTM Principles and Standards for Mathematically Talented Students

    or standard forms of reasoning, to approach mathematical problems; and • ability to transfer mathematical reasoning to new and untaught situations. It should be noted that computa-tional skill and accuracy are not men-tioned on talent checklists. Yet, the majority of CBAs evaluate math cal-culation skills rather than math reason - ing ability.

  19. PDF Implementing the NCTM Standards: Hopes and Hurdles

    1This paper was prepared for a conference on Telecommunications as a Tool for Educational Reform: Implementing the NCTM Standards, December 2 -3, 1991, Aspen Institute, Wye Conference Center, Queenstown, Maryland. 2Deborah Loewenberg Ball, associate professor of teacher education at Michigan State University, is a senior researcher with the ...

  20. PDF National Council of Teachers of Mathematics (NCTM)

    SPA Standards: Standard 1. Knowledge of Problem Solving. Candidates know, understand and apply the . process of mathematical problem solving. 1.1 Apply and adapt a variety of appropriate strategies to solve problems. 1.2 Solve problems that arise in mathematics and those involving mathematics in other contexts . 1.3 Build new mathematical ...

  21. Solar Eclipse Math Word Problems, Solar Eclipse 2024 Math ...

    This engaging collection of math word problems is designed to challenge students' problem-solving abilities while immersing them in real-life scenarios related to solar eclipses. Aligned with grade-level math standards, this product features 10 thoughtfully crafted word problems covering a range of mathematical operations, including addition ...

  22. The Biggest Policy Challenges Schools Are Facing Right Now

    Many middle school students do not know their career trajectory; having the option to delay acceleration until junior year and take a compressed Algebra 2/precalculus course would allow more ...