In our years of teaching geometry, the greatest challenge has been getting students to STUDENT REASONING IMPROVING IN GEOMETRY

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1 IMPROVING STUDENT REASONING IN GEOMETRY Parallel geometry tasks with four levels of complexity involve students in writing and understanding proof. Bobson Wong and Larisa Bukalov In our years of teaching geometry, the greatest challenge has been getting students to improve their reasoning. Many students have difficulty writing formal proofs a task that requires a good deal of reasoning. Proof is a problem-solving activity, not a procedure that can be done routinely (Cirillo 2009). We wanted to find ways to scaffold our instruction to prepare students for harder problems. In planning geometry lessons, we noticed that many problems that we selected could be arranged from very straightforward (often a simple diagram with a missing quantity) to very complex (such as a detailed formal proof). Tiering the lessons that is, creating multiple pathways for students to understand the goals of a lesson might be the best strategy (Pierce and Adams 2005). The work of van Hiele (1986) and others building on van Hiele s work, such as Burger and Shaughnessy (1986) and Clements (2003), was extremely helpful in suggesting that geometry requires higher-order thinking and that students need more experience with lower levels of thinking before they can succeed at higher levels. Following the recommendation of Artzt et al. (2008), we wanted to present problems in a way that was accessible enough for students to use their prior knowledge but also challenging enough so that they could extend their learning. 54 MATHEMATICS TEACHER Vol. 107, No. 1 August 2013 Copyright 2013 The National Council of Teachers of Mathematics, Inc. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

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3 Although sequencing problems properly is important, we knew that sequencing alone would not guarantee that all students would be able to improve their reasoning. We wanted all students to have an equal opportunity to improve, and we wanted to avoid the dangers of tracking students by ability. In tracked classes, lower-level students often have limited exposure to a high-quality mathematics education (Useem 1990). We wanted a system that was fair and flexible. Our solution was to divide the lessons into parallel tasks, allowing students with different levels of understanding of a topic to work on the same task simultaneously (Small and Lin 2010). We organized problems into four levels of complexity but allowed students to select their own level and move freely between levels. By having access to all levels, students could monitor their own progress and know what they needed to do to move to the next level. We fit our lessons within a familiar three-part framework: whole-group introductory discussion, guided independent practice, and whole-group summary. At the same time, our model was simple enough for teachers, students, and parents to understand. THE FOUR LEVELS OF COMPLEXITY Our organizational model divides problems into four levels with questions of increasing complexity. These four levels were inspired by the four-point system used to report our statewide exam questions as well as the four Depth of Knowledge levels (Webb, Vesperman, and Ely 2005). Table 1 contains a brief, general description of levels of problems in our model, along with a sample problem for each level for a lesson on the Pythagorean theorem. Level 1, the simplest level, consists of problems that can be solved by directly applying a fact, method, or formula. Problems at this level typically do not require students to use precise mathematical language. For example, the level 1 problem in table 1 is a straightforward application of the Pythagorean theorem. From the diagram, students can immediately identify the two legs and the hypotenuse of the triangle and apply the formula a 2 + b 2 = c 2 without having to name the theorem or explain why triangle ABC is a right triangle. Other examples of level 1 problems include finding the midpoint of a line segment given the coordinates of its endpoints or stating the properties of a parallelogram. Table 1 Levels of Complexity for Problems Level Characteristics 1 Student solves problem by directly recalling a fact, method, or formula. Sample Problem for Lesson on Pythagorean Theorem Find the value of x in simplest radical form. 2 Student solves problem through an additional step beyond a level 1 problem. Typically, some guidance about the method needed for solving is provided. 3 Student solves problem by selecting the appropriate pieces of information independently (usually two or more definitions, theorems, formulas, or methods). 4 Student solves problem by using deductive reasoning to prove mathematical statements. The lengths of three sides of a triangle are 25, 7, and 24. Determine whether the triangle is a right triangle. In an isosceles trapezoid, the lengths of the bases are 14 in. and 30 in. The length of each of the nonparallel sides is 10 in. Find the length of the altitude of the trapezoid. Explain how the diagram at right can be used to prove the Pythagorean theorem. 56 MATHEMATICS TEACHER Vol. 107, No. 1 August 2013

4 Level 2 problems require an additional step beyond a level 1 problem. For example, level 2 problems may require students to draw an accurately labeled diagram when none is provided. A level 2 problem may also require translating a simple word problem into an algebraic representation. To solve the level 2 problem shown in table 1, students must recognize that if the given triangle is right, then the side length of 25 must be the hypotenuse, the longest side of a right triangle. Level 2 problems typically provide some instruction about the method required to solve the problem, such as Determine whether the triangle is a right triangle or Use the midpoint formula to determine whether the quadrilateral is a parallelogram. Level 3 problems require students to determine independently what information is needed for a solution typically, several formulas, theorems, or facts. The level 3 question in table 1 requires students to integrate several ideas: drawing an appropriately labeled diagram, recognizing that the altitudes divide the trapezoid into right triangles and a rectangle, and then applying the Pythagorean theorem and properties of rectangles to find the length of the altitude. Level 3 problems can also include interpreting complex multistep diagrams, a task that requires the application of several theorems. Problems at this level do not require applying deductive reasoning to write formal Euclidean proofs. However, level 3 problems can ask students to select appropriate solution methods and justify their calculations by citing appropriate definitions or theorems. For example, a level 3 problem could give the coordinates of a quadrilateral and ask students to prove that it is a parallelogram. Level 4 problems require students to use deductive reasoning to prove mathematical statements. Students must have reasoning skills that are strong enough for writing formal proofs. Problems at this level include formal Euclidean proofs. SAMPLE LESSON WITH PARALLEL TASKS: INSCRIBED ANGLES By dividing classwork into four levels, we provide multiple entry points for students. Problems from all four levels were included on one activity sheet distributed to all students so that they could see questions from all levels. However, students would have to master one level before moving on to the next. Following Small and Lin s (2010) advice, we allow students to select the level that they feel is most appropriate for their readiness for the lesson. We believe that allowing students to select the appropriate starting point for their work empowers them. Many weaker students told us that they did not feel stigmatized by starting at lower levels because they were able to get more practice to work Fig. 1 This sample activity sheet includes questions from all four levels. up to higher levels. Although many students who selected a level beyond their understanding for that topic soon chose a lower level, others found that they could handle a more challenging level than they originally thought. Whenever possible, we encourage students to start at a higher level if they find the lower-level questions too easy. To give a better idea of what typical level 4 classwork looks like, we have included a sample activity sheet (see fig. 1). Samples of student work are also included to illustrate the level of detail and type of thinking expected at each level. This activity sheet provides not only practice problems but also enough guided questions (such as nos. 4, 6, and 10) so that students can learn new information without much direct instruction. Students who started at levels 3 or 4 were encouraged to review the lowerlevel problems to make sure that they knew how to do them and get relevant information (such as theorems and formulas) for higher-level problems. To help students work independently and make the class flow more smoothly, we designed the Vol. 107, No. 1 August 2013 MATHEMATICS TEACHER 57

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