Cognitive Abilities Test A ie o Tees

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A Guide for Teachers

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Cognitive Abilities Test™ FORMS 7/8

Cognitive Abilities Test™

David F. Lohman • Joni M. Lakin

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Contents

Part 1 Basic Information about the Cognitive Abilities Test ..........1Purpose of the Test ..................................................................................................................1Structure of the Test ................................................................................................................1

Grades K–2 (Levels 5/6–8) .....................................................................................................2Grades 3–12 (Levels 9–17/18) ...............................................................................................4

Using the Test Results ..............................................................................................................4Understanding Ability Profiles ................................................................................................5

Score Levels ...........................................................................................................................6Score Patterns .......................................................................................................................6Relative Strengths and Weaknesses ....................................................................................8Ability Profile Examples .......................................................................................................8

Part 2 General Principles for Differentiating Based on Individual Differences .....................................................................9

Common Myths about Differentiating Instruction ................................................................9On Students and Classroom Environments .......................................................................11Important Characteristics of Students ...............................................................................11Important Characteristics of Classroom Environments ...................................................11

General Principles of Instructional Differentiation ..............................................................12A Note on Opportunity to Learn...........................................................................................15

Part 3 Instructional Suggestions for Students of Different Ability Levels .................................................................19

Instructional Suggestions Based on Overall Ability Level ....................................................19Below Average Reasoning Abilities (Stanines 1–3) ..........................................................20Average Reasoning Abilities (Stanines 4–6) ......................................................................22Above Average Reasoning Abilities (Stanines 7–8) ..........................................................24Very High Reasoning Abilities (Stanine 9) ........................................................................25

Part 4 Adapting Instruction to Build on Relative Strengths .........27Relative Strength in Verbal Reasoning (V+) .....................................................................28Relative Strength in Quantitative Reasoning (Q+) ...........................................................30Relative Strength in Nonverbal Reasoning (N+) ...............................................................30

Part 5 Adapting Instruction to Shore Up Weaknesses ..................33Relative Weakness in Verbal Reasoning (V–) ....................................................................33Relative Weakness in Quantitative Reasoning (Q–) .........................................................35Relative Weakness in Nonverbal Reasoning (N–) .............................................................37

ii CogAT Forms 7 and 8—A Guide for Teachers

Part 6 Adapting Instruction for Mixed Ability Profiles ................39Achievement Test Performance .........................................................................................39Adapting Instruction for Students with Mixed Ability Profiles .......................................39

Part 7 Case Studies and Instructional Examples ...........................41

Appendix Acknowledgments and References .............................53

1Basic Information about the Cognitive Abilities Test

Part 1 Basic Information about

the Cognitive Abilities Test

Purpose of the TestForms 7 and 8 of the Cognitive Abilities Test™ (CogAT®) appraise the level and pattern of verbal, quantitative, and spatial (nonverbal) reasoning abilities for students from kindergarten through grade 12. These abilities reflect the overall efficiency of cognitive processes and strategies that enable individuals to learn new tasks and solve problems. Because these abilities are closely related to an individual’s success in school in virtually all subjects, CogAT test results are helpful in planning effective instructional programs and adapting instruction in ways that enhance the student’s chances of success in learning.

Structure of the TestEach of the three CogAT batteries—Verbal, Quantitative, and Nonverbal—has three subtests. The abilities appraised are those that enable students to acquire, organize, store in memory, and recall information; to make inferences; to detect relationships; to comprehend and analyze problem situations; to form concepts; to discover and remember sequences; to recognize patterns; to classify or categorize objects, events, and concepts; to infer rules and principles; and to relate and use previous experience to accomplish new learning tasks or solve novel problems. All three of the batteries have been designed to appraise not only both general inductive and deductive reasoning abilities, but also specific reasoning abilities that are unique to each battery.

The test is divided into 10 levels that increase in difficulty from Level 5/6 (typically used in kindergarten) through Level 17/18 (appropriate for late high school).

Subtests by Battery

Verbal Quantitative Nonverbal

Picture/Verbal Analogies

Sentence Completion

Picture/Verbal Classification

Number Analogies

Number Puzzles

Number Series

Figure Matrices

Paper Folding

Figure Classification

Levels of CogAT

Form 7/8 Level

Typical Grade Level

5/6 K

7 1

8 2

9 3

10 4

11 5

12 6

13/14 7–8

15/16 9–10

17/18 11–12

2 CogAT Forms 7 and 8—A Guide for Teachers

Grades K–2 (Levels 5/6–8)Levels 5/6, 7, and 8 are designed for students in kindergarten through second grade. The questions in each battery are divided into three subtests, each of which has a different item format. (See the sample items in Figure 1.) No reading is required on the part of the student. All directions are read aloud by the proctor or recorded narrator. The proctor paces students through the test so that they do not rush heedlessly or labor needlessly.

In the Verbal Battery, the Picture Analogies subtest and the Picture Classification subtest are comprised entirely of picture-based questions that measure verbal reasoning processes without tying questions to a specific administration language. The Sentence Completion test is the only subtest that requires spoken prompts. On this test, the proctor or recorded narrator reads a question in English and/or Spanish and the students choose the picture that best answers the question. The Sentence Completion subtest can be omitted or not scored for children whose native language is not English or Spanish. Omitting Sentence Completion results in an Alternative Verbal (or Alt-V) score.

The Quantitative Battery consists of three subtests that have been adapted for young students by framing quantitative reasoning challenges in engaging and accessible formats. The Number Analogies subtest for primary-grade students relies on picture-based quantitative concepts rather than numeral representations. The Number Puzzles subtest presents equations as either two trains which must carry the same number of objects (Levels 5/6 and 7) or numbers and functions (Level 8). Finally, the Number Series subtest is presented as an abacus in which students search the beads looking for patterns. All of these formats have been extensively tried out with students and found to be engaging, and all three subtests tap into important quantitative reasoning skills.

The three subtests in the Nonverbal Battery at the lower levels are just like those at the upper levels and did not require much adaptation for young students. The Figure Matrices test contains three figures in an analogy (A➝B: C➝?) that the student must complete. The Paper Folding subtest requires students to determine how a folded, hole-punched paper will appear when it is unfolded. The Figure Classification subtest presents three figures in the stem, and the student chooses the fourth figure that belongs to the set.


Figure 1: Item Formats Implemented in CogAT Forms 7 and 8

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Picture Format (Levels 5/6-8) Text/Standard Format (Levels 9-17/18)V

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“Which one swims in the ocean?”


Grades 3–12 (Levels 9–17/18)Levels 9 through 17/18 of CogAT are designed for students in grades 3 through 12. Level 9 transitions from the picture-based, teacher-paced, verbal and quantitative subtests used with students in grades K–2 to the text- and numeric-based, timed, verbal and quantitative subtests used at the upper grades. Children are allowed to pace themselves through each subtest, but all subtests have a 10-minute time limit. The student must read individual words on two subtests in the Verbal Battery (Verbal Analogies and Verbal Classification) and a sentence in the third (Sentence Completion). The three subtests in the Quantitative Battery are Number Series, Number Analogies, and Number Puzzles. The first two formats are commonly used on ability tests. The latter is a novel format that requires students to determine the values of one or more geometric shapes that make a simple equation true. The three subtests of the Nonverbal Battery are Figure Matrices, Paper Folding, and Figure Classification, which were all described above for the Levels 5/6, 7, and 8 tests.

Using the Test ResultsThe three primary uses of CogAT scores are (1) to guide efforts to adapt instruction to the needs and abilities of students, (2) to provide a measure of cognitive development, and (3) to identify students whose predicted levels of achievement are markedly discrepant from their observed levels of achievement. A brief discussion of each use follows.

The first and most important use of CogAT scores is to help classroom teachers of all kinds adapt instructional goals, methods, and materials to the individual needs of students. Part 3 of this guide explains how to make principled adaptations of instruction and discusses why CogAT scores are especially useful for guiding this process. Part 4 offers specific suggestions for building on student’s strengths, Part 5 for how to shore up weaknesses, and Part 6 for assisting students with mixed Ability Profiles that have both significant strengths and weaknesses.

The second use of CogAT is to provide a measure of each student’s level of cognitive development that captures important information not represented in school grades or in other measures of school achievement. For example, CogAT scores help identify academically gifted students. Less than half of the students who score in the top 3% on the Iowa Assessments also score in the top 3% on CogAT. This means that CogAT will identify many students as academically gifted who otherwise would not be so identified based on academic achievement alone. CogAT scores show that most low-achieving students are able to reason at higher levels than their academic performance suggests. In fact, the lower the students’ scores on an achievement test, the greater the probability that their CogAT scores will be at significantly higher levels. Their relative strengths and weaknesses can also provide valuable insight into what types of adaptations will increase their classroom performance. This use is explained in more detail in the Score Interpretation Guide.

The third use of CogAT scores is to identify students whose levels of academic achievement are substantially lower or higher than expected given their CogAT scores. Students whose achievement is markedly below expectations should be checked for other problems such as learning disabilities, poor vision or hearing, disrupted formal education, language barriers,

The first and most important use of CogAT scores is to help classroom teachers of all kinds adapt instructional goals, methods, and materials to the individual needs of students.


the need for more assistance in completing school lessons, markedly different preparation for the current grade level, or the need for a different instructional program. On the other hand, students whose academic performance is better than would be expected from their CogAT scores should also be looked at more carefully. These students have learned well the specific skills taught in school but are less successful in solving unfamiliar problems. Such students might profit from tasks that emphasize transfer and innovation. This use is also explained in more detail in the Score Interpretation Guide.

Understanding Ability ProfilesA CogAT composite score provides the average of a student’s scores on the three CogAT batteries. However, the Ability Profile is a far more informative and useful index. The Ability Profile captures two characteristics of the student’s scores (see Figure 2):

• level – the typical magnitude of scores on the three batteries

• pattern – whether some scores are significantly higher or lower than other scores

Figure 2: A Sample Ability Profile

8 B (Q-)

Pattern

Level

Figure 3: Levels and Patterns


Score LevelsA stanine indicates one of nine broad score groupings on a normalized standard score scale. Stanines range from 1 (lowest) to 9 (highest). (See Figure 4.) The most commonly used stanine is based on age norms. Age-based, rather than grade-based, norms are more appropriate when making interpretations about students’ reasoning abilities.

Each CogAT Ability Profile begins with a number that represents the student’s median stanine. For example, if the student has age stanines of 6, 3, and 8 on the Verbal, Quantitative, and Nonverbal batteries, respectively, the student’s median stanine is 6 (the middle stanine of the student’s three stanines).

In a student’s Ability Profile, the median stanine indicates a level of reasoning ability. However, it is often useful to describe a student’s CogAT results in terms of one of the five reasoning ability levels (very low, below average, average, above average, or very high, all shown in Figure 4) rather than in terms of the stanine number.

Figure 4: Median Stanine by Reasoning Ability Level

14%

Verylow

Stanine scoreFrequency

Interpretation

Normal Distributionof Scores

Belowaverage

Average Aboveaverage

Veryhigh

27%

312%

417%

520%

617%

712%

87%

94%

Score PatternsThe graph of a student’s score for each CogAT battery includes an estimate of the margin of error, displayed on the score report as a confidence band (shaded rectangle) around the age percentile rank (APR) score for each of the three batteries. These margins of error vary by battery and student. Unusually wide confidence bands indicate that the student’s scores on the subtests or items in the battery were inconsistent and so the score on the battery probably should not be used. The profile is then also suspect.

Based on the relative position of these confidence bands, Ability Profiles are classified as A, B, C, or E profiles. The List of Student Scores excerpts below show examples of these profiles and their confidence bands.


In an A profile, all three confidence bands overlap, meaning the student’s Verbal, Quantitative, and Nonverbal Battery scores are roughly at the sAme level. About 44% of all students have this profile.

In a B profile, two of the confidence bands overlap. The third score is a relative strength or weakness, significantly aBove or Below the other two. About 33% of students have a B profile.

In a C profile, two scores Contrast. The student shows a relative strength and a relative weakness. About 12% of students have a C profile.

An E profile indicates Extreme score differences. At least two scores differ by 24 or more points on the standard age score (SAS) scale. About 10% of students have an E profile.


Relative Strengths and WeaknessesAn Ability Profile also indicates any relative strengths and/or weaknesses evident in the student’s battery scores.

+ V, Q, or N followed by a plus sign (+) indicates a relative strength on the Verbal, Quantitative, or Nonverbal Battery, respectively.

– V, Q, or N followed by a minus sign (–) indicates a relative weakness on the Verbal, Quantitative, or Nonverbal Battery, respectively.

For example, an Ability Profile of 4B (V+) means that the student’s median stanine is 4 and that the student’s score on the Verbal Battery was significantly higher than (aBove) the student’s scores on the two other batteries.

Ability Profile ExamplesA variety of Ability Profiles are explained in the examples below.

Profile Interpretation Most helpful guidance section

9A Very high scores on all three batteries Part 3 on overall ability, starting on page 19

8B (Q–) Generally high scores but a relative weakness on the Quantitative Battery

Part 5 on relative weaknesses, starting on page 33

2B (N+) Generally belowaverage scores but a relative strength on the Nonverbal Battery

Part 4 on relative strengths, starting on page 27

5C (V+ N–) Generally average scores but a relative strength on the Verbal Battery and a relative weakness on the Nonverbal Battery

Part 6 on mixed Ability Profiles, starting on page 39

8E (V–) Generally high scores but an extreme relative weakness on the Verbal Battery

Part 5 on relative weaknesses, starting on page 33

In general, the number (the median stanine) carries the most information in the interpretation of A profiles, less for B profiles (now we must also consider the strength or weakness), still less for C profiles (because we must consider a strength and a weakness), and the least information for E profiles.

9General Principles for Differentiating Based on Individual Differences

Differentiating instruction means using instructional planning to create learning environments that are based on student characteristics. A substantial body of research exists to guide decisions of how best to educate different students. However, very little of this information has found its way into curriculum guides and other materials that teachers use. The following is a review of some of these general principles for adaptation of instruction, beginning with some common myths about instructional adaptation.

Common Myths about Differentiating Instruction

Myth 1: All students are pretty much alike.

Although few educators would agree with this statement, many well-meaning but uninformed educational policy makers presume it to be true by asserting that there is one best way to teach science, mathematics, or reading. On occasion the presumption stems from a failure to appreciate the range of individual differences that teachers must accommodate in their classrooms. For example, it is not uncommon for students in a class to differ by 4 or more grades in achievement levels.

The belief that all students are the same ignores these individual differences and their implications for instruction. What is good for the least-able student is assumed to be good for the most-able student or vice versa. What works for the student who reasons well with images but poorly with words is assumed to be just as effective for the student with the opposite profile.

Myth 2: Every student is unique.

Those who do not subscribe to Myth 1 sometimes subscribe to the opposite myth—namely, that every student is unique. It is helpful to keep in mind that, in some respects, every student is like all other students, like some other students, and like no other student. Generalizations about teaching and learning are possible only to the extent that the first and second statements hold. If every student is considered unique, then no generalizations can be offered. A good educational program, then, is faithful to all three aspects—the universality, the commonality, and the uniqueness of each student.

Myth 3: Differentiation should be based on self-reported learning styles.

The third myth is that effective instructional differentiation should be based on students’ self-reported learning styles or learning preferences. When educators refer to a student’s “learning style,” they typically imply something about the student’s ability to reason only in a particular symbol system. Labels that describe students as “visualizers” or “verbalizers” or as “auditory learners” or “visual learners” may be helpful in assisting students to understand themselves better, but such measures have not proven useful for helping teachers adapt their teaching methods and materials in ways that help more students succeed. Rather, the critical information for understanding how students learn is given by the profile of their reasoning abilities. Because of this, the CogAT Ability Profile provides a system for instructional differentiation that actually works.

Part 2 General Principles for

Differentiating Based on Individual Differences


Myth 4: If the method is right, the outcome will be good.

Another myth is that if we somehow knew more, we would be able to specify exactly how to arrange conditions to maximize the learning and motivation of every student. This ignores the inherent unpredictability of human behavior. It assumes that behavior can be understood with the same causal models we use for predicting the flight of a golf ball or the reaction of two chemicals. Yet even physical systems such as the weather can be described only in terms of probabilities. Improvements in our ability to measure winds and moisture and to create ever more sophisticated computer models of the weather will reduce this uncertainty, but they will never eliminate it. Educators are in a similar position when they attempt to apply principles of learning to individual students. Some efforts will fail, but if the research that guides these efforts is solid, educators will, on average, make better decisions than if they had not made the adaptations in instruction.

Myth 5: Individualization requires separate learning tasks.

Some early attempts to individualize instruction implied that each student should work on a different task, one uniquely matched to her or his needs. Most of these efforts were based on behavioral theories of learning that viewed development as a ladder with many small steps, each of which needed to be reinforced. In the extreme, students ended up working alone (in cubicles) on workbooks or computers. Teachers were reduced to paper shufflers and monitors, occasionally dispensing instruction but rarely engaging the group as a whole.

We now know better. We know that students learn by observing and interacting with other students and adults. Groups are especially important for learning how to think. We learn to think in new ways by observing others as they solve problems and then verbally or physically reenacting the process ourselves. With practice, what is at first social and external becomes personal and internal.

Development occurs along many dimensions, not just one. Lower-level skills need not always be learned before higher-level skills. Therefore, instead of searching for the one task that uniquely matches the student’s needs, educators must more often search for tasks that can simultaneously appeal to students at many different levels. In other words, the goal should be to find broad activities that engage many students at once rather than to find many narrow tasks that uniquely fit the needs of each student.

What do broad tasks look like? Consider, for example, classic stories. Students of different ages can enjoy the same story because it allows entry at multiple levels. The youngest child may attend only to the pictures and to some of the action. An older child may understand the plot, and the adult who is reading the story may appreciate the broader theme. Different students can meaningfully engage a story at different levels or from different perspectives to learn from it. Thus, adaptation does not mean that students should work alone or even that they should be separated into groups.

Myth 6: Cognitive strengths will appear in standard ways for all students.

The research on the use of teacher rating forms for the identification of gifted students shows that teachers tend to identify only a subset of academically gifted children in their classes. Students who are not well behaved or do not make perfect grades are not seen as having potential talents as often as other students. This is why multiple sources of information are needed for gifted program identification, as well as an awareness of student differences in opportunity and challenges. Reasoning abilities may manifest differently for English language learner (ELL) students, cultural or linguistically diverse students, or twice-exceptional students (e.g., those with specific learning disabilities but also academic talents).

General Principles for Differentiating Based on Individual Differences 11

On Students and Classroom EnvironmentsSuccessful adaptation of instruction requires an understanding of how different kinds of classroom environments can have different effects on students with different characteristics. These effects depend on the characteristics of the students (such as their abilities) and the characteristics of classroom environments (such as the amount of structure provided).

Of the hundreds of ways in which classrooms differ from one another, a handful of ways have repeatedly emerged as important sources of harmony or discord. Similarly, of the hundreds of ways in which students differ, some are much more important than others. Instructional environments differ in their demands and in their resources through the characteristics of the classroom. Students differ in their abilities to meet the demands and their sensitivity to the requirements. Whether a quiz is perceived as a challenge or as a threat depends in large part on the characteristics of the students, such as whether they enjoy and seek out competition. In general, however, some students will be more in tune with a situation than others. For some there will be harmony, for others discord. The key to making effective instructional adaptations is knowing the major dimensions along which these interactions occur. The section that follows begins first by summarizing characteristics of students and then turns to characteristics of environments.

Important Characteristics of StudentsSuccessful learning depends on many personal and social factors. Some important factors that teachers may want to consider are persistence, affect (including anxiety), interests, exposure to enrichment or formal education (opportunity to learn), and school and home resources.

Among the many cognitive factors, two that seem to matter the most are the students’ knowledge and skills in a content area and their abilities to reason in the symbol systems used to communicate new knowledge in that domain. Symbol systems include written language/letters, numbers, mathematical symbols, shapes, and figures. Classroom assessments report on students’ knowledge and skills; they provide information about what students need to learn. But successful adaptation requires knowing how students tend to learn best. CogAT measures reasoning abilities in the three major symbol systems required for academic learning, which is why it is so helpful in guiding efforts to adapt instruction. For example, it is not primarily the ability to generate visual images that matters for academic learning, but the ability to reason with and about those images. Similarly, it is not the ability to remember words or to speak fluently that matters more in some instructional treatments than in others, but rather the ability to reason about the concepts that the words signify.

Important Characteristics of Classroom Environments There is no objective way to classify environments. A student’s typical patterns of knowing, feeling, and choosing help to determine the types of school environments they perceive and seek out. For example, students who are highly anxious will tend to perceive class presentations, tests, and other situations in which they must demonstrate competence quite differently than students who are generally not anxious. There are many characteristics of classrooms that are important for learning. The focus here is on those features of instructional methods that affect students differently depending on their abilities and personalities.

CogAT measures reasoning abilities in the three major symbol systems required for academic learning, which is why it is so helpful in guiding efforts to adapt instruction.


Structure. Instructional programs differ in the amount of structure they provide. In general, students who have poorly developed reasoning abilities in a domain do better when the instructional program offers greater structure in terms of activities, outcomes, and behavior management. Students with higher reasoning abilities generally do better with less structure. These students often do better in discovery-oriented environments. Structure is a variable that also describes the way classrooms are organized. In general, anxious and impulsive students struggle more in unstructured classrooms compared to orderly and predictable classrooms.

Novelty/Complexity/Abstractness. Students with poorly developed reasoning abilities may struggle when the curriculum consists of tasks that are unfamiliar, complex, or require abstract thinking. The development of reasoning abilities requires environments that challenge students with novelty, complexity, and the need for abstraction. Tomlinson (How to Differentiate Instruction in Mixed-Ability Classrooms, 2001) provides in-depth examples of teaching that is simple vs. complex, familiar vs. novel, and concrete vs. abstract.

Dominant Symbol System. Instructional environments differ in the extent to which they require students to process information using different symbol systems. The three most important symbol systems in academic learning are verbal, quantitative, and figural (or spatial). One of the most effective ways to adapt instruction is to allow students to use their better-developed abilities in one symbol system to scaffold learning in another. For example, a student with good verbal but poor quantitative reasoning abilities can improve the latter by learning to verbalize their reasoning behind a math solution.

Delivery Formats and Decisions. Classroom environments differ in many ways, especially when we consider the increasing role of technology. Classrooms can differ in the extent to which they allow students to work with others or to work by themselves. Some classrooms have autonomous online learning while others use only physical teaching materials. Characteristics of the classroom that matter for differentiation include pacing, teacher- and student-centered strategies, group vs. solo work, and competitive vs. collaborative work, among many other features.

General Principles of Instructional DifferentiationInstructional differentiation is an important skill that all teachers use to provide appropriate instruction to their wide diversity of students. The following are some general principles that guide our recommendations for differentiation.

Build on Strength. When a student is weak in one area but strong in another, a general rule is to build on the strength. Students are better able to process information more elaborately and at higher levels when tasks emphasize the type of thinking they do best.

When adapting instruction to build on strengths, follow these guidelines:

• Instruction geared to a strength should challenge that strength. It should encourage students to go beyond the information given, not merely register it.

• Students must learn to perform tasks that they do not do well. In such cases, emphasize aspects of the tasks that avoid their weakness until the students have established a foothold.


For example, consider students who have difficulty learning computation skills but who show strength in verbal reasoning. Using group oral recitation and encouraging students to talk through math problems will emphasize their verbal strength more than silent practice on computation. This strategy builds a weaker area using a strength.

Focus on Working Memory. One of the most pervasive findings in all research on instruction is that students with stronger reasoning ability do better when instruction allows them to do things in their own way. Likewise, students with weaker reasoning skills do better when given greater instructional support. Instruction that scaffolds, sequences, and otherwise reduces the burden of information processing generally helps students with weaker skills. The critical factor here is the burden placed on working memory. When helping students with weaker skills, the key is not so much to reduce the need for thinking as it is to reduce the burden on working memory.

When students are required to remember and do more things than they are capable of remembering and doing at one time, they generally fail. In cognitive terms, their working memory is overloaded.

As they learn, students must understand, temporarily store, and then transform new information in some way. All three of these processes require working memory, which is a limited resource. For example, students who must calculate their multiplication facts rather than simply recall them will struggle more with long division than other students because they are holding more numbers in memory at once.

Effective use of working memory is critical for successful reasoning. Students cannot make inferences about how two or more ideas are connected if they cannot hold the ideas in their working memory while trying to compare them.

Indicators that a student’s working memory is overloaded include the following:

• inability to recall and complete a list of oral instructions

• skipping or repeating parts of a task

• task abandonment and frustration

When helping students who are unfamiliar with a task or who have difficulty learning, aim to reduce the burden on working memory while maintaining the integrity of the lesson objective.

Two important questions for educators to ask are:

• “What are the major demands that this activity places on the students’ working memories?”

• “Which of these memory requirements can be offloaded or scaffolded?”

Scaffold Wisely. Whenever students try to solve problems, there are many processes that must be executed simultaneously in working memory. Scaffolding wisely means reducing memory requirements and processes that are not the focus of the instructional activity. As skill increases, those memory requirements can be added back in.

For example, the demands of spelling and grammar can easily overwhelm the working memory resources of a beginning writer. Scaffolding these processes (such as planning to write a rough draft and then correct errors) temporarily frees the student to construct a connected narrative.


Similarly, one of the last steps in the acquisition of skills is learning to monitor one’s own performance (metacognition). Especially in the early stages of skill acquisition, monitoring functions can be supported by prompting students to check their work. Writing things down, drawing pictures, and practicing a skill until it can be performed automatically also reduce demands on working memory.

When working with students who have difficulty making inferences, deductions, and elaborations, avoid the temptation to make things easy by eliminating the reasoning requirements of the tasks. This strategy works well in the short run but leaves students increasingly unprepared to face future challenges of school learning. When reasoning is an essential part of a task, find ways to support and guide learners while maintaining their need to reason.

Emphasize Strategies. Psychologists who study reasoning distinguish between two types of reasoning processes:

• Tacit reasoning processes occur outside awareness. They typically do not require much attention and are performed quickly and intuitively. They are acquired through practice.

• Intentional reasoning processes require conscious awareness. Intentional thinking is often described as effortful and rule-based. These processes may be novel and unpracticed.

For example, skilled readers use tacit reasoning processes to understand much of what they read. They retrieve word meanings quickly and automatically build mental images that help them keep track of the meaning of a passage as they move from one sentence to the next. Beginning readers, on the other hand, use intentional reasoning processes to understand the meaning of both individual words and of the sentences that they make, often relying on illustrations rather than their own mental imagery.

Reasoning processes are most useful when students learn to use them strategically. At the lowest level, this means simply having a strategy that one can consciously use when necessary. At an intermediate level, it means having multiple strategies available for possible use. At a more advanced level, it means knowing under which circumstances each strategy is best used. And at the highest level, it means becoming strategic and reflective in one’s thinking.

Instructional adaptations are most effective over the long term if they help learners become more intentional and self-regulated in their learning. Encouraging students to use and monitor the effectiveness of different strategies helps them better leverage their strengths and avoid, or scaffold, their weaknesses.

When Grouping, Aim for Diversity. It is generally not wise to group students by score levels or by score profiles into uniform groups. Students are most likely to improve their ability in a domain if they have the benefit of learning from classmates whose skills and approaches to problems differ from their own. Aim for diversity and flexibility in grouping students, whether the grouping is done within a classroom or between classrooms.

Working with students of different ability levels is particularly important for students who have a marked deficit in one area. Improvement is more likely if such students have high- quality interactions with individuals who have a relative strength in the same area than if they are constantly paired with other students who, like themselves, have difficulty in that domain.

Higher-ability students benefit from such groups to the extent that they are asked to provide explanations and assistance while still growing their own skills and understanding. Aiming for


flexibility in grouping can also allow average and above-average students the opportunity to be the “big fish” sometimes. (See “Big Fish/Small Pond” on page 16.)

Note that academically talented students can benefit from being grouped with other high-ability students in classes that offer advanced or accelerated learning. Not all differentiation can or should be offered in a single classroom. In fact, research shows that when a single teacher is tasked with differentiating for all students, the high-ability students are much less likely to receive appropriate adaptations than the low-ability or low-achieving students. This leaves such students bored and prevents them from pursuing the achievement levels of which they are capable.

A Note on Opportunity to LearnIn the recommendations that follow, we sought to include information that may help teachers adapt instruction for student populations that are culturally and linguistically diverse. This means classrooms where a range of socioeconomic levels are present, students who are not native English speakers are present, or students vary from each other in terms of racial, ethnic, and cultural backgrounds, among others.

When considering the needs of students and their demonstrated abilities, be sure to consider the relative amounts of Opportunity to Learn (OTL) that students may have. Teachers must not engage in stereotyping or overgeneralization, but it is important to recognize that parental influence and access to educational resources will play a much larger role in students’ current skill level when they enter school than later in their education. In making instructional differentiation decisions for younger students, it is important to keep in mind the greater role of home resources in achievement levels. Flexible grouping and retesting is essential for younger students.

An obvious case of OTL effects is an English language learner student. A typically developing child in the United States will have around five years of English OTL when they enter kindergarten. A student who begins to learn English in kindergarten has had much less OTL. We should clearly assume that differences in verbal skills for these students in English are mostly due to OTL rather than differences in verbal ability. Comparing students with similar OTL can reveal better insight into verbal ability. For example, two ELL students who have been in the same ESL class for the same length of time will likely show differences from each other in English acquisition, where the student with greater English skills is probably higher in verbal ability. Comparing students only when they have similar OTL is critical to making interpretations about ability. That is why subgroup and local norms are strongly recommended.

When administering CogAT, we encourage teachers to consider whether all students have roughly similar opportunities to develop skills related to test performance or if the students should be compared to those with similar backgrounds. Opportunities to practice the test formats (with free practice materials) and collecting information about educational background can help teachers consider OTL appropriately.

…When a single teacher is tasked with differentiating for all students, the high-ability students are much less likely to receive appropriate adaptations.


Big Fish/Small Pond (written with Barbara Kozak)

Here’s an example of how the clustering of students can create growth opportunities.

When you enter Mr. Dowdell’s classroom, you see students working in groups and you immediately notice that Sam and Tina are constantly telling their groups what to do, answering all the questions, and in general running the students around them, telling them “what” and “how” to think—because each thinks his or her idea is best! They have taken over all of the roles within the group and hijacked the group’s thinking. How does the teacher create an opportunity for all of the students to participate and stretch their thinking? One way is to take the students of very high ability, Sam and Tina, out of their respective groups and have them work together leaving the above-average students in the groups to fill the “big fish” hole that was just left by Sam or Tina. Classrooms often have students who are capable of great thinking and leadership but are constantly overshadowed by the “smart one.” Or, on the flip side, they have learned to sit back and let the “smart one” do the work instead of speaking up.

So what happens when Mr. Dowdell pulls Sam and Tina out of their groups? Now Sam and Tina have a challenge: each other. No longer will they automatically get their way or answer all of the questions. They will have to defend their arguments and thinking to the other person because they both think that they are correct or have the best way. Since they are used to being the sole big fish in their respective groups, they will now have to navigate how to coexist in that role. Sam and Tina’s former groupmates now have the opposite challenge to contend with. With no big fish in the group, a new leader (or leaders) will have to step up. For above-average students (stanines 5–7) this can be a refreshing change. When Tina’s best friend, Lily, no longer has Tina there overshadowing her thinking and ideas, she feels the confidence to speak up and express her ideas in the group more often. She has been enabled to go from being a passive learner into taking a more active role simply by modifying the ability grouping in the classroom.


Cluster Grouping to Decrease Ability Variation (written with Barbara Kozak)

Narrow grouping based on ability can restrict the development of all students. However, this doesn’t mean that grouping for reduced variability is not a valuable clustering method. Classrooms can be adjusted to have manageable ranges of abilities in one classroom. Katherine Fosnot has termed this “optimal mismatch.”

Ms. Smith is about to start a new school year and, upon receiving her class roll, realizes she has groups of students in all of the ability levels: below average, average, above average, and very high. How will she manage to differentiate effectively for all of the different instructional needs across subjects and abilities? Perhaps the one or two very bright students could engage in peer tutoring so they don’t get bored?

Rewind to the summer when classes were made, and then bring in the concept of cluster grouping. Instead of students in four ability groups, where the very high ability students often “teach” students in lower groups, cluster groups of very high ability students together to allow for them to be challenged and to reduce the variability in each classroom. Cluster grouping can place students in ability levels that are in stanines adjacent to each other, such as having a classroom with stanines ranging 1–6 and another with 4–9. This provides students with examples of classmates with a variety of strengths to learn from, but also reduces frustration of students working with peers who are moving quickly through material they may struggle with. A very high ability student (stanine 9) and a below-average student (stanines 1–3) often find frustration when working together. The very high ability student can be bored and will not grow in their thinking, and the below-average student will often not understand the reasoning of the high student at the speed that they process.

For Ms. Smith, having students in two or three of the four ability groups allows her to consciously group students based on their strengths and areas of need so that they are learning at their own level while stretching their thinking and learning. An above-average student can be a great model to an average student when their strengths are taken into consideration. The very high ability students also need opportunity to encounter challenges that are a stretch for their ability levels. Without this opportunity for challenge, very high ability students will not learn to persevere and may struggle with attempting problems they see as difficult because they require actual thinking!


19Instructional Suggestions for Students of Different Ability Levels

Instructional Suggestions for Students of Different

Ability LevelsPart 3

There is both unity and diversity in cognitive abilities. Unity is reflected in the substantial correlation between measures of verbal, quantitative, and figural reasoning abilities. Students who obtain a high score in one domain are likely to be above average in the other two domains. Cognitively, it means that reasoning tasks share common attention, memory, and other processing resources. On CogAT, unity is estimated by the overall height, or level, of the score profile. This is captured by the median stanine.

Diversity of abilities is reflected in the fact that although tests of verbal, quantitative, and figural reasoning are correlated, these correlations are much lower than the reliabilities of the three reasoning tests. Cognitively, it means that students differ in their abilities to reason with verbal, quantitative, and figural symbols. On CogAT, diversity is reflected in the pattern of the scores across the Verbal, Quantitative, and Nonverbal batteries.

The implications for instruction of any score profile must take into account both the overall level as well as the pattern of the three scores. In this part of the guide, we consider differences in the overall level of the profile. These differences are divided into four groups based on the median, or middle, age stanine:

Stanines 1–3 Below Average

Stanines 4–6 Average

Stanines 7–8 Above Average

Stanine 9 Very High

Parts 4 and 5 of this guide consider some of the major differences in the pattern of scores. Part 4 discusses how to capitalize on relative strengths in reasoning abilities. Part 5 considers the problem of providing scaffolds to shore up specific weaknesses.

Instructional Suggestions Based on Overall Ability LevelFor students who do not have an A profile, refer to “Adapting Instruction to Build on Relative Strengths” on page 39 and “Adapting Instruction to Shore Up Weaknesses” on page 33 for additional guidance based on a student’s particular strength or weakness.

You can obtain more information on a specific Ability Profile by using the Interactive Ability Profile Interpretation System. This tool is available at https://www.riversideinsights.com/apps/cogat.

In the descriptions below, you will see some guidelines marked with CLD Tip. These are specific considerations for Culturally and Linguistically Diverse students.

https://www.riversideinsights.com/apps/cogat


Below Average Reasoning Abilities (Stanines 1–3)

Principle Guidelines

Learner Characteristics • Difficulty learning abstract concepts

• Minimal or ineffective strategies for intentional learning and remembering (They tend to approach learning tasks in a trial-and-error fashion.)

• Tendency to spend little time planning before attempting to solve a problem (As a result, they frequently do not transfer knowledge and skills learned in one context to another context unless prompted to do so.)

• Difficulty detecting relationships, similarities, and differences that go beyond appearances

• Tendency to be easily distracted by salient but irrelevant details in problems

• Interest or passion may be evident, but generally shows limited creativity or detail in products

• CLD Tip: May struggle more with cultural context and code-switching than other students with similar backgrounds

Build on Strengths Look for strengths in terms of specific interests and achievements. Even more than other students, those who are behind their peers in reasoning abilities often learn more and sustain their efforts longer if the teacher discovers and builds on their interests.

• To the extent possible, tailor learning to a student’s interests; it will lead to greater effort and a generally more sophisticated outcome.

• Identify and emphasize other competencies these students have, especially when students are working in groups.

• Students who feel that they are participants (rather than observers) have higher levels of motivation and engagement in a task.

• Gather information on students’ interests, specific skills, and areas of confidence.

• Pre-assessments are especially helpful to target their starting level in a skill domain.

Focus on Working Memory

Students with poor reasoning skills can reap great benefits when you can reduce the demands on their working memory capabilities. These students are easily overloaded by too many concepts, images, sounds, and words that must be held in mind. A lesson may start out meaningfully but soon degenerate into an anxious search for surface features of tasks that suggest a solution. Reduce burdens on working memory with instructional methods such as these:

• Eliminating the need to remember ideas, even temporarily, can greatly assist these students. Keeping all information on one page or the board, but not both, is helpful.

• Use familiar concepts and make analogies to students’ everyday experiences.

• Provide lists of items to be remembered or scaffold processes that must be performed simultaneously.

• Provide ample structured practice so that skills such as writing, typing, or calculating become automatic.

Continued on the following page...

Instructional Suggestions for Students of Different Ability Levels 21

Below Average Reasoning Abilities (Stanines 1–3) (continued)


Scaffold Wisely Students with poorly developed reasoning abilities often have difficulty identifying what is important to learn and judging where they should focus their attention in a learning situation. Be aware of these limitations:

• Directions: These students need attention-getting and specific directions before they start a task.

• Structure: They learn more effectively in structured learning environments that make fewer demands on their cognitive resources and provide more direct guidance, coaching, and support.

• Pacing: They tend to process information slowly and need a slower pace of instruction than do students with higher stanine scores. For these students, “doing it” works better than talking about it.

• Risk: Provide low-risk activities like writing down an answer or think-pair-share rather than expecting students to hazard a guess in front of the whole classroom.

• Instructional strategies likely to be more effective than verbal explanations include:

– teacher- or peer-modeling of cognitive and metacognitive skills

– concrete representations of abstract concepts

– demonstrations and hands-on activities

– pictures, other types of illustrations, videos, and three-dimensional models

– peer review or accountability for more frequent feedback

– preview materials with vocabulary lists or graphic organizers

– more opportunities for review and practice with skills

Refer to page 13 for details on scaffolding without eliminating the development of important reasoning skills.

Encourage Strategic Thinking

Because these students often have considerable difficulty identifying appropriate situations in which to use a particular strategy, follow these guidelines for teaching them learning strategies:

• Modeling: Use modeling and demonstration during ongoing learning situations in the classroom.

• Feedback: Provide frequent opportunities for feedback or checking their work to prevent them from practicing errors.

• Reflection: To help students become more reflective in their learning, focus on a few good strategies rather than on a detailed list of rules.

• Transfer: Once students have learned how to apply a strategy in a particular context, provide opportunities for them to apply it in other contexts.

When Grouping, Aim for Diversity

These students should not be segregated in classes or groups consisting solely of other low-scoring students. Those who have difficulty reasoning when alone typically learn more effectively and have higher levels of achievement when they have many opportunities to interact with higher-ability peers. If grouping is based on ability, provide opportunities for students to move between groupings frequently as the content changes and as their skills increase.


Average Reasoning Abilities (Stanines 4–6)


Learner Characteristics • Often use words that are correct but do not precisely describe a concept or relationship (assuming they are native English speakers)

• Likely to use only previously learned methods when faced with new tasks

• Difficulty transferring knowledge and skills when tasks look different from those already learned

• Vulnerable to learned helplessness and may not exert effort even when they are capable of the task

Build on Strengths Although these students have good strategies, they often have difficulty applying what they know when learning a new task, particularly when it looks different. Consider the following instructional adaptations for these students:

• Recognize that their strengths will primarily be evident in their interests and somewhat in their levels of achievement in different domains.

• Find ways to encourage and acknowledge the particular academic mastery of these students.

• Help them develop the habit of analyzing new tasks to detect relationships with previously learned tasks. Do this by modeling the process for them.

• Have these students experience productive levels of challenge in the learning process and encourage them to exert effort before asking for help.


These students are frequently working at the limits of their working memory capabilities. Reducing the burden on working memory can have a significant effect on their success in learning.

• Put all the needed information on a single sheet of paper.

• Use familiar, concrete concepts rather than unfamiliar, abstract symbols.

• Provide extensive practice so that students master skills in problem solving and comprehension.

• Provide reminders and opportunities to check their work in groups or with feedback. Reducing the need for self-monitoring can be especially effective early in the process of acquiring a new skill or strategy.

Burdens on working memory change dramatically as these students gain proficiency with a skill. What is initially overwhelming can be well within a student’s range with practice.

Scaffold Wisely Students with average reasoning abilities tend to learn most effectively in the following conditions:

• school environments that are somewhat, but not highly, structured

• instruction that is moderately paced and provides frequent monitoring and feedback on their progress

• instruction on metacognitive and study skills to support their learning

Provide students with enough support in the form of strategies, memory prompts, and task structure so they can infer, deduce, connect, and elaborate (in short, so they can understand and think for themselves).



Average Reasoning Abilities (Stanines 4–6) (continued)



When these students learn to be more strategic, memory burdens are reduced and thinking leads to better results. Common challenges encountered and instructional methods to help students overcome them include:

• Frequent errors in implementing learning strategies

– Provide frequent monitoring when the students are learning a new strategy so that errors can be corrected early and not practiced.

– Model correct implementation of a strategy rather than describing it.

• Lack of effective study skills

– Provide direct instruction in study skills such as note taking, outlining, diagramming, and planning use of time.

– Formulate questions to guide their study.

• Inability to solve complex problems

– Show students how to break up complex problems into simpler units.

– Provide tools and methods for tracking progress in solving complex problems.

– Help students become mindful of their strengths and weaknesses and the effectiveness of different strategies in different contexts.


Many cognitive skills are learned first by observing other students interacting and then by gradually learning to participate in the same sort of exchanges. Plan group activities with these guidelines in mind:

• Try to structure group interactions so that all students have an equal opportunity to participate. Research shows that students with average abilities are often left out of group problem-solving efforts.

• Structure groups so that higher-ability students model higher-order skills (via student conversations) before group members practice the skills. Only after extensive practice can a skill be fluently performed and then executed without supports.

• If grouping is based on ability, provide opportunities for students to move between groups frequently as the content changes and as their skills increase.


Above Average Reasoning Abilities (Stanines 7–8)


Learner Characteristics • Ability to learn relatively quickly

• Good memory and effective learning strategies

• Typically need less practice to master a skill compared with average students

• CLD Tips:

– May show detailed solutions or products that reflect deep thinking but in nonverbal formats or a native language

– May show advanced skills in a narrow area of interest or in nontraditional domains

– Cultural expectations may lead students to speak less and not show off their accomplishments

Build on Strengths • Recognize that these students generally benefit when allowed to discover relationships themselves. Guided discovery methods work better than more structured teaching methods.

• Challenge them with materials, projects, and problems that are somewhat more difficult than those used for the typical student.

• Improve their reasoning skills by encouraging them to precisely, rather than approximately, describe the relationships among concepts or the rules that sequence them.

• Encourage these students to follow their interests, and reward perseverance on long-term projects.


• Complexity: Teach these students to understand how highly complex tasks can be broken into a series of simpler tasks or skills. Provide focused practice on these components, building up to the complex task.

• Record Keeping: Teach students how to monitor their own thinking and problem solving by recording their thought process on paper. Show them how studying the written record allows them to focus, reflect, revise, and clarify their thinking.

Scaffold Wisely These students typically have effective learning strategies in place and are generally good at recognizing when they need help in order to accomplish a task. They can benefit from:

• instruction that helps them plan the use of different strategies in different contexts

• working with higher-ability peers, particularly on difficult problems or learning tasks

• guidance on using more effective strategies or implementing strategies correctly



Above Average Reasoning Abilities (Stanines 7–8) (continued)



Able students are quick to acquire different learning strategies. The following approaches are suggested for these students:

• Expose them to alternative strategies, especially if modeled by respected adolescents and adults. Help students appreciate the value of different strategies for different purposes and problems.

• Encourage students to try each modeled strategy and help them keep track of the results. As students progress beyond middle school, encourage them to expect changes in strategies that work best for learning.

• Provide these students with enough challenge that they must think strategically and persist in the challenge to learn new skills.

• Encourage proactive planning to manage frustration and other emotions during problem solving.


Above-average students are generally excellent group participants, especially if the group is structured so that no one can dominate the discussion or be left out of it. These students can learn well in groups by explaining, by helping to summarize discussions, and by modeling higher-order thinking skills for other students.

Very High Reasoning Abilities (Stanine 9)


Learner Characteristics • Preference for discovery learning rather than highly structured learning environments. When adapting instruction for these students, realize that good discovery learning need not be a solitary task.

• Need for the company of other learners who model new ways of understanding a problem and who challenge these learners to improve their current understanding.

• CLD Tip: Some cultures have norms that may lead students to speak less or not call attention to their accomplishments.

Build on Strengths The single greatest need of these students is for academic challenge at a level commensurate with their abilities and achievements. Consider the following instructional adaptations for these students:

• Carefully select challenging instructional materials, special projects, or other enrichment activities.

• Offer instruction, particularly in mathematics, at a level that may be several years advanced.

• Provide opportunities to explore interests and skills in independent, open-ended projects.


When helping these students acquire new academic skills, consider these adaptations:

• Encourage mindful and self-regulated learning, even for students in early primary grades.

• Let them try different skill-acquisition strategies and monitor the effectiveness of each.



Very High Reasoning Abilities (Stanine 9) (continued)


Scaffold Wisely These students need access to instruction that allows and encourages them to develop their academic skills. Some also need help coping with negative feelings, such as anxiety. Learning to persist in the face of difficulty can also be an important affective or motivational issue for very able students. Working with an older and more experienced student (or adult) can be especially beneficial.

Be sure that adapted instruction for these students is truly advanced and not just busy work or extra work. Quality, not quantity, should be the goal of adaptations.


These students are generally receptive to activities that allow them to discover how they can best use their cognitive resources.

For students in the early primary grades, this can mean learning not only that there are different ways to attain competence in performing a skill, memorizing poetry, or solving problems, but also that learners must discover which methods work best for them.

For older students, the emphasis should be on developing a willingness to expand their reasoning abilities in these ways:

• Reflect on existing knowledge to compare, contrast, and internalize new information.

• Shift perspectives and consider alternative opinions and evidence.

• Entertain increasingly sophisticated theories of what counts as knowledge and evidence.


These students can benefit from group interactions when they are able to explain difficult concepts to other students, but they learn more when they are able to participate as learners as well.

When grouping these students with other students, try to devise groups that provide them with opportunities to be learners, not just explainers. They will also be challenged by a diversity of perspectives among peers and opportunities to practice social negotiation.

27Adapting Instruction to Build on Relative Strengths

Approximately half of the students who take CogAT show a relative strength or a relative weakness in one of the three test batteries. Understanding this provides the opportunity to adapt instruction to build on the student’s strengths and shore up any weaknesses.

Ability profiles with a V+, Q+, or N+ indicate a relative strength on the Verbal, Quantitative, or Nonverbal Battery, respectively.

Profiles that show a relative strength are more common for low scores (median stanines of 1, 2, or 3) than for high scores (median stanines of 7, 8, or 9).

Profiles are especially important for understanding the abilities of the students with weaker reasoning skills. Extreme profiles that show an extreme strength are most common for students with a median stanine of 1. In fact, for students with a median stanine of 1, profiles are evenly split between those that show a significant or extreme strength and those that show a relatively flat (A) profile. Both occur for about 45% of students nationally. The information that follows offers suggestions on adapting instruction to build on a relative strength indicated by a student’s CogAT Ability Profile.

Relative Strength Cognitive Domain Page

V+ Verbal 28

Q+ Quantitative 30

N+ Nonverbal 30

Ability Profiles with a V–, Q–, or N– indicate a relative weakness on one of the three CogAT batteries. Guidance on how to shore up weaknesses begins on page 33.

Part 4 Adapting Instruction to

Build on Relative Strengths


Relative Strength in Verbal Reasoning (V+)

Research-based Principle Guidelines for Adapting Instruction

Learner Characteristics These students typically obtain higher-than-expected achievement test scores in all areas except mathematical computation. The differences between observed and expected achievement are smallest at the primary level and largest at the secondary level. A strength in verbal reasoning has this broad effect on achievement because verbal reasoning abilities are important for success in virtually all school subjects.

CLD Tip: Facility with “code switching” and adapting to the language of others may be a sign of verbal strength. Teachers should seek to find evidence of verbal creativity and flexibility while remaining open to how these skills are expressed. English learners may show strong ability to find cognates and relationships between their multiple languages.

Relative Strength • These students generally do best when they are encouraged to talk and write about what they are attempting to learn.

• These students often have remarkably good memories for arbitrary sequences of sounds, letters, words, and events. As a result, they often excel in spelling, knowledge of syntax and grammar, ability to learn other languages, and ability to remember dialogue, prose, and poetry.

• These students should have opportunities to demonstrate these skills through writing and one-on-one conversations with the teacher, not just in class discussions.

Building on Strength • Offer greater challenges in areas of the curriculum that involve reading, writing, and speaking.

– At the elementary level, this includes providing special reading or writing assignments that are more demanding than the assignments given to other students.

– At the secondary level, if scores on the Verbal Battery are high (stanine 8 or 9), this includes placement in honors or advanced-placement classes.

• Encourage these students to use their verbal reasoning skills in other curricular areas, particularly in mathematics.

– Restate mathematical expressions verbally and explain them to others.

– Provide the student with an incorrect answer and ask them to explain the errors made.


Adapting Instruction to Build on Relative Strengths 29

Relative Strength in Verbal Reasoning (V+) (continued)


Building on Strength (continued)

• Students with relatively strong verbal abilities often find it easier to memorize formulas than to build more abstract, often spatial, mental models. The abstract model leads to long-term retention of mathematical concepts and to the ability to transfer mathematical knowledge to unfamiliar domains.

– Graphing data and using visualization software can help discourage these students from simply memorizing formulas.

– Use learning materials and test problems that allow these students to use their excellent verbal reasoning skills when learning mathematics.

– Make the application of math facts more effortful to discourage rote memorization.

• Encourage the habit of creating a mental model and coordinating it with a verbal description. These students sometimes have difficulty creating a visual mental model of the scenes depicted in a story.

– When reading aloud to these students, especially in science and social studies, pause to respond to their questions or to ask what they envision.

– Have them sketch models and concept maps.

– Select texts with illustrations and ask students to make explicit connections between the text and the illustration.

– Visual representations may include scientific data or graphs, lab plans, map directions, historical timelines, or diagrams.

• For young students, allow them to make a model of the situation described in the story and then have them alter the model as changes occur in the text. Their goal is to learn how to create a visual mental model that allows them to keep track of the persons and events described in the text.

• CLD Tip: Ask ELL students to describe a diagram or a story they have written. With prompting, they may reveal more about their verbal reasoning than the artifact itself.

– Use think-pair-share strategies in math and science class so that ELL students can refine their English answer before sharing it with the whole class.

Developing Verbal Strengths for Students Who Avoid WritingSurprisingly, a large number of students who demonstrate strong verbal reasoning skills (V+ profiles) show reluctance in writing tasks. It may be necessary to help them develop a love of writing by providing meaningful writing assignments with low stakes attached. Here are some ideas:

• Creative writing journal

• Captioning cartoons or creating comic books online

• Funny writing prompts

• Writing for an audience on a topic of interest

• Writing prompts that ask students to use creativity, like writing in dialect or poetry formats

• Participating in web-based writing communities


Relative Strength in Quantitative Reasoning (Q+)


Learner Characteristics Students are capable of abstract thinking from an early age. Students in the primary grades who show a strength in quantitative reasoning tend to score somewhat higher than expected (on the basis of their verbal and nonverbal reasoning abilities) on both the mathematics and language portions of standardized achievement tests because of their skills at pattern recognition. By the elementary years, however, this advantage is confined to mathematics and persists through the high school years.

Relative Strength • At lower ability levels (median stanine), a quantitative strength may be apparent in the student’s abilities with the computational aspects of mathematics rather than the conceptual aspects.

• These students typically excel in identifying patterns and then reasoning by using their abstractions. Examples include learning base number systems other than base 10.

• They often learn computer skills more readily than their peers, especially skills such as organizing data, creating graphs, using computational logic in robotics. They may not exhibit strengths in computer programming languages.

• Students who excel at pattern recognition often show strong knowledge of grammar.

• May enjoy math puzzles, such as Sudoku, and creative math challenges.

Building on Strength • Some students may require acceleration or honors classes while others benefit from enrichment activities such as math clubs.

– Selecting appropriate strategies requires knowledge of a student’s level of achievement in mathematics and of personal factors such as anxiety about working with older students.

• These students may benefit from presenting math solutions or data interpretations verbally.

• Collaborative projects can build verbal reasoning and support math interests. Activities include investigative math projects, using statistics with data, and exploring evidence and claims.

• Select activities that are cooperative rather than competitive.

Relative Strength in Nonverbal Reasoning (N+)


Learner Characteristics Students who show a relative strength on the Nonverbal Battery can be either good at reasoning with spatial representations or just effective at solving novel problems that are unlike those encountered in school. Students with strong spatial abilities often experience difficulties in verbal fluency or in remembering sequences of words or letters (as in spelling). On the other hand, these students often excel at drawing, sculpting, and other visual and mechanical arts. Deciding which skill a student is demonstrating requires information from observing the student or asking about extracurricular activities or career interests.

Because of their relative weakness in verbal and quantitative reasoning abilities, these students need activities both in and out of school that will develop these reasoning abilities. For suggestions on improving these areas, see “Adapting Instruction to Shore Up Weaknesses,” beginning on page 33.


Adapting Instruction to Build on Relative Strengths 31

Relative Strength in Nonverbal Reasoning (N+) (continued)


Relative Strength The suggestions in this section are based on the interpretation that the N+ profile represents a strength in spatial thinking.

• Students tend to prefer visual mental models when solving problems.

• Learning is easiest when these students can readily connect each new concept or relationship with a mental or physical model (e.g., a schematic drawing) of the situation.

• These students respond well to texts that contain difficult graphics and prefer maps to verbal directions.

• At younger ages, these students learn most readily when the concepts described in textbooks and other media have previously been experienced concretely and can subsequently be applied concretely.

Building on Strength • For young students, provide reading texts that contain detailed illustrations for unfamiliar content for which the students cannot form their own mental model.

• In all areas of the curriculum, but especially in science and mathematics, use metaphors, analogies, and real-world examples to help students connect unfamiliar, abstract concepts to more familiar objects or experiences.

• When material is presented verbally at a rapid rate, provide pauses or allow students to control the rate at which the information is presented, such as pausing and replaying a video presentation. When multiple sources of information are provided, allow time for review of each source.

• Encourage students to create drawings when solving problems in mathematics, concept maps when taking notes, or mental models of a scene when reading a text.

• Provide a hands-on approach to learning. Relate traditional academic subjects to students’ interests and offer physical applications for problem solving.

• When teaching writing, encourage these students to try descriptive rather than narrative prose. Provide examples of good descriptive prose.

– Have them first envision the scene they would like to describe before they attempt to describe it to someone else.

– For young students especially, ask, “What do you see?” and allow them to describe a mental picture. With older students, ask them to illustrate the scene.

• Encourage application of these students’ spatial reasoning and thinking abilities. These students are often quite skilled in the visual arts and can excel in trades such as carpentry, landscaping, product design, and computer graphics.


Spatial Reasoning in School

Most achievement tests do not measure spatial reasoning. A strength in and preference for spatial reasoning creates obstacles in the predominantly linear and verbal thinking required by conventional schooling. Therefore, students with N+ profiles will benefit from efforts by the teacher to incorporate the skills of these students.

Although much effort is directed toward the development of students’ verbal and quantitative reasoning abilities, little effort is made to develop their spatial reasoning abilities. Yet these abilities routinely play an important role in high-level learning and in creative contributions in mathematics, science, engineering, and the visual arts. Like verbal and quantitative reasoning abilities, spatial reasoning abilities respond to instruction. This lack of direct instruction on spatial reasoning skills has been implicated in the disparity of women entering college majors that require spatially intensive courses, such as physics, engineering drafting, or organic chemistry.

Recognizing Strengths and Weaknesses for English Language Learners (written with Barbara Kozak)

When presented with a diverse classroom, teachers are faced with the challenge of not only diversifying lessons, challenges, and expectations based on academics and Ability Profiles, but also students’ needs as they relate to their home culture, home language, and socioeconomic standing. English language learner (ELL) students often fall in multiple categories of diversity, bringing a host of challenges when meeting their needs academically and socially.

In an elementary classroom, these strengths and weaknesses may appear in a variety of ways. For example, ELL students will often excel in activities that involve a creative component, such as drawing and illustrations or creative writing. These activities can allow opportunities for ELL students to flexibly choose how to express their ideas. Observing detailed and creative products in a story-telling context can be an indicator to the teacher that verbal strengths are present but may be obscured by language proficiency. Verbal and social language skills will often be much higher than written or academic language skills; especially in their nonnative language.

For example, a second grade student was faced with a blank page during a writing activity. The teacher understood that he spoke Spanish and could not write well in English—yet. It was obvious from the student’s drawing, which they were supposed to be writing a story about, that he had an elaborate story in his head that he had illustrated on paper! This was confirmed when he explained his story aloud. When the teacher asked if he could write his story in Spanish he vigorously nodded with a huge grin on his face. The paper was soon filled with a complete story, in Spanish, about aliens visiting from another planet! The problem was not coming up with the story; it was finding the “correct” words to express the story.

Teachers must recognize that an ELL student who is leaving the classroom for language intervention can still be a high-ability learner.

When ELL students are given the opportunity to learn content, do they meet the challenge? Can they show their thinking in nontraditional ways? Is their thinking detailed and sophisticated? In the example above, the young boy had not been looked at as a high-ability student because he could not complete some specific activities. Alternate activities without a language barrier (drawing about a topic) allowed this student to tell his story in his native language, showing his creative thinking and elaborate ideas.

Recognizing weaknesses may also take time and careful attention from teachers. An ELL student who learns conversational English more slowly than his ELL peers or who has greater trouble writing in English or his native language than peers may have a relative verbal weakness. Identifying this need as soon as possible is key to providing appropriate instruction to this ELL student.

33Adapting Instruction to Shore Up Weaknesses

Ability Profiles with a V–, Q–, or N– indicate a relative weakness on the respective CogAT battery. When a student displays a significantly lower score on one of the three batteries, it typically indicates a preference for thinking in one cognitive domain (verbal, quantitative, or nonverbal) rather than another. Areas of weakness often cause frustration and avoidance. Effective teaching will help students build perseverance and avoid learned helplessness.

Extreme profiles that show weakness are most common for students with a median stanine of 9. Indeed, for students with a median stanine of 9, profiles that show a significant or extreme weakness are almost as common as relatively flat (A) profiles. This is one reason why the CogAT authors discourage use of the overall CogAT composite score to identify academically talented students.

The information that follows offers suggestions on adapting instruction to shore up a weakness indicated by a student’s CogAT Ability Profile.

Relative Weakness Cognitive Domain Page

V– Verbal 34

Q– Quantitative 35

N– Nonverbal 37

Relative Weakness in Verbal Reasoning (V–)


Learner Characteristics These students prefer nonverbal (visual) or quantitative reasoning and often find it difficult to translate their thoughts into words. Over time, this preference causes a lag in their development of verbal abilities of all sorts.

Students with this profile often have lower scores on achievement tests than would be expected on the basis of their median stanine (ability level). These weaknesses may be more apparent in writing than in spoken communication. It may also appear as a reluctance or low motivation for reading activities.

Some students who exhibit relatively poor verbal skills do not routinely participate in conversations that involve formal language structures or meaningful dialogues. They may need exposure to a greater variety of reading materials and multimedia.

Rule out disabilities like dyslexia and support students with disabilities to develop their verbal skills despite their challenges. For English learners, recognizing a verbal weakness can be difficult to identify. Try teaching strategies tailored to disabilities and to ESL to determine which leads to greater gains.


Part 5 Adapting Instruction to Shore Up Weaknesses


Relative Weakness in Verbal Reasoning (V–) (continued)


Relative Weakness • Activities that involve more verbal demands may reduce students’ performance even in areas in which they excel.

• Common sources of difficulty are directions that are too wordy and test formats with excessive reading demands.

• Minimize competing sources of verbal information. For example, these students will find it difficult to view a rapidly paced video presentation and take notes at the same time.

Shoring Up the Weakness • To improve performance and reduce frustration, reduce the demands placed on verbal working memory.

– Do not expect these students to keep in mind a verbal statement and apply it at the same time. Allow the student to use a prompt, such as a written statement of the concept or strategy needed for the work at hand.

– Allow many opportunities to practice a new strategy in diverse contexts.

– Help students who scored at lower stanine levels to identify the conditions that cue possible use of a new reasoning strategy. The goal is for students to learn to apply different procedures as circumstances demand and not rely on fixed strategies in all cases.

– Use videos and preview worksheets to introduce vocabulary words ahead of a lesson. This strategy, also helpful for English language learners, can reduce memory requirements during a content lesson.

• Offer a broad language curriculum that combines reading, writing, and speaking as well as opportunities to practice and receive feedback on each.

• Introduce students to unfamiliar ways of conversing by asking them to imitate the speaking and writing styles of individuals they admire. Drama, poetry, and storytelling are particularly useful.

• Provide reading assignments and follow-up discussions or activities designed to build verbal comprehension. Emphasize learning vocabulary in context, not as a standalone activity.

• These students may benefit from morphology lessons across content areas, studying the origins of words and patterns in grammar.

Adapting Instruction to Shore Up Weaknesses 35

Relative Weakness in Quantitative Reasoning (Q–)


Learner Characteristics These students tend to score somewhat lower across all portions of standardized achievement tests, especially at the primary level. The difference is largest on the mathematics, computation, and language tests.

There are many causes of a relative weakness in quantitative reasoning. Some students have difficulty creating, retaining, and manipulating symbolic representations of all sorts. For some students, this problem seems confined to numbers; for others, it stems from a fundamental difficulty in thinking with abstract concepts and number sense.

A relative weakness in quantitative reasoning abilities generally has a broader impact on the achievement of students than does a relative strength in quantitative reasoning. The connection between lower achievement on the computation and language tests could reflect a common difficulty in learning rule-based systems, or it could reflect a lack of instruction in both areas. Gather information on the students and the educational curricula to make this judgment.

Relative Weakness • Some students prefer concrete modes of thinking and often fail to think abstractly when using verbal concepts. For example, a student may understand how to use money but not understand how to perform math problems with currency concepts.

• For other students, the difficulty lies in the failure to develop an internal mental model that functions as a number line. For these students, solving even basic computations such as adding 2 to a given number is a challenge. When performing computations, such students often make substantial errors that they do not detect unless prompted—and even then, they may not notice the errors.

• A final explanation is that the weakness represents lack of experience in thinking and talking about quantitative concepts. This occurs in the primary grades and sometimes occurs at the secondary level among those who avoid mathematics. At the middle school and high school levels, math anxiety can also be a significant issue.



Relative Weakness in Quantitative Reasoning (Q–) (continued)


Shoring Up the Weakness Based on the source of the deficit, select strategies from the following list that seem most appropriate for the student and the learning situation:

• If students have difficulty reasoning abstractly, help them focus on the quantitative aspects of a problem while ignoring more compelling verbal and figural features. For example, even foundational concepts in mathematics are abstractions. When counting objects, students must recognize that the number 3 in “3 oranges” means the same thing as the number 3 in “3 automobiles.”

• If students have not established a mental model for representing numeric quantities (e.g., number sense), give them practice in drawing a number line and then have them use a mental number line to solve basic addition and subtraction problems. Substantial practice is needed to automatically use a mental number line to solve problems, but this ability to reason and use number sense is crucial for later learning.

• Help these students discover how to use their better-developed verbal and spatial reasoning abilities for solving mathematical problems. Especially in middle school and high school, encourage these students to develop the habit of restating mathematical expressions in words. Ask them to talk about mathematical concepts rather than silently solving problems on worksheets or computer screens.

• If consistent with desired learning outcomes, encourage students to use computers and other tools to offload lower-level computation processes and to focus instead on higher-level concepts. This is often best done using graphic representations of geometric and algebraic concepts.

• Ask them to create drawings that represent essential aspects of a problem. Show them how drawings can range from concrete depictions of the objects described in the problem to increasingly abstract representations that capture only the essential aspects of the problem.

• If the difficulty is a lack of experience or the presence of anxiety, provide greater structure, reduce or eliminate competition, eliminate time pressures, and allow students greater choice in the problems they solve. Experiencing success will gradually reduce anxiety; experiencing failure will cause it to spike higher.

• Engage students in authentic expanded mathematics projects to create relevance and personal interest.

• Look for signs of learned helplessness, especially among girls and CLD students. Avoid reinforcing a belief that there are people who are naturally good at math. All students need to develop their math reasoning skills.

• Engage less motivated students in authentic expanded projects in mathematics to bring personal relevance and interest into math practice. This can also concretize abstract concepts in verbal and visual representations.

Adapting Instruction to Shore Up Weaknesses 37

Relative Weakness in Nonverbal Reasoning (N–)


Learner Characteristics At the primary and elementary levels, these students tend to have lower scores on standardized achievement tests in the areas of reading and mathematics. At the secondary level, the deficit is largest in the area of science.

A weakness in nonverbal reasoning ability has more noticeable and negative consequences for average-ability students than for students who score in the high (stanines 7–8) or very high (stanine 9) range on CogAT.

Be sure that all students have an equal opportunity to prepare for taking CogAT. Request the freely available practice tests to help all students gain familiarity with the test format. This is especially important for the Nonverbal Battery because these tests are often the most novel for students.

Relative Weakness There are two explanations for a relative weakness in nonverbal reasoning: Either the student has difficulty reasoning with figural-spatial stimuli or the student has difficulty solving unfamiliar problems. Before adapting instruction for these students, try to identify the source or cause of the deficit.

For most students, N– profiles are caused by difficulty with figural-spatial stimuli. Although only moderate levels of spatial reasoning abilities are required for success in school, students with weak spatial reasoning abilities encounter difficulties in many areas of the curriculum, especially science and mathematics.

Sometimes the N– profile indicates a difficulty solving novel problems. If this is the case, you are likely to notice a consistent decline in performance as the student moves from school-like tasks to unfamiliar tasks. For example, in the verbal domain, the student performs best on a language achievement test, somewhat lower on a reading achievement tests, lower still on the CogAT Verbal Battery, and lowest on the CogAT Nonverbal Battery. A similar progression would be apparent in the quantitative domain.

Difficulty in solving novel problems is also suggested when the student works diligently, even obsessively, at school tasks. Such students often become anxious when placed in situations that lack clear guidelines on what they are expected to do or how they will be evaluated.

Shoring Up the Weakness Spatial reasoning abilities can improve with instruction. Educational planning for students with N– Ability Profiles should include training in the specific types of spatial thinking required by the curriculum. For example, provide explicit training on interpreting diagrams and reading graphs. In other learning situations, it will be easier for the students if instruction compensates for, or scaffolds, their poor spatial reasoning abilities.

• Provide printed or digital formats that students can mark on.

• Allow students to inspect and physically manipulate objects if necessary.

• Start with concrete objects and physical models of concepts used in the curriculum. Then teach students to draw the model from memory. For example, the act of drawing a map from memory will result in greater retention of the images than having students view the map without any drawing.



Relative Weakness in Nonverbal Reasoning (N–) (continued)


Shoring Up the Weakness (continued)

• Replace the question “Do you see…?” with the more informative “What do you see?” Have students create descriptions or inferences from visual information.

• Provide simple drawings that encapsulate the essential features of the visual mental model required by the problem. Then give students time to examine the drawing and to label it or coordinate it with the text.

• When possible, do not require the students to shift their attention between two different locations, such as a drawing displayed on the board or LCD projector and a description of the problem in a textbook or workbook. Place the text and drawing in view together or allow students to study the drawing while you read the problem aloud or explain it to them rather than requiring students to read the text themselves.

• Use computer graphics or physical models to display problems that require transformation of images such as imagining how the drawing would appear from another perspective or following a dynamic transformation. This can be especially helpful in mathematics.

• When teaching strategies, summarize them in short verbal statements or mnemonic devices that can be rehearsed and committed to memory. When practicing strategies, encourage these students to repeat (aloud) the statements as they perform each step.

• Reduce the need for students to visualize by providing drawings, using computer graphics, or having students work in groups in which a partner performs this part of the task.

If you believe the N– score pattern seems to reflect a difficulty solving problems unlike those encountered in school rather than a relative weakness in spatial reasoning, the following strategies are suggested:

• Engage the students in discovery learning or provide structure to their explorations. (See Part 7 for a description of science inquiry.)

• A student’s problem-solving skills should proceed in small increments on unfamiliar, less-structured situations.

• Encourage and reward engaging in tasks that are less familiar and increasingly less structured. This approach gives students practice in assembling and reassembling strategies to solve new problems.

39Adapting Instruction for Mixed Ability Profiles

Part 6 Adapting Instruction for

Mixed Ability Profiles

C profiles show a significant contrast between the student’s highest and lowest battery scores. The general pattern for C profiles is one high score (a relative strength), one middle score, and one low score (a relative weakness). E profiles are a more extreme form of this pattern.

In a CogAT report that graphs a student’s battery scores, scores that differ significantly have confidence bands that do not overlap. If the bands around two scores overlap, those scores do not differ significantly from one another. In the example, the Verbal and Quantitative scores differ significantly. For this student, Quantitative is a relative strength and Verbal is a relative weakness.

Achievement Test Performance

The achievement test scores of students who have C profiles generally fall midway between the scores for the two corresponding B profiles. For example, students with the Ability Profile 4C (V+ Q–) show achievement levels that are approximately midway between those shown by the students with 4B (V+) and 4B (Q–) profiles. This means that the consequences for achievement test scores for students with C profiles are smaller and less easily summarized than those for students with B profiles.

Adapting Instruction for Students with Mixed Ability ProfilesStudents with C (mixed) Ability Profiles are the most challenging to assist with planned interventions. It is often difficult to know when particular instructional methods or materials will capitalize on the students’ strengths or compensate for their weaknesses. For example, students who have difficulty creating and reasoning with mental models often perform much better if given a concrete model or a line drawing when attempting to understand a problem. If the model or graphic is too complex, however, then it may require spatial reasoning that exceeds a student’s capabilities. These effects differ among students depending on the complexity of the model, a given student’s familiarity with it, and the level of each student’s spatial or figural reasoning abilities.

When a student has both a relative strength and a relative weakness, as in a C profile, it becomes very difficult to know how a given intervention will be perceived and processed by the student. Plan a strategy based on your knowledge of the student’s learning preferences and challenges and your experience with the curricular materials.


Ultimately, the learners’ success as they try to navigate a lesson, a unit, and, eventually, a course will help you determine whether a strategy is working as planned. Therefore, although all learners should be encouraged to develop strategies for regulating their own learning, such self-monitoring and self-reflection are particularly important for students with mixed patterns of cognitive strengths and weaknesses. Help these students understand that the process of learning, using, and then evaluating different strategies is similar to the process of trying on different articles of clothing to see how they fit. Explain that, like clothing, the strategy that fits best now may change as they mature or as the context varies.

41Case Studies and Instructional Examples

In this part of the guide, we provide a series of case studies on how the principles described in this guide might be applied to each content area.

• CogAT Classroom Summary Worksheet (for organizing student scores by profile)

• Adapting to Interests without Overextending Yourself

• Engaging All Students in Rich Mathematical Tasks

• Adjusting Scientific Inquiry Based on Profile Level

• Addressing a V– Profile in Reading

• Targeting Higher Reasoning Abilities in a Social Studies Classroom

• Using Investigative Math to Better Understand Quantitative Reasoning

• Playing Games and Reducing Demands on Working Memory to Increase Mathematical Thinking

CogAT Classroom Summary WorksheetThe CogAT Ability Profile captures two characteristics of the student’s scores:

• level – the typical magnitude of scores on the three batteries

• pattern – some scores are significantly higher or lower than other scores

14%

Verylow

Stanine scoreFrequency

Interpretation

Normal Distributionof Scores

Belowaverage

Average Aboveaverage

Veryhigh

27%

312%

417%

520%

617%

712%

87%

94%

Each CogAT Ability Profile begins with a number that represents the student’s level using a median stanine. A stanine indicates one of nine broad score groupings on a normalized standard score scale. Stanines range from 1 (lowest) to 9 (highest).

The letters that follow the median stanine describe the profile pattern.

In an A profile, the student’s Verbal, Quantitative, and Nonverbal Battery scores are roughly at the sAme level. About 44% of students have this profile.

In a B profile, two of the confidence bands overlap. The third score is a relative strength or weakness, significantly aBove or Below the other two. About 33% of students have a B profile.

Part 7 Case Studies and

Instructional Examples


In a C profile, two scores Contrast. The student shows a relative strength and a relative weakness. About 12% of students have a C profile.

An E profile indicates Extreme score differences. At least two scores differ by 24 or more points on the standard age score (SAS) scale. About 10% of students have an E profile.

Stanine level

Profile shape 1–3 4–6 7–8 9

sAme

aBove + (strength)

Below – (weakness)

Contrast

Extreme

Case Studies and Instructional Examples 43

Stanines 1–3 Verbal Quantitative Nonverbal

Names + for strength, – for weakness





Stanine 9 Verbal Quantitative Nonverbal



Example of completed table:

Stanine level

Profile shape 1–3 4–6 7–8 9

sAme Ro 2ASusan 4A

Liza 5AChris 7A

Pat 9A

Rita 9A

aBove + (strength) Cindy 2B V+ Ann 6B N+ Eva 8B N+

Below – (weakness)Todd 6B V–

Dev 5B V–Ivsa 7B N– Joe 9B Q–

Contrast Mika 8C V– N+

ExtremeLee 1E Q+

Torv 3E V+

Todd 6B V–

Dev 5B V–Riva 9E N–

Adapting to Interests (without Overextending Yourself) As we stated earlier, it is common to think that adapting instruction means creating unique lessons for each student. This is neither necessary nor feasible. Here are a few ways our collaborating teachers have created opportunities for students to follow their interests.

Create opportunities for self-selectionMs. Bass lets students choose their own independent reading books several times each year (in addition to grouped or whole-class assignments). She uses this reading time to informally ask students about their books and track their progress. Older or more able students may be able to peer monitor and set goals for themselves rather than have frequent teacher check ins. If the teacher sets clear expectations, autonomy is important for these students.

Allow students to choose their product to show mastery with different degrees of challenge possible

This is a well-known strategy, but it can be helpful in the context of adapting instruction. Letting students choose their product is an effective way to build on strengths.

Ms. Kozak finds that students usually pick products appropriate to their strengths and overall ability. If students consistently choose the same products or low-challenge activities, the teacher can encourage variety in products to create more challenge and develop areas of weakness.

Group products can also help students engage in activities that may require them to develop weaker skill areas. Working with other students allows them to shore up each other’s weaknesses.


Engaging All Students in Rich Mathematical Tasks Effective Mathematics Teaching Practices (National Council of Teachers of Mathematics, 2014) encourages implementing tasks that promote reasoning and problem solving. These tasks have multiple entry points and encourage various strategies for solving, which allows teachers to differentiate their instruction in order to support students with high quantitative reasoning (Q+ profiles) and students with low quantitative reasoning (Q– profiles). Ms. Horn implemented the following task in her fourth-grade classroom:

Problem: The latest season of Sam’s favorite show was just released on Netflix, and he wants to watch the entire season before any of his friends accidentally tell him what happens first, spoiling the surprise. There are twenty-four episodes in the new season, and Sam begins to wonder if he can finish the entire season in one day. If each episode is twenty-five minutes long, how long will it take Sam to watch the entire season if he doesn’t take any breaks? Provide evidence to support your thinking.

Enrichment (Q+) adjustment: Sam quickly realizes he will not be able to make it through the series without taking breaks. He takes a break for 1/3 of an hour after every six episodes to make a snack. He takes a break for 1/12 of an hour after every four episodes to use the restroom. How many hours did it take Sam to finish the series including the breaks?

Support for Weaknesses (Q– or V+/N+): Start the problem by identifying and pulling out key features of the problem through close reading of the problem. Circling key words may also help. If students have a verbal strength, encourage them to explain the problem and then their solution to a peer. If students are having difficulty, and have a relative nonverbal strength, encourage them to draw a model.

Example Solution StrategiesStudents with high quantitative reasoning (Q+) may use efficient strategies such as doubling and halving. Using this strategy, students double one factor and halve the other in order to create an easier problem with the same product.

25 x 24

Doubling 50 x 12 Halving

100 x 6

Because 100 x 6 = 600, students know 25 x 24 = 600 as well. These students could be challenged to write an explanation to accompany their solution before working on the enrichment activity that is designed to further extend their learning. Verbalizing solutions can also allow students with verbal strengths to leverage that skill in their mathematical understanding.

Students with a weakness in quantitative reasoning (Q–) may use strategies such as repeated addition that are more likely to contain errors due to the large volume of numbers added together. When students explain their strategies to the class, the teacher could encourage students to look for relationships between the strategy below and the doubling/halving strategy.

25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25+25

50

100 + + + + + = 600100 100 100 100 100

50 50 50 50 50 50 50 50 50 50 50


This task provides an entry point for all students to develop problem solving strategies. Students with a weakness in quantitative reasoning should be encouraged to use any strategy such as visual models, verbal rules, or concrete manipulatives regardless of inefficiency. It is important for them to develop confidence in their ability to reason and make connections to more efficient strategies to progress their learning.

Adjusting Scientific Inquiry Based on Profile LevelThe levels of scientific inquiry map well onto the levels of challenge that may be appropriate to students with different overall profile levels (i.e., stanine scores) or within a content area based on a relative strength or weakness. Scientific inquiry can be boiled down to a simple definition with three parts: it involves answering a scientific question (1) by analyzing (2) data (3) that has been collected.

Level 1 Confirmation Inquiry involves a confirmation study where the question, data collection procedure, and analysis is determined by the teacher. Usually the students and teachers know what will happen in the study. Each higher level of inquiry involves releasing one of these constraints, ultimately leading to Level 4 Open Inquiry where the question, procedures, and analysis are all determined by the students doing the investigation.

For students in the same grade level, providing structure can be important for adjusting the scientific exploration for students’ strengths and weaknesses. Pairing students with more able classmates can be one solution. Providing clues, suggestions, or limitations on the inquiry process can also help adjust the difficulty. For example, when conducting a science fair project in middle school grades, students in stanines 1–3 may be most successful when they identify an existing project, such as those on websites or in books, and make small adjustments to the procedure to create a personally interesting investigation. For students in stanines 4–6, they may still start with an existing project idea but may be comfortable coming up with a novel analysis approach or applying the procedure to a wholly new topic. Students in stanines 7–9 should be able to tackle more novel scientific questions and design appropriate (although possibly basic) analyses of their data to reach a scientific conclusion.

Four levels of inquiry

Inquiry LevelScientific Question

Data Collection Procedure

Solution or Interpretation

(1) Confirmation X X X

(2) Structured X X

(3) Guided X

(4) Open

Adapted from Banchi & Bell (2008). Retrieved from https://my.nsta.org/resource/6335/the-many-levels-of-inquiry

Note of Caution. It is always important to consider the developmental nature of reasoning. Although young students or students who struggle with reasoning skills may start with confirmation inquiry tasks, as their content knowledge or their strategic knowledge increases, all students should be exposed to less structured inquiry environments over time, allowing them to have greater autonomy and experience of the true “open” scientific inquiry process while still receiving the support needed to be successful.

https://my.nsta.org/resource/6335/the-many-levels-of-inquiry


Addressing a V– Profile in ReadingAfter lunch, the third graders walk into the classroom and arrange themselves on the carpeted area in the front of the classroom. Mr. Stephson begins the read-aloud period by pointing out the cover illustration, title, and author. He invites the students to predict what the book is about and discuss any connections with a partner.

After a few minutes of observing students’ conversations, he notices Emma is looking around the room and not talking with her partner. Mr. Stephson begins to read the first three pages. Mr. Stephson asks students to share what they think will happen next. After several students have shared, he strategically calls on Emma. Emma states, “My predict that dog run away.” Mr. Stephson responds, “Your prediction is the dog will run away.” Emma nods in agreement. Mr. Stephson responds, “Well that is an interesting idea, how did you think of that?” Emma repeats, “My predict that dog run away.” “That’s one possibility, let’s see what the author has in mind,” Mr. Stephson replies.

It is the third week of school and it is clear to Mr. Stephson that Emma’s verbal reasoning skill is low. He inspects her CogAT profile and sees she has a 3B V–.

What will the rest of the year look like for Emma?

Mr. Stephson will employ the following instructional strategies to address low verbal reasoning:

• Preteach or preview vocabulary, then repeat key terms with emphasis or repetition in the lesson that follows.

• Activate prior knowledge.

• Encourage the use of graphic organizers and drawing models from readings that can be updated as the story progresses.

• Provide a word bank of important terms.

• With chapter books and group or independent reading, use sticky notes to summarize the main points of each chapter. This builds a timeline of the text the student can refer to.

• Readers’ Theater has been shown to be effective in building fluency through repetition.

• Make book available in audio so individual students can reread as needed.


Targeting Higher Reasoning Abilities in a Social Studies ClassroomWhen working with students of higher reasoning skills in a middle school social studies classroom, attempt to make the quality of the assignment challenging while avoiding increasing the quantity of work for those students. For example, suppose that the classroom focuses on five of the social studies disciplines: civics, government, history, economics, and geography. In every historical event studied, it is important for students not just to be able to recall dates and events but to be able to apply the social studies disciplines to one another and understand the relationship between them. Below are examples of ways to differentiate for learners of higher reasoning abilities in such a social studies classroom.

Historical Theater: While storytelling a historical event, a script (provided or written) can aid students’ understanding of the role the event played on government, economics, geography, or civics. In addition, students can conceptualize (verbally or with diagrams) the role each of these disciplines played in the event. Students can engage in Reader’s Theater as an author or an actor. Some quantitative historical effects (e.g., the Great Depression) can be concretized through acting out the script. Students can also capture events through time lapse where they are given specific time periods and describe or draw what has changed between those periods.

Write Your Own Definition (Context Clues): When introducing new vocabulary, begin using the word frequently in context in class discussions. Once you have repeated use of the term, students will examine it in the text. After hearing and seeing it demonstrated, students will write a working definition of the term. (Scaffolds Verbal weakness)

Timelines: Ideas for timelines include representations in reverse chronological order, vertical order, time-lapse explanations, or applying demographic comparisons to timelines (e.g., population increases). Build Nonverbal/Quantitative skills by emphasizing spatial ratios and time lapses. For Verbal weakness, emphasize complete labels with full sentences (who/what/when/where/why details).

Economics: Many historical events can be represented or enriched with tracking numerical facts or implications. Some ideas would be using data to estimate the economic performance of a product, examining fluctuations in product cost when considering supply and demand, or inferring and explaining how buying on margin contributed to the Great Depression. (Verbal/Quantitative strengths to support the other weakness)

Geography: Examples here could include using the map scale to measure geographic distances, building models such as dioramas, distinguishing geographic patterns in satellite imagery (e.g., greater illumination means greater population), detecting variations in elevations on topographic maps, and explaining how crop rotation can diminish or intensify the effects of a drought. Many innovative Geographic Information Systems technologies can be included here as well, such as online locational and directional mapping, satellite imaging, map overlays, and classroom geocaching. (Entry points for Verbal/Quantitative/Nonverbal)

Primary Sources: In examining primary sources, students may infer the intent of an artist (e.g., editorial cartoons) or the mindset of the art’s subject (e.g., Great Depression photography by Dorothea Lange, music by Woody Guthrie). Students may predict an author’s intent or create imagined contemporary sources. (Scaffold Verbal weakness)


Using Investigative Math to Better Understand Quantitative Reasoning (written with Lori Shaw)Mathematical learning can be most effective in a problem-centered, inquiry-based environment. Given properly designed instructional tasks within a culture of inquiry, students learn to construct, revise, and refine their concepts and theories to make sense of and solve problems. Difficulties that students face and the way that they overcome these difficulties lead to their mathematical discovery. When developing powerful mathematical thinking, instruction must carefully guide and support a students’ personal construction of a concept and ways of reasoning while the student tries to make sense of a situation. Yet, to be effective, this guidance and support must be grounded in detailed analyses of a child’s mathematical experiences and the individual processes by which that child develops his or her mathematical knowledge.

Case Study: Multiplication of multidigit numbersThis task must be built upon a student’s understanding of the structure of multidigit numbers and how he or she can apply that understanding when adding and subtracting. Now I can ask similar questions about multiplication: How can large numbers be decomposed and recombined in order to multiply them? What ideas about place value are significant and need to be tracked as students multiply?

It becomes important to prepare yourself for the reasoning that students use, first by trying to use various representations of multiplication yourself. For example, how would you arrange manipulatives, draw a diagram, to verbally represent 17 x 19? Do your representations suggest ways in which you might decompose the numbers to make the multiplication more manageable? When preparing for students to vary in their quantitative reasoning skills, you might ask:

• What do your students already understand about the topic?

• How can I help students to extend their ideas and to build concepts that will support their future work?

• What earlier mathematical ideas were developed that support the current work?

• What errors arise and how do the students sort them?

In a fourth grade classroom as part of multiplication exploration, I have asked students to work with arrays: to play games with them and to study them. We have discussions about the operations and how to use them flexibly, for example, by choosing to use addition and multiplication in the same problem, if possible. We use base 10 blocks and graph paper. We talk about multiplying two-digit numbers, and where addition and subtraction come into play. The children write story problems based on math sentences I have given them, and they have responded with math sentences based on story problems I have written.

Recently, I gave some students a task that asked them to solve 28 x 4 in two different ways. In their written work, I found significant clues about what they understood at that time. (See the table that follows.) Doing this assignment allowed my students to (a) practice checking their own work, (b) show comprehension and not just memorization, (c) develop more flexible number sense, and (d) understand the “why” behind algorithms.


Student answers What I know about student’s reasoning skill

Student A:

Solution #1: I added all the 20s, which got to 80, and then I added four eights and got 112.

Solution #2: I added the 20s but then I realized a shorter way than adding. I just did 4 x 8, which is 32, so 80 + 30 = 110 plus 2 more is 112.

This student’s strength with skip counting by 20s, memorized multiplication facts, and decomposing numbers allowed her to manipulate the multidigit multiplication problem.

Student B:

Solution #1: I know that 2 x 28 = 56, so 56 + 56 = 112

Solution #2: First I added up all the 20s and got 80, and then I added up all the eights and got 32, and then I got 112.

This student flexibly used the doubling strategy to arrive at the answer. This student also demonstrated the relationship with repeated addition and multiplication (2 x 28).

Student C:

Solution #1: I did the work in my head, and I thought 28 + 28 = 56, so I added 56 + 56 and got 112. I added 28 and 28 because I know it is half of 4 x 28, so I simply put the two halves together and I got 112.

Solution #2: I know that 28 is 20 + 8, so I multiplied 4 x 20 and I got 80. Then I added 4 x 8, which is 32, and I got 112.

This student built upon his addition skill strength, yet demonstrates an understanding about the relationship between addition and multiplication. This student is also able to decompose numbers and use addition to flexibly arrive at an answer.

Student D:

Solution #1: I know 2 x 28 = 56 and 2 + 2 = 4, so 4 x 28 = 112 because 56 + 56 = 112. I found out that 2 x 28 = 56, and then I added 56 + 56.

Solution #2: 20 x 4 = 80, 8 x 4 = 32, so I did 80 + 32.

This student’s multiplicative reasoning is somewhat stronger than the previous students, using multiplication as the starting point. Yet this student also relies on the doubling strategy to arrive at the answer.

Before moving on, I would like to understand that each child in my class is as solid in their thinking as this student. To help the students work on their developing multiplication skills, the responses above suggest that I group them based on their current status:

1. Students who primarily relied on addition need to work on their comprehension of partial products as well as fluency in multiplication facts.

2. Students who used partial products need rehearsal on algorithms.


Playing Games and Reducing Demands on Working Memory to Increase Mathematical Thinking (written with Lori Shaw)People of all ages love to play games that are fun and motivating. Games give students opportunities to explore foundational number concepts, such as the counting sequence, one-to-one correspondence, and computation strategies. Engaging mathematical games can also encourage students to explore number combinations, place value, patterns, and other important mathematical concepts that challenge their quantitative reasoning skills. Further, they also provide opportunities for students to deepen their mathematical understanding and reasoning. Teachers should provide repeated opportunities for students to play games, allowing the mathematical ideas to emerge as students notice new patterns, relationships, and strategies. Games are an important tool for learning in elementary school mathematics classrooms.

• Playing games encourages strategic mathematical thinking as students find different strategies for solving problems and deepen their understanding of numbers.

• When played repeatedly, games support students’ development of computational fluency and working memory.

• Games present opportunities for practice, oftentimes without the need for teachers to provide the problems. Teachers are free to observe or assess students and work with individuals or small groups of students.

• Games have the potential to allow students an opportunity to develop familiarity with the number system and with “benchmark numbers” (such as 10s, 100s, and 1,000s) and engage in computation practice, building a deeper understanding of operations.

• Games can also support a school-to-home connection. Parents can learn about their children’s mathematical thinking by playing games with them at home.

Games can be useful in any content area or any grade level. They also don’t have to cause behavioral management issues if introduced effectively.

Provide checklists and/or written game directions to offload working memory. Introduce the games step by step through (1) whole-group modeling, (2) whole-group play, and then (3) solo play guided by written instructions. This allows the students to free themselves from having to remember the directions and allows more room to process the learning. Teachers can also take complex games and break the directions into smaller chunks. When you ask students to focus on just one part at a time, you increase the likelihood that their working memory will be up to the task of processing the information successfully. When you assign a game for your students, consider the working memory required by the activity. Build in working memory support, break the task into smaller chunks, and provide frequent feedback, preferably so that students can track themselves or self-assess. By making conscious choices in our instructional methods that reduce the burden on working memory, we can make a significant impact on student achievement through the use of mathematical games.

Note of Caution. Sometimes teachers use games solely to practice number facts. These games are not engaging for children for long because they are based on students’ recall or memorization of facts. Some students are quick to memorize; others need a few moments to compute a related fact. When students are placed in situations in which recall speed determines success, they may associate being “smart” in mathematics with getting the correct


answer quickly instead of valuing the process of thinking. Consequently, students may feel incompetent when they use number patterns or related facts to arrive at a solution and may begin to dislike mathematics because they are not fast enough.

53Acknowledgments and References

AcknowledgmentsWe gratefully acknowledge the contributions of the gifted educators who contributed to updating and extending this resource as part of a series of workshops in 2017:

• Mrs. Lori Shaw, Northside Intermediate School, Opelika, AL

• Mrs. Beth Bass, Wrights Mill Road Elementary, Auburn, AL

• Dr. Pamela Scott-Williams, Pick Elementary School, Auburn, AL

• Ms. Faith Burns, St. Clair County School System, Asheville, AL

• Dr. Patti Wood, Professor of Curriculum and Instruction at Samford University in Birmingham, AL

References• Banchi, H., and Bell, R. (2008). The Many Levels of Inquiry. Science and Children, 46(2),

26-20.

• National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. Author.

• Tomlinson, C. A. (2001). How to Differentiate Instruction in Mixed-Ability Classrooms. Association for Supervision and Curriculum Development.

Appendix Acknowledgments and References

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Cognitive Abilities Test A ie o Tees

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