191 Chapter 5: Task Classification In this chapter, I analyze two assignments using the classification explained in Chapter 4. This is intended to demonstrate how the classification can be used to describe an assignment and compare it to other assignments. For the purposes of classification, a questionnaire was developed that helps to determine which categories apply and which instances of these categories apply to the assignment. This questionnaire can be found in Appendix C. Each question corresponds to a category or subcategory and the answer codes can be appended to the code in the right margin to give the numbers corresponding to the instance of the category. The codes resulting from each answer can be recorded in the corresponding box on the Coding Results Sheet in Appendix D; the first column lists the codes found in the right margin of the questionnaire and the second has spaces to enter the code(s) for the answer(s). Therefore the resulting numbers correspond to instances of categories found in the task according to the numbering in Appendix B. In the analysis, I also include comments from the teachers that describe why certain choices were made. These comments may suggest ways in which different choices in designing a writing task might affect student learning. Two assignments were chosen for analysis: Adams’ use of TIPS [Appendix A: 25.1] and Favata’s Olympic Games [Appendix A: 10]. These two assignments were chosen because significant detail and context were collected for these assignments and they varied in a number of ways. Also, the teachers were available and willing to respond to questions about the analysis. TIPS is a short daily assignment for middle school students, graded based on completion. Olympic Games is a longer assignment within a series of assignments for AP Statistics students, graded based on correctness. These two assignments were analyzed using the questionnaire, and additional
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191
Chapter 5: Task Classification
In this chapter, I analyze two assignments using the classification explained in
Chapter 4. This is intended to demonstrate how the classification can be used to
describe an assignment and compare it to other assignments. For the purposes of
classification, a questionnaire was developed that helps to determine which categories
apply and which instances of these categories apply to the assignment. This
questionnaire can be found in Appendix C. Each question corresponds to a category
or subcategory and the answer codes can be appended to the code in the right margin
to give the numbers corresponding to the instance of the category. The codes
resulting from each answer can be recorded in the corresponding box on the Coding
Results Sheet in Appendix D; the first column lists the codes found in the right margin
of the questionnaire and the second has spaces to enter the code(s) for the answer(s).
Therefore the resulting numbers correspond to instances of categories found in the
task according to the numbering in Appendix B. In the analysis, I also include
comments from the teachers that describe why certain choices were made. These
comments may suggest ways in which different choices in designing a writing task
might affect student learning.
Two assignments were chosen for analysis: Adams’ use of TIPS [Appendix A:
25.1] and Favata’s Olympic Games [Appendix A: 10]. These two assignments were
chosen because significant detail and context were collected for these assignments and
they varied in a number of ways. Also, the teachers were available and willing to
respond to questions about the analysis. TIPS is a short daily assignment for middle
school students, graded based on completion. Olympic Games is a longer assignment
within a series of assignments for AP Statistics students, graded based on correctness.
These two assignments were analyzed using the questionnaire, and additional
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information was sought from the teachers who used them in order to complete the
analysis. The Coding Results Sheets for each of these assignments can be found in
Appendix E. This analysis is explained in more depth below. The Coding Results
Sheets and the analysis below were shared with Adams and Favata to ensure accurate
reflection of the writing tasks. Any differences in understanding of the assignment
and categorization were discussed, with preference in any disagreement given to the
teacher’s perspective. After the analysis of the two tasks separately, the two tasks are
compared to show how the classification can highlight differences between
assignments.
Each analysis is organized based on the three main groupings in the categories:
Teacher Intentions and Goals, Assignment, and Response and Assessment. Each
choice of instance within the categories is explained in the context of the assignment
and tied to the classification by the number of the instance, which can be found in
parentheses in bold. These numbers refer to the numbering system found in Appendix
B. Also, each section will be summarized with a table giving the numbers and names
of the categories and applicable instances that are determined by the questionnaire.
Following the analysis and a comparison of the tasks, a few challenges in using
the classification are considered.
Olympic Games as used by Laura Favata
This writing task was given to an AP Statistics class near the middle of the
school year. The students who take this class are near the end of their high school
careers and are usually intending to attend college. The summary from Appendix A
can be found below in Figure 2. Students are given the handout found in Figure 3,
along with the “Thoughts Beforehand” page in Figure 4. The rubric used to grade the
assignment is in Figure 5.
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Figure 2: Olympic Games Assignment Summary from Appendix A
10. Olympic Games: analysis of long-jump information (Favata) This assignment is from Bock, Velleman & DeVeaux’s Stats: Modeling the
World (2004) and is handed out to students on a sheet of paper. AP Statistics students are given data on the gold medal long-jump distances in all the Olympic Games. They are asked to individually (a) do numerical and graphical analyses, (b) discuss the trend in long jumps based on a linear model, (c) explain the decisions made in creating the model with historical analysis of gaps in the data and departures from the trend, and (d) predict the distance that will win in the 2004 Games, including their faith in that prediction. This assignment was done after these topics had already been covered in lecture and in group activities. The rubric specifies what is desired, defining graphical analysis as a scatterplot, and numerical analysis as finding a linear model, defining variables, calculating r and r2, and a description of the direction, form, scatter and meaning of r2. The latter description is 5 of 12 points in this section. In terms of (c), they were expected to note gaps for the World Wars, and big jumps between the first two Olympics and in 1968, as well as their analysis of linearity. Finally, there were points for the prediction. The analysis was more than half the points. Some bonus points were available for work beyond what the project asked for; for example, some students searched for the athlete who was the outlier. The assignment was graded out of 30 points. This assignment is part of a series of assignments, so students have an understanding of the requirements for the write-up. Students do a similar project at the end of most chapters, for a total of 7-10 assignments of this type throughout the course. Projects like this make up 25% of the course grade. Students are expected to be clear and concise and answer all the questions, as well as including all the graphs and data and supporting all their conclusions. They have seen what is acceptable from previous assignments and have the opportunity to ask any questions during the week they are doing the project or after the project is graded. They also have a sample grading rubric they can use as a checklist. The writing affects the grading only as a tool for communicating. If the statistics is done accurately and in context and communicated clearly, this is all that matters. Favata reports that in this assignment, students have already had enough experience in the class so that “they are handing in projects that are well written and edited” (personal communication, July 13, 2005). Students are expected to recognize that communication is a major part of the work of statistics. Students are given freedom regarding the form in which they write the projects; for example, they can add extra graphs or write it in the form of a newspaper article. The length is determined by the information that they need to communicate. Each paper is commented on extensively in accordance with the rubric in the areas of content and clarity. The grading is intended to model an AP grader’s rubric. The comments are intended to help students correct their mistakes and misconceptions.
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Figure 3: Handout for Olympic Games task
AP Statistics Name _________________________ Project Date ___________________
Olympic Long Jumps
The modern Olympic Games, a modified revival of the ancient Greek Olympian Games, were inaugurated in 1896. Since then, the Games have been held nearly every four years at various sites around the world, and have become a major international athletic competition.
Based on the gold medal distances shown, write a report about the men’s long jump. In your report be sure to:
• Include appropriate graphical and numerical analyses • Discuss the trend in long jump performances, based on an appropriate
linear model • Explain the decisions you made in creating your model, with some
historical analysis of gaps in the data and departures from the trend • Predict the distance that will win the men’s long jump in the 2004 Games
in Greece, with comments on your faith in that prediction. Year Distance
Bock, Velleman, DeVeaux Stats: Modeling the World, 2004
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Figure 4: “Thoughts Beforehand” Handout for Olympic Games Task This handout was intended to give guidelines for what students should include in the report and how the reports would be graded.
Thoughts as you prepare your clear, concise and complete statistical analysis:
Think
Create a good linear model: • Use a subset of the data • Justify modeling decisions
Show
Visual: The scatterplot . . . • Has correct explanatory/response variables • Is accurate and clearly labeled • Shows the regression line
Numerical: The analysis . . . • Has correct r-squared • Has correct slope and y-intercept • Uses the proper notation
Tell
Interprets the model: • Evaluates the model with residuals • Describes trend in distances • Interprets the slope in context • Distinguishes model from reality
Makes a prediction for 2004: • Makes correct prediction from model • Expresses caution based on r-squared • Expresses caution about extrapolation
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Figure 5: Rubric for Olympic Games Assignment
I. Graphical Analysis
______ [5] Correct scatterplot of the data
II. Numerical Analysis
______ [3] Constructed a reasonable linear model for the data
______ [2] Defined variables
______ [1] Calculated r to measure strength of association
______ [3] Commented on direction, form, and scatter (strength)
______ [1] Calculated r2
______ [2] Described r2 in the context of the problem
III. Discusses the trend in long jump performances
______ [2] Made a statement about the trend in the data in context
IV. Explanation of decisions made in creating your model
Historical analysis of gaps in the data
______ [1] Gap between 1912 and 1920 (WWI)
______ [1] Gap between 1936 and 1948 (WWII)
Mention/justification of any departures from the trend (outliers/influential points)
______ [2] Big jump from 1896 to 1900
______ [2] 1968
Linear Model
______ [1] Constructed a residuals plot
______ [1] Correct assessment of linearity
V. Prediction
______ [2] Made correct prediction from model
______ [1] Commented on faith in your prediction
VI. Bonus Points
______ ___________________________________
___________________________________
___________________________________
Score ______
30
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Teacher Intentions and Goals
This task is intended to help students apply the collection of skills that they
have learned to real-world data. This is consistent with the intention to help students
learn procedural (1.4.2) and conceptual (1.4.3) mathematical content. Part of the
culture of the class is that answers need to be communicated well, and that answers are
more than a number or picture, but need “clear, concise and complete” explanations
(personal communication, July 13, 2005). This emphasizes both the need to
communicate carefully in an acceptable mathematical style (1.6.2) and the
mathematical need to make logical arguments (1.4.5). The AP culture of the class also
emphasizes specific forms of communication, and preparation for the AP Statistics test
is part of the rationale for why the task is done and graded as it is. Therefore, the
instance of communication in preparation for standardized tests (1.6.3) should be
included in classifying the intentions. Laura Favata uses these assignments to assess
whether students understand the concepts and skills they will need for the AP test, and
does individual remediation based on them, in part through the comments made on
student work. This exemplifies the instance of using the task to individually assess
students (1.1.2). These instances and their titles are listed in Table 50.
Table 50: Olympic Games Teacher Intentions and Goals
1. Intended Area of Change/Purpose
1.1 Teacher Assessment
1.1.2Assess individual student learning
1.4 Mathematical content
1.4.2Procedural understanding
1.4.3Conceptual understanding
1.4.5Logical argument or proof
1.6 Communication ability
1.6.2 "Math-talk" - disciplinary specific standards
1.6.3Standardized test-preparation
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Assignment
I will consider the categories that describe the writing assignment itself within
the classes of the classification: Practical Product Specifications, Theoretical Product
Specifications, and Process and Context.
Practical Product Specifications
In this section I will consider the product students are to turn in and who is
involved in the completion. The instances applicable to this task from this class of
categories can be found in Table 51.
Table 51: Olympic Games Practical Product Specifications
Laura intended this assignment to mirror much of the process and challenges
of the AP Statistics exam, and therefore required that students work individually. This
implies that the individual is both the writer (2.1.1a) and the only collaborator, or
person who has direct input into the written product (2.1.2a). The teacher is the only
intended reader (2.1.4.2), and she tries to model AP standards for responses. The
2.1.1a Writer(s): Individual Student
2.1.2a Collaborators: Individual Student
2.1.3a Audience: Not specified
2.1.4.2 Reader: Teacher
2.1.5 Sources
2.1.5.2Individual learning of class content
2.1.5.4Textbook
2.2.1.1a Units of Writing: Not specified
2.2.1.2 Length Restrictions: None (N/A)
2.2.2b Grammar and SpellingRequirements: Complete sentences required
2.2.3c Breadth of Topic: Multiple concepts or a large topic
2.2.4c Use of Visuals and Graphics: Required
2.2.4c.4Mathematical content: graph, table, etc.
2.2.5a Multiplicity of Answers and Methods: One answer, one method
2.3g Level of Student Manipulation: Analyzing information from the class
2.4c Student Choice: Choice of major parameter of assignment
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audience for the written work is not specified (2.1.3a) and students have the freedom
to choose different audiences. For example, Laura had some students choose to write
the report as a newspaper article. This shows a choice that students have which can
determine a significant parameter of the assignment (2.4c). Since students are
working individually and using content they have already learned in class, their only
sources are their individual learning of the content (2.1.5.2) and their text (2.1.5.4).
The written product is intended to communicate the content clearly, which
means, given the high caliber of students in the class, that the writing is generally of
good quality and significant length. Laura reports that by this assignment students
“are handing in projects that are well written and edited” (personal communication,
July 13, 2005). However, these standards are more implicit than explicit in the
assignment itself. While students turn in reports with multiple paragraphs which
amount to three or four pages including data and graphs, the actual length of the
written report is not specified (2.2.1.1a). In this case, not specifying the length implies
more student responsibility because they have to decide when they have included
enough information. In the same way, students are given no length restrictions, so the
category of Length Restrictions (2.2.1.2) is not applicable. Students are expected to
write in complete sentences (2.2.2b) since they are writing a report, but no particular
grammar and spelling or format restrictions are made. Again, these formatting issues
are taken for granted since the students are advanced; however, these elements do not
affect the grade directly and so the task is not classified as requiring them.
Students are writing about a particular example in order to apply the methods
of statistical analysis that they have learned, so it would appear that this is how the
breadth of topic should be classified (2.2.3a). However, students are applying a
combination of methods to this example, and in this sense are integrating many
different skills. These skills are the true content of the task, not the example.
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Therefore, according to the intentions of the category, this should be classified as
multiple concepts (2.2.3c). The methods students are expected to use are standard
statistical methods on a well-defined problem, and therefore students will use one
basic method, resulting in one answer (2.2.5a). Even though the required explanation
may vary slightly in wording, it will still be directed toward the same meaning. Some
of the methods include standard graphical analyses. These graphics are required and
part of the mathematical content (2.2.4c.4). As an AP class, students are expected to
reach a high level of performance, carefully analyzing the data and working with it in
a number of ways. Therefore, there is significant student manipulation of the data
given to them, so this is classified as analyzing information from the class (2.3g).
Theoretical Product Specifications
In this class of categories, the cognitive demands of the assignment are
classified, with a specific focus on what is required to complete the assignment
satisfactorily rather than what may result from the task. The categories and applicable
instances of this class of categories can be found in Table 52.
The function of this task is clearly to convey information. The use of the word
“report” in the assignment signals this, as do key verbs like “explain” and “predict.”
These imply specific information and an audience that wants to understand this
information. This is not an artistic performance to please the writer, as poetic writing
would be, and is not expressive of the writer’s personal thoughts and feelings, as
expressive writing would be. It is transactional (2.5c), and intended to convey the
results of the data analysis to the reader. The rhetorical demands of this specific task
include some generalization as students analyze the data for a general pattern and use
this pattern to make a prediction. Therefore, the task demands at least low-level
analogic work. This task is best classified as analogic (2.5c.e), because the nature of
the report and the methods used require that this generalization be well organized
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Table 52: Olympic Games Theoretical Product Specifications
and justified. This is a better choice than the analogic-tautologic or tautologic levels
because the result is the generalization itself rather than discussion of the
generalizations. The use of prediction might appear to confuse this; however, the
prediction required by this task is the use of the generalization – the equation for a line
modeling the length of the winning long jump – rather than prediction of further
generalizations.
When the cognitive demands of the task are determined according to Bloom’s
Taxonomy, it is evident that the processes of analysis are used as students compare the
data and determine outliers and gaps. This assignment also combines many of the
methods and concepts that students had been learning in class and therefore requires
synthesis. Since Bloom’s Taxonomy is considered a hierarchy, this task should
therefore be classified as synthesis (2.6e).
2.5c.e Function of Writing: Transactional: Analogic
2.6e Bloom's Taxonomy: Synthesis
2.7.1 Level of Language
2.7.1.1Particular procedural
2.7.1.2Particular descriptive
2.7.1.4Generalized descriptive
2.7.2 Aspects of Mathematics
2.7.2.2Logical
2.7.2.3Algorithmic
2.7.2.4Methodological
2.7.2.5Conventional
2.8 Connections
2.8.1 Concepts with procedures, symbols and objects
2.8.3 Current learning with other experiences
2.8.4cConcepts and procedures with related examples
Example(s) followed by student generalization
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The language of this task is varied and complex because many different
aspects of the concepts and the example are integrated. The category Level of
Language (2.7.1) describes the type of statements that students will need to make and
whether they are particular or general, procedural or descriptive. The rubric in Figure
5 can help determine whether some of these types of statements are required. In the
procedures required, particular procedural language (2.7.1.1) should result when
students construct a linear model, calculate r, and calculate r2 because they must use
specific language about these procedures relating to the example. As they comment
on trends found in this example, particular descriptive language (2.7.1.2) would occur
when r2 is described in the context of the problem. Students may make statements that
are generalized procedural if they discuss the procedures they are using in general, but
this is not required of the task. In order to make “a statement about the trend in the
data in context” students will have to use generalized language, since they are
discussing a trend and this language will also be descriptive (2.7.1.4) rather than
procedural. Students are also required to use a variety of Aspects of Mathematics
(2.7.2) in their writing. Again, the rubric can direct the classification through its clear
statements of what is required for successful completion of the task. In order to define
variables, students must use conventional language (2.7.5). The specific methods and
procedures they follow in finding a linear model and computing variables will require
algorithmic language (2.7.3), and the explanation of methods and conclusions will
require logical language (2.7.2). Discussion of outliers and gaps is not concerned with
a specific procedure, and therefore is methodological (2.7.4) in nature; students will
make general statements about outliers that they will also interpret within this specific
context.
There are also various connections needed in this assignment. Students must
connect the various concepts and procedures that they have been studying to each
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other (2.8.1). They also need to connect their analysis in the project to their
knowledge of history in order to understand the gaps caused by World War I & II.
This is an experience that is outside the mathematics class (2.8.3). Finally, students
need to connect the concepts and procedures they know to this specific example
(2.8.4). It is difficult to determine in this instance how the examples are sequenced
with explanation. It is the nature of statistics to work within the context of a given
example, so this can be considered a case where all explanation is related specifically
to the example and there is no general explanation (2.8.4a). On the other hand, one
could consider the nature of statistics as inherently generalizing from examples, which
results in models and future predictions. In this case, students must generalize from
the particular data they are given and explain this generalization (2.8.4c). This shows
how these cognitive categories depend on the perceptions of the subject and the task
brought by the person categorizing them. However, I believe that the argument for
generalization is stronger since students are required to make statements about trends,
which are generalizations, and therefore classify this accordingly. Laura agrees with
this, noting that students also need to distinguish the model they are working with
from reality, as noted in the “Thoughts Beforehand” handout in Figure 4.
Process and Context
In this section, the greater context of the task within the classroom is analyzed,
including issues of related activities, repetition and communication of the task, and
time, place and sequencing of the task. The categories and applicable instances of this
class of categories can be found in Table 53.
As noted above, this assignment is intended as an individual assessment and no
other groupings were used after the completion of the task, so the only grouping is that
of the individual (2.9.1.1). Therefore, the category Sequencing of Activities (2.9.2)
does not apply to this task. Since the task is individual, it is completed entirely at
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Table 53: Olympic Games Process and Context
home, or outside the classroom (2.13.1e), and therefore receives no time within the
classroom (2.13.3a) and the category of Timing Within Class Period (2.13.5) does
not apply. Similarly, no revision is required for this task, so the categories related to
revision (2.10.1) and (2.10.2) do not apply. Some students may choose to
2.9.1.1 Groupings: Individual
2.9.2 Sequencing of Activities: None (N/A)
2.10.1 Modifying Writing Within the Same Structure: None (N/A)
2.10.2 Adapting Writing into a New Structure: None (N/A)
2.11.1c Task Frequency: Once a Unit
2.11.2cImportance/Centrality of Assignment Type
Regular, high value series of assignments
2.11.3b Overlap of Material with Other Assignments
Content repeated in other activities
2.11.3b.1Lecture/class discussion
2.11.3b.3Group work
2.11.3b.5Exercises (routine)
2.11.3b.6Tests/summative evaluation
2.11.4c Placement in Learning Sequence: Follow up for introduced material
2.11.4c.bNovel context
2.11.5b Relationship Between Words and Symbols
Explaining symbolic work as it is completed
2.12.1.3 Mode of Assignment: Given in writing on a sheet of paper
2.12.2b Modeling of Assignment: Some modeling of assignment
2.12.2b.1Processes of assignment modeled
2.12.2b.3Example(s) of assignment assessed
2.13.1e Place of Completion: Entirely at home
2.13.2d Time to Complete Assignment: Until 5-9 class periods later
2.13.3a Amount of Classroom Time: No time
2.13.4d Expected Time at Home: 2-4 hours
2.13.5 Timing Within Class Period: (N/A)
2.14.5 Relevance of Task Context: Real-life setting
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do some self-modification through editing and this is encouraged, but it is not
required.
Some categories describe the type of writing tasks that Olympic Games
exemplifies. Most units have a similar project, so the task frequency is once a unit
(2.11.1c). Students are given about a week to complete each of these tasks (2.13.2d),
expected to spend multiple hours on it (2.13.4d), and 25% of the course grade is
devoted to projects like these. Therefore, this is a central and important part of the
course structure, constituting a regular, high value series of assignments (2.11.2c).
This task took place at the end of the unit that contained the skills used in the
assignment. Therefore the task was follow-up for material that had been introduced
(2.11.4c), allowing students to integrate the various methods and concepts and apply
them in a real-life setting (2.14.5). In particular, this task gives students a novel
context to apply their learning (2.11.4c.b). Students were to apply the skills they had
learned without any further expansion of the concepts, so this is an example of content
being repeated in other activities (2.11.3b). Students had participated in group
activities (2.11.3b.3) regarding this content; one activity required that each student
find a linear equation that goes through two points of the data and then residuals were
calculated for each of these lines to see which was the best fit to the data. The topics
had also been covered through lecture and class discussion (2.11.3b.1), homework
exercises that required application of the concepts and methods (2.11.3b.5), and were
covered in tests (2.11.3b.6). Students have already done a significant amount of
symbolic work throughout the unit, as well as related explanation. Since the course
emphasizes explaining findings in context, symbolic work is usually explained
concurrently. In this task, they are explaining symbolic work as it is being completed
(2.11.5b), although the order is not as easy to determine here since students have a
significant amount of time to do the assignment and the order may vary.
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The task is communicated by passing out a sheet of paper to each student
(2.12.1.3). The processes of the assignment have been modeled in previous
assignments of this type and throughout the unit as students have learned the content
(2.12.2b.1). Students also have their own graded work from similar tasks completed
earlier, giving them insight into how the task is assessed (2.12.2b.3).
Response and Assessment
This group of categories concerns what happens after the task is completed,
considering both response to the task in the form of grading and comments and any
use of the assignment in future learning. The categories and applicable instances of
this class of categories can be found in Table 54.
Table 54: Olympic Games Response and Assessment
3.1.1 Teacher Comments
3.1.1.1.1Form: Written
3.1.1.2.1Group: Individual
3.1.1.3Type
3.1.1.3.1 Focus: content and clarity
3.1.1.3.2 Specificity: detailed by rubric
3.1.2.1 Feedback Frequency: Every written product
3.1.3a Peer Response: None
3.1.4b.c Collection of Writing: Turned in: At due date
3.1.5b Availability to Students: Kept at discretion of student
3.2 Use of Completed Assignments
3.2.1 Source for grade
3.2.3 For assessment of individual progress
3.3.1c Standards for Grading: Graded based on correctness
3.3.1c.1a Content vs. Expression: Content mostly
3.3.1c.2e Standards for Justification: by Proof
3.3.2b Grading method: Distribution of Points
3.3.3.3 Grader: Teacher
3.3.4 Percentage of grade: 3% (75% for type)
3.3.5b Student Awareness of Assessment Parameters
Students have access to grading scheme
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After the task is completed and collected at the due date (3.1.4b.c), Laura reads
each student’s work carefully and grades it (3.3.3.3) according to a rubric that
specifies a distribution of points (3.3.2b). These grades are based on whether the
content is correct (3.3.1c) rather than on effort. Most of the grade is based on the
content (3.3.1c.1a); points may be lost if the expression obscures the content, but this
is not included in the rubric and therefore is not directly affected. Students are
expected to justify and explain their work according to statistical standards, so the
standards for justification are equivalent to those of proof or logical argument
(3.3.1c.2e). Students had the Thoughts Beforehand handout in Figure 4 that
corresponds closely to the rubric so that they knew how the task would be graded
(3.3.5b). This particular task would account for about 3% of their grade, with similar
tasks accounting for 25% of the course grade (3.3.4).
Laura makes comments on the tasks that are individual (3.1.1.2.1) and written
(3.1.1.1.1). The distribution of points in the rubric is an aid in making these comments
since they give students specific information (3.1.1.3.2) about where they lost points.
The comments focus on both content and clarity (3.1.1.3.1). Such comments are made
on every assignment of this type (3.1.2.1), and no peer response is made, so the
instance in the category of Peer Response is none (3.1.3a). These tasks are returned
to students, who keep them at their own discretion (3.1.5b). Overall, the tasks are
used both as a source for a grade (3.2.1) and for assessment of individual progress
(3.2.3) that allows for remediation.
Summary
Overall, this analysis of the assignment shows that it requires significant
independence and responsibility on the part of students to complete this assignment.
Students do the work individually, it is graded individually, and the task is intended to
help individual assessment and prepare for a test that students must take individually.
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This is understandable and expected within the culture of a college preparatory class
and the need to perform individually on the AP Statistics examination. A number of
aspects of this assignment are also implicit rather than explicit, and therefore
underemphasized by the categories. Students are to submit excellent work, but since
these are advanced students and this is the middle of the course, these expectations are
not explicit in the assignment. For example, no explicit requirements for grammar,
spelling or format are made, and no minimal length is imposed. These elements are
taken for granted in excellent assignments. Finally, this assignment makes a variety of
cognitive demands on students that are in keeping with an advanced course and the
desire to have students perform at discipline standards. There are a number of
questions that require a high level of both transactional writing and cognitive
processing as measured with Bloom’s Taxonomy. The assignment also expects many
different types of language and connections. Discipline standards can be seen in these
cognitive standards, as well as in the general expectations for writing, the requirement
of graphics, and analysis as part of student manipulation.
TIPS as used by Kimberlee Adams
This task was used as a typical Problem of the Day (POD) to begin an 8th grade
mathematics block on January 11, 2005. I observed this assignment and have
conversed with Kimberlee about her daily use of similar tasks. The task summary
from Appendix A can be found in Figure 6.
Teacher Intentions and Goals
This assignment was used to review ideas about solving linear equations that
had been learned in the previous class. Kimberlee writes that the “idea is for the kids
to understand and express their learning” (personal communication, July 13, 2005).
The first goal of understanding is related to students learning mathematical content
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Figure 6: TIPS Assignment Summary from Appendix A
(1.4), while the second goal of expression refers to helping students to communicate
about mathematics (1.6). The content is procedural by nature and therefore the
improvement of procedural understanding (1.4.2) was central in Kimberlee’s
intentions. The task also included elements of conceptual understanding in the larger
context of solving equations and the various properties of operations. Therefore,
conceptual understanding (1.4.3) was also a goal of this task. This task is intended to
improve students communication ability (1.6) for a variety of reasons. The school
25. TIPS (Spriggs, Adams) In the Think-Ink-Pair-Share (TIPS) process, students are asked to think about a problem, write about it, share it with a partner and then share with the whole group. This is a pattern that fits into the school emphasis on writing and the use of the Collins Writing Process (see Note 3 after [34]). 25.1: (Adams) Problem Of the Day (POD)
Students are regularly given a problem of the day that is done using the TIPS process. On the day observed, students were asked to solve a problem based on the previous day’s lesson: “Solve: 7N + 4 = 32. Using words, explain how you would solve this equation.” They are reminded that they need to write in complete sentences for their explanations - using words, not numbers - and should write 3-4 sentences. Students are asked to write as if they were writing an E-mail or letter to a friend explaining how to do the problem. Students are given about 30 seconds to think, 2 minutes to write, 1-2 minutes to share with a partner, and then each table of four students was asked to have one person read exactly what they wrote. While students are working, Adams would read what they were writing and ask questions to help them answer the problem. More time can be given or steps repeated if students have difficulty. Students were asked to write in such a way that it would be a good explanation for someone else in the class. After some students read to the class, all were given a short period of time to Re-Ink, that is, revise what they had written. Adams goes over the POD after students have written and shared about it, asking questions until they reach the answer. This type of task was recorded in their class notebooks. At this point in the course, students have already done a number of hands on activities related to linear equations, as well lecture and class discussion and homework exercises. This assignment is usually used to check understanding on the most recent lesson or to introduce a new lesson. It could be used for individual assessment, in which case Adams would draw a line after what students had written individually and then allow them to write below the line after interaction with peers and the class.
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district is emphasizing the Collins Writing Process [Appendix A, Note 3 after [34]]
and therefore the Writing Across the Curriculum (1.6.4) movement is a significant part
of the motivation for this task. Also, Kimberlee notes that these questions were
intended to prepare them for tests (personal communication, July 13, 2005), including
standardized tests (1.6.3). Finally, at the time of observation, Kimberlee noted that
she wanted students to use this task as a way to gather their thoughts for the class
period and use their own words to express their learning rather than having the teacher
telling them how to do it. This shows how she wants students to use their own words
and everyday normal speech to explain their work (1.6.1). These instances of Teacher
Intentions and Goals are found in Table 55.
Table 55: TIPS Teacher Intentions and Goals
Assignment: Practical Product Specifications
In this section I will consider the writing product students complete in this task
and who is involved in the completion. The categories and applicable instances of this
class of categories can be found in Table 56.
Since this task was an opportunity for students to express their personal
understandings and to struggle with the work, the individual was central in the actual
production of the original written work. The individual student was both the writer
(2.1.1a) and the only collaborator (2.1.2a), and they draw only on their individual
learning of class content as a source (2.1.5.2). The written product was intended to
communicate to a peer how to solve the equation, so the audience is that of a peer
1. Intended Area of Change/Purpose1.4 Mathematical content
2.9.2.2 Sequencing of Activities: Grouping after writing2.10.1 Modifying Writing Within the Same Structure
2.10.1.1Self-modification2.10.1.2Responding to peer comments2.10.1.3Responding to teacher comments
2.10.2 Adapting Writing into a New Structure: None (N/A)2.11.1f Task Frequency: Daily2.11.2d Importance/Centrality of Assignment Type
Regular, low value series of assignment2.11.3b Overlap of Material with Other Assignments
Content repeated in other activities2.11.3b.1Lecture/class discussion2.11.3b.2Activity/exploration2.11.3b.3Group work2.11.3b.5Exercises (routine)2.11.3b.6Tests/summative evaluation2.11.3b.7Other writing activities
2.11.4c Placement in Learning Sequence2.11.4c.bFollow-up for introduced material: Novel context
2.11.5b Relationship Between Words and SymbolsExplaining symbolic work as it is completed
2.12.1 Mode of Assignment2.12.1.1Given verbally2.12.1.2Given in writing to class as a whole
2.12.2b Modeling of Assignment: Some modeling of assignment2.12.2b.1Processes of assignment modeled2.12.2b.2Example(s) of assignment given
2.13.1a Place of Completion: Entirely in class2.13.2a Time to Complete Assignment
Task given and done in same class period2.13.3c Amount of Classroom Time: 5-20 minutes2.13.4a Expected Time at Home: No time2.13.5a Timing Within Class Period: Beginning of Class Period2.14.2 Relevance of Task Context: Pure mathematical work
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Although this task is written individually, a significant amount of work is done
in the class to process the written product that results. Students share their written
work in pairs, and then about a quarter of the students read their work to the whole
class. Therefore, a number of groupings are involved: the individual (2.9.1.1), pairs
(2.9.1.2) and the class (2.9.1.4). All of these groupings, other than the individual
student, occur after students have written their answers (2.9.2.2). Revision is part of
this structured processing of the written work. Kimberlee gave them the opportunity
to Re-Ink on this particular day, allowing them to make changes to their work. This
revision did not change the form of the written product, so the category Modifying
Writing Within the Same Structure (2.10.1) applies, while the alternate category of
Adapting Writing into a New Structure (2.10.2) does not. Each student revised
their own work, and the theory of Collins Writing views this self-modification
(2.10.1.1) a response to reading the written product aloud. Since they had read to a
peer, there was a structured opportunity for peer comments which could affect the
revision (2.10.1.2), and they also were responding to teacher comments (2.10.1.3),
given both to individual students as Kimberlee circulated throughout the classroom
and to the class as a whole as she reviewed what she had heard when students read
their work to the class.
This type of task, the Problem of the Day (POD), occurred daily (2.11.1f), and
Kimberlee had been using the TIPS process as part of the POD for about a month.
These paragraphs were required regularly, but did not have a high value, forming only
a few points in the notebook check that occurred every four to six weeks. The low
grade value combined with the time spent on this task every class, signifies that this
type of task is important, but not very important. Therefore it is classified as a regular,
low value series of assignments (2.11.2d). The content in this task applies content
addressed in other activities of the class in the same form, and therefore is repeated
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(2.11.3b) rather than expanded in these activities. The variety of activities that
address this content is wide and varied, and is supported by a curriculum that involves
many different activities. Students had learned about solving linear equations the
previous day through lecture and class discussion (2.11.3b.1) and hands-on
explorations in the curriculum (2.11.3b.2). These activities are done in the context of
groups (2.11.3b.3) and reinforced through routine homework exercises (2.11.3b.5). In
Kimberlee’s class structure, students will have written a class summary [Appendix A:
31] about this learning the previous day, so the content has been addressed through
other writing activities (2.11.3b.7). After this task, solving linear equations will be
part of a test or other summative evaluation (2.11.3b.6). From this description, it is
clear that this task is placed near the end of the learning sequence for this content, as
follow-up for the introduced material in the novel context of a new linear equation to
solve (2.11.4c). In this particular task, students are explaining their symbolic work as
they are completing it (2.11.5b). Some students may break out of this pattern and
solve the equation symbolically and then write the paragraph, but this is not the
intention of the task. In contrast, in the previous class students completed similar
symbolic work and then wrote a class summary about this topic, so the writing task
clearly followed symbolic work. Although there is explanation involved, this is
standard algebraic manipulation and not connected to any context, so the relevance of
the task would be classified as pure mathematical work (2.14.2).
This assignment was communicated and modeled for students in a number of
ways. The prompt was written on the overhead projector for the whole class
(2.12.1.2) and the parameters of the assignment, such as length and audience, were
given verbally (2.12.1.1). There was some modeling of the assignment (2.12.2b), both
through previous uses and through explanation at the time. The processes of the task,
both of the task structure and content, have been modeled some for students
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(2.12.2b.1). Students had already used the TIPS process, and the content of the task
was taught on the previous day and therefore the content processes have been modeled
as well. Since students have used this process over time, they have also seen examples
of similar completed tasks (2.12.2b.2). In fact, the very structure of the task requires
that students read examples to each other.
The task is given about 10-15 minutes (2.13.3c) at the beginning of the class
period (2.13.5a). These tasks are entirely within the class period (2.13.1a), both given
and completed within the class (2.13.2a). Therefore, students are not expected to
spend any time at home (2.13.4a).
Response and Assessment
This group of categories concerns what happens after the task is completed,
considering both responses to the task in the form of grading and comments and any
use of the assignment in future learning. The categories and the instances that apply to
this use of TIPS can be found in Table 59.
Since this was a daily assignment, Kimberlee did not comment on student
work in writing because of time constraints. Instead she comments verbally to
students (3.1.1.1.2). These comments are made to students individually (3.1.1.2.1) as
she circulated around the classroom as students were working and to the class as a
whole (3.1.1.2.2) after the students have shared with each other and before they
revised their work. These comments usually took the form of questions intended to
help the students understand the content of the task. This means that students get
verbal responses as a class to every written product (3.1.2.1) and will get individual
responses occasionally (3.1.2.2). These PODs are compiled in one section of student
notebooks, which also receive the feedback of a grade (3.1.2.3). Therefore, the
assignment are both compiled (3.1.4b.2) and turned in after the day they are
completed (3.1.4b.1b). Since this compilation is done by students (3.1.5c.a) in their
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Table 59: TIPS Response and Assessment
class notebook, it will be available to students at any time. Peers also responded to
student writing verbally through pairings (3.1.3b.1.1). This particular use of TIPS did
not include class discussion of the student work, even though students shared written
work with the whole class, since the students were getting the main ideas, and the
teacher only had to summarize these ideas.
After the whole task has been completed, the written products are used
primarily as a source for a grade (3.2.1). These tasks are graded based on effort and
completion (3.3.1b), so the categories of Content vs. Expression (3.3.1c.1) and
Standards for Justification (3.3.1c.2) do not apply. With this in mind, the tasks are
graded holistically (3.3.2a), with only 5 points given for all the PODs each time
notebooks are collected. Other class activities recorded in the notebook are given
3.1.1 Teacher Comments3.1.1.1.2Form: Verbal
3.1.1.2Group3.1.1.2.1Individual3.1.1.2.2Class
3.1.2.3 Feedback Frequency3.1.2.1Every written product3.1.2.2Occasional response3.1.2.3Compilations of writing
3.1.3b.1.1 Peer Response: Verbal feedback: Pairs or small groups3.1.4b Collection of Writing
3.1.4b.bTurned in: Later3.1.5c.a Availability to Students: Compiled: By student
3.2.1 Use of Completed Assignments: Source for grade3.3.1b Standards for Grading: Graded on effort and completion
3.3.1c.1 Content vs. Expression: (N/A)3.3.1c.2 Standards for Justification: (N/A)