# National Council of Teachers of Mathematics

Mathematics

Exercise 1:
Introduction:

In the twenty-first century, mathematics education must take on a different form. Teachers are no longer expected to simply be a dispenser of information; new models of teaching and learning are needed that align with best practices and the standards of the National Council of Teachers of Mathematics (NCTM).

Requirements:
A. Provide an accurate, secondary-level problem appropriate for a problem-based learning activity in numbers or algebra by doing the following:
1. Provide a traditional procedural mathematics problem that could be improved by being transformed into a discovery problem.
a. Discuss shortcomings of using this problem when teaching for understanding.
2. Modify the problem you provided in part A1 so that it better serves students when teaching for understanding.
a. Explain how the modified problem promotes student engagement.
b. Explain how the modified problem promotes problem-solving or reasoning skills.
c. Explain how the modified problem promotes a deeper understanding of the mathematical concepts.

B. Discuss how teaching mathematics for understanding provides opportunities for all students to become mathematically proficient, as it relates to your modified problem from part A2.

C. When you use sources, include all in-text citations and references in APA format.

Exercise 2:
Introduction:

Teaching through problem solving is an exciting, effective strategy. This approach is advocated by the National Council of Teachers of Mathematics (NCTM) and is used by many mathematics teachers.

Requirements:
A. Plan how you would implement a secondary-level, problem-based lesson covering a topic related to functions by doing the following:
1. Provide an appropriate problem that you selected, created, or modified.
2. Describe an effective way to make the problem relevant to your students.
3. Describe an effective way to help students connect the problem to their prior knowledge.
4. Recommend solution strategies that students should use to approach the problem.
a. Describe different mathematical representations (e.g., tables, equations, pictures) that students should use while solving the problem.
5. Provide three meaningful questions to support a student who is struggling (without removing the challenge).
a. Provide an example of a hypothetical dialogue in which you use the three questions from part A5 to support a student who is struggling.
6. Provide two questions or prompts to effectively challenge students who finish their work early.
7. Describe how you would engage students in meaningful discourse by doing the following:
a. List the solution strategies from part A4 in the order in which you would utilize them during a whole-class discussion.
i. Justify your sequencing of the solution strategies.
b. Provide a specific example of how to help students make connections between different strategies.
c. Describe two strategies to effectively engage all students.
8. Discuss two objections to using a problem-based approach to teaching mathematics.

B. When you use sources, include all in-text citations and references in APA format.
For this task, you will create a problem-based lesson plan and then provide a brief reflection on your lesson plan.

Requirements:
A. Modify or design a problem-based lesson in secondary geometry by doing the following:

Note: You may use the attached “WGU Problem-Based Lesson Plan Template,” or you may create your own lesson plan format to meet the requirements of this task.

1. Determine the mathematics content and learning goals by doing the following:

a. Identify a state standard or a standard from the Common Core State Standards for Mathematics that will guide your lesson.
b. Create a measurable learning objective (condition, behavior, criterion) aligned to the standard.
2. Consider your students’ needs by doing the following:
a. Describe the prior knowledge necessary for a student to understand the lesson.
b. Describe an appropriate strategy or strategies, specific to this task, for accommodating the needs of any English Language Learners (ELLs).
c. Describe an appropriate strategy or strategies, specific to this task, for accommodating the needs of any students who struggle or have special needs.
3. Select or design a worthwhile task that is integrated throughout the phases of the lesson.
4. Design lesson assessments by doing the following:
a. Describe observations that will be used to determine how well students are meeting the objective.
b. List questions that will be used to determine how well students are meeting the objective.
5. Describe how to effectively introduce the problem to the students in the “before” phase of this lesson.
6. Describe how to effectively monitor student progress and provide hints while students are engaged with the task in the “during” phase of this lesson.
7. Describe how to effectively connect the task to the learning goals in a whole-class discussion in the “after” phase of this lesson.

B. Provide a reflection on the lesson you just created by doing the following:

1. Check for alignment within the lesson by doing the following:
a. Describe the alignment between the objective, assessments, and questions asked in the “during” and “after” phases of the lesson.
b. Describe how the flow of the lesson builds in sophistication.
2. Discuss misconceptions or barriers students might have and how to appropriately respond.
3. Justify that the questions used in the “during” and “after” phases of the lesson are higher-level questions.

C. When you use sources, include all in-text citations and references in APA format.
Assessment is an integral part of the learning and teaching of mathematics. A quality assessment plan uses a variety of strategies and data in order to provide evidence of mathematics understanding and proficiency, feedback to students necessary to assess their own learning, and invaluable feedback to help inform instructional decisions.

Requirements:

Create summative assessment strategies for assessing student understanding of a probability or statistics topic appropriate for a secondary mathematics class by doing the following:
A. Create a performance-based assessment task.
1. Develop a rubric for the task that includes performance indicators (i.e., task-specific statements that describe what performance looks like for each level of the rubric).
2. Justify how the task reflects the full range of mathematics (i.e., conceptual understanding, procedural proficiency, and mathematical processes and practices).
B. Create an open-ended writing prompt that can be used to gather self-assessment data.
1. Describe how the writing prompt encourages students to become active learners.
C. Create a quiz (suggested length of 3–5 items).
1. Provide an accurate answer key for the quiz.
2. Justify how the quiz assesses computational skills as well as a conceptual understanding of the process.
D. When you use sources, include all in-text citations and references in APA format.

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