CHAPTER 3 MOTIVATING STUDENTS TO ENGAGE IN MATHEMATICAL LEARNING ACTIVITIES 98

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in Case 3.10 utilizes presentation techniques de­signed to obtain and maintain student engagement.

CASE 3.9 Mr. Johnson's 26 seventh-grade mathematics students are. sitting at their desks. Nine have paper and pen poised to take notes; others are involved with their own thoughts as he begins, "Today, class, we're going to study a measure of central tendency called the 'arithmetic mean: Some of you may have already heard of it." Turning away from the class to write on the whiteboard, he continues to speak, "The arithmetic mean of N numbers equals the sum of the numbers divided by N." Turning his side to the class, he says, "For example, to compute the mean of these numbers-15, 15, 20, 0, 13, 12, 25, 40, 10, and 20-we would first add the numbers to find the sum. Right?" He looks at the class but does not notice whether students appear to respond to his question, turns back, and adds the numbers on the board. Turning to the class, he says, "So the sum is 170. Now, because we have 10 numbers, N in the formula is 10 and we divide 170 by N, or 10. And what does that give us? It gives us 17.0. So, the arithmetic mean of these numbers is 17.0. Is that clear?"

Mr. Johnson stares at the class momentarily, notices Armond nodding and softly saying, "Yes." With a smile, Mr. Johnson quickly says, "Good! Okay, everybody, the arithmetic mean is an important and useful statistic. Suppose, for example, I wanted to compare the following group of numbers (from his notes he copies the following numbers on the board: 18, 35, 30, 7, 20) to these over here." He points to the previous data sequence.

Mr. Johnson: What could we do, Ramon? Ramon: Compute the arithmetic mean you told us about. Mr. Johnson: That's right! We could compute the

arithmetic mean. 18 + 35 + 30 + 7 + 20 = 110, and 110 -;- N - which in this case is 5-okay?-is 22.0. Okay? Now that means this second data set has a higher average than the first, even though the first sequence has more numbers. Any questions? [pause] Good! [pause] Oh, okay, Angela?

Angela: Why do you write "17.0" and "22.0" instead of "17" and "22"? Aren't they the same?

Mr. Johnson: Good question! Hmmm, can anybody help Angela out? [pause] Well, you see in statistics the number of decimal places indicates something about the accuracy of the computations, and for that matter, the data-gathering device. So that one decimal point indicates that the statistics are more accurate than if we had written just "17" and "22" and not as as accurate if we had written, say "17.00000" or "22.00000." Got it? That was a good question. Do you understand now?

Angela: I guess so.

Mr. Johnson: Good! Now, if there are no more questions, there's some time left to get a head start on your homework.

CAStE 3.10 Ms. Erickson's 27 seventh-grade mathematics students are quietly sitting ~t their desks, each ready with paper and pen. She has previously taught them how to take notes during large-group presentations so that they record information during the session on paper and then, after the session, organize the notes and transfer them into their required notebooks.

After distributing a copy of Exhibit 3.5's tasksheet to each student, she faces the class from a position near the overhead projector and says, 'Tm standing here looking at you people and I just can't get one question out of my mind." Very deliberately, she walks in front of the fourth row of students and quickly, but demonstratively, looks at their feet (see Exhibit 3.6). Then she moves in front of the first row and repeats the odd behavior with those students. "I just don't know!" she says shaking her head as she returns to her position at the overhead.

She turns on the overhead, displaying the first line of Exhibit 3.5, and says, "In the first blank on your copy please write: Do the people sitting in the fourth row have bigger feet than those in the first row?" She moves closer to the students to monitor how they follow directions. Back at the overhead as they complete the task, she says, "Now, I've got to figure a way to gather data that will help me answer that question." Grabbing her head with a hand and closing her eyes, she appears to be in deep thought for a few seconds and then suddenly exclaims, "I've got it! We'll use shoe sizes as a measure. That'll be easier than using a ruler on smelly feet!" Some students laugh, and one begins to speak while two others raise their hands. But Ms. Erickson quickly says, "Not now please, we need to collect some data." She flips an overlay off the second line of the transparency, exposing "Data for Row 4."

Ms. Erickson: "Those of you in the fourth and first rows, quickly jot down your shoe size on your paper. If you do not know it, either guess or read it off your shoe if you can do it quickly. Starting with Jasmine in the back and moving up to Lester in front, those of you in the fourth row call out your shoe sizes one at a time so we can write them down in this blank at our places." As the students callout the sizes, she fills in the blank on the transparency as follows: 6, 10.5, 8, 5.5, 6, 9. Exposing the next line, "Data for Row 1," on the transparency she asks, "What do you suppose we're going to do now, Pauline?" Pauline: "Do the same for Row 1." Ms. Erickson says, "Okay, you heard Pauline; Row 1, give it to us from the back so we can fill in this blank." The numbers 8.5, 8, 7, 5.5, 6.5, 6.5, 9, and 8 are recorded and displayed on the overhead.

99 ENGAGING STUDENTS IN LARGE-GROUP PRESENTATIONS

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An Experiment

Question to be answered:

Data for Row 4:

Data for Row 1:

Treatment of Data for Row 4:

Treatment of Data for Row 1:

Treatment to compare the two data sequences:

Results:

Conclusions:

Exhibit 3.5 Tasksheet Ms. Erickson Uses During a Large-Group Presentation.

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CHAPTER 3 MOTIVATING STUDENTS TO ENGAGE IN MATHEMATICAL LEARNING ACTIVITIES 100

• Exhibit 3.6 Why Is Ms. Erickson Looking at Students' Shoes?

Ms. Erickson: "Now, I've got to figure out what to do with these numbers to heLp me answer the question." SeveraL students raise their hands, but she responds, "Thank you for offering to heLp, but I want to see what I come up with." Pointing to the appropriate numeraLs on the transparency, she seems to think aLoud saying, "It's easy enough to compare one number to another. Jasmine's 6 from Row 4 is Less than RoLando's 8.5 from Row 1. But I don't want to compare just one individuaL's number to another's. I want to compare this whoLe bunch of numbers (circling the set of numbers from Row 4 with an overhead pen) to this bunch (circling the numbers from Row 1). I guess we couLd add aLL the Row 4 numbers together and aLL the Row 1 numbers together and compare the two sums­the group with the greater sum wouLd have the bigger feet."

·. ! AcoupLe of students try to interrupt with, "But that won't work because-" but Ms. Erickson motions them to stop speaking and asks, "What's the sum from Row 4, Lau­Chou?"

Lau-Chou: 45. Ms. Erickson: Thank you. And what is the sum for Row 1,

Stace? Stace: 59. Ms. Erickson: Thank you. So Row 1 has bigger feet

because 59 is greater than 45. She writes, "59 > 45" on the transparency.

Ms. Erickson: I'LL pause to hear what some of you with your hands up have to say. EvangeLine?

Evangeline: That's not right; it doesn't work.

Ms. Erickson: You mean 59 isn't greater than 45, EvangeLine? Evangeline: 59 is greater than 45, but there are more feet

in Row 1. Ms. Erickson: ALL the peopLe in Row 1 have two feet each

just Like the ones in Row 4. I carefuLLy counted. Now that we've taken care of that concern, how about other comments or questions? Brooke.

Brooke: You know what Evangeline meant! There are more peopLe in Row 1. So what you did isn't right.

Ms. Erickson: ALright, Let me see if I now understand EvangeLine's point. EvangeLine said we don't want our indicator of how big the feet are to be affected by how many feet-just the size of the feet. So, I've got to figure a way to compare the sizes of these two groups of numbers when one has more numbers than the other. I'm open for suggestions. [pause] Kip?

Kip: You couLd drop the two extra numbers from Row 1; then they'd both have six.

Ms. Erickson: That seems Like a reasonabLe approach. I Like that. But first Let's hear another idea-maybe one where we use aLL the data. Myra?

Myra: Why not do an average? Ms. Erickson: What do you mean? Myra: You know, divide Row 4's totaL by 6 and Row l's by 8. Ms. Erickson: How will that dividing heLp? Seems Like just

an unnecessary step. Tom. Tom: It evens up the two groups. Ms. Erickson: Oh, I see what you peopLe have been trying

to teLL me! Dividing Row 4's sum of 45 by 6 counts each number 1/6. And dividing Row l's sum of 59 by 8 counts each number by 1/8. And that's fa"ir, because

101 ENGAGING STUDENTS IN LARGE-GROUP PRESENTATIONS

Topic: Proof by induction Date: 3/7

Presentation Outline I. Review of familiar methods of proving theorem

A. Direct B. By contradiction

II. Types of theorems to which proof by induction applies III. Logic of a proof by induction

A. Sequential cases B. Is it true for one case?

.C. If it is true for one case, will it be true for the next case? IV. Some everyday examples of the induction principle

A. Playing music B. On soccer field C. In the kitchen D. Eating food E. Computer programming n

V. An example with an arithmetic series, :L = n(n + 1)/2 VI. Formalizing the process ;=1

A. Show the statement is true for i = a. B. Show that if the statement is true for some value of i, then it must also be

true for; + 1. C. Draw a conclusion.

VII. Proof of the following theorem: n

:L;2 = (n/6)(n + 1)(2n + 1) ;=1

VIII. Summary

• Exhibit 3.7 Example of an Outline Distributed to Students for a Large-Group Presentation.

six one sixths is a whole just as eight one eighths is a whole. How am I doing, Jasmine?

Jasmine: A lot better than you were.

Removing another overlay on the transparency, Ms. Erickson displays the next two lines from Exhibit 3.5 and continues.

Ms. Erickson: Let's write: "The sum of Row 4's numbers is 45." 45 -;- 6 is what, Lester?

Lester: 7.5. Ms. Erickson: Thanks. And on the next line we write "59

-;- 8." Which is what, Sandy? Sandy: 7.375. Ms. Erickson: Since 7.5 is greater than 7.373, I guess we

should say that the feet in Row 4 are larger than the feet in Row 1. That is, of course, if you're willing to trust this particular statistic-which is known as the "MEB." Any questions? Yes, Evangeline.

Evangeline: Why the MEB? Ms. Erickson: Because I just named it that after its three

inventors, Myra, Evangeline, and Brooke. They're the ones who came with the idea of dividing the sum.

Ms. Erickson shifts to direct instruction to help .tudents remember the formula, practice using it, and

-- remember its more conventional name, "arithmetic mean."

Ten Points About Large-Group Presentations

Consider the following thoughts when designing large-group presentations:

• Students are more likely to be engaged during a presentation if the teacher has provided clear di­rections for behavior. Students need to have learned how to attend to a presentation. Ques­tions about how to take notes, if at all, should be answered before the presentation begins.

• Some sort ofadvanced organizer to direct students' thinking helps them actively listen to the speaker. Ms. Erickson used Exhibit 3.5's tasksheet to focus students' attention and structure the activity. Con­sider taking that idea a step further by having an outline of the presentation (see Exhibit 3.7) or a session agenda (see Exhibit 3.8) in the hands of students or displayed on an overhead trans­parency. You can then use it to direct attention and provide a context for topics and subtopics. Having such advanced organizers in students' hands facil­itates their note taking and helps you monitor their engagement (e.g., by sampling what they write on the form in Exhibit 3.9). By using transi­tional remarks such as "Let's move on to Item 4" in