Study Guide

Field 125: Middle Level/Intermediate Mathematics 
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 will be asked to prepare a written response of approximately  300 to 600  words on the assigned topic.

Read the assignment carefully before you begin your response. Think about how you will organize your response. You may use the erasable sheet(s) to make notes, write an outline, or otherwise prepare your response.  start bold and italics However, your final response to the assignment must be either:  end bold and italics 

  1. typed into the on-screen response box,
  2. written on a response sheet and scanned using the scanner provided at your workstation, or
  3. provided using both the on-screen response box (for typed text) and a response sheet (for calculations or drawings) that you will scan using the scanner provided at your workstation.

 start bold Instructions for scanning your response sheet(s) are available by clicking the "Scanning Help" button at the top of the screen. end bold 

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 by merely reciting factual information.

Your responses to the assignments will be evaluated based on the following criteria.

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

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 response must be your original work, written in your own words, and not copied or paraphrased from some other work.

Be sure to write about the assigned topic. Remember to review your work and make any changes you think will improve your response.

Any time spent responding to the assignment, including scanning the response sheet(s), is part of your testing time. Monitor your time carefully. When your testing time expires, a pop-up message will appear on-screen indicating the conclusion of your test session. Only response sheets that are scanned before you end your test or before time has expired will be scored. Any response sheet that is not scanned before testing ends will  start uppercase NOT end uppercase  be scored.

Sample Constructed-Response Assignment

subarea roman numeral 5 
Pedagogical Content Knowledge

 start bold Use the data provided to complete the task that follows.  end bold 

Using the data provided, prepare a response of approximately  300 to 600  words in which you:

 start bold Class Context end bold 

Students are working on calculating probabilities using the fundamental principle of counting. In previous class sessions, they practiced systematic methods of counting the number of possible outcomes. Currently, they are applying those concepts to calculating probabilities. The teacher asks students to practice the concepts they have learned through the completion of a worksheet. One student's work is shown in an exhibit.

 start bold Excerpt from Lesson Plan end bold 

 start bold Standard: 6.D.2 end bold 

"Use probability to solve real-world and mathematical problems; represent probabilities using fractions and decimals."

 start bold Lesson Objective: 6.D.2.2 end bold 

"Determine the sample space for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations."

 start bold Essential Question: end bold 

How can we use our understanding of counting principles to find the probability of getting various outcomes when flipping coins?

 start bold Previous Knowledge Needed: end bold 

Using systematic methods to count the number of possible outcomes

 start italics Source: Oklahoma Academic Standards  2016 to 2017  Oklahoma State Department of Education end italics 

 start bold Student Work Sample end bold 

Practice Calculating Coin Flip Probabilities

Calculate the probabilities:

  1. Flipping 2 coins and getting exactly 1 head Graphic shows 3 groups of 2 circles. Each circle is defined T or H.
    group 1
    group 2
    group 3

    probability: 1 third

  2. Flipping 3 coins and getting exactly 2 heads Graphic shows 4 groups of 3 circles. Each circle is defined T or H.
    group 1

    group 2

    group 3

    group 4

    probability: 2 fourth equals 1 half

  3. Flipping 3 coins and getting exactly 3 heads Graphic shows 4 groups of 3 circles. Each circle is defined T or H.
    group 1

    group 2

    group 3

    group 4

    probability: 1 fourth

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 student has a partial understanding of probability, which is defined as the ratio of desirable outcomes to possible outcomes. This is evident by the incorrect answers, which are based on the limited number of possible outcomes listed (see student work sample, responses 1, 2, and 3). Another strength is shown in the candidate's ability to reduce a fraction. In the student work sample, the answer to number two, shows the student knowns how to reduce a fraction from  2 fourths  to  1 half. 

In looking at the student work sample, it is clear that the student needs to understand that order matters. This need is evident in all three parts of the student work sample. There is no identification of HT (Heads, Tails) as different from TH. Therefore, all answers in the student work sample are incorrect.

To intervene, the student could be prompted to make a systematic list or diagram that would generate all possible outcomes. For example:

systemic list of possible coin flip outcomes.
1st Coin 2nd Coin 3rd Coin
 Heads   Heads   Heads 
 Heads   Heads   Tails 
 Heads   Tails   Heads 
 Heads   Tails   Tails 
 Tails   Heads   Heads 
 Tails   Heads   Tails 
 Tails   Tails   Heads 
 Tails   Tails   Tails 

OR

(Third Coin added to each outcome)
diagram of possible coin flip outcomes.
 this cell intentionally left blank.  First Coin
 Heads   Tails   
Second 
Coin 		 Heads   Head Heads Heads, Heads Heas Tails   Tails Heads Heads, Tails Heads Tails 
 Tails   Heads Tails Heads, Heads Tails Tails   Tails Tails Heads, Tails Tails Tails 

OR

A tree diagram, as shown below, would work as would flipping two different coins. By using any of these systematic lists or diagrams to show all possible outcomes, the teacher would be building on the student's understanding of probability and would simultaneously be helping the student learn how to understand that order does matter.

The first level of the tree diagram is a fork to the 2 possibilities of heads and tails. At the next level, heads forks to heads heads and heads tails. Tails forks to tails heads and tails tails. At the next level, heads heads forks to heads heads heads and heads heads tails. Heads tails forks to heads tails heads and heads tails tails. Tails heads forks to tails heads heads and tails heads tails. Tails tails forks to tails tails heads and tails tails tails.

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 response fulfills the purpose of the assignment (refer to the instructions for the assignment) by identifying the student's partial understanding of probability, citing evidence from the student work sample (i.e., the student showed how to reduce a fraction from  2 fourths  to  1 half ), and by identifying the student's misunderstanding of sample space due to the lack of recognition that order matters. The purpose of the assignment is largely achieved, as the response also suggests a variety of ways to provide intervention and support to address the student's need for improvement (i.e., a systematic list, diagrams, hands-on activity). Overall, the response reflects an adequate understanding of the student's weaknesses and includes an adequate plan for intervention. A stronger response would demonstrate a comprehensive understanding of the topic and would provide further high-quality, relevant examples for each part of the prompt.

Sample Weak Response to the Constructed-Response Assignment

The given standard for this assignment is 6.D.2, "Use probability to solve real-world and mathematical problems; represent probabilities using fractions and decimals." We should look at the student work sample in order to identify the mathematical strengths and needs of this student.

The given student does show some strengths in the student work sample. First of all, the student has a beginning idea of possible outcomes and can illustrate it. This is evident in the sketches on the student work sample, in responses 1 to 3.

Unfortunately, the student also shows some significant areas of need. For instance, the student does not fully understand possible outcomes. Not all possible outcomes are included in the student work sample.

To build on the student's strength, have the student flip two different coins and record the results and see how many different variations are tossed. The student can record the results in a variety of ways. This will help the student be able to see all of the possible outcomes.

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 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 the assignment is not achieved. While the response attempts to identify the student's strengths and weaknesses, the evidence cited was not specific. In fact, the response does not actually state a significant mathematical strength, it simply alludes to work shown in the student work sample, yet fails to give any specific evidence. The response states the student does not understand "possible outcomes" for the area of need, but provides no relevant examples to support the identified need, again failing to provide any relevant examples. Moreover, the plan for intervention is not well developed and lacks detail. Overall, the response reflects little understanding of the student's weakness and intervention needed.

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.