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Mathematical Practices - Explained with Videos

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Mathematical Practices - Explained with Videos

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Providing instructional and assessment tasks, lesson plans, and other resources for teachers, assessment writers, and curriculum developers since 2011.

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CCSS - Questions for Each Mathematical Practice

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CCSS - Questions for Each Mathematical Practice

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CCSS Questions for Each Mathematical Practice

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Integrating Social and Emotional Learning and the Common Core State Standards for Mathematics

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Integrating Social and Emotional Learning and the Common Core State Standards for Mathematics

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Mathematical Practices - Excellent resource for teaching the mathematical practices

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Mathematical Practices - Excellent resource for teaching the mathematical practices

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Anchor Charts, Graphic Representations, Posters, and Standards

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The standards for mathematical practices are the same for K-12 grade. Many times teachers need to know “How do I teach practice #1 at my grade level. The link above shows examples of the math…

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Awesome Website - Variety of Materials for Teaching Students to Think Mathematically

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Awesome Website - Variety of Materials for Teaching Students to Think Mathematically

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Box It or Bag It is a math program for K-2 teachers who want to surround children with a language-enriched, activity-centered learning environment. Students enjoy hands-on experiences with a variety of materials and are encouraged to learn from each other as well as the teacher.

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Model Lesson:  Elementary Connecting Representations Routine

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Model Lesson:  Elementary Connecting Representations Routine

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In this video, Chelsea & Giuliana, from PS 10 in Manhattan, are enacting Contemplate then Calculate with their elementary school students. The task is intended to support students in understanding order of operations through connecting symbols to words. The task in this video was designed by Chelsea and Giuliana.

You can watch a longer version of this video here.

Resources for enacting Connecting Representations are available here or you can watch another video.

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Watch Chelsea & Giuliana enact Connecting Representations - New Visions Math Curriculum

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In this video, Chelsea & Giuliana, from PS 10 in Manhattan, are enacting Contemplate then Calculate with their elementary school students. The task is intended to support students in understanding order of operations through connecting symbols to words. The task in this video was designed by Chelsea and Giuliana.

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Make sense of problems and persevere in solving them | Think Math!

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One way to help students make sense of all of the mathematics they learn is to put experience before formality throughout, letting students explore problems and derive methods from the exploration. For example, students learn the logic of multiplication and division-the distributive property that makes possible the algorithms we use-before the algorithms. The algorithms for each operation become, in effect, capstones rather than foundations.

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Each Teach - Applying the Routines for Reasoning in Science and Math

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Each Teach - Applying the Routines for Reasoning in Science and Math

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New Mexico Public Education Department - Math Standards and Resources

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New Mexico Public Education Department - Math Standards and Resources

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Common Core Math Resources

New Mexico Math Framework 1.0

Below are a variety of resources to support educators and administrators in the implementation of the Common Core State Standards for Mathematics and the Standards for Mathematical Practices

Foundations of Common Core Math

Appendix

Math Appendix A - Designing high schools courses based on the Common Core State Standards for Mathematics

Learning Progressions

Understanding the Standards

Tools

Open Educational Resources (OER)

What Can We Help You Find?

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New Mexico Math Framework 1.0 New Mexico Math Framework Below are a variety of resources to support educators and administrators in the implementation of the Common Core State Standards for Mathematics and the Standards for Mathematical Practices Foundations of Common Core Math Common Core State Standards for Mathematics Key Shifts in Mathematics Appendix

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Inside Mathematics is a professional resource for K-12 teachers, coaches, and administrators.

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Building Classroom Climates for Mathematical Thinking 

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Building Classroom Climates for Mathematical Thinking 

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Videos, iBooks, and PDFs regrading a variety of topics around mathematical thinking in the classroom.

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In this series of video slideshows and downloadable practitioner guides, Inside Mathematics invites you to explore the teaching practice of two engaging practitioners, Mia Buljan (2nd grade) and Patty Ferrant (8th grade). Neither teacher considers herself a perfect practitioner, but an educator who continues to learn from and with her students and colleagues each year. Through the documentation of their classrooms multiple times during the school year, they open up new conversations for us- around the daily work to help children begin to see themselves as mathematical thinkers who can draw on their own strategies and those of others to understand and find solution pathways in various problem settings. No matter what day it is when you read this, for you, it can be Day One for you. Today, you're deciding to explore some other teachers' practices so you can rethink your own. Welcome! Each of the guides can be accessed either as a PDF or as an iBook with embedded videos.

Mia (2nd grade): "I love my job!"

Patty (8th grade): "You have to believe that every single student can learn."

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Building Classroom Climates for Mathematical Learning | Inside Mathematics

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In this series of video slideshows and downloadable practitioner guides, Inside Mathematics invites you to explore the teaching practice of two engaging practitioners, Mia Buljan (2nd grade) and Patty Ferrant (8th grade). Neither teacher considers herself a perfect practitioner, but an educator who continues to learn from and with her students and colleagues each year.

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Colorado Department of Education - Math Standards and Resourcees

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Colorado Department of Education - Math Standards and Resourcees

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Academic Standards

The mathematics standards compel us to make mathematics relevant to students by moving beyond mere answer getting to doing the work of mathematicians. The standards emphasize the development of students' abilities to use mathematics to represent their lived experiences and to simplify and explain complex phenomena.

Curriculum Support

Article IX, Section 16 of Colorado's constitution states, "Neither the general assembly nor the state board of education shall have power to prescribe textbooks to be used in the public schools." However, CDE can still serve as an advisor to help districts and schools adopt, adapt, develop, or use the materials of their choice.

Instructional Support

Instructional support is grounded in CDE's shared approach to Best, First Instruction as well as practices and strategies supported by professional organizations and scholars at the leading edge of mathematics education.

Community

There are many opportunities to connect and grow as a professional educator with the larger mathematics education community, including mailing lists, professional organizations, and events.

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The mathematics standards compel us to make mathematics relevant to students by moving beyond mere answer getting to doing the work of mathematicians. The standards emphasize the development of students' abilities to use mathematics to represent their lived experiences and to simplify and explain complex phenomena.

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Teacher Moves that Engage Students in Mathematical Practices

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Teacher Moves that Engage Students in Mathematical Practices

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Practices i-ready Webinar - this might be a good starting point in our thinking around creating the mathematics practicum manual

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Model Lesson:  Middle School Connecting Representations Routine

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Model Lesson:  Middle School Connecting Representations Routine

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In this video, Kit enacts Connecting Representations with his middle school students from the NYC Lab School. The task selected is intended to support students in naming the part to part or part to whole relationship in a ratio. Kit does an excellent job in this video of letting students do the heavy mathematical lifting while simultaneously pressing them to reconsider an idea they all held. The task in the video was designed by Kit.

You can watch a longer version of this video here.

Resources for enacting Connecting Representations are available here or you can watch another video.

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Watch Kit enact Connecting Representations - New Visions Math Curriculum

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In this video, Kit enacts Connecting Representations with his middle school students from the NYC Lab School. The task selected is intended to support students in naming the part to part or part to whole relationship in a ratio.

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Check out as a resource - Math in Practice Sampler

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Check out as a resource - Math in Practice Sampler

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Model Lesson:  Secondary Connecting Representations Routine

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Model Lesson:  Secondary Connecting Representations Routine

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Watch a pair of high school mathematics teachers, Harris and Maria, enact Connecting Representations with their 9th grade students. You can watch a longer version of this video here.

Resources for this task, Visualizing Distribution, are available on the Fostering Math Practices website (free registration required).

The instructional routine Connecting Representations was developed by Amy Lucenta and Grace Kelemanik of Fostering Math Practices; with their support we have developed resources so that teachers can use this routine in their classrooms.

Connecting Representations has these five main parts:

1. Launch

2. Connect Representations

3. Share and Study Connections

4. Create a Representation

5. Meta-Reflection

Additional resources for getting started with enacting Connecting Representations are available below.

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Connecting Representations - New Visions Math Curriculum

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Watch a pair of high school mathematics teachers, Harris and Maria, enact Connecting Representations with their 9th grade students. You can watch a longer version of this video here. Resources for this task, Visualizing Distribution, are available on the Fostering Math Practices website (free registration required).

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The Myth Fueling Math Anxiety - Education Week

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Math and the Mind's Eye (MME) units are a great way to introduce and extend visual models. This versatile collection can be taught in sequence or used individually as needed to supplement any curriculum. Hands-on explorations are easily adapted to fit changing classroom situations, diverse student backgrounds, and various grade levels. Each unit contains four to twelve activities that use models, manipulatives, sketches, and diagrams to develop and reinforce mathematical concepts and processes.

For more information about manipulatives and optional items see the Materials List. Also see the free Virtual Algebra Pieces app developed by Dr. Laurie Burton of Western Oregon University.

NEW: View overall correlations to the Common Core State Standards for Mathematical Content. Product pages for each unit include more detailed correlation information. Math and the Mind's Eye downloadable PDFs were developed from printed archival masters. As a result, some PDF functionalities, such as editing, copying, and text search, are not available.

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Math and the Mind's Eye (MME) units are a great way to introduce and extend visual models. This versatile collection can be taught in sequence or used individually as needed to supplement any curriculum. Hands-on explorations are easily adapted to fit changing classroom situations, diverse student backgrounds, and various grade levels.

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Mentor Teacher / Examples from the Classroom - Grades 9/10

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Mentor Interview

Cathy's lesson is posed as a "real-world" situation, which motivates a context for her students' investigations. There are many truly real-world applications of mathematics (e.g., the mathematics of global warming). These applications are especially important for students to experience in order to understand the relevance and importance of mathematics in their lives. During these two days of geometry investigations, Cathy's students work individually and in small groups to develop convincing arguments (for themselves, for a partner, for a skeptic) about the properties of quadrilaterals.

Throughout their conversations, her students need to make use of tools strategically. Knowing how to use a tool is very different from knowing when to use it, and students need a lot more practice with
problems that require them to dig deep to find, from all that they know, the tools and ideas that will help them solve a new and unexpected problem.

See how Cathy's students demonstrate many mathematical practice standards in one lesson:

Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, she models how to gather information for the tinkering stage of the investigative process and emphasizes the importance of the process of private individual thinking prior to the group addressing a task. This clip is also indicative of MP6.
See the video in the context of an entire lesson
Continuing into the group section of the task, Humphreys' students work in teams of four to engage in extended discussion about the properties of kites made by a factory. Their discussion of the properties of quadrilateral shapes proves less complex for some shapes (square, rectangle, rhombus) than for others (non-isosceles trapezoid.) Humphreys circulates around the classroom as the students work, and in her commentary she notes students' use of imprecise or inaccurate language to defend their thinking. This clip is also indicative of MP3, MP6, and MP7.
See the video in the context of an entire lesson
In the closure of the first day of Humphreys' students' exploration, she orients them to the next phase, which will require students to justify and prove their findings about the diagonals of the kites. In this clip, she asks them to "convince yourself, convince a friend, convince a skeptic" with their justifications.
See the video in the context of an entire lesson
On the second day of Humphreys' exploration of the properties of quadrilaterals, students work through their understandings of congruent triangles, the triangle postulates, parallel lines, transversals, and other geometric properties to apply those to create proofs for the quadrilaterals. Humphreys illustrates how to use selected notation in a proof as students work through the logic and reasoning. In small groups, Cathy Humphreys' students discuss and debate proof arguments. At selected times, she pulls the class together to share findings, ideas, or sample justifications. After sharing ideas or arguments with the entire class, students then return to working in their small groups.
See the video in the context of an entire lesson
Cathy Humphreys works with her students as they get closer to being able to present their group defenses of the various properties of quadrilaterals. She reminds them about the difference between the properties of a square and a non-square rhombus, and that the groups should keep track of them development of their thinking as they go along. A group of students defines their quadrilateral, then say "Now let's prove it... we need to not just say that it's a rhombus, but prove it." This clip is also indicative of MP3 and MP6.
See the video in the context of an entire lesson See MP 1 examples from all teachers
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In these clips, students engage in the first of two block-length explorations of their proofs. The students are in groups of four. Each group has chosen a group member to perform these roles: a team captain, resource manager, recorder, and facilitator. The groups have access to the problem (one page per group), two packets of manipulatives, and other resources that they can retrieve, including a page of definitions of quadrilaterals. In some of the groups, students work individually for a while. In other groups, students work in pairs, and in still others, the entire group of four is collaborating. Humphreys' commentary notes subtleties in the students' discourse that either advance or impede the development of their thinking. This clip is also indicative of MP 1, MP 6, and MP7.
See the video in the context of an entire lesson
In the closing of the group work, Humphreys refers her students to the idea of "mathematical friends." This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking. This clip is also indicative of MP6, MP7, and MP 8.
See the video in the context of an entire lesson
On the second day of Humphreys' exploration of the properties of quadrilaterals, students work through their understandings of congruent triangles, the triangle postulates, parallel lines, transversals, and other geometric properties to apply those to create proofs for the quadrilaterals. The students move between group work and whole class interaction throughout the lesson. In small groups, the students discuss and debate proof arguments. At selected times, the teacher pulls the class together to share findings, ideas, or sample justifications. After sharing ideas or arguments with the entire class, students then return to working in their small groups. This clip is also indicative of MP6.
See the video in the context of an entire lesson
Once each group has worked through the reasoning of the proof, Humphreys checks in with the group and instructs them to begin designing a poster that will display the proof they had created. Groups are instructed to design a poster that contains a drawing of the figure, the conjecture of what is to be proved, a list of the given from the conjecture, and what needs to be proved. The students can use a two column or a flow chart format of the proof. This clip is also indicative of MP1 and MP6.
See the video in the context of an entire lesson See MP3 examples from all teachers
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, Humphreys introduces the task by posing a problem as a real-life investigation in which a kite manufacturer who "only manufactures quadrilateral kites", and needs to know the properties of convex quadrilaterals that will always result in a given kite shape, saying "how to do the sticks is the issue." The students work in groups to give prototype advice to this manufacturer, so that any time an order comes in, the manufacturer will always know "what kind of sticks to put in the kit and how they are to be put together."
See the video in the context of an entire lesson See MP4 examples from all teachers
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, Humphreys circulates around groups of students as they use rulers, protractors, and sheets of paper to make different kinds of quadrilaterals and examine and "guarantee" their properties. A group of students debates whether or not one student's assertion about the properties of a trapezoid hold in all circumstances, using the kite "sticks" to illustrate their points. This clip is also indicative of MP1.
See the video in the context of an entire lesson
In this clip, Humphreys closes day one of the Properties of Quadrilaterals lesson, and orients students to the second part of the investigation in which they will justify and prove their findings about the diagonals of the kites, saying "convince yourself, convince a friend, convince a skeptic" to describe for students the level of precision necessary to justify their conjectures. The students use definitions, postulates, and theorems to develop a proof about the diagonals of a quadrilateral and how they constrain the type of figure that is formed. Humphreys moves between groups, checking in on the progress students are making in developing their justifications. At the close of the period, she employs the resource manager to make sure all the manipulatives and materials are collected and stored.
See the video in the context of an entire lesson
In this clip, Humphreys articulates the focus for day two of the group investigations of the properties of quadrilaterals: to prove that their conjectures are actually true. She shares examples of how students "kept track" of their own thinking, helping students in their meta-cognitive efforts at understanding how they're thinking and how to document their understandings. The students mark a figure drawn from the given to help reason through the proof. This clip is also indicative of MP6.
See the video in the context of an entire lesson
In this clip, groups are actively engaged in creating statements about the properties of their quadrilaterals that they can defend. The facilitator asks the resource manager to go get markers, paper, a ruler, and a compass for the group so that they can make their thinking visible. They continue to use the "kite sticks" from Tuesday's lesson to experiment with different points of intersection. This clip is also indicative of MP6.
See the video in the context of an entire lesson See MP5 examples from all teachers
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In this clip, she orients students to the task and explains how they are to communicate their ideas clearly to one anotherduring their group work. This clip is also indicative of MP1.
See the video in the context of an entire lesson
In these clips, Humphreys' students engage in the first of two block-length explorations of their proofs. The students are in groups of four. Each group has chosen a group member to perform these roles: a team captain, resource manager, recorder, and facilitator. The groups have access to the problem (one page per group), two packets of manipulatives, and other resources that they can retrieve, including a page of definitions of quadrilaterals. In some of the groups, students work individually for a while. In other groups, students work in pairs, and in still others, the entire group of four is collaborating. Humphreys' commentary notes subtleties in the students' discourse that either advance or impede the development of their thinking. This clip is also indicative of MP1, MP3, and MP7.
See the video in the context of an entire lesson
In the closing of the group work on the first day, Humphreys refers her students to the idea of "mathematical friends." This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking.
See the video in the context of an entire lesson
In the opening of the second day, Humphreys shares students' work sheets to illustrate how the students were thinking while investigating the quadrilateral. After sharing work, she has students write to a prompt about how well they keep track of their thinking.. The students then share with the class their individual reflections. The class is still learning how to prove a conjecture. Each group begins to settle on the quadrilateral they will formally prove. The students use their prior knowledge of parallel lines and congruent triangles to approach the proofs of the quadrilaterals. This clip is also indicative of MP5.
See the video in the context of an entire lesson
Continuing their explorations of the properties of quadrilaterals, Humphreys' students work through their understandings of congruent triangles, the triangle postulates, parallel lines, transversals, and other geometric properties to apply those to create proofs for the quadrilaterals.The students move between group work and whole class interaction throughout the lesson. In small groups, the students discuss and debate proof arguments. At selected times, the teacher pulls the class together to share findings, ideas, or sample justifications. After sharing ideas or arguments with the entire class, students then return to working in their small groups. This clip is also indicative of MP3.
See the video in the context of an entire lesson
Humphreys' student groups begin to design a poster that contains a drawing of the figure, the conjecture of what is to be proved, a list of the given from the conjecture, and what needs to be proved. The students may use a two column or a flow chart format of the proof. In this clip, a group of students works together to develop their proof. One student summarizes, "So. The definition's here, the picture and conjecture are done." They turn to creating a poster to represent their thinking.
See the video in the context of an entire lesson See MP6 examples from all teachers
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In these clips, students engage in the first of two block-length explorations of their proofs. Humphreys observes, "The square, rectangle, and rhombus appeared to be the most straightforward for the students. Mathematically, if two of the diagonals form right angles, then at least a pair of sides of the quadrilateral will be equal in length. If the diagonals intersect at the midpoint of both diagonals, then the figure formed will be some parallelogram. In order for two diagonals to form a non-isosceles trapezoid, the following relationships must hold true: If AB is one diagonal and DE is the other diagonal, then trapezoid ADBE is formed only if the diagonals intersect at point P, which is not the midpoint, and AP/PB = DP/PE. This relationship was quite difficult for the students to investigate and conclude. The students did not choose to measure the diagonals with rulers, and therefore did not pick up on the proportional aspects of the diagonals in a non-isosceles trapezoid." This clip is also indicative of MP1, MP3, and MP6.
See the video in the context of an entire lesson
In the closing of the group work on the first day, Humphreys refers her students to the idea of "mathematical friends." This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking. Humphreys asks to meet with the students who are playing the role of "facilitators" in their groups to ensure that the Burton and Mason argumentation structure is followed. This clip is also indicative of MP3, MP6, and MP8.
See the video in the context of an entire lesson See MP7 examples from all teachers
In the closing of the group work on the first day, Humphreys refers her students to the idea of "mathematical friends." This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. This framework helps them test the reasonableness of their results. Humphreys asks to meet with the students who are playing the role of "facilitators" in their groups to ensure that the Burton and Mason argumentation structure is followed. This clip is also indicative of MP3, MP6, and MP7.
See the video in the context of an entire lesson See MP8 examples from all teachers

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Cathy Humphreys (9th/10th Grade) | Inside Mathematics

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Cathy's lesson is posed as a "real-world" situation, which motivates a context for her students' investigations. There are many truly real-world applications of mathematics (e.g., the mathematics of global warming). These applications are especially important for students to experience in order to understand the relevance and importance of mathematics in their lives.

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75 digital tools and apps teachers can use to support formative assessment in the classroom

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75 digital tools and apps teachers can use to support formative assessment in the classroom

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75 digital tools and apps teachers can use to support formative assessment in the classroom

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We’ve compiled an extensive list of 75 digital tools that can help teachers use formative assessment to elicit evidence of student learning.

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Practice Guide for Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students. 

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Practice Guide for Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students. 

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practice guide; wwc; teaching; algebra; middle school; high school

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WWC | Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

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This practice guide provides three recommendations for teaching algebra to students in middle school and high school. Each recommendation includes implementation steps and solutions for common roadblocks. The recommendations also summarize and rate supporting evidence. This guide is geared toward teachers, administrators, and other educators who want to improve their students' algebra knowledge.