Unit 3: Instructional Strategies

Traditional problem-solving lessons include a teacher explaining the math and students practicing the math, after which they apply those skills to solving related math problems. Which answer explains why this rarely works? 1. It assumes detailed explanations produce understanding. 2. Problem-solving is too wrapped up in the lesson. 3. It begins where the students are. 4. Students do not have enough time to practice in advance.

1. It assumes detailed explanations produce understanding. Such an approach can help with instrumental but not relational understanding. Constructivism argues that people produce knowledge and form meaning based on their experiences, which help them make connections. A student may understand a concept and yet still be unable to assimilate the new information if it cannot be connected to prior knowledge and experiences.

Which TWO statements describe common mathematical learning theories? 1. New ideas are accommodated in the brain by reworking old ideas. 2. New ideas are assimilated into the brain by memorizing procedures. 3. New ideas are made accessible through peer and teacher support. 4. New ideas are built through teacher modeling of efficient algorithms.

1. New ideas are accommodated in the brain by reworking old ideas. 3. New ideas are made accessible through peer and teacher support.

Know what to do and why 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

1. Relational Understanding

Asking the question, "Was there something in this problem that reminded you of another problem you've done?" 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

1. Relational Understanding Know what to do and why

Giving your students a problem and asking them to explain how they arrived at the solution. 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

1. Relational Understanding Know what to do and why

Using problem-based learning 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

1. Relational Understanding Know what to do and why

Select TWO learning theories that describe effective math learning. 1. Students learn new ideas through collaboration with their teacher and peers. 2. Students should be presented with step-by-step procedures. 3. Students learn by rearranging previously learned concepts and making connections. 4. New math concepts should be taught in isolation.

1. Students learn new ideas through collaboration with their teacher and peers. 3. Students learn by rearranging previously learned concepts and making connections.

An eighth-grade class has been exploring volume by using centimeter cubes to build rectangular prisms. They have developed the volume formula of V = lwh. Which activity is a recommended follow-up to this lesson? 1. A practice activity—students identify geometric solids in the classroom and practice finding their surface area. 2. A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each. 3. A conceptual activity—students explore the area of various 2D shapes using manipulatives. 4. A drill activity—students practice their multiplication facts.

2. A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each. The objective of the lesson was to find the volume formula for a rectangular prism. Any task associated with the objective must align. Because students have developed a conceptual understanding of the material, it is okay to use a drill activity in this situation. A drill activity should only be used after students develop a conceptual understanding. Drill activities should never be used to teach concepts. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas

Which TWO strategies teach for the understanding of mathematics? 1. Reteach using a standard procedure to solve all problems 2. Allow use of a variety of manipulatives to model the problem 3. Encourage use of verbal descriptions in problem-solving activities 4. Correct student errors by solving the problem using the correct method

2. Allow use of a variety of manipulatives to model the problem 3. Encourage use of verbal descriptions in problem-solving activities

To create an environment for doing mathematics, what is the teacher's role? 1. Test ideas and conjectures and tell students the results 2. Create a spirit of inquiry, trust, and expectations 3. Demonstrate mathematical procedures 4. Focus on posing problems

2. Create a spirit of inquiry, trust, and expectations In a constructivist math class, the teacher should act as a facilitator and ask guiding questions. This process takes time, and the teacher must use class time effectively to build a classroom culture of respect and trust. Once this is established, students will feel comfortable talking about errors and learning from their mistakes.

Know what to do, but don't know why 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

2. Instrumental Understanding

Having students compete to see who can solve the most problems. 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

2. Instrumental Understanding Know what to do, but don't know why

Having your students complete a 10-question quiz at the end of the unit. 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

2. Instrumental Understanding Know what to do, but don't know why

Having your students memorize the multiplication table to be able to do their work faster. 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

2. Instrumental Understanding Know what to do, but don't know why

Which statement is not included in relational and conceptual understanding? 1. It requires less to remember. 2. It eliminates poor attitudes and beliefs. 3. It enhances memory. 4. It improves problem-solving abilities.

2. It eliminates poor attitudes and beliefs. Thinking and learning processes occur in two main ways known as relational and instrumental understanding. Instrumental understanding is knowing how to get from Point A to Point B. It gives a step-by-step process that can be followed. Students with instrumental understandings know how to do a mathematical process; however, they may not understand why they are doing it.

A group of students is given the equation 6/8 = 3/4. They are asked to explain the relationship. Which student response demonstrates relational understanding of the concept? 1. It makes the pieces smaller. 2. It's like 2 quarters equals 5 dimes. 3. You reduce and the numbers get smaller. 4. You take out the 2 from top and bottom.

2. It's like 2 quarters equals 5 dimes.

A fifth-grade class is exploring the relationship between various volume formulas and are asked to fill a rectangular prism, cube, and triangular prism with water. Select TWO strategies that could help increase the effectiveness of this activity. 1. Watch students work on the task and stop them if they are doing something wrong 2. Review the properties of 3D shapes prior to the activity 3. Encourage students to collaborate and discuss their findings and strategies after the activity has been completed 4. Give students step-by-step instructions for the activity

2. Review the properties of 3D shapes prior to the activity 3. Encourage students to collaborate and discuss their findings and strategies after the activity has been completed Not number 4 because you want them to figure out for themselves

Even though constructivism and sociocultural theory are two different learning theories, they can both be used simultaneously in teaching mathematics. Imagine that you are teaching your students about fractions. You instruct them to explain how to divide five brownies between three friends. Which TWO scenarios illustrate these two theories working together? 1. One student is working by herself and remembers a time when she actually did divide three brownies between five friends. This helps her know how to proceed. 2. Students are working in groups of three and are figuring out how to solve this problem using some of the concepts about fractions that they have already learned. They must then write their answer using fractions and share them with the class. 3. One student is working by herself but cannot figure out how to proceed. She then asks the teacher for help and the teacher says, "Let's first begin by writing 5 over 3 and see what happens." Students are working by themselves to solve this problem and must write out their solution using fractions. 4. One student is working by himself to figure out how to solve this problem. He is not sure how to proceed and the teacher assists him by asking, "Have you ever divided candy between your friends before?" The student has. He completes the problem and can then show other students how he did it.

2. Students are working in groups of three and are figuring out how to solve this problem using some of the concepts about fractions that they have already learned. They must then write their answer using fractions and share it with the class. 4. One student is working by himself to figure out how to solve this problem. He is not sure how to proceed and the teacher assists him by asking, "Have you ever divided candy between your friends before?" The student has. He completes the problem and can then show other students how he did it.

Students in a sixth-grade class are using models to explore how to find percents. Which question should help foster critical thinking? 1. Will a percent always be out of 100? 2. What patterns have you noticed when modeling percents? 3. Will a percent always be between 1% and 100%? 4. Can you model 50%?

2. What patterns have you noticed when modeling percents? Critical thinking can be evoked by asking open-ended questions. When asking students to explain the observed patterns, the teacher is asking an open-ended question. The other choices are closed-ended questions and can be answered with a simple yes or no. In a constructivist classroom, teachers should ask open-ended questions that cannot be answered with a simple yes or no.

Which process demonstrates a conceptual understanding of solving 32 + 48 = _____? 1. Students use a traditional algorithm and then check their work with a calculator. 2. A student uses a peer-assisted approach when solving the problem by using a manipulative. 3. A student first adds 8 + 2 to get 10 and then adds 30 + 40 + 10 to get a sum of 80. 4. A student collaborates with a classmate to solve the problem with a calculator.

3. A student first adds 8 + 2 to get 10 and then adds 30 + 40 + 10 to get a sum of 80.

Comprehension of mathematical concepts, operations, and relations 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

3. Conceptual understanding

Meaningfully learned and well-integrated knowledge about a topic, including many logical connections among specific concepts and ideas 1. Relational Understanding 2. Instrumental Understanding 3. Conceptual understanding

3. Conceptual understanding

How is a conceptual understanding beneficial to students when learning mathematics? 1. Students are able to reflect on and evaluate their work. 2. It allows them to use their prior knowledge to make connections to the current content they are learning. 3. It allows the student to apply mathematical principles in different contexts. 4. When students approach a problem, they are able to apply different strategies.

3. It allows the student to apply mathematical principles in different contexts. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas

Which of the following are not benefits of relational understanding? 1. It requires less to remember. 2. It improves problem-solving abilities. 3. It helps students gain proficiency in a single method. 4. It enhances memory.

3. It helps students gain proficiency in a single method. Relational understanding refers to knowing both what to do and why - an understanding of all of the parts, how they relate, and why they are applied in the manner they are.

What is an environment that is not desirable for mathematical instruction? 1. Students are testing ideas, making conjectures, developing reasons, and offering explanations. 2. Reasoning is celebrated as students defend their methods and justify their solutions. 3. Students are working through practice problems to learn the mathematical strategy. 4. The focus is on students actively figuring things out.

3. Students are working through practice problems to learn the mathematical strategy. Practice problems should only be used after conceptual understanding. Practice problems do not help students build problem-solving strategies. Problem-solving strategies should be encouraged and taught by using rich problems and building a classroom of respect. Teachers should use class time to discuss students' strategies and allow students time to collaborate while working through rich tasks. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas

What is an important characteristic of the process of teaching and learning mathematics? 1. Symbolism is the most powerful way to communicate mathematical ideas. 2. The concept of proof should be first taught in high school geometry courses. 3. Students of all ages should frequently be asked to provide rationale for their answers. 4. The learning of mathematics should be connected to science, but making connections to art and social studies is not necessary.

3. Students of all ages should frequently be asked to provide rationale for their answers.

Which type of information should NOT be provided by math teachers? 1. Mathematical conventions 2. Clarification of students' methods 3. The preferred method 4. Alternative methods

3. The preferred method In a constructivist classroom, students should be encouraged to explore multiple strategies and methods to solve a problem. The teacher should avoid using step-by-step instructions and reverting to a single process or method. Although the teacher may have a preferred method, students should be encouraged to explore alternative problem-solving approaches.

A student is asked to add two fractions and writes the following: 1 over 2 plus 1 over 3 equals 2 over 5. The student is unsure whether this solution is correct. Which action shows strategic competence? 1. The student explains that 1 + 1 = 2 and 2 + 3 = 5 are correct, so 2/5 is correct. 2. The student asks another student to show what she did to complete the problem. 3. The student uses fraction strips to model adding 1/2 and 1/3 and notes that it is equivalent to 5/6, not 2/5. 4. The student asks the teacher for help and is told to use a common denominator, which prompts the student to the correct answer.

3. The student uses fraction strips to model adding 1/2 and 1/3 and notes that it is equivalent to 5/6, not 2/5.

Traditional problem-solving lessons involve a teacher explaining the math, and then students practice that math. This is followed by applying mathematics to solve problems. Why does a lesson set up as described rarely work? 1. Problem-solving is too wrapped up in the lesson. 2. It begins where the students' knowledge and skills are at. 3. Students do not have enough time to practice in advance. 4. It assumes wonderful explanations produce understanding.

4. It assumes wonderful explanations produce understanding. Constructivist learning theory is a philosophy that enhances students' logical and conceptual growth. The underlying concept within the constructivism learning theory is the role that experiences—or connections with the adjoining atmosphere—play in student education. The constructivist learning theory argues that people produce knowledge and form meaning based on their experiences. A wonderful understanding of a concept does not always mean the student will assimilate the new information. In a constructivist classroom, students need a variety of learning opportunities to explore and build a relational understanding.

2/5 - 1/3 = 1/2 A student solves and submits the above fraction problem. How can this student demonstrate strategic competency? 1. The student collaborates with a peer; the peer informs him that a common denominator was not used and the error is corrected. 2. The student explains to another student that 5 - 3 = 2, so the answer is 1/2. 3. The student works with other students to show them how to solve fractions. 4. The student uses a manipulative and realizes that a common denominator was not used.

4. The student uses a manipulative and realizes that a common denominator was not used.