Standard 1: Make Sense of Problems & Persevere in Solving Them

standard 1: make sense of problems & persevere in solving them

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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.

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.

connections to classroom practices

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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. 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.

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.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.... They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

In this clip, Mia Buljan works with her students on a number talk, using a mental math approach to multiplying a two-digit number and a one-digit number. She invites her students to share their approaches to the problem, and probes them with questions to identify rationales for their reasoning.

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?"

As her 3rd-grade students work on a multiplication/division word problem, Buljan works with students Enmy and Esbin to dig deeper into their ways of making sense of the problem. She encourages Enmy to persevere and Esbin to let her do so, saying “Just let her try and figure it out. No coaching.”

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?”

Mathematically proficient students… make conjectures about the form and meaning of the solution …

On the first day of the learning segment, Michelle Makinson engages her learners in a math talk focused on unit fractions, combining into wholes, “parts of,” and the idea of equivalence, using manipulatives to create and explain a visual representation of a contextualized representation / word problem. Her students then share with their partners, explaining their approach. This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others), standard 5 (use appropriate tools strategically) and standard 6 (attend to precision).

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

As Erika Isomura’s 5th-grade students work in pairs, she spends several minutes with one pair of students who appear to be stuck on the question of how you might divide 3 by 100, into “teeny tiny little pieces.”

Erika comments “I think this would be a good place to draw,” inviting her students to think about the “story of the problem.” This clip also relates to standard 8 (look for & express regularity in repeated reasoning)

Mathematically proficient students …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.

Erika Isomura’s 5th-grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Erika invites her students to compare their work to their work for prior investigations and other problems. She reminds each partner to contribute equally to the pair work. She asks “How did you get that? Can you show it to me in a picture or with numbers?” She challenges partners to contribute to the pair’s mutual work: “Make sure he's proving it to you. Don't just let him talk it out.” As partners finish their work, Erika invites them to do a “gallery walk” of other pairs’ work to check and compare their work. This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others), standard 6 (attend to precision) and standard 7 (look for and make use of structure).

Mathematically proficient students …monitor and evaluate their progress and change course if necessary.…Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"

As her 5th-grade students finish their card sorts and defend their thinking, Erika Isomura increases the rigor, inviting pairs to go beyond just solving the problem: “Make something challenging, make something that the other kids are going to be like ... then after they think about it like ‘oh yes, I got it.’ Give them a little bit of a push, okay?” She connects their thinking back to the previous lesson with fractional lengths of strings. Erika closes the lesson by praising students’ perseverance and the value of mistakes —“We like mistakes because they give us something to learn from…think of this as your prototype, this is our test run, we're trying it out, once we learn a little bit more, once we become more experts we can always go back and make some changes.” This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others).

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.”

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution... They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, she asks her students to understand the approaches of other students to representing mixed numbers and fractions.

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?"

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In these clips, students engage in a card sort, first doing work in pairs and then sharing their pair work to match mixed numbers, area models, and line segment models.

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).

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 standard 2 (reason abstractly and quantitatively).

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).

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.

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).

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).

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."

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.

Antoinette Villarin begins her lesson on graphing constant rates of change by reviewing the learning goals and mathematical practices, naming Standards for Mathematical Practice 1, 3, 6, and 7. She notes that it is important that her students understand how to build a mathematical argument, and she shares sentence frames and key vocabulary that the students will use as they build their arguments.

Antoinette presents a model of two bottles attached to each other so that fluid can flow between them, and she asks her students to make sense of the problem by describing what they see happening.

Students share that as the amount of fluid in the top container/prism decreases, the amount in the bottom container/prism increases.

This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others), standard 4 (model with mathematics), standard 6 (attend to precision), and standard 7 (look for and make use of structure).

Mathematically proficient students …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.

Antoinette Villarin asks her students to engage in a “turn and talk” with a fellow student, with the purpose of describing how they would know how many centimeters of liquid will be in the bottom container, given the number of centimeters in the top container.

In the “turn and talk,” Antoinette’s students use the vocabulary and sentence frames she provided to make sense of the problem. Antoinette gathers the students back together and asks the pairs to share their findings with the whole group.

Antoinette describes the constraints for the problem. She then asks students to look at a graphical representation of the problem and respond to the prompt "I think this graph represents ..." on their whiteboards.

Her students then share their statements with their partners. After the students share with each other, Antoinette asks pairs to report out on their conversations.

This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others).

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.

Antoinette Villarin asks her student pairs to share their discussions with the whole group. She models academic language — constraints, rate of change, initial value, starting situation — that she expects her students to use.

Once a student pair has shared, Antoinette asks the larger group to add on additional details that helped them identify a matched pair of graphs that show the flow of liquid between a given pair of containers.

Antoinette refers back to her anchor chart of the lesson vocabulary and sentence frames that she expects the students to use, and she names and reinforces students’ use of academic language.

This clip also relates to standard 6 (attend to precision), and standard 7 (look for and make use of structure).

Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"

Antoinette Villarin reminds her student pairs that — once they are confident about a match between graphs — they should record their matches and explain their justification, ensuring that both partners in each student pair understand and can explain their reasoning.

Student pairs discuss and record their thinking on their recording sheets. Antoinette circulates among the pairs and engages them in explaining their thinking — for example, reminding them to identify the starting situation to help guide them.

This clip also relates to standard 6 (attend to precision).

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.

Patty introduces her lesson by charging students to identify the “big ideas” they should be thinking about when they work with right triangles. Students pair-share their ideas, and Patty notes when they are making reference to available tools and supports, such as anchor charts, around the room. In her commentary, Patty notes that this lesson is intended to develop students’ capacity to engage in modeling mathematical situations. Students identify the Pythagorean Theorem, and Patty prompts them to attend to precision and communicate precisely. In a whole-group sharing, she engages all students to add on to, critique, extend, and clarify each other’s thinking. Students deepen their capacity to make sense of the problem or situation. Patty presents student work from a previous assessment and asks students to critique the person’s strategies and precision, giving advice to each exemplar learner about how to improve their approach. This clip also relates to standard 3 (construct viable arguments & critique the reasoning of others), standard 4 (model with mathematics), and standard 5 (use appropriate tools strategically).

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.

Patty’s students give advice to their peers about perseverance and strategies they can use to help themselves in an upcoming MARS performance assessment task. She asks “What do you do when you’re stuck? What strategies should you try?” Students think, write, then share their strategies with each other. They identify strategies using anchor charts and calculators, drawing pictures, consulting a peer, taking their time, double-checking their work, and ensuring that it makes sense.

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).

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).

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.

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).

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).

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).

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.