Standard 7: Look for & Make Use of Structure

standard 7: look for & make use of structure

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classroom observations:
Teachers who are developing students’ capacity to "look for and make use of structure" help learners identify and evaluate efficient strategies for solution. An early childhood teacher might help students identify why using "counting on" is preferable to counting each addend by one, or why multiplication or division can be preferable to repeated addition or subtraction. A middle childhood teacher might help his students discern patterns in a function table to "guess my rule." A teacher of adolescents and young adults might focus on exploring geometric processes through patterns and proof. Visit the video excerpts below to view multiple examples of teachers engaging students in identifying and making use of mathematical structure.

the standard:
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 x2 + 9x + 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.

connections to classroom practices

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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.

Liz O'Neill works with her first grade students engaging them in composing and decomposing numbers within twenty, focusing on ways that numbers combine to produce other numbers. She begins by having her students rename a "target" number in as many ways as possible in their math message book. They are then given a 2 second quick look at 3 ten frames and asked to determine that number (23) mentally. Using sentence frames, students share 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. The main lesson activity is a game called "How Many are Hiding?" Student pairs were given a bag with 10 cubes, a paper plate, and the "How Many Are Hiding Recording Sheet". 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. One partner takes some of the cubes and "hides" them under the plate. The remaining are placed on the top. The second partner uses sentence frames to answer the questions "What number do you see?", "How many are hiding?", "How do you know __ are hiding"? In addition, the answers are recorded. Roles are then reversed. The partner game gives students practice in composing and decomposing numbers within ten.

Mathematically proficient students look closely to discern a pattern or structure.

Mia Buljan’s 3rd-grade students engage in defending their thinking in a number talk. Students work with Buljan to connect the ideas of “switching over from adding, adding, adding, adding, adding, to thinking about multiplying.” Buljan connects and contrasts two students’ approaches to help identify different patterns to inform problem-solving.

They also can step back for an overview and shift perspective.

After her 3rd-grade students work individually on various problems, Buljan invites her students to identify a card on the board that represents the problem they worked.

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.

Becca Sherman works with 4th grade students in a “number talk” to connect the basic components of the Singapore Bar Model with students’ original thinking, thus front-loading students with several applications of the Bar Model as a representation of equal parts. In the exploration problem the words “three times” becomes a division problem or a missing factor problem. The intermediary step of drawing a “math picture” or model of the problem, poses a challenge for many students who have limited exposure to models.

Mathematically proficient students look closely to discern a pattern or structure.

On the second day of the learning segment, Michelle Makinson's students return to their sentence starters and discussion sheets. They examine their work, looking for a set of cards that they matched with a particularly strong justification that they can “save-the-planet-level defend.” Pairs then share their justifications with the whole class, and Michelle challenges her students to communicate precisely and to evaluate their own justifications. She uses a “turn and talk” strategy to generate suggestions to improve the clarity of a justification. She models academic language: “Does anyone have a different strategy for enhancing the justification?” Students consider their own justification cards to make improvements: “Is all the evidence you need there? If not, add it.” This clip also relates and standard 6 (attend to precision).

Mathematically proficient students look closely to discern a pattern or structure.

Michelle Makinson introduces a new card, a contextualized or word problem representation. She begins by having students create a visual representation matching a contextualized representation, and a verbal representation identifying the mathematical quantities. She emphasizes the importance of the discussion between students that proves that the cards match. She circulates around the classroom, engaging pairs in conversation about their discussion and their representations. Students use tape loops on the back of their cards so that they can rearrange them as needed. She explains the importance of students debating and discussing their representations. They engage in identifying patterns, structures, and connections between the representations.

Mathematically proficient students look closely to discern a pattern or structure.

Students continue to create sets of cards that match each other (Green: Set or Area Model, White: Verbal Representation, Dark Blue: Contextual / Word Problem, Yellow: Justification). Michelle Makinson asks pairs to explain the connections between the cards. She challenges students to persevere in grappling with a scenario working with 22/12. The class works with incomplete sets of 12 and incomplete wholes, recognizing that students seek to make connections to real life and become confused about how an improper fraction works in reality. The students create connections between the representations based on their observations of patterns and structures. This clip also relates and standard 6 (attend to precision).

Mathematically proficient students look closely to discern a pattern or structure.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, a pair of students takes stock of their representation, observing that 6/12 is the same as ½.

They also can step back for an overview and shift perspective.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, she recognizes that some pairs of students may have changed their approaches as a result of their conversations, and encourages them to note these changes on their papers for sharing out.

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 closure section of a number talk, Fran Dickinson works with his 5th/6th grade students to identify many different ways of representing the rule: x3 – 3, times 3 minus 3, 3x – 3. The students discuss the rule and the best way to represent it, making connections to their mathematics textbook in their conversations. This clip is also indicative of standard 3 (construct viable arguments & critique the reasoning of others), standard 6 (attend to precision), and standard 8 (look for & express regularity in repeated reasoning).

Mathematically proficient students look closely to discern a pattern or structure.

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 standard 1 (make sense of problems & persevere in solving them), standard 3 (construct viable arguments & critique the reasoning of others), and standard 6 (attend to precision).

Mathematically proficient students look closely to discern a pattern or structure.

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 standard 3 (construct viable arguments & critique the reasoning of others), standard 6 (attend to precision), and standard 8 (look for & express regularity in repeated reasoning).

Mathematically proficient students look closely to discern a pattern or structure...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.

Carlos Cabana works with his high school English language learning students on algebraic reasoning and multiple representations around parabolas. In this clip, his students work together in a group, clarifying each other's process and thinking. The female students clarify accurate steps for the male student. They discuss how to use the x- and y-intercepts to support their process. They step back and ask themselves what they are seeking in their work.