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Rigor & Mathematical Practices
Gateway 2 - Partially Meets Expectations | 62% |
|---|---|
Criterion 2.1: Rigor | 5 / 8 |
Criterion 2.2: Math Practices | 5 / 8 |
Criterion 2.1: Rigor
Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.
The instructional materials reviewed partially meet the expectations for Rigor and Balance. The materials display a balance between procedural skills and applications. The materials give students sufficient opportunities to utilize mathematical concepts and skills in engaging applications as students complete problems in real-world contexts and engage with non-routine, contextual problems. The materials also provide intentional opportunities for students to develop procedural skills as there are sets of practice problems for each lesson. The materials do not develop conceptual understanding of key mathematical concepts as students do not get opportunities to work with multiple representations of concepts or explain their reasoning about concepts in different formats.
Indicator 2a
Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.
The instructional materials reviewed for Walch Integrated Math Series do not meet the expectations for giving attention to conceptual understanding. The materials rarely develop conceptual understanding of key mathematical concepts where called for in specific content standards or cluster headings. The materials rarely offer opportunities for students to engage with concrete and semi-concrete representations, as well as verbalization and writing, when developing conceptual understanding.
The following examples indicate where the materials lack the opportunity to develop conceptual understanding:
- Mathematics III Unit 2A Lesson 2A.3.3 (A-APR.B): The cluster is intended to build an understanding of the relationship between zeros and factors of polynomials. The materials do not connect the concepts of zeros, factors, and the shapes of the graphs. The connection between zeros and factors of polynomials is an area that lacks conceptual problems and conceptual discussion questions. The majority of the problems focus on finding factors of a problem to sketch a graph. The text also provides step-by-step directions for how to graph polynomial functions.
- Mathematics II Unit 5 Lesson 5.8 (G-SRT.6): The instruction portion of the lesson (page 502, TR) leads the student to see the connection between ratios of sides in similar triangles and the definition of trigonometric ratios. Then in Example 3 on page 510 the materials state that “without drawing another triangle, compare the trigonometric ratios of Triangle ABC with those of a triangle that has been dilated by a factor of K=3.” The remainder of the section, including any examples and practice exercises, are just procedural in nature.
- Mathematics I Unit 2 Lesson 2.4 (F-LE.1): Students are not given the opportunity to distinguish between linear and exponential situations; they are usually directed toward a particular model. Mathematics I Lessons 2.5.1 and 2.5.2 also have students work with linear and exponential models. In the Problem-Based Task for Lesson 2.5.2, students needed to determine whether a scenario would best be modeled by a linear or exponential function and then create that model and use that model to answer questions. However, problems in Practice 2.5.2 tell students to “(w)rite an exponential function to model the scenario” rather than providing them the opportunity to determine whether a linear or exponential model is most appropriate based on the scenario. Students are not given independent opportunities to demonstrate their conceptual understanding of this standard.
- Mathematics III Unit 3 Lesson 3.1.3 (F-TF.2): While guided practice connects the unit circle to the coordinate plane, students are not asked to explain or expand on the connection. Whereas problem 10 on page 69 (TR) does ask students to sketch the unit circle and label, this could be memorized and practice problems are procedural in nature.
- Mathematics I Unit 5 Lesson 5.6.2 (G-CO.8): Students are asked to explain why two triangles cannot be deemed similar. The example provided in the guided practice (Example 4) demonstrates what to do if something is not similar. The indicated correct response indicates that the triangles are not similar because “no congruence statement that allows us to state that the two triangles are congruent based on the provided information.” This is a missed opportunity where the text could have developed conceptual understanding geared towards use of transformations.
- Mathematics III Unit 2A Lesson 2A.4.1 (A-REI.11): Problems are procedural in nature and require students to verify if the intersections are solutions. The indicated correct responses in the teacher's’ resource indicate that if the solutions are solutions to the original equations then the answers are solutions. However, students are not asked to explain why, but rather it is provided in the materials.
The following examples indicate where the materials develop conceptual understanding:
- Mathematics I Unit 3 Lesson 3.1 (A-REI.1): Students work with solving linear equations by explaining the connection between each step in solving and a property of equality. In Mathematics I Lesson 3.1.1, the Problem Based Task has students answer a magic number problem: “Think of a number. Then double it. Now add 6. Take half of that number. Finally, subtract the number you started with. Your answer is 3.” As students consider why this works and how this is possible, they are analyzing their reasoning as they progress from one step to the next (A-REI.A).
- Mathematics I Unit 2 Lesson 2.1 (A-REI.10): Through the examples and the practice problems, the students are given several opportunities to discuss how the graph represents all of the solutions.
- Mathematics I Unit 2 Lesson 2.1.3 has students consider whether a relation is a function using multiple means (mapping, analysis of coordinates, and the vertical line test) as they seek to better conceptually understand what a function is, how it is represented, and what it looks like (F-IF.A).
Indicator 2b
Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.
The materials meet the expectation for providing intentional opportunities for students to develop procedural skills and fluency. Within the lessons, students are provided with opportunities to develop procedures for solving problems that begin to develop fluency. A practice set that includes 10-15 problems is present for each lesson. These practice sets are often “naked number” problems with no context and provide students the opportunity to practice procedural skills.
Some highlights of strong development of procedural skills and fluency include the following:
- A-APR.1: Mathematics II Unit 1 Lesson 1.2.1 and Mathematics III Lesson 2A.1.2 provide opportunities for students to add, subtract, and multiply polynomials. Mathematics III questions extend students’ procedural fluency from those problems students were exposed to in Mathematics II by using larger exponents and more terms in a polynomial expression.
- A-SSE.2: Mathematics II Unit 3 Lesson 3.1 on pages 63-104 reinforces vocabulary and concepts of the parts of expressions and develops skill with writing expressions in different ways in the practice tasks. Mathematics III Unit 2A (pages 46-91) and Unit 2B (pages 5-79), in the Station Activities Set 1, have students build on previous concepts of simplification to rewrite complicated expressions. These stations develop procedural skills as students are required to work from both representations of expressions.
- F-BF.3: There are opportunities provided throughout the series for students to identify the effect of replacing f(x) by f(x)+k, k f(x), f(kx), or f(x + k). Mathematics I Unit 2 Lesson 2.8.2 provides practice with linear and exponential functions; Mathematics II Unit 4 Lessons 2.4.1, 2.4.2, 2.6.1, and 2.6.2 provide practice with quadratic, square root, cube root, and absolute value graphs; and Mathematics III Lessons 3.3.1 and 4B.2.1 provide practice with trigonometric, quadratic, exponential, logarithmic, and linear functions.
- G-GPE.4: Mathematics II Unit 5 Lessons 5.7.1 and 5.7.2 provide opportunities (examples, problem-based task, and practice exercises) for students to use the slope formula, distance formula, and midpoint formula to classify quadrilaterals. Several cases are considered (not only proving a quadrilateral is a parallelogram), and students also work with multiple problems during the practice and guided practices to develop the procedural steps required to prove geometric theorems using coordinates in Mathematics I Unit 6 Lesson 6.1.2.
Indicator 2c
Attention to Applications: The materials support the intentional development of students' ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.
The materials meet the expectation of the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Students work with mathematical concepts within a real-world context. Each lesson contains a problem-based task at the end of the lesson. The problem-based task includes material found in each lesson in a contextualized situation. Single-step and multi-step contextual problems are used throughout all series’ materials. Non-routine contextual problems are also present within the materials. The problem-based tasks also require greater levels of problem solving sophistication as the series progresses.
Examples of mathematical concepts found in application are as follows:
- G-SRT.8: In Mathematics I Unit 5 Lesson 5.9.3, students use trigonometric functions to solve angle of elevation and depression problems. Practice problems include word problems where students must sketch a diagram of the situation and then solve the problem. The problem-based task for the lesson requires students to complete two trigonometric functions and then subtract to find the answer, therefore creating a multi-step contextual problem. In Mathematics II Unit 5 Lesson 5.8, students are asked to determine the dimensions of a ramp using right triangle trigonometry.
- G-MG.2: In Mathematics III Unit 4B Lesson 4b.5.2, a non-routine contextual problem is located on page 363 as students relate the density of ice to a graph and table and determine an equation to represent the data.
- A-SSE.3: In Mathematics I Unit 1 Lesson 1.2.1 Problem Based Tasks, scaffolding practice, and student practice stress the application of mainly linear relationships. For example, in Guided Practice 3, students must create linear equations to determine when two cars will meet. In Mathematics II Unit 3, the unit has various application problems where station activities and large group discussions provide for application scenarios.
- F-IF.4: In Mathematics I Unit 2 Lesson 2.4, students are asked to use information about the purchase of a car to construct a graph of the value of the car over time and identify key features of the graph.
- A-CED.4: In Mathematics III Unit 4B Lesson 4b.1, students are given formulas which relate the frequency and length of the strings on stringed instruments and asked to create a combined formula to determine the tension on the string.
Indicator 2d
Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.
The materials partially meet the expectations for not always treating the three aspects of rigor together and not always treating them separately. Each lesson has an application warm-up exercise, procedural concept development section (guided practice), a problem-based task, and procedural individual practice, regardless of the standards addressed in the lesson. Materials rarely incorporate conceptual understanding into a lesson. The three aspects are not balanced with respect to the standards being addressed as minimal evidence of conceptual understanding can be found throughout the content. The majority of lessons are heavily focused on procedural skill and application. Instructional materials balance procedural fluency problems with application problems throughout the entire series. Procedural skills are enhanced when practiced within the context of an application problem. Instructional materials missed opportunities to incorporate conceptual-based problems throughout the series, thus preventing the balance of all three.
Criterion 2.2: Math Practices
Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice
The instructional materials partially meet the expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified in the implementation guides for the problem-based tasks. A spiral reference notebook is provided that lists the MPs, but it does not connect the MPs to the materials. The materials give students opportunities to develop overarching, mathematical practices, reasoning, modeling with mathematics, and seeing structure. The materials do not support the intentional development of explaining, using tools, and generalizing.
Indicator 2e
The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.
The instructional materials meet the expectation for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6). Overall, the instructional materials develop both MP1 and MP6 to the full extent of the mathematical practice standards. Accurate and precise mathematical language and conventions are encouraged by both students and teachers as they work with course materials. In each of the units there is also a set of station activities that includes a discussion guide. These discussion guides prompt the instructor to ask discussion questions to help students to make sense of the task and to provide responses including precise vocabulary. Emphasis is placed on using units of measure and labeling axes throughout the series (explicit instruction in Mathematics I Lesson 1.2.1 and expectations of using correct units carried throughout the rest of the series). Making sense of answers within the context of a problem is also emphasized. Students also persevere in problem solving in each problem-based task at the end of each lesson.
- Mathematics II Unit 3 Lesson 2.3. (A-SSE.2): Students solve a quadratic equation and determine whether both solutions make sense in the context of a throwing a basketball.
- Mathematics I Unit 2 Lesson 10.1 (F-LE.5): Students need to interpret what the parameters represent in the context of a problem in order to determine whether a solution makes sense. In the implementation guide for the problem-based task with this lesson, teachers are reminded to "check to make sure that students understand how the pricing on the cell phone plans works. Ask them to determine how much someone would pay under each plan for a given number of minutes."
- Mathematics I Unit 5 Lesson 6.1 (G-CO.7): Students identify corresponding parts of congruent triangles as they are introduced to symbolic notation and markings used to represent congruent side and angles in geometric figures. In the implementation guide for the problem-based task with this lesson, teachers are reminded to "encourage students to discuss their prior knowledge of angle pairs formed by a transversal that intersects parallel lines."
- Mathematics III Unit 2 Lesson 2.2: Students use A-REI.2 and the Pythagorean Theorem (G-SRT.8) to solve problems involving radicals. Students are asked to solve applications in both the warmup and problem-based task that require sense-making and perseverance to initiate and precision of units and language to solve.
- Mathematics III Unit 4A Lesson 4a.3: Students work on F-IF.7e. The problem-based task involves creating functions that model the voltage in a three phases of AC coming from a generator. Students will need to persevere to begin the problem and will need to be precise in mathematical language to finish the problem.
Indicator 2f
The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.
The instructional materials partially meet the expectations for supporting the intentional development of reasoning and explaining (MPs 2 and 3).
The materials develop MP2 as students are provided opportunities in which they can develop their mathematical reasoning skills. Examples of the materials providing opportunities for students to develop MP2 include:
- In Mathematics I Unit 1 Station Activities Set 2 Station 4, students are asked to match inequalities to real-world situations. After completing this matching task, students are asked to “explain the strategies you used to match the inequalities to the situations.” In this activity students decontextualize a situation to represent it symbolically, and they contextualize the symbolic representations by considering if the calculated quantities make sense in the given real-world situations.
- In Mathematics II Unit 5 Lesson 6.4, the implementation guide for the problem-based task with this lesson reminds teachers that "students will reason abstractly as they make sense of the information represented in the scenario ... and will reason quantitatively as they calculate the midpoints and slopes of each side length of the triangle."
- In Mathematics III Unit 4A Lesson 2.3, students reason abstractly by determining how to organize data presented to them in a paragraph so that they can find a logarithmic function that models the data. Students also reason quantitatively by determining if the corresponding exponential function models the given data.
The materials do not develop MP3 to its full extent. In the materials, students construct viable arguments, but students are not prompted to critique the reasoning of others. As students complete problem-based tasks, they construct arguments to explain their solutions, but there are no questions or directions in the prompts for students to critique the reasoning of other students regarding the task. In addition, teachers are provided with general instructions to have students discuss their own arguments with each other and explain their own reasoning if disagreements arise while students complete problem-based tasks. Examples of how students do not have to critique the reasoning of others include:
- In Mathematics I Unit 2 Lesson 4.1, students construct an argument that supports their conclusions about an exponential graph, and the accompanying function, for the depreciating value of a car based on given information. The implementation guide for the problem-based task has teachers "encourage students to share their thoughts and ideas with others and to defend their ideas should disagreements arise."
- In Mathematics II Unit 3 Lesson 6.1, students construct an argument to determine which of two options is better for investing $20,000 over 5 years. The implementation guide for the problem-based task has teachers "encourage students to discuss their arguments with each other and explain their reasoning if they do not agree with each other."
- In Mathematics III Unit 2A Lesson 1.3, students construct an argument while determining an expression that can be used to represent the area of a walkway. The implementation guide for the problem-based task has teachers "encourage students to describe their own solution methods, listen to and critique their classmates’ methods, and discuss which method, if any, is best." The implementation guide also states that students could possibly answer "we agreed that there was no best method because all of our methods worked."
Indicator 2g
The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.
The instructional materials partially meet the expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5).
The materials fully develop MP4 as students build upon prior knowledge to solve problems, and they create and use models in the problem-based tasks provided with most lessons. The materials pose problems connected to previous concepts and a variety of real-world contexts. Students are provided meaningful real-world problems to model using mathematics as they identify important relationships when there are opportunities to compare and contrast or draw conclusions. In the implementation guides for the problem-based tasks, references to MP4 typically describe how students will translate the context of the tasks into either an algebraic or graphical model.
The materials do not fully develop MP5 as students are not given the opportunity to choose their own tools, but rather, tools are provided to them. The materials do not encourage the use of multiple tools to complete investigations even though tools are incorporated throughout the series as students engage in mathematics; for example, students use a compass and patty paper to perform geometric constructions, a graphing calculator and pencil/paper to graph equations, and a graphing calculator to determine statistics of a data set. While tools are appropriately modeled throughout the series (step-by-step instructions are provided), limited opportunities exist for students to discuss their benefits/limitations and when to use one tool over another. In the implementation guides for the problem-based tasks, references to MP5 typically describe how students will use a form of graphing technology to help them complete the given task.
- In Mathematics I Unit 3 Station Activities Set 3 Station 3, students use a graphing calculator to solve a system of equations rather than allowing students to use the method they choose to solve the system of equations.
- Mathematics I Unit 5 has a strong emphasis on performing geometric constructions. While lessons include step-by-step instructions on how to perform constructions with a compass and straightedge in addition to using patty paper, all examples, problem-based tasks, and practice exercises have students use a compass and straightedge to perform constructions. Directions throughout the unit explicitly state, “Use a compass and straightedge to …” Mathematics I Unit 5 Station Activity Set 2 Station 1 has students show that two triangles are congruent using a ruler and then show the same two triangles are congruent using a protractor. In doing so, students can compare the tools, but the materials do not support students to evaluate the benefits or limitations of each tool or which tool is “better” to use in the given context.
- In Mathematics I Unit 6 Station Activities Set 2 Station 2, the materials state, “Do NOT use your protractor” to determine whether or not two lines are perpendicular. Students are instructed to “...use your protractor to determine whether or not the lines are perpendicular” in the second part of the station task.
- In Mathematics II, Unit 5 Station Activities students use the provided tools, rather than choosing their own, to investigate angles formed by parallel lines.
- In Mathematics II Unit 6 Station Activities Set 2, the materials state students "will be given a ruler, a compass, a protractor, and a calculator" as they work in groups to answer questions investigating and drawing conclusions about secant and tangent lines to circle theorems.
- In Mathematics III Unit 1 Lesson 1.1, students engage with S-ID.4 as they determine the mean and standard deviation of data sets. Every example has the students follow a set of steps on a TI calculator to solve for the values. The materials offer little opportunity for students to choose an appropriate tool.
Indicator 2h
The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.
The instructional materials partially meet the expectations for supporting the intentional development of seeing structure and generalizing (MPs 7 and 8).
The materials develop MP7 as students are provided opportunities in which they can look for and make use of structure. Examples of the materials providing opportunities for students to develop MP7 include:
- In Mathematics I Unit 1 Lesson 4.1, students look for and make use of the structure of the information provided about two types of skates to create two linear inequalities in two variables. Students use the linear inequalities to determine possible combinations of the two types of skates that could be made.
- In Mathematics II Unit 5 Lesson 3.1, students use the structure of similar figures to determine the two possible locations for a vertex of a triangle on the coordinate plane.
- In Mathematics III Unit 2A Lesson 5.1, students look for and make use of the structure of the information provided to write an explicit formula for a geometric sequence that models the number of people that will hear a positive restaurant experience n weeks after the positive experience was had.
The materials do not develop MP8 to its full extent. In the materials, there are many tasks where students engage in repeated calculations or reasoning, but students do not use the repeated calculations or reasoning to make mathematical generalizations. Examples where students do not use repeated calculations or reasoning to make mathematical generalizations include:
- In Mathematics I Unit 2 Lesson 3.1, the implementation guide for the problem-based task states that students engage in MP 8 by "noticing that the same calculations are performed repeatedly in order to achieve the desired results and recognize that the same domain value is used in order to evaluate the sequences for all three species of trees." The repeated calculations are used to answer questions about the diameters of trees and determine which type of trees should be purchased, but the repeated calculations are not used to make any mathematical generalizations. The calculations were made using general formulas provided for each species of tree.
- In Mathematics II Unit 4 Lesson 1.3, the implementation guide for the problem-based task states that students engage in MP 8 by "using repeated reasoning as they determine a pattern of possible outcomes when two coins are tossed and using the repeated process of calculating probabilities for each event." The pattern of possible outcomes and calculated probabilities are not used to make any mathematical generalizations.
- In Mathematics III Unit 4A Lesson 1.1, the implementation guide for the problem-based task states that students "express regularity in repeated reasoning as they explain and justify their steps involved in determining the inverse of the function representing the motion of the overhang of rocks." The inverse function that is created is specific to the problem-based task and does not represent any mathematical generalization.