# Chapter 2 – Learning Theory, Curriculum, and Assessment

Ed Begle said, “We have learned a lot about teaching better mathematics but not much about teaching mathematics better.” When perusing the history of teaching mathematics, it appears as if we constantly look for a magic method or strategy that will serve all students. There is no enchanted formula that fits every learner. We need to realize that a variety of methods and strategies are necessary to meet the mathematical learning needs of all students. Our task, as teachers of mathematics, is to determine which method will be most beneficial to the mathematical learning of each student and when these strategies will be most effective. If a teacher uses a strategy with no successful result from the learner, at what point is that method abandoned in favor of another? The decision is most often influenced by our own background, training, experiences, bias, and current curriculum.

## Exercises

### Exercise 2.1

- What are the major characteristics you would ascribe to a positive mathematics classroom? Which of these would you control? Which would be dependent on your students? Which of these would be dependent on administration?
- Was the mathematics learning environment you experienced in your secondary school years constructivist or behaviorist based? Describe your experiences to amplify your selection.
- List three similarities and three differences found in the behaviorist and constructivist mathematics classroom.
- Constructivism and behaviorism are idealistic. Can either of them be effective in the real world of secondary schools? Why or why not? Should there be a blending of the two?
- Describe your mathematics classroom of the future.

### Exercise 2.2

- Select an activity from one of the books in the Addenda Series. Critique the activity. Will it appeal to students? Why or why not? Would you change it before using it? Why or why not? Are the readiness expectations necessary for the activity clear? Will the students be able to connect the activity to something in their world? Why or why not?
- List the titles of the books in the Addenda Series. Just using the list, which one sounds the most appealing to you and why?
- Select any book from the Addenda Series. Select any chapter description and summarize it. Be sure to include the commentaries listed as “Assessment Matters,” “Teaching Matters,” “Try This,” and so on for the whole chapter.

### Exercise 2.3

- The Professional Standards (pp. 135–139) list courses that should be a part of the background for teachers of mathematics in Grades pre-K–12. Focus on the courses for Grades 5–8 or Grades 9–12. Compare the coursework in your background with those listed. Elaborate on any differences and describe whether or not you feel they are significant.
- Select a vignette from the Professional Standards that you believe to be a description of a good classroom situation. Highlight the strong points of the vignette and describe your impression of the strengths.
- Select a vignette from the Professional Standards that you believe to be a description of a poor classroom situation. Highlight the weak points of the vignette and describe your impression of the weaknesses along with what could be done to strengthen them.

### Exercise 2.4

- Read the “What’s Next?” (pp. 81–83) section of the Assessment Standards. What do you think? Are the descriptions accurate? Were there items that should be added to or deleted from the discussion? Do you think you could build this type of assessment program in your classes and school when you begin teaching? Why or why not?
- Summarize each of the six standards (pp. 11, 13, 15, 17, 19, and 21 in the Assessment Standards). Indicate any section you feel is particularly useful or of little benefit, stating why you feel as you do.
- React to the highlighted description of the process used by Ms. Lee, Mr. Jackson, and Ms. Romario as they made decisions as described on pp. 36–39 of the Assessment Standards. If you had been in the situation, would you have made other suggestions or rejected any they made? Why or why not?
- The Assessment Standards contain a list of suggested major shifts from some things to others as a part of an effective assessment practice (p. 83). What do you think? Are there items that should have been added to or deleted from the discussion? Do you think you could build this type of assessment program in your classes and school when you begin teaching? Why or why not?

### Exercise 2.5

- Compare and contrast the problem-solving and communication standards for Grades 9–12 in the Standards and Standards 2000. Is there a significant difference in the theme behind these standards?
- Review focal points for Grade 6, 7, or 8. Describe how the focal points enhance that particular grade for Standards 2000.
- Look at the standards for Grades 5–8 in the Standards and Standards 2000. Rank order the standards in each document as to their importance in the mathematics curriculum. Is there a significant difference in the order in which NCTM proposed them? Does it matter?
- Look at the standards for Grades 9–12 in the Standards and Standards 2000. Rank order the standards in each document as to their importance in the mathematics curriculum. Is there a significant difference in the order in which NCTM proposed them? Does it matter?

### Exercise 2.6

- How do the Common Core Standards differ from Standards 2000 for Grades 6, 7, and 8?
- How do the Common Core Standards differ from Standards 2000 for Grade 3? Explain the significant difference you discover.
- How do the Common Core Standards differ from Standards 2000 for Algebra?
- How do the Common Core Standards differ from Standards 2000 for Geometry?
- How do the Common Core Standards differ from Standards 2000 for Statistics and Probability?

### Exercise 2.7

- Examine the problem sections of a secondary mathematics textbook. How many of the problems come from the world viewed by a student? Of all the problems, how many appear to be designed to appeal to girls?
- How much paint does it take to paint all the segments on a football field? Is this a good problem-solving problem? Why or why not? Does this problem reflect equity bias? If yes, what could be done to eliminate it? If no, why not?

### Exercise 2.8

- Is it reasonable to have a test of major proportions (like the national Algebra I test that was discussed) at some point during the school year? Why or why not? Describe the impact on the curriculum.
- Does your state have a
*mandatory*mathematics test as a graduation requirement? If so, what is the content level of the test? Is the test based upon the highest mathematics course required by the state for graduation? - If your state has required Algebra I test for graduation, is there a correlation between student scores on the test and grades given in the course? Should there be a strong correlation? Does it matter?

### Exercise 2.9

- Is a national curriculum or text-driven curriculum desirable? Why or why not?
- If the curriculum framework requires a specific concept taught that is not addressed in the textbook assigned to the student, will you teach the concept? How will you supplement the text?
- Examine several textbooks for a given mathematics concept. Describe their similarities. Are there any significant differences? Is there a text that is notably different from the rest? If there is a different text, rationalize why it should or should not be available for adoption. If there is no different text, discuss why they are all the same.
- Are the mathematics concepts taught in high school driven by the state curriculum frameworks or the textbook used in the classroom? Justify your response.

### Exercise 2.10

- Should you, the teacher, as the local authority on your class, solely determine the material to be covered in a given class? If yes, why? If no, how much outside influence should be acceptable and why?

### Exercise 2.11

- If you have been to a local, regional, or state mathematics contest, describe the emotional and general atmosphere of the setting. If you have not seen a competition, visit one as an observer and then describe what you noticed.
- Locate a secondary student who is currently a member of a school mathematics team or who has participated in a mathematics competition in the past year. Describe that student’s reactions and feelings about the associated experiences.
- Locate a teacher who is sponsoring or has sponsored a mathematics competition team. Ask why they do it and what is in it for them.

### Exercise 2.12

- Determine the error pattern the student made in each of the following problems:

Describe the error the student is making. List the steps you would employ to assist the student in learning how to do the problem correctly and avoid repeating the same error. Could this error have been caused because the student is not accustomed to seeing addition problems written horizontally?

- Determine the error pattern the student made in each of the following problems:

4567 389 2468 3421

+7968 +964 +3517 +2476

14635 1453 7085 5897

- Determine the error pattern the student made in each of the following problems:

46.325 + 234.56 + 13.567 + 2.7964 = 0.000000111312

3.579 + 54.32 + 684.2 = 0.015853

35.234 + 67.531 = 0.102765

4.8 + 32 + 0.79 + 7.8 = 0.0237

Describe the error the student is making. List the steps you would employ to assist the student in learning how to do the problem correctly and avoid repeating the same error.

- Define an error pattern you think a student would make. State the grade level. Provide sufficient examples. Give your error pattern to a peer to solve. Describe your discussions with your peer about the error pattern, and how to correct it.
- Develop a continuum of concepts necessary for a student to be able to solve a problem like . Assume the students are able to work successfully with the four arithmetic operations.

## Problem Solving Challenges

### 1. Pyramid of Numbers

$$\begin{array}{c}1\\ 11\\ 21\\ 1211\\ 111221\\ 312211\end{array}$$What are the next two lines?

## Answer

Answer/solution: 13112221 and 1113213211.

The next line is determined by reading the preceding one. Look at the top line “1.” Reading it, you could say “There is one one.” Looking at the second line, you see “one one.” Looking at the second line you would say “There are two ones.” Read that as “two ones” and you have the third line. Reading the third line, you would say “There is one two and one one” which describes the numbers on line four. The fifth line is determined by reading the fourth line “one one, one two, two ones” or 111221. The fifth line indicates that numbers that are read as digits appear and are grouped by the same digits in that order. The sixth line comes from reading the fifth as “Three ones followed by two twos which are followed by one one” or 312211. Thus, the seventh line would be 13112221, and the eighth line would be 1113213211.

### 2. The Letter Mania

Based on the following two groups of letters, place each letter from the following letters in its appropriate group: I, J, K, L:

Group 1: A, E, F, H

Group 2: B, C, D, G

.

## Answer

Answer/solution: Group 1 letters are all straight segments while Group 2 contain curves. So, I, K, and L go into Group 1 while J goes into Group 2.

## Additional Learning Activities

**1. Why are manhole covers round?**

**Extension**

The text discussions and exercises should help you understand why manhole covers are round. They will not fall down the hole (assuming the diameter of the lid is slightly larger than the diameter of the hole). The circle is not the only shape that can serve as a manhole cover. A Reuleaux triangle in Figure 2.1 will work, too.

Construct an equilateral triangle, ABC. Using A as the center and B as the radius end, construct a minor arc between B and C. Place a point, E, on the arc, making segment AE. Repeat this construction for the remaining two sides of the original triangle to create segments BG and CJ. Segments AE, BG, and CJ are all the same length since they are radii of congruent circles. Thus no lid would fall down this hole.

This Reuleaux triangle water main cover was spotted on one of the back streets of St. Augustine, FL in July 2004. The historic district of St. Augustine is many square blocks. Check a few blocks off of the main street in front of the Methodist church, for sure. There are several others throughout the historic district.

Could a Reuleaux square serve as a manhole? Construct a square. Select two adjacent vertices and make a circle. The arc is not outside the square so this cannot be used to create a Reuleaux square. Select one side and construct its midpoint. Use the midpoint as the center of a circle whose radius length is determined by one of the vertices not on the midpoint’s segment. That circle passes through two vertices. But the radius of the circle will be $$\frac{s\sqrt{3}}{2}$$ while the diagonal length is $$s\sqrt{2}$$. The diagonal is longer and the lid could fall down the hole. So, this construction in Figure 2.3 cannot be used to create a Reuleaux square.

Could a Reuleaux regular pentagon serve as a manhole? Create a regular pentagon. Join a vertex with a vertex from a non-adjacent side and construct the short arc between two adjacent vertices, putting a point on the arc. Repeat this for the other four sides as shown in Figure 2.4. The five segments are all the same length since they are radii of congruent circles. Thus no lid would fall down this hole.

**Your Turn**

- Create a Reuleaux figure from a regular heptagon as was done with the regular pentagon. What conclusions can be made about Reuleaux figures generated from regular n-gons when n is odd?

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