Math Methods Chapter Three

20) This chapter pointed out the difference between drill and practice. What do these differences look like in a classroom?

1. Drill is used to increase facility of an already learned procedure 2. Drill is used to help the retention of facts and procedures 3. Practice is used to develop alternative and flexible strategies 4. Practice can deepen conceptual knowledge through more useful connections

19) Problem solving provides opportunities for students to demonstrate mathematical proficiency. Identify two ways that teaching through problem solving works for all students.

1. approaches mathematical learning through inquiry 2. explore real contexts, problems, situations and models 3. learning through doing shifts the focus on the students 4. problems have multiple entry and exit points 5. links to other disciplines

9) Researchers have found that teachers' benefit from using an intentional selection process to determine whether a task they chose is worthwhile. One of the criteria of the selection guide is "worthwhile features." Which statement below represents one of the categories in "worthwhile features"? A) Relevant context. B) Problematic task. C) Requires tools and models to represent the mathematics. D) Mathematically interesting.

A) Relevant context.

2) Which statement about the teaching through problem solving approach is most accurate? A) Requires a four-step approach to problem solving. B) Rarely requires students to use alternative methods for problem solving. C) Can only be used with certain problem solving tasks. D) It is closely associated with teaching for problem solving.

A) Requires a four-step approach to problem solving.

16) The act of writing is a reflective process and involves students in metacognition? What statement is an example of a students' conscious monitoring of how and why they are doing something? A) Taking a timed multiplication test. B) Taking two numbers in a word problem and adding them because your class worked on addition problems the previous day. C) Looking back at problems previously worked incorrectly to examine the mistakes. D) Students are playing a game of Integer War,during which they each flip a card and the student who calls out the higher of the two numbers first wins the round.

A) Taking a timed multiplication test.

6) Which is the most accurate statement regarding posing a worthwhile problem? A) Teachers should select the problems that will help make relationships between mathematical concepts explicit for students. B) Any problem used should have the potential to be solved by students using a memorized procedure. C) There should be agreement between students that there is one correct answer. D) Problems should involve words.

A) Teachers should select the problems that will help make relationships between mathematical concepts explicit for students.

12) Facilitating classroom discourse requires that the teacher be aware of what factors that will affect the level of mathematical talk in the classroom. What statement below would impact the mathematical talk? A) The level of English proficiency of the students in the classroom. B) The depth of mathematical concepts required to meet the curriculum guidelines. C) The level of teacher knowledge of the mathematical concepts. D) The depth of the student questioning skills.

A) The level of English proficiency of the students in the classroom.

14) Teachers can prompt productive classroom discussion through the use of five teacher talk actions. What statement below reflects the type of prompt to elicit student reasoning? A) You used a hundreds chart. B) Ricardo said that he used a draw a picture strategy. C) Who also used similar strategy as Ricardo? D) Let's take some time to think about the strategies.

C) Who also used similar strategy as Ricardo?

17) Teaching through problem solving provides opportunities for all students to become mathematically proficient. Teaching through problem solving benefits all students in what way? A) Focusing students on ideas and sense making. B) Focusing on the single strategy need to solve most problems. C) Focusing on procedures that will effectively find answers. D) Focusing on the technology that will guide them to solutions.

B) Focusing on the single strategy need to solve most problems.

18) Asking effective questions requires planning and teachers become more adept at asking quality questions throughout their careers. What common pattern of questioning fosters a greater chance of classroom discussion? A) Funneling. B) Focusing. C) Initiating, responding, feedback. D) Correcting.

B) Focusing.

7) Which statement below best represents worthwhile features of tasks or problems for learning mathematics? A) Problematic, memorization, no connection. B) Problematic, concepts and or misconceptions, relevant. C) Problematic, require correct answer, rules and formulas. D) Problematic, algorithmic, routine.

B) Problematic, concepts and or misconceptions, relevant.

11) Drill of skills has a place in helping students improve procedural knowledge but not conceptual knowledge. Before committing to a solution of "just drill" what do you need to know? A) The amount of time the student has put in on learning basic facts. B) The type of drill that will build understanding. C) The student's attitude toward mathematics learning. D) The level of procedural knowledge the student has demonstrated.

B) The type of drill that will build understanding.

4) Problem solving tasks that have multiple entry and exit points exhibit what characteristics below? A) Are focused on producing correct answers. B) Varying degrees of challenge and methods to approach a solution. C) Are best done independently to address diverse learning styles. D) Varying levels of written explanation are required.

B) Varying degrees of challenge and methods to approach a solution.

10) Drill and practice is present to some degree in every classroom. The definitions of drill and practice demonstrate how they are different and how they link. One of the statements below is representative of practice more than drill. A) An increased facility with a procedure. B) A focus on a singular method. C) An increased opportunity to develop conceptual ideas. D) A focus on rule oriented view of mathematics.

C) An increased opportunity to develop conceptual ideas.

1) Which statement best reflects the approach of teaching for problem solving? A) It is the method rarely used in traditional textbooks. B) It usually involves students exploring alternative methods of solving a story problem. C) It frequently results in the instructor explaining a skill and providing practice and application of the skill. D) The approach has traditionally been a very effective way to help students gain conceptual understanding.

C) It frequently results in the instructor explaining a skill and providing practice and application of the skill.

3) Selecting problem solving tasks that require higher levels of cognitive demand should include which one of the following? A) Use of a specific procedure. B) Are routine and students can use previously learned procedure. C) Use of complex and non-algorithmic thinking. D) Are straightforward with little ambiguity of what needs to be done to solve.

C) Use of complex and non-algorithmic thinking.

13) Effective use of classroom discussion has the teacher monitoring the students' strategies and asking questions. What question below would require the student to reflect on their specific strategy? A) What did you do to make sense of the problem? B) What numbers or information in the problem did you not need to use? C) What mathematical terms, symbols or tools did you use to find a solution? D) What other strategies did your classmates use to solve the problem?

C) What mathematical terms, symbols or tools did you use to find a solution?

8) Which of the following is an instructional example of teaching through problem solving? A) Providing students with a list of area formulas and asking them to find the area of a given rectangle. B) Teaching students the algorithm for fraction division and then asking them to find out how many servings of 1/3 pizza could be made from 11/3 pizzas .C) Having students develop their own word problems that use a recently learned algorithm. D) After students have conceptual understanding of the area of a rectangle, asking them to find the area of a triangle that was constructed by cutting a given rectangle in half and then to generalize their process to how they might find the area of any given triangle.

D) After students have conceptual understanding of the area of a rectangle, asking them to find the area of a triangle that was constructed by cutting a given rectangle in half and then to generalize their process to how they might find the area of any given triangle.

15) How much to tell or not tell is a constant dilemma for teachers. What statement would be the type of information that teachers do need to tell? A) The level of cognitive demand that the students need to demonstrate. B) Specific problem solving strategies needed to solve the problem. C) Information needed to keep students from struggling to solve the problem. D) Help students clarify their ideas and point out related ideas.

D) Help students clarify their ideas and point out related ideas.

5) Children's literature can be a method of creating tasks with context students can connect to other subjects. What statement below demonstrates how a teacher made a connection? A) Today we are going to learn about doubling numbers. B) Today we are going to compute using doubles. C) Today we are going to solve problems with numbers that double. D) Today we are going to find out how numbers double with a Chinese magical pot.

D) Today we are going to find out how numbers double with a Chinese magical pot.