High School - Gateway 1
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Focus & Coherence
| Score | |
|---|---|
Gateway 1 - Does Not Meet Expectations | 38% |
Criterion 1.1: Focus & Coherence | 7 / 18 |
Criterion 1.1: Focus & Coherence
Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).
The instructional materials reviewed for the Glencoe Traditional series do not meet the expectations for focus and coherence within the CCSSM. For focus, not all the standards, including the modeling standards, are taught to the depth that allows all students to fully learn each standard. Each course has two independent resources, a textbook and an Interactive Student Guide. There is no guidance on how to use the two resources in collaboration, which detracts from the coherence of the intended curriculum.
Indicator 1a
The materials focus on the high school standards.*
Indicator 1a.i
The materials attend to the full intent of the mathematical content contained in the high school standards for all students.
The instructional materials reviewed for the Glencoe Traditional series meets the expectation that materials attend to the full intent of the mathematical content contained in the high school standards for all students. Correlation documents for the materials in the series were provided to show where in the materials primary instruction of each standard is addressed. The lessons and subsequent lessons from the provided correlation documents for the materials were then examined for evidence of each standard and the extent to which the full depth was met. There was an additional Interactive Student Guide reviewed, as well, that did not provide any correlation document. Overall, there was no evidence found for some aspects of some standards. Those specific standards and aspects include:
- A-SSE.1.B: There was no evidence that students were required to interpret complicated expressions by viewing one or more of their parts as a single entity.
- S-IC.6: This is the topic of Lab 11-1 of the Algebra 2 material; however, the report is not evaluated for for its data, as directed by the standard.
- G-CO.4: Some of the definitions are given on pages 296 and 623 of the geometry textbook, but reflections, rotations and translations are not developed in terms of angles, circles, perpendicular lines, parallel lines and line segments as stated in the standard.
Indicator 1a.ii
The materials attend to the full intent of the modeling process when applied to the modeling standards.
The instructional materials reviewed for the Glencoe Traditional series do not meet the expectation for attending to the full intent of the modeling process when applied to the modeling standards. All of the modeling standards were reviewed for evidence of the modeling process. While there are examples of the use of tangible models, real-world situations, tables, equations, expressions, and graphical representations, there is no evidence of students making sense of a real-world situation to (1) identify essential aspects, (2) interpret and represent the situation mathematically, (3) manipulate the model, and then (4) analyze the quantitative information in terms of the original situation. Students are told the modeling tool or representation to use for many of the situations. Although students do manipulate the model, they are not asked to analyze the results within the context of the situation represented to determine if the model is appropriate. Listed below are examples with components of the modeling process that are attended to and places where the modeling process fell short:
- G-SRT.8: There are application problems for this modeling standard, but there are limited opportunities for students to explore problems using multiple pathways or to think creatively or "formulate" their own strategy to solve. Also, problems frequently bypassed the modeling process by providing clearly drawn and labeled pictures or listing examples that students could reference. Section 8-3 in Geometry lists problem 3 on page 550 and problem 43 on page 553 as modeling for standard G-SRT.8, but these are not fully modeling because they are extensively scaffolded.
- Problem 26 on page 563 is a modeling question, except the material directs the students to an example problem that directs them through the process. Problem 56 on page 575 could match the modeling standard G-MG.3 (though there is no CCSSM stated) and is close to the full modeling (involving units) process, but a picture is provided requiring no creative thinking.
- Lab 11-2 in Geometry for G-MG.2, regarding density, provides examples and asks students to repeat the shown process in an applied situation, but the lab does not attend to the full modeling process. Problem 3 on page 796 is another example for this standard, and does not engage students in the modeling process because the exact steps are given in activity two and leaves little room for students to find multiple approaches to solve the problem.
- Algebra 1 and 2 have data labs (Algebra 1: 2-6; Algebra 2: 4-8, 7-8) that have the potential to engage students in the modeling process. Students collect data using graphing calculators, but they do not solve any type of problem. They are looking at a pattern of data. They do have to answer some interpreting questions, but the students do not have to formulate anything as the book walks them through the data collection and equation-creating processes.
- The activity of finding the limit of a geometric sequence in Extend 10-4 for standard A-SSE.4 in the Algebra 2 book could be adapted for modeling, but the materials give too much information to allow students to fully engage in the modeling process.
- Frequently, there are examples directly next to the student exercises that show students exactly how to work through the problem. One example is problem 9 on page 36 of Algebra 2, which is labeled as a CCSS modeling problem using writing and solving an inequality. Example 4 is listed next to the problem, directing students to follow the completed process made available in this example.
- A-REI.11: There was no evidence of modeling found to support this modeling standard of the CCSSM.
- The “formulate” part of modeling is consistently lacking in the materials. The CCSSM states that students should be formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables. There was limited evidence found that students had to come up with a strategy to solve an open-ended problem with multiple solution paths or were required to formulate a process for solving any problems or work through the modeling process on their own.
- The Interactive Student Guide attempts to provide more modeling problems, but the problems do not attend to the full modeling process as they are either application problems or heavily scaffolded problems.
- Problem 4 on page 367 of the Geometry Interactive Student Guide addresses standard G-MG.2, but it is an application problem rather than a modeling problem.
- The amusement park problem on page 396 of the Algebra 2 Interactive Student Guide gives the model, and though the model is a very complicated trigonometric function with lots of reading, the problem is scaffolded to provide information that doesn't allow the student to experience the modeling process.
Indicator 1b
The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.
Indicator 1b.i
The materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers.
The instructional materials reviewed for the Glencoe Traditional series partially meet the expectation for allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers (WAPs). For this Indicator, the Table of Contents and Chapter Planners throughout the series were reviewed along with their alignment to the CCSSM. The following were considered distracting and took away from time focused on the WAP's: Lessons explicitly marked as optional, direct teaching of Middle School standards that were not being developed specifically into High School standards, plus standards, lessons teaching specifically the practice standards, and standards not included in the WAPs. Overall, the materials do not spend the majority time on the WAPs and incorporate distracting topics. The Interactive Student Guide does offer extra time on the WAPs, but there is no accounting or pacing guide to help plan for what should be left out of the main materials and incorporated into from the Interactive Student Guide. It would also be very difficult to teach everything in the textbook and the Interactive Student Guide in one year making the intended design of these materials unclear and distracting. There were some prerequisites from Grades 6-8 that were seen as helpful and not distracting. For example, in Algebra 1, the Grade 8 standards on functions are included and are used to help develop the high school standards. However, there were many distracting topics and chapters including:
- Arithmetic with rational functions found in both Algebra 1 (Chapter 11) and Algebra 2 (Chapter 9) is distracting because this is a plus standard in the CCSSM.
- The full development of conic sections in Chapter 9 of Algebra 2 is distracting because it is a plus standard.
- The formal development of sequences and series in Chapter 10 of Algebra 2 goes beyond the CCSSM. For example, section 10-7 proves limits of sequences using mathematical induction.
- The matrices sections in both Algebra 1 and Algebra 2 are distracting because they are plus standards.
- Algebra 1 includes several lessons that are specifically teaching middle school standards without tying them to high school standards. Chapters 0 (Preparing for Algebra); 1 (Expressions, Equations and Functions); 2 (Linear Equations); and parts of Chapter 3 (Linear Functions) are not high school standards, and they are not used to develop high school standards.
- All of Chapter 0 of Geometry is distracting material based on prerequisites and review of middle school content.
- Chapter 12 of Geometry has a mix of middle school standards relating to surface area and volume as well as topics that go beyond the CCSSM related to surface area. One example of the advanced topic in this chapter is spherical geometry.
- Chapters 7 and 8 of Geometry also have a mix of middle school standards and plus standards. For example, lessons 3, 6 and 7 as well as the extend of lesson 4 in Chapter 8 are plus standards. Lessons 1, 3 and 7 of Chapter 7 are middle school standards.
- Algebra 2 spends most of Chapters 0-4 on middle school standards and topics that were already taught in the Algebra 1 materials.
- Lessons 4, 5 and 9 of the Trigonometry chapter in Chapter 12 of Algebra 2 are plus standards that distract from the non-plus standards in F-TF.
- All of Chapter 13, Trigonometry, in Algebra 2 is either plus standards or content beyond the content of the CCSSM that detract from the essential F-TF standards of the CCSSM. For example, this chapter includes full sections on (1) Sum and Differences of Angles Identities; (2) Double-Angle and Half-Angle Identities; (3) Solving Trignometric Equations; and (4) Trigonometric Identities (verifying and using the identities to solve problems).
Indicator 1b.ii
The materials, when used as designed, allow students to fully learn each standard.
The instructional materials reviewed for the Glencoe Traditional series do not meet the expectation that students are provided with opportunities to work with all non-plus standards and do not distract students with prerequisite or additional topics. Lessons were examined for evidence that, when used as designed, they would enable all students to fully learn each standard. There are many standards where students are not given an opportunity to fully learn all aspects of the standard. In addition, many standards are only taught in the Interactive Student Guide, which provides no guidance on its appropriate use. Listed below are standards that are not fully taught in the series and standards that are primarily found in the interactive student guide.
- A-APR.4: There was one example of proving polynomial identities in Algebra 2 on page 350. There were a few examples using a calculator, but there is not enough practice for students to fully meet the depth of this standard.
- N-Q.1: Choosing and interpreting units consistently in formulas is found in Section 2.7 of the Algebra 1 Interactive Student Guide.
- G-CO.3: Example 3 on page 656 asks students to describe a single transformation that maps a figure onto itself after copying and reflecting the figure; however, there was no evidence found to describe the rotations and reflections that carry a polygon onto itself.
- G-CO.12: When formal geometric constructions are addressed, the materials provide students with the opportunity to practice each construction at most two times. For instance, the Extend Lab 1-5 shows students the steps to construct perpendicular lines two different ways using a compass and straightedge, then has the students practice each construction once. Even though there are several sections within the Geometry material where students work with perpendicular lines (including Chapter 3 on parallel and perpendicular lines), the students are not required to construct perpendicular lines.
- G-SRT.4: The proof on page 547 of the geometry material uses geometric mean instead of similar triangles to prove the Pythagorean theorem as referred to in the standard.
- G-C.2: Angles, radii and chords are defined in the geometry materials, but there was no evidence found of describing the relationships among inscribed angles, radii and chords as the standard states.
- G-GPE.6: There were some opportunities for students to develop an understanding of midpoint, but not of any other points on a directed line segment that partitions the segment in a given ratio that is not 1:1. Problem 69 on page 34 of the geometry material does offer some understanding of partitioning in thirds, but there was no other evidence of partitioning other than halfway.
- G-GPE.7: Perimeters of triangles and squares are found, but are not generalized to polygons to go beyond the middle school standards.
- G-GMD.1: Informal arguments of the area and volume formulas are not explicit for each aspect of this standard, especially with regard to informal limit arguments.
- S-ID.7: There is brief reference to the the meaning of the slope in context, but there are no exercises that ask students to interpret the slope in terms of the context of the data.
- S-ID.8: There is no evidence of interpreting the correlation coefficient.
- S-CP.6: There was only one problem that fully met the "interpret" part of this standard, and that was problem 6 on page 955 in the Extend 13-5 of the Geometry text.
- In Algebra 1, many standards, including N-Q.2, N-Q.3, A-SSE.3c, A-REI.7, S-ID.5, and S-ID.9, are not addressed and only appear in the Extend lessons, which does not provide students with extensive opportunity to work with those non-plus standards. For example, the N-Q.2 standard appears in the Extend 2-6 for Algebra 1, which only contains four exercises, and only one of the exercises has students defining an appropriate quantity for a modeling purpose.
Additionally, there are several standards for which evidence was found in the supplemental Interactive Student Guide, but either not in the student textbook or minimally in the student textbook. Using only the textbook would cause all of these standards to not be fully developed. There is no teacher guidance or correlation documents to help interpret and know how and when to use the Interactive Student Guide. Some of these standards include:
- A-APR.1: Understanding that polynomials are closed under multiplication was found in section 5.1 of the Interactive Student Guide of Algebra 2.
- A-REI.4b: The Algebra 2 Interactive Student Guide, section 4.6, has exercises that asks students to recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
- G-CO.2: Evidence of comparing transformations that preserve distances and angles to those that do not is not explicitly taught. The concept is there, but it is not explicit in the material but is addressed briefly in the Interactive Student Guide throughout Chapter 1.
- G-C.5: The Geometry Interactive Student Guide includes this standard in lesson 9 - 2.
- G-SRT.7: There are many examples that allow students to engage fully in using the relationship between the sine and cosine of complementary angles; however, problem 64 on page 576 of the materials was the only evidence of asking students to explain the relationship between the sine and cosine of complementary angles, and this one opportunity does not lead students to fully learn this standard. In the Interactive Student Guide, there was a multi-part investigation plus three exercises to further develop the idea.
- G-MG.1: This was taught minimally in the materials and covered a little more in detail on pages 364-366 of the Geometry Interactive Student Guide.
- S-ID.3: Interpreting is addressed in problem 10c of lesson 12-3 in the Algebra 1 material. Additionally there are a few problems in Chapter 12 of the Interactive Student Guide. The problems in the Interactive Student Guide use standard deviations to interpret the differences whereas the standard asks to use shape, center and spread.
- S-ID.4: Recognizing that there are data sets for which such a procedure of fitting a data set to a normal distribution is not appropriate was found only in the Algebra 2 Interactive Student Guide in section 9.7 which states to use after section 11-5 in the textbook.
- S-IC.2: There were a few problems in the Algebra 2 Interactive Student Guide on page 318 that allowed students to experience deciding if a specific model is consistent with results from a given data-generating process; however, more problems are needed to fully develop this standard.
- S-IC.3: The Algebra 2 Interactive Student Guide provides evidence of the aspect of identifying the characteristics of different study types for this standard in section 9.1, which states to use after 11-1 in the textbook. Recognizing the purposes of each of these study types is not explicitly taught, nor is explaining how randomization relates to each of these study types.
- S-IC.4: The Algebra 2 Interactive Student Guide has problems that use the maximum error of estimate to find a confidence interval to estimate a population parameter from a random sample and develops a margin of error in section 9.8 to be used after 11-6 in the textbook.
- S-IC.5: The Algebra 2 Interactive Student Guide has two problems in section 9.1 that compare two treatments. This section states to use this lesson after 11-1 in the textbook.
- S-ID.5: There were only a couple of problems in section 11-5 of the Algebra 2 Interactive Student Guide, and this standard needs more problems for students to master the standard from the aspect of possible associations in the data.
- S-ID.9: There is one example and five total problems for this standard (correlation and causation) in an extend section (4.5 extend on page 254) in Algebra 1. In Algebra 2 there are seven questions in an Algebra lab on page 99. Both of these sections are at the end of the chapter tied to optional extend/lab/explore sections.
- F-LE.3: There was only one problem in the Interactive Student Guide of Algebra 1 in section 10.7 that allowed students an opportunity to observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or as a polynomial function. Students need more practice to master this standard.
- F-TF.8: The Algebra 2 Interactive Student Guide includes some conceptual development and the proof of the Pythagorean identify in Section 10.7, which is stated as being used after section 13.1 and 13.2 of the textbook.
Indicator 1c
The materials require students to engage in mathematics at a level of sophistication appropriate to high school.
The instructional materials reviewed for the Glencoe Traditional series partially meet the expectation that the materials require students to engage in mathematics at a level of sophistication appropriate to high school. The materials generally meet the depth of the non-plus standards; however, all students are not given extensive work with non-plus standards.
- There are many additional practice worksheets available through the online component, for each lesson in the series. These include:
- Study Guide and Intervention
- Practice
-
- Word Problem Practice
- Enrichment
- Interactive Student Guide. This guide also introduces new objectives in some lessons and is not referenced in the teachers resources, so there is not an indication of how to use it in lesson planning for differentiated instruction.
- Each lesson has a differentiated homework option in the teacher edition that has recommended homework for students who are (1) approaching level, (2) on level or (3) beyond level. One example of this is in Lesson 2-6 of Algebra 1. There are 53 problems for students approaching grade level, 51 problems for students on grade level and 38 problems for students beyond grade level. The approaching-level homework skips most all of the word problems. The on-level homework mainly does odd problems. The beyond-level includes 3 optional skills review problems that are not included in either the approaching or on-level recommended homework. This is typical of most every lesson in the series.
- Lower performing and advanced students are not getting the same opportunities to engage in non-plus standards experiences. Though advanced students are afforded the opportunity to engage in more word problems, those word problems do not necessarily engage students deeply in the non-plus standards.
- In Geometry, G-CO.9, G-CO.10, and G-CO.11 are addressed, but based on the suggested differentiated homework options, the approaching-grade-level students write out only a few actual proofs related to the lesson topic. For instance, lesson 2-5 on postulates and paragraph proofs suggests students approaching grade level do only proof problems 30 and 31. Then, in lesson 2-8 on proving angle relationships, it is suggested that students approaching grade level do only two proof problems (14-15). Students on level and beyond level are doing several more proof problems in each lesson, however the extra proof problems are not at a more advanced level.
Indicator 1d
The materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the Standards.
The instructional materials reviewed for the Glencoe Traditional series partially meet the expectation that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the standards. Overall, materials partially foster coherence through meaningful connections in a single course and throughout the series. However, conceptual categories are not always used together to foster coherence.
- The coherence between algebra and functions is disconnected in these instances:
- Transforming equivalent expressions of quadratics among standard, vertex and factored form is taught as separate entities throughout different sections in Chapters 8 and 9 of Algebra. There is not an opportunity for students to gain experience in converting among the forms appropriate to the context. The materials teach these in isolation instead of seeing the structure in each expression and understanding which form is simplest to use in given various contexts.
- The development of polynomials in high school should parallel the development of integers from earlier grades. The domain of Arithmetic with Polynomials and Rational Expressions provides little coherence among the standards. For instance, polynomial division and the remainder theorem lacks coherence to linear factors and zeros. There is no evidence of this coherence. Synthetic division is taught as a process instead of a technique in understanding factors and zeros in Section 5-2 of Algebra 2.
- Coherence across the conceptual categories of Functions and Modeling is weak. Functions and Modeling work together to allow a student to find a function that best fits an observed relationship between quantities. Since modeling is weak in the materials, students do not have the opportunity to develop coherence between the conceptual categories of functions and modeling.
- The materials do not make connections to lessons and units that develop in a systematic way to meet the full depth of the high school standards. The Teacher Edition includes a vertical alignment section within the Focus on Mathematical Content at the beginning of each chapter; however, the alignment does not always develop in a way that meets the full depth of the CCSSM. Each lesson in both the Teacher Edition and Student Edition begins with a description of student's prior learning and what they will be learning in that lesson to explicitly make connections between prior and current learning.
- One example of where the connections across courses is weak is with regard to using function notation. Although students learn about functions and use function notation in Chapter 1 of the Algebra 1 materials, function notation is not carried through the material into Chapters 3 and 4 on linear functions or Chapter 7 on exponential functions. Function notation is used in lesson 4-7 only to explain finding the inverse of linear functions.
- Chapter 3 of Algebra 1 has a section regarding arithmetic sequences and how they connect to linear functions. Chapter 7 (Exponential Functions) has a section regarding geometric sequences and how they connect to exponential functions. While this section does ask students to determine if a sequence is arithmetic or geometric, the section does not provide students with questions and problems to draw connections (similarities and differences) between linear and exponential functions using sequences. The geometric sequences section makes no reference to linear functions of the related concept that was developed in Chapter 3.
- In Algebra 1, students are given a brief lab (between 10-5 and 10-6) to connect similarity with trigonometry, which is a connection for students to develop a conceptual understanding of trigonometry. The lab is designed to help students "discover" the connection between similar triangles and side length ratios, but it has three problems in which students are to discover this connection. Immediately following this lab in Section 10-6 titled "Trigonometric Ratios," there are no exercises that require students to use the conceptual understanding that was discovered/developed in the lab. Instead, students spend their time finding angles and side lengths in right triangles which does not necessarily require students to use the connections developed in the lab. The Geometry section on trigonometry briefly mentions how similarity helps form the trigonometric ratios, but again, the exercises for students do not further promote this connection.
- Each lesson in the materials contains a vertical alignment section in the teacher's portion of the text. This includes related topics before the chapter in the same course and in previous grades, related topics in the same chapter, and topics this chapter prepares students for in future lessons or courses. For example, in section 8-4 of Algebra 2 the vertical alignment states that students should be able to graph reciprocal functions before this lesson, will graph rational functions in this lesson, and will learn to solve rational equations by graphing in a future lesson.
Indicator 1e
The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.
The instructional materials reviewed for the Glencoe Traditional series do not meet the expectations that the materials explicitly identify and build on knowledge from Grades 6-8 to the High School Standards. Overall, the materials include Grade 6 - 8 standards; however, they are not identified as such.
- The online materials more clearly identify Grade 8 concepts than the print materials do. Although they indicate topics from Grade 8, they do not clearly identify the standards. There was no evidence of connections being made and/or articulated between the middle school and high school standards. The Grade 8 material is presented, but there is no evidence of how it is extended or built upon to develop high school standards.
- There is no building on knowledge of Geometry from the Grade 8 CCSSM for students and teachers. For example, the standard G-CO.7 is in the cluster "Understand congruence in terms of rigid motions." There is evidence of the converse of this, but there was no evidence found of explicitly using rigid transformations. Students learn the definition of congruence through corresponding parts and not through rigid transformations.
- Other than Chapter 0, any indication of teaching standards from Grades 6-8 is not labeled. All lessons indicate that they align to one or more non-plus standards even if the content in the lesson does not meet the standard indicated. While Chapter 0 in the materials is labeled clearly as containing topics from previous courses, the only standards indicated, if indicated at all, in the Chapter 0 lessons are the Mathematical Practice Standards and the plus standards.
- In Algebra 1 lessons 1.2 on order of operations, 1.4 on distributive property, 2.6 on ratios and proportions, and 7.4 on scientific notation all align to middle school standards; moreover, the problems are mostly routine and neither the rigor nor the depth approach high school standards.
Indicator 1f
The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.
The instructional materials reviewed for the Glencoe Traditional series partially meet the expectations that the materials explicitly identify the plus standards and coherently support the mathematics which all students should study in order to be college and career ready. Overall, the materials include aspects of many plus standards which include: N-CN.3,N-CN.6, N-CN.9, N-VM.7, N-VM.8, N-VM.9, N-VM.10, N-VM.12, A-APR. 7, A-REI.9, F-IF.7d, F-BF.1c, F-BF.4b, F-BF.5, F-TF.7, F-TF.9, G-SRT.10, G-SRT.11, G-GPE.3, S-CP.9, S-MD.2. However, the plus standards are not always clearly labeled and sometimes distract from the full depth of the non-plus standards.
- The only plus standards indicated in Geometry are in Chapter 0, Chapter 8, Chapter 10 and Chapter 13 in both the Teacher Edition and Student Edition. Connections between non-plus and plus standards are not always clearly articulated, i.e. it is explicit in lesson 8-6 but not in lesson 10-5 of Geometry.
- There is no reference to advanced courses in regard to the plus standards for teachers. The material does not identify the plus standards as plus standards. Section 8-6 (law of sines and cosines) in Geometry supports the learning of the chapter; however, section 10-5 (constructs tangents - G.C.4) does not support the learning of that chapter.
- Matrices, rational expression operations and conic sections in Algebra 1 and Algebra 2 lessons do connect to the algebra and function standards; however, they distract from the overall learning of the non-plus standards. For example, in Algebra 2 chapters 8 and 9 are devoted to rational functions and conic sections, both of which are plus standards. Chapters 10 and 11 teach sequences and series as well as probability and statistics. Because there are two full chapters of plus standards before essential non-plus standards, it is distracting to these standards that follow them in the materials.
- The plus standards are identified in the correlation materials at the beginning of each teacher edition, but not within the materials or the on-line resources.