Study Guide

Field 182: Elementary Mathematics Specialist 
Sample Constructed-Response Assignment

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Test Directions for the Constructed-Response Assignment

This section of the test consists of one constructed-response assignment. You are to prepare a written response of approximately 300–600 words on the assigned topic. You should use your time to plan, write, review, and edit your response to the assignment.

Read the assignment carefully before you begin to write. Think about how you will organize your response.

As a whole, your response must demonstrate an understanding of the knowledge and skills of the field. In your response to the assignment, you are expected to demonstrate the depth of your understanding of the content area through your ability to apply your knowledge and skills rather than merely to recite factual information.

Your response to the assignment will be evaluated on the basis of the following criteria:

 start bold PURPOSE: end bold  the extent to which the response achieves the purpose of the assignment
 start bold SUBJECT MATTER KNOWLEDGE: end bold  accuracy and appropriateness in the application of subject matter knowledge
 start bold SUPPORT: end bold  quality and relevance of supporting details
 start bold RATIONALE: end bold  soundness of argument and degree of understanding of the subject matter

The constructed-response assignment is intended to assess subject matter knowledge and skills, not writing ability. However, your response must be communicated clearly enough to permit valid judgment of the scoring criteria. Your response should be written for an audience of educators in this field. The final version of your response should conform to the conventions of edited American English. Your written response must be your original work, written in your own words, and not copied or paraphrased from some other work.

Sample Constructed-Response Assignment

Competency 0011 
Analyze student work and assessment data related to a student's development of mathematical knowledge and abilities and plan appropriate instructional strategies for the student.

 start bold Use the information provided in the exhibits to complete the assignment that follows. end bold 

Using your knowledge of developmentally appropriate evidence-based instructional practice and citing evidence from the exhibits provided, write a response of approximately  300 to 600  words in which you:

Be sure to utilize all of the exhibits in your response.

Exhibit 1: Lesson Plan

Students are working on collecting and analyzing data. In previous class sessions, they have used tally marks to keep track of counts and have represented counts with bar graphs. They have worked with pictographs using a key of one to one. Currently, they are applying their knowledge of skip-counting to their pictographs. The teacher asks students to practice the concept individually after a whole-class discussion and teacher modeling. One student's work is shown in an exhibit.

Oklahoma Academic Standard (OAS): 2.D.1.2. Organize a collection of data with up to four categories using pictographs and bar graphs with intervals of 1s, 2s, 5s, or 10s.

Lesson Objective: Students will be able to accurately record and interpret data using pictographs.

Student Learning Goal: I can use a pictograph to answer questions.

Vocabulary:

Previous Knowledge Needed:

Exhibit 2: Work Sample

Students work on a pictograph activity. One student's work is shown below.

Pizza Pictograph

a diagram of a student's worksheet

A student worksheet labeled Pizza Pictograph. There is a key indicating one circle equals 2 pizzas. There are 4 types of pizza listed. The cheese has 6 full circles. The pepperoni has 4 full circles. The mushroom has 3 full circles. The combination has 4 full circles and 1 half circle.

1.   How many pepperoni pizzas were sold?     4

2.   How many cheese pizzas were sold?     6

3.   Which pizza was ordered the most?     cheese

4.   Which pizza was ordered the least?     mushroom

5.   Did pepperoni or combination get ordered more?     Combo

6.   Did cheese or combination get ordered more?     Cheese

Sample Strong Response to the Constructed-Response Assignment

 start bold Please note: The sample response provided below is for review purposes only and should not be used in a response on an operational exam. Use of the exact words and phrases presented in this sample response will result in a score of "U" (Unscorable) due to lack of original work. end bold 

The lesson plan shown in Exhibit 1 states that students have previously worked with pictographs on a one-to-one key. The class is now beginning to apply their skip-counting skills to pictographs which have other ratios for basis of the key. While this response does describe what the students have worked on so far in this unit of study, the actual learning strategy they are practicing is unclear. A whole-class discussion and modeling are mentioned but may not be enough for this student to effectively learn from. Often, students may require additional direct instruction or a more hands-on approach to their practice.

While some may require additional practice, this is still a developmentally appropriate strategy in other ways. First, the progression of skill building shown by starting with one-to-one pictographs and later introducing other keys is quite appropriate for young learners. Additionally, the connection to skip-counting skills may help students activate prior knowledge in this area.

The student work sample in Exhibit 2 shows that the student has not grasped the idea behind each picture representation equaling two pizzas. It appears that is the concept that caused the errors shown in their calculation. Perhaps more practice and direct instruction in this area would be advisable. One way the teacher can do this is to develop a game where students will use pictographs, perhaps starting with just using one-to-one, one-to-two, and one-to-three keys in the game.

First, the teacher should start with a quick review of the ideas behind the key and the pictorial representations. This student lacks a fundamental understanding of why pictures are being used to represent numbers. The student requires clarification and a visual aid to use while practicing. A good visual aid in this case might be anchor charts that were created together as a class or in a small group of students requiring this reinforcement of concepts. These anchor charts would show something like the pictograph used by the student in the work sample shown in Exhibit 2. They would show the different keys and clearer color-coded pictures to represent the number of pizzas (or other objects). Creating these together and displaying them during further discussions or activities may be helpful for our more visual learners. Using these anchor charts, the class could play a commerce-based game together in small groups. This would give students the opportunity to practice “trading” two pizzas for one picture, card, or whatever key was decided to be used at the time. Ideally, the game might be entertaining and engaging, helping students remember the basic mathematic skills needed to play.

An effective formal assessment method the teacher could use to evaluate student progress for this additional learning module would be a retest. Moreover, the teacher could use the informal assessment of observation — through observation of the gameplay, the teacher is likely to see if a student has managed to grasp the concepts needed. The teacher might combine these by observing students during gameplay and taking notes or checkmarks for certain skills they observe the student doing (correctly or incorrectly). Additionally, the teacher could create a new assessment very similar to Exhibit 2 but exchange cookies for pizzas. This way, there is a new bit added to the previously partially understood pictograph situation shown in the student work sample and hopefully after the reteaching is completed, the teacher will observe the student has mastered how to use pictographs to answer questions.

Rationale for the Sample Strong Response

Please note that the response is evaluated based upon the four performance characteristics of Purpose, Subject Matter Knowledge, Support, and Rationale. Please also note how the score point descriptions are based upon how the examinee attends to the performance characteristics. You should be very familiar with the CEOE performance characteristics and score scale and refer to them when reviewing this rationale.

The purpose of this assignment is largely achieved. This writer points out that it is likely that the teaching methods used may have been ineffective for the student(s). This is an assertion that they support when they describe that some students may require more direct instruction or hands-on work for successful mastery of new mathematical concepts. This is often necessary as concepts become more abstract in nature and harder for students to picture themselves. In this case, it is likely that manipulatives or some other tactile activity for skip-counting might have been more effective.

The writer demonstrates a generally accurate and appropriate application of subject matter knowledge throughout the response. This is shown through their identification of the student's misconceptions shown in the work sample. The student made what looks like miscalculations at first glance. However, looking deeper, one can see that the mistakes made are all connected to the same concern of improper skip-counting. The writer of the response shows that they are knowledgeable enough to also use this information to plan some sort of additional instruction to assist the student.

The response contains evidence that generally supports the discussion using some relevant examples to back up claims and other assertions made by citing the exhibits and the student work sample in the response. The writer references where their information is coming from and why they think something to be true about the student. The response effectively explains the writer's observations about the student work. Details given are relevant to the claims made.

Rationale is explained throughout the response when instructional strategies are introduced by the writer. Strong connections are made between instructional strategies and the study skills they address. The response reflects a generally accurate understanding of the topic.

Sample Weak Response to the Constructed-Response Assignment

We have a short explanation of previous related lessons in the Exhibit 2 materials. There we can see that the teacher has taught them just the beginnings of pictograph instruction. It was maybe effective that the teacher used modeling to show the students how to do this before asking them to do it on their own. That is appropriate for this grade level and most likely for these kids in my class. These students already have started to learn about this type of data collection before. They will be able to use what they know from the previous related lessons to connect to this pictograph lesson. They have been introduced to skip-counting and hopefully are all proficient at that. That will be a needed skill for this kind of activity. The student who completed the sample work is probably not sure about this lesson concept. They have issues understanding the math behind the pictures. They seemed to make mistakes when doing the skip-counting or repeated addition needed. It seems like they may not know to multiply by two or add twice for the pictures shown in the graph. A new lesson or reteach needs to help this student to learn the parts they misunderstood in the previous examples. They made mistakes in the first work sample we see. We will show them how to do the appropriate operations and then they will do better on the next test. After we do this, the student will be able to better complete the activity like that shown in the exhibit. We will then do a retest for the student and some or all their peers to see if any do better now on the test. This way we will know how effective the lesson was. If we do a retest just like the old test, we can easily see the progress, if any is made.

Rationale for the Sample Weak Response

Please note that the response is evaluated based upon the four performance characteristics of Purpose, Subject Matter Knowledge, Support, and Rationale. Please also note how the score point descriptions are based upon how the writer attends to each of the performance characteristics. You should be very familiar with the CEOE performance characteristics and score scale and refer to them when reviewing this rationale.

The purpose of this assignment is partially achieved. While the writer appears to attempt to answer most, or even all, of the prompt's bullet points, they do insufficiently and in a limited manner. There are misconceptions shown by the writer regarding the student work sample and the lesson plan. They seem to understand the student's mistakes somewhat but are not connecting the mistakes to a much greater misunderstanding the student is displaying.

The writer of this response does not display an accurate understanding of the subject matter. They may not have sufficient subject matter knowledge themselves to properly assist this student with this specific content. This is evident in their argument presented, which was that the student just made some computation errors, while it is evident that there is a more widespread concern to address in this student's work.

Relevant support is not shown in all parts of this response. Support would likely have come from citing the exhibits specifically as well as presenting the writer's own ideas with a well-reasoned rationale. This was not shown in this response to an appropriate degree.

The degree of understanding displayed in this response does not prove to be effective in completing the tasks assigned. The writer gives us some argument for their claims at times, but it is not sound information. Overall, the response reflects a limited, poorly reasoned understanding of the topic.

Performance Characteristics

The following characteristics guide the scoring of responses to the constructed-response assignment.

Characteristics that guide the scoring of responses
Purpose The extent to which the response achieves the purpose of the assignment
Subject Matter Knowledge The accuracy and appropriateness in the application of subject matter knowledge
Support The quality and relevance of supporting details
Rationale The soundness of argument and degree of understanding of the subject matter

Scoring Scale

Scores will be assigned to each response to the constructed-response assignment according to the following scoring scale.

Score Scale with description for each score point.
Score Point Score Point Description
4  start bold The "4" response reflects a thorough knowledge and understanding of the subject matter. end bold 
  • The purpose of the assignment is fully achieved.
  • There is a substantial, accurate, and appropriate application of subject matter knowledge.
  • The supporting evidence is sound; there are high-quality, relevant examples.
  • The response reflects an ably reasoned, comprehensive understanding of the topic.
3  start bold The "3" response reflects a general knowledge and understanding of the subject matter. end bold 
  • The purpose of the assignment is largely achieved.
  • There is a generally accurate and appropriate application of subject matter knowledge.
  • The supporting evidence generally supports the discussion; there are some relevant examples.
  • The response reflects a general understanding of the topic.
2  start bold The "2" response reflects a partial knowledge and understanding of the subject matter. end bold 
  • The purpose of the assignment is partially achieved.
  • There is a limited, possibly inaccurate or inappropriate application of subject matter knowledge.
  • The supporting evidence is limited; there are few relevant examples.
  • The response reflects a limited, poorly reasoned understanding of the topic.
1  start bold The "1" response reflects little or no knowledge and understanding of the subject matter. end bold 
  • The purpose of the assignment is not achieved.
  • There is little or no appropriate or accurate application of subject matter knowledge.
  • The supporting evidence, if present, is weak; there are few or no relevant examples.
  • The response reflects little or no reasoning about or understanding of the topic.
U The response is unscorable because it is illegible, not written to the assigned topic, written in a language other than English, or lacking a sufficient amount of original work to score.
B There is no response to the assignment.