math instruction case study 1 - david orcutt

8/8/2019 Math Instruction Case Study 1 - David Orcutt

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Module 1 - Case 1: David Orcutt

This mini-case provides an introduction to the use of cases as a reective professional developmenfor sustained use. This also uses student work examples to explore understandings and misconceptpercents, and decimals.

INTRODUCTION AND CONTEXTDavid Orcutt is one of two 7th grade mathematics teachers in the lone junior high school for thisdistrict. The district serves students from a largely rural agricultural and recreational area whichincludes two villages. The school is a 7-8 school in a small school building next to the districtshigh school. In fact, a number of teachers are on the faculty of both schools to provide appropriatecoverage for topic areas. David has four classes among his other duties as the 7th grade advisorand a track coach.

In his three years of teaching, he has learned that students coming in from the two K-6 schools inthe district (as well as a small but growing migrant labor population that is becoming a morepermanent xture in the area) often have varying skills and understanding in mathematics. Tounderstand each of the students abilities and conceptions about basic topics, he has devised atwo week introduction to his course which addresses a different topic from the grade 4-6standards each day or two, and uses this to establish norms for classroom participation, work expectations, etc. The following sample of classroom interaction starts by asking students to takeout the homework task from the previous day, which was really a pre-assessment of sorts tounderstand student knowledge of decimals, percents, and fractions.

CLASSROOM ACTIVITIESDavid starts class by greeting all students at the door as they come in, and has a problem on theboard, which he reminds students to get a paper out and copy the problem down after they havetaken their homework out from the previous day. Meanwhile, he checks attendance and missingassignments from the previous day, and then begins wandering through the aisles to see whatstudents are doing with the problems on the board, and whether they have their homework out.He quickly scans the homework for each student, noting whether they have all twenty problemsdone, and whether they have them numbered, the problem written down, and the answerunderlined for each. Most do, which results in him writing a 10 on the top of the page, but acouple did not nish, receiving 5 and 7 points respectively, and three others had 3 points deductedfrom these for not organizing their work properly. For these, David underlined a few of the answersthey had in their work that were not already underlined, and had jotted down the words show yoursteps on some of these papers. While doing this, he marked on a copy of a grade sheet thepoints for the homework assignment for each student.

Following this fairly quick review (which took four minutes from the time he started moving aroundthe room), he told the students they would review the answers of the homework. He circled theroom as he called out problem numbers, and would look around the room to see who was lookingat him (or not) and would call out the names of students to state what their answer was. Once onestudent gave the answer, he would call on two other students and ask if they came up with the

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same answer as the original student, or if they had something different. At every problem in whichall students agreed on the answer, he would quickly ask if any other answers were out there, andunless a quick response came, he would say correct and repeat the problem number andanswer and move on. When students disagreed, he would quickly survey students in the room tosee which of the stated answers other students got, or, what other answers people came up with,and unless it seemed that one was an outlier, would note that problem number of the whiteboard,so that the class could go through it after checking homework. Six of the problems were noted onthe board, and he they asked, problem by problem, if there were any volunteers to go to the boardand do the problem. Two of the problems had no volunteers, so he asked one student whatanswer they got for the problem, then asked if anyone had a different answer, and had both (ormore if several different answers arose) go up to the board to write their explanation or proceduresfor the problem.

One of the two problems that had contested answers was the following:Emma was asked to order the following numbers from smallest to largest: .43, 8%, and .7Emmas order was: .7, 8%, .43Is she correct? Why or why not?

Two students wrote their answers on the board initially as shown below.Student D: No because .43 is just about half and .7 is almost full and 8% is like 8 1s. .43 .7 8%

Student F: She is correct because 7 is the smallest and 43 is the biggest

The following dialog is taken from this activity:

DO: So, what do we think everyone. We have two answers here. What do we think?

Student H: [D] is right. Emma didnt get the right answer.

DO: And why is that?

H: Well, sort of right. Emma didnt get the right answer, but [D] didnt get it right either.

DO: [F], what you you think? You said Emma got the right answer. Explain what you said.

F: Well, the numbers get larger, um, in Emmas order, and, um, the dots and percents are thesame cause you can change from dots to percents and so I, um put them in order, and so, um, 7is smallest, then 8, then 43.

H: But they arent the same. Dots are two places different.

DO: [D], what do you think? You said Emma wasnt right, just like [H], but she said you werenteither. What do you think?

D: I was just trying to see what they are close to, and .43 is close to .5, which is a half. .7 isbigger. It is nearly a whole thing, and denitely more than half. The percents dont have thedecimals, so I thought 8% is like 8 whole things. But I think [H] is kinda right, um, cause you haveto do move the dot two places.DO: Lets see what someone else says. [G], how about you? What did you say?


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G: I said Emma was wrong. It should be 8%, .43, .7 in that order because I put them all inpercents.

DO: Aha. There we go. You put them all in percents. All in the same units. That is exactly whatwe want to do when we have decimals and percents together is put them in the same units. [H], isthat what you meant? Is that what you did?

H: Yeah, I made them all the same, but I didnt do percents. I changed percents to fractions, sothey were all some part of 100.

DO: Excellent. There we go. We want to change them all to the same, and the best way is tochange them to fractions. Since we have percents, we should change them to parts of 100. Thatis what percents really are. They are parts of 100. So, when you have all of your test right, forinstance, you have 100%. You get everything out of 100. So, how do we want to change these tofractions of 100?

C: (called on after raising hand) If it is one place. like .7 was, that is 7 out of 10, because the rstplace is tenths. Then hundredths. so we could add a zero to the end of that, because .7 is thesame as .70, and that is seventy out of a hundred.

DO: Great. Thats exactly it. Are we okay? Can we move on?

No responses, so they go on to the next question. Shortly thereafter, David moves through theother answers, and to the boardwork task. This task is written on the board already. It wasmodied by David from a task he had seen in a workshop focusing on differentiation, which wasaddressing visual learners. The original task from the workshop is below.

Shade 10 of the small squares in the rectangle shown below. Using the diagram, explain how to determine

each of the following: a) the percent area that is shaded, b) the decimal part of the area that is shaded, andc) the fractional part of the area that is shaded.

Davids modied version that is on the board is the following:

Shade 10 of the boxes in the rectangle shown below (same rectangle). Find the percent area that is shaded.

David says that, in the interest of time, he is going to go through it, and asks students to watch.He shades in 10 of the rectangles, picking them at random, and shading individual rectangles.


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DO: So, it really doesnt matter which ones I pick, it will be the same. What I really care about ishow many total ones we have. [A], how many total boxes are there?

A: 40

DO: And how did you get that?

A: I counted ten across, and there are four rows, so it was four times ten.

DO: Exactly... or you could count everyone of them if you didnt gure that out. So, what next(looking at A)?

A: Well, it is a quarter. There are 10 out of 40, and if we write that as a fraction (DO pauses A witha hand gesture and writes this on the board as the fraction 10/40, and then motions for him toproceed)... so yeah, thats it. And then you can cross out the zeros, cause 10 out of 40 is like 1out of 4, and thats a quarter. And a quarter is always 25%.

DO: Exactly. Does everyone see that? Once [A] got it to a fraction, he could easily change it to apercent. If it was a fraction you didnt know already, like... suppose we had 12 shaded boxesinstead? You could make it 12 out of 40, and then cross multiply to gure out the number out of 100 (as he draws on the board 12/40 = n/100 and then proceeds to write, 12 x 100 = n x 40),and so in this case you could multiple 12 and 100...[A], what is that?

A: Twelve and a hundred? Thats one thousand two hundred.

DO: and divide that by 40 and we would get 30. Thirty percent... if it was twelve out of 100. Doyou all see that?

The class seems to agree quietly, and David moves on to the next part of class...

math instruction case study 1 - david orcutt

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