Preparation Manual

Score Point 4

Benjamin demonstrates a lack of foundational skills for applying multiplication concepts in word problems as specified in the objective and the TEKS (Exhibit 4). He has demonstrated mastery of automaticity in multiplication facts, as shown in the "Mathematics Fact Timed Test" (Exhibit 5), where he scored 100%. However, Benjamin was unable to give evidence of his understanding of multiplication concepts when solving one- and two-step word problems in the Independent Practice Activity (Exhibit 6). In the first problem, Benjamin created an addition equation rather than a multiplication equation. He did not identify the mathematics vocabulary necessary to solve the word problem. Therefore, he struggled with creating an array to fully illustrate how to solve the problem. In the second problem, he created an array, but again used an addition equation to illustrate one of the steps rather than multiplication, which resulted in an incorrect answer. In the third problem, Benjamin did not create a picture, array, or an equation. He relied on an incorrect interpretation of vocabulary and solved the problem incorrectly.

A developmentally appropriate instructional strategy the teacher could implement would be a "Think Aloud," verbalizing thoughts while teaching a logical process, using the mnemonic device "CUBES." The teacher would read the problem with Benjamin, circle the numbers, underline the question, box the key words, evaluate and draw, then solve and check (circle, underline, box, evaluate, solve). In the first problem, while thinking aloud, they would circle 8 and 5 and underline the question. They would box "chairs in each row". Next, the teacher would demonstrate using manipulatives to create an array, which is a concrete representation of the multiplication problem. They would create 8 rows of 5 using manipulatives and a drawing. Lastly, they would determine the correct operation to solve the problem. They could discuss that addition could be used, but it would need to be a multi-step problem. They could also discuss how multiplication is a way to do addition more quickly. They would then write the equation to solve the problem. The teacher would model additional word problems, guiding Benjamin in the steps. Finally, Benjamin will solve problems on his own.

This strategy is effective because direct, explicit instruction in the use of step-by-step problem-solving with manipulatives, pictures, and writing the equation is a progressive process, moving from the concrete to the abstract. The teacher is modeling the process which will provide Benjamin the opportunity to understand the steps involved in solving the problem successfully. Practicing solving word problems using this strategy consistently, Benjamin will begin to see similarities in other problems. This practice illustrates the Gradual Release of Responsibility model (I do, we do, you do).

The teacher can monitor Benjamin's progress with a checklist and anecdotal notes based on regular observation of Benjamin's work. The checklist should include specific steps for Benjamin to follow to solve word problems using CUBES. The teacher's observations of and discussions with Benjamin about his problem-solving approach will allow the teacher to verify mastery and make necessary adjustments to instruction.

To support and extend learning, the teacher could ask Benjamin to write his own mathematics story problems. The teacher could provide a template with characters, conflict, and resolution. This activity could strengthen Benjamin's problem-solving skills by giving him an opportunity to apply mathematics vocabulary and apply his knowledge of multiplication concepts through the story elements of a mathematics word problem. He could share the story with peers and ask them to solve the problem. He can verbalize how to use the steps to solve the equation; thus, providing more reinforcement. This extension lesson integrates math vocabulary, reading, writing, and speaking.

Rationale for the Score of 4

This response reflects a thorough understanding of the relevant content knowledge and skills. The response fully addresses all parts of the assignment and demonstrates an accurate, highly effective application of the relevant content knowledge and skills. The response provides strong, relevant evidence; specific examples; and well-reasoned explanations.

Completion: Notice that each of the five tasks presented in the assignment are answered completely, and in the order presented in the prompt. The response fully described and explained Benjamin's academic need citing relevant evidence from the exhibits provided. The candidate fully described a developmentally appropriate instructional strategy that effectively targets Benjamin's needs using professional knowledge and in a sequential, logical order. The response then offers a sound rationale of why the strategy will be effective, which reflects appropriate pedagogical knowledge of teaching mathematics to early childhood students. An informal assessment to effectively monitor Benjamin's progress toward the skill is thoroughly described. The candidate described a cross-curricular activity that could be integrated into the curriculum as an extension.

Application of Knowledge: As you read the response, note the accurate, current application of professional knowledge about teaching young children mathematics. The candidate interpreted the data provided accurately to identify Benjamin's need, used that information to design an appropriate strategy to address his needs, explained why the strategy would be effective using professional knowledge pedagogy, and continued the skills into another lesson via cross-curricular integration. The teacher is explicitly modeling each step through the "Think Aloud." Manipulatives, pictorial representation, and sequential learning are developmentally appropriate and relevant to addressing Benjamin's academic need of understanding and applying multiplication concepts to solve word problems. Checklists combined with anecdotal notes are effective progress monitoring tools. The extension strategy incorporates other curricular areas.

Support: The response supports assertions with specific, relevant evidence from the exhibits provided. The strategy, rationale, progress monitoring, and extension activity have specific examples. The response cites evidence from the exhibits provided. The strategy is clearly presented with specific supporting details for each step. The rationale for the strategy's effectiveness reflects sound reasoning. The extension lesson has specific examples.

Score Point 2

One area of academic need that the third grade student, who is working on his math skills, has is how to solve multiplication word problems. On the practice activity, Benjamin didn't answer any of the questions correctly. He substituted addition for multiplication and couldn't explain to his teacher how he solved his problems. He drew some pictures and one array but still was not able to correctly solve the problems. When the teacher was talking with him, it seemed like Benjamin knew how to multiply, but he also seems confused on the difference in multiplying, dividing, adding, and subtracting.

The teacher should instruct Benjamin in drawing pictorial arrays to represent the information he needs to solve each word problem. Benjamin drew eight X's to represent eight rows, but he did not draw five chairs in each row. If the teacher provided manipulatives or some other kind of real world tool, Benjamin would be better able to draw pictorials. She could have even used her classroom to help him understand. How many rows of desks? How many desks in each row? Then Benjamin would have a real world connection.

Exit tickets are a very effective way to formatively assess students' knowledge. The teacher can create exit tickets with word problems and then Benjamin would complete one ticket each day. He could have a chart with each day and then each time he was correct, he could put a sticker or check mark. At the end of the week, the teacher could have an incentive to reward him for getting at least 3 correct. By offering an incentive like station time, Benjamin will definitely increase his effort and ultimately his success. Since the problem he is working on is about cookies, the teacher could use toy cookies or pictures of real cookies to make it more of a real world math problem.

The teacher can work with other teachers on her team to create cross curricular activities. This could be in science, art, language arts and physical education. The PE teacher could have the students make life size arrays, in art they could create pictures of arrays with paints, and so forth. There are so many ways to make real world connections for our students. Creating real world activities to help Benjamin understand the math problems will increase his success rate.

Rationale for the Score of 2

The "2" response reflects a limited understanding of the relevant content knowledge and skills. The response partially addresses some of the parts of the assignment and demonstrates a limited application of the relevant content knowledge and skills. The response provides limited evidence and examples or explanations, when provided, are only partially appropriate.

Completion: Notice that the response does not fully respond to each portion of the assignment. The response describes an academic need, but unlike a score point "4" or "3" response, the evidence cited is presented in a less specific manner with partial evidence from the exhibits. The response provides an instructional strategy without the specifics of how to implement the strategy reflecting limited pedagogical knowledge of teaching mathematics to early childhood students. The response partially describes an informal assessment and a cross-curricular activity. This differs from the score point "4" or "3", which would have fully responded to each part of the assignment with more specificity and strong pedagogical knowledge.

Application of Knowledge: As you read the response, note the lack of accurate, current application of professional knowledge about teaching young children mathematics. The response demonstrates a partially accurate, limited application of the relevant content knowledge and skills. The response identifies a need related to multiplication, but the response does not explain why the strategy would be effective. This response begins with some information from the exhibits to determine an academic need. The academic need is partially identified. A strategy to address the need is provided, however, the response lacks a clear description of how the strategy would be implemented, the steps the teacher would take, and the order of how the instruction would be presented. It focuses on helping Benjamin make real-world connections, which is important, but does not address how the teacher will specifically instruct Benjamin to draw the arrays and then apply that knowledge to solve multiplication word problems. The informal assessment focuses on rewards and incentives for motivation and real-world connections and not on the more relevant, important information. The response demonstrates limited content knowledge regarding creating appropriate assessments to monitor student progress. This differs from the score point "4" or "3" response, which demonstrate an accurate, highly effective application of the relevant content knowledge and skills.

Support: Notice how the response provides limited evidence and examples or explanations, when provided, may be only partially appropriate. The response supports assertions with limited evidence from the exhibits provided. The strategy, rationale, progress monitoring, and extension activity are partially described. The explanations contain little or no evidence from the text or of relevant pedagogical knowledge. The explanation for the strategy's effectiveness is limited and reflects weak reasoning. The focus is not directly linked to Benjamin's specific academic need. It discusses motivation strategies without supporting that need with any evidence from the exhibits. The cross-curricular activity primarily focuses on real-world connections and not on how the activity specifically relates to extending Benjamin's knowledge of using multiplication to solve word problems. This differs from the score point "4" or "3" responses that provide relevant evidence, a higher degree of specific examples, and well-reasoned explanations supporting the major points in the assignment.