Standard 1: Make sense of problems and persevere in solving them

The Standard:
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.

Classroom Observations:
Teachers who are developing students’ capacity to “make sense of problems and persevere in solving them” develop ways of framing mathematical challenges that are clear and explicit, and then check in repeatedly with students to help them clarify their thinking and their process. An early childhood teacher might ask her students to work in pairs to evaluate their approach to a problem, telling a partner to describe their process, saying “what [they] did, and what [they] might do next time.” A middle childhood teacher might post a set of different approaches to a solution, asking students to identify “what this mathematician was thinking or trying out” and evaluating the success of the strategy. An early adolescence teacher might have students articulate a specific way of laying out the terrain of a problem and evaluating different starting points for solving. A teacher of adolescents and young adults might frame the task as a real-world design conundrum, inviting students to engage in a “tinkering” process of working toward mathematical proof, changing course as necessary as they develop their thinking. Visit the video excerpts below to view multiple examples of teachers engaging students in sense-making and mathematical perseverance.

## Connections to Classroom Practices

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution...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...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...

Liz O'Neill works with her first grade students engaging them in composing and decomposing numbers within twenty. She begins with two warm-up activities engaging students in working with numbers less than 20. Using sentence frames, students shared with their partner what number they saw and how they saw it. A variety of ways were discussed as a whole group after everyone had a chance to share with their partner. Students then played the game "How Many are Hiding?" Student pairs were given a bag with 10 cubes, a paper plate, and the "How Many Are Hiding Recording Sheet". The partner game gives students practice in composing and decomposing numbers within ten. In addition, sentence frames were posted on the board so students could produce academic language using structured student talk and convince their partners with oral justification.

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They monitor and evaluate their progress and change course if necessary...Mathematically proficient students check their answers to problems...

Tracy Lewis leads a re-engagement lesson on the language of word problems, helping students to use word clues to identify mathematical operations. In this clip, she creates a “safe place” for students to examine their own errors and think about how to improve their own work. The students ask each other how they arrived at the answers to the problem and discuss different approaches.

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Mathematically proficient students can explain correspondences between equations, verbal descriptions…or draw diagrams of important features and relationships… Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.

Becca Sherman leads a formative assessment on understanding division in story problems. In this clip, her students use original pictorial representations; the Singapore Bar Model is used as another representation to further understanding of the problem and of division. She asks her students, “how could you add to your math picture, to show this idea?”

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They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Hillary Lewis-Wolfsen invites students to examine a problem about proportions and ratios with a strategy used by a student to organize the information in the problem. In this clip, she invites students to compare the strategy to another used previously. A student describes the second strategy as “easier to see.”

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They analyze givens, constraints, relationships, and goals… 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.

Fran Dickinson leads a number talk on an input/output table and graph, asking “What’s my rule?” In this clip, he generates an output number after students offer an input number. The students discuss whether or not 0 is a possible input for the table and graph.This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others).

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Mathematically proficient students can explain correspondences between equations...They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Fran Dickinson leads a lesson on numerical patterning, in which learners are asked to investigate a numeric pattern and to generalize what they see happening as the pattern grows. In this clip, the learners individually review a copy of two students' work on the MARS "Buttons" task and then discuss what they know or a question that they have about the sample work.This clip is also indicative of standard  2 (reason abstractly and quantitatively).

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Mathematically proficient students start by explaining to themselves the meaning of a problem... Mathematically proficient students can explain correspondences between equations, [and] verbal descriptions...Younger students might rely on using concrete objects or pictures to help conceptualize... They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Fran Dickinson’s students sketch and color code their pattern drawing and use tiles to show what is staying the same and what is changing. In this clip, one student explains to another how blue tiles represent the “old tiles,” or the tiles from the previous stage.This clip is also indicative of standard 5 (use appropriate tools strategically).

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Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution...They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt...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.

Joe Condon works with his 6th grade students to identify strategies for comparing unit rates. In this clip, his students first think about their strategies on their own, then share in groups.

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They monitor and evaluate their progress and change course if necessary...

Jacob Disston leads a lesson on connections between ideas about equations, inequalities, and expressions, helping students to use mathematical vocabulary for a purpose to describe, discuss, and work with these symbol strings. In this clip, he asks his students to group themselves by the common features of the symbol strings, and justify their groupings to the class.This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others) and standard 6 (attend to precision).

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They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Cecilio Dimas leads a lesson on constructing, communicating, and evaluating student-generated tables while making comparisons between three different financial plans, helping students use multiple representations of mathematical problems: verbal, tabular, graphical, and algebraic generalization. In this clip, Dimas asks his students to examine a table comparing DVD rental plans, and ask themselves, “Does this make mathematical sense? Why or why not?” This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others).

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Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution...They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.

Jake Disston works with his middle school students to deepen their understanding of the connections between different representations of functions—graphs, t-charts, and equations. In this clip, he begins the period by challenging his students to work at their tables to group the cards they have in front of them, creating as many different groupings as they can, and devising language to describe each group. Disston's colleague Jesse Ragent then asks the students to do a "matching game." He passes out sets of tables and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set."

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Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, ... 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...

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 standard 6 (attend to precision).

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They monitor and evaluate their progress and change course if necessary. 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. Mathematically proficient students …continually ask themselves, “Does this make sense?”

Continuing into the group section of the task, Humphreys’ students work in teams of four 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 standard 3 (construct viable arguments and critique the reasoning of others), standard 6 (attend to precision), and standard 7 (look for and make use of structure).

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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 monitor and evaluate their progress and change course if necessary.

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.

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Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution... They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.

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. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others) and standard 6 (attend to precision).

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They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

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. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others) and standard 6 (attend to precision).

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They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt...

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 standard 3 (construct viable arguments and critique the reasoning of others) and standard 6 (attend to precision).

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9th - 12th

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...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.

Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In these clips, his students build sketches of parabolas and link them to a T-table. His students have had some exposure to factoring and multiplying, but it's not automatic for any of them yet. They see how to find x and y intercepting with lines at a procedural stage. The students have previously made tables for parabolas and they can find the vertex in the table, and are developing capacity to use the vertex to calculate other parabolic properties.

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