K-9 Math Teachers Implementation of BreakThrough...

K-9 Math Teachers Implementation of BreakThrough Mathematics: Fall 2007 to Spring 2008 Evaluation Report to the Ohio Department of Education

Prepared by Kelli Millwood, Ph.D.Manager of Evaluation

Pearson Education September 2008

2008 ODE BreakThrough Math Annual Evaluation Report

- i -

Table of Contents Executive Summary .................................................................................................................................... ii Introduction................................................................................................................................................. 1 Evaluation Measures................................................................................................................................... 2 Teacher Background and School Climate Survey Results ......................................................................... 3

Teaching History..................................................................................................................................... 3 Teacher Training..................................................................................................................................... 6 School Climate...................................................................................................................................... 10

End-of-Course Survey Results.................................................................................................................. 12 Teacher Perceptions of BreakThrough Math Courses .......................................................................... 12

Quality of the course......................................................................................................................... 12 Gains in content knowledge.............................................................................................................. 17

School Climate for Sharing and Sustaining Gains from this Professional Development..................... 18 Application to the Classroom................................................................................................................ 20

Impact on Teaching and Student Learning ....................................................................................... 20 Changes in Instructional Practices Described by Teachers .............................................................. 23

Teachers’ Confidence in Mathematics Results......................................................................................... 27 Change in Pedagogical Content Knowledge Results................................................................................ 30

Pre- and Post-Course Thinking Tasks................................................................................................... 31 Concepts students need to understand to solve problems................................................................. 32 Different solutions methods to solve a problem ............................................................................... 33 Most common errors students make while solving a specific problem............................................ 34 Concepts that a student used to solve a problem .............................................................................. 35 Student reasoning used to solve a problem....................................................................................... 36

Further and Final Tasks ........................................................................................................................ 37 Aggregate Results ............................................................................................................................. 37 Fraction Concepts ............................................................................................................................. 38 Numbers & Numeration.................................................................................................................... 39 Multi-Digit Whole Number Computation ........................................................................................ 41

Summary & Recommendations ................................................................................................................ 43 Appendix A. Frequency report for Teacher Background and School Climate Survey ............................ 45 Appendix B. Frequency report for End-of-Course survey........................................................................ 62 Appendix C. Pre- and Post Course Thinking Tasks ................................................................................. 78

Concepts of Measurement..................................................................................................................... 79 Equality and Inequality ......................................................................................................................... 81 Fraction Concepts ................................................................................................................................. 83 Multi-Digit Whole Number Computation ............................................................................................ 85 Numbers & Numeration........................................................................................................................ 87 Perspectives on Problem Solving (K-5)................................................................................................ 89 Perspectives on Problem Solving (6-8) ................................................................................................ 90 Rational Numbers & Operations........................................................................................................... 91 Solving Equations ................................................................................................................................. 93 Statistics, Data Analysis, and Probability ............................................................................................. 95

Appendix D. Further and Final Thinking Tasks ....................................................................................... 97 Fraction Concepts ................................................................................................................................. 98 Numbers & Numeration...................................................................................................................... 102 Multi-Digit Whole Number Computation .......................................................................................... 105

- ii -

EXECUTIVE SUMMARY BreakThrough Math courses were open for enrollment to any certified teacher teaching in Ohio. Teachers enrolled in a course through one of the three participating universities: Ohio State University, Miami University, and Wright State University. Select professors from these universities were trained facilitators of specific BreakThrough Math courses. Teachers were offered the courses at a discounted rate and they earned course credits that counted toward “highly qualified” status. Prior to the start of the course, and upon completion of the course, teachers completed online surveys and tasks. Surveys included a preliminary Teacher Background and School Climate Survey and an End-of-Course survey. Embedded in both of the surveys was a measure of teacher confidence. The tasks teachers completed were open-ended tasks to measure changes in pedagogical content knowledge as a result of the course. Below are the major findings from the aforementioned surveys and tasks. • Extensive teaching experience. Eighty-two percent of the teachers have taught for more than 5

years and 71% have been teaching math for more than 5 years. Most have been teaching in their current school (80%) and grade level (81%) for more than 5 years. Most of the enrolled teachers (80%) taught Kindergarten – 6th grade and 20% taught at grades 7, 8, or 9. The majority of teachers (67%) indicated that their personal level of satisfaction with teaching math was “Good” or “Excellent,” while 33% of teachers indicated that their level of satisfaction with teaching math was “Average” or lower.

• Strong educational background. All teachers who participated in the BreakThough Math courses

had either a Bachelor’s degree (23%) or a Master’s degree (77%). The majority of teachers (79%) were not math majors in college. Most teachers (64%) held a teaching credential in Elementary Education. In the last 2 years, 66% of the teachers have taken a math course or training.

• Large percentages of teachers have achieved “highly qualified” status. The majority of teachers

(approximately 60%) achieved “highly qualified” status prior to taking the course. As a result of taking the course, 27% of those not previously “highly qualified” report that they now achieved “highly qualified” status.

• Supportive school culture for instructional change. Most of the teachers do not teach at a multi-

track school (75%), and approximately half (47%) of the teachers teach at Title I schools. Most teachers (61%) indicate that they feel supported by their school administration. As such, a large percentage of teachers indicated that they are involved in planning strategies for improving student achievement (77%), and the entire school is engaged in an effort to improve student achievement through a focus on improving classroom instruction (69%). Most teachers (68%) feel supported by the school to apply different instructional approaches in the classroom. The largest percentages of teachers indicate that they are implementing specific instructional practices in the classroom that benefit students. Almost all teachers (96%) participate in instructional practices workgroup meetings at their school. Of these teachers, all 97% agree that these meetings enhance their teaching in the classroom.

• High levels of satisfaction from the BreakThrough Math courses. Most of the teachers report

that the BreakThrough Math course either met or exceeded their expectations. Almost all teachers (86%) indicated that the facilitator was either exceptionally prepared (44%) or quite prepared (42%)


- iii -

to facilitate the course and could not have done anything more to facilitate learning. The majority of teachers (80%) indicated that they thought the BreakThrough Math course was either above average or excellent. Additionally, 91% of teachers responded that they would probably or definitely recommend a BreakThrough Math course to others. The largest percentage of teachers (92%) selected the following as the major strengths of the BreakThrough Math course: The convenience of the online portion (17%), Discussions (15%), and Pedagogical learning (15%). The largest percentage of teachers (51%) selected “None” for the areas that could use improvement. The remaining teachers (49%) selected the following top three areas for improvement: (a) Eliminate technology problems, (b) Reduce workload, and (c) Organization of the course. The largest percentage of teachers (55%) selected “Convenience of an online course” as being the most unique feature of the BreakThrough Math course.

• Gains in content knowledge. Almost all teachers (89%) who completed a BreakThrough Math

course agreed that they gained at least some new content knowledge as a result of completing the BreakThrough Math course. Most-cited changes included questioning techniques, anticipating student responses, and allowing for students to struggle with important math concepts and/or content. The majority of teachers (76%) indicated having experienced an “aha!” moment while taking the course. Most of the “aha!” moments focused on either new math content knowledge (43%) or new strategies for teaching math concepts (57%).

• Change in classroom practice and student learning. Most teachers (n = 114, 77%) agreed that

the BreakThrough Math course resulted or will result in a change in classroom practice. If teachers indicated that they taught the content covered in the BreakThrough Math course, then they were asked to rate if they noticed any changes in student learning that they attribute to the course. Almost all teachers (95%) indicated that they noticed changes in student learning. Most teachers commented on the types of changes that they noticed; however, approximately half (49%) of the changes were focused on changes in teacher practice rather than changes in student learning (51%).

• Sharing and sustaining knowledge from the BreakThrough Math course. Teachers were just as

likely to enroll in the BreakThrough Math course with other teachers from their school (47%) as they were to enroll independently (53%). Of the 47% of teachers who enrolled in the course with other teachers from their school, 73% indicated that they had opportunities to discuss the course and/or network with other teachers during the BreakThrough Math course. Almost equal percentages of teachers indicated that there was (48%) or was not (52%) a system in place to share what was learned in the BreakThrough Math course. The largest percentage of teachers (54%) indicated that they planned on telling other teachers about what they learned in the BreakThrough Math course through casual conversations.

• Minimal changes in teachers’ confidence levels. At the scale score level, teachers showed no

changes in their confidence in math. There was a slight and negative effect on teachers’ beliefs about correct answers and beliefs about operations. At the individual item level, for several items, the change in score from pre- to post-course was in the desired direction, indicating that, on average, teachers made a detectable move in the expected direction for each of these measured beliefs.

• Increase in pedagogical content knowledge. After completing a BreakThrough Math course

teachers had a better understanding of the concepts that students need to know in order to solve problems, as well as a better understanding of the most common errors that students make while

- iv -

solving a specific problem. Teachers’ analysis of student work samples indicated that after teachers completed a BreakThrough Math course they were better able to determine the type of reasoning that students use to solve a problem.


- 1 -

INTRODUCTION BreakThrough Math courses were open for enrollment to any certified teacher teaching in Ohio. Teachers enrolled in a course through one of the three participating universities: Ohio State University, Miami University, and Wright State University. Select professors from these universities were trained facilitators of specific BreakThrough Math courses. Teachers were offered the courses at a discounted rate and they earned course credits that counted toward “highly qualified” status. A total of 327 teachers enrolled in the BreakThrough Math courses during the 2007-2008 academic year. Prior to the start of the course, and upon completion of the course, teachers completed online surveys and tasks. Surveys included a preliminary Teacher Background and School Climate Survey and an End-of-Course survey. Embedded in both of the surveys was a measure of teacher confidence. The tasks teachers completed were open-ended tasks to measure changes in pedagogical content knowledge as a result of the course. The following sections include the details of the evaluation measures, major findings from the measures, and a summary with recommendations.

- 2 -

EVALUATION MEASURES Several evaluation measures were used to generate the information for this report, including the Teacher Background and School Climate Survey, the End-of-Course Survey, and Pre- and Post-Course Thinking Tasks. The Teacher Background and School Climate Survey is a self-report questionnaire that teachers complete at the beginning of the course. Forty-one items assess the following dimensions:

Teaching history (e.g., grade level/s taught, years teaching, years at current school, etc.) Teacher training (e.g., highest degree, college major, additional math training, etc.) School climate (e.g., instructional support, professional learning communities, etc.)

The End-Of-Course Survey is a self-report questionnaire that teachers complete at the end of the course. Forty-seven items assess the following dimensions:

Quality of and satisfaction with the course What was gained by taking the course Whether and how content learned in the course was applied

Pre- and Post-Course Thinking Tasks are tasks developed specifically for and embedded within each of six courses, designed to assess change in pedagogical content knowledge. Depending on the course, there are 7-14 items to complete both prior to the start of the course and at the end. Additionally, items developed by Deborah Ball and colleagues (e.g., Hill, Schilling, & Ball, 2004)1 that assess mathematical pedagogical content knowledge and that were selected based on their similarity to the content focus of a given course were also embedded in the following three courses: Fraction Concepts, Multi-Digit Whole Number Computation, and Numbers & Numeration.

1 Hill, H., Schilling, S., & Ball, D. (2004). Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal, 105, 11-30.


- 3 -

TEACHER BACKGROUND AND SCHOOL CLIMATE SURVEY RESULTS A total of 327 teachers enrolled in the BreakThrough Math courses during the 2007-2008 academic year. The survey to measure teacher background and school climate was launched in October 2007 and closed on June 18, 2008. As of June 20, 2008, 82% (n = 268) of the teachers enrolled in the BreakThrough Math courses had completed the survey. Frequency reports for every item on the Teacher Background and School Climate Survey can be found in Appendix A. The following three sections – Teaching History, Teacher Training, and School Climate—summarize the results from the survey.

Teaching History Teachers enrolled in BreakThrough Math courses from October 2007 – June 2008 can be characterized as follows: Teachers from 64 school districts in Ohio participated in BreakThrough Math courses. More than 50% of the teachers came from the following school districts:

Mason City (9.7%) Lakota Local (9.3%) London City (5.6%) Milford Exempted Village (5.6%) Oakhills Local (5.6%) Loveland City (4.9%) Cincinnati City (4.1%) West Clermont Local (4.1%) Forest Hills (3.0%) Princeton City (3.0%)

The majority of teachers (n = 145, 58.2%) met “highly qualified” status prior to taking the course. Of the teachers who did not meet highly qualified status more than half (55.8%) indicated that they were taking the course to achieve highly qualified status. As a result of taking the course, 27% of the teachers (n = 67) not previously “highly qualified” report that they now achieved “highly qualified” criteria. The teachers taking these courses are in general an experienced group: 82% taught for more than 5 years and 18% taught for fewer than 5 years. Most have been in their current school (80%) and grade level (81%) for more than 5 years. Additionally, 38% have been teaching math for more than 10 years (see Figure 1).

- 4 -

Figure 1. Teaching experience of BreakThrough Math participants

2.3%

.4%

.0%

1.5%

14.1%

7.0%

7.3%

9.6%

36.6%

18.6%

20.8%

26.8%

29.0%

34.1%

31.3%

33.3%

13.0%

19.4%

21.2%

18.4%

3.1%

9.3%

10.8%

5.4%

1.9%

11.2%

8.5%

5.0%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Years teaching

Years in current school

Years teaching in currentgrade level

Years teaching math

< 1 year

1-2 years

3-5 years

6-10 years

11-20 years

21-30 years

>30 years

Teachers were asked to select all of the grade levels that they taught. On average, teachers taught one grade. The majority of the enrolled teachers (80%) taught Kindergarten – 6th grade. The remainder of teachers (20%) taught grades 7, 8, or 9 (see Figure 2).


- 5 -

Figure 2. Grade level(s) taught by BreakThrough Math participants

9% 9%13% 12% 13% 15%

9%6%

3%

11%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Kindergart

en 1st 2nd

3rd 4th 5th

6th 7th

8th 9th

In general, this group of teachers is satisfied with teaching math. The majority of teachers (67%) indicated that their personal level of satisfaction with teaching math was “Good” or “Excellent” (see Figure 3), while 33% of teachers indicated that their level of satisfaction with teaching math was “Average” or lower.

- 6 -

Figure 3. BreakThrough Math participants’ level of satisfaction in teaching math

1%4%

28%

46%

21%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Not at all satisfied A little satisfied Average Good Excellent

Teacher Training All teachers who participated in the BreakThough Math courses held either a Bachelor’s degree (23%) or a Master’s degree (77%) (see Figure 4). The majority of teachers (79%) were not math majors in college (see Figure 5). Teachers were asked to select from a list all areas they are credentialed to teach. The majority of teachers selected one area. Elementary Education was the most common credential, selected by 64% of teachers (see Figure 6). In the last 2 years, 66% of the teachers have taken a math course or training (see Figure 7). The majority of teachers (83%) participated in some form of professional development that focused on improving math teaching (see Figure 8). Ongoing professional development programs that focus on improving teaching in the classroom are evident in the majority of teachers’ schools (63%). Of the teachers who participate in their school’s ongoing professional development to improve classroom teaching, 45% described the effectiveness of their school’s program as “quite a bit” or “very much so” (see Figure 9). The largest percentage of teachers (54%) labeled their school’s professional development program as “a little” or “somewhat” effective.


- 7 -

Figure 4. Highest degree achieved by BreakThrough Math participants

23%

77%

0%0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Bachelor's Master's Doctorate

Figure 5. College major of BreakThrough Math participants

79%

8%12%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Did not major or minor in math Minored in math Majored in math

- 8 -

Figure 6. Areas BreakThrough Math participants are credentialed to teach1

64%

24%

14%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Elementary Mathematics Special Education

1. Note: categories do not sum to 100% because teachers were able to select all of the areas they are credentialed to teach. Figure 7. Most recent math professional development course taken by BreakThrough Math participants

53%

13% 14%21%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

In the past year 1-2 years ago 2-5 years ago Over 5 years ago


- 9 -

Figure 8. Number of math professional development courses taken by BreakThrough Math participants

17% 14%

41%

22%

6%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

None 1 2-4 5-10 More than 10

Figure 9. BreakThrough Math participants’ perceived effectiveness of ongoing professional development program

1%

46%

37%

8%8%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Not at all A little Somewhat Quite a bit Very much so

- 10 -

School Climate The majority of teachers (n = 192; 75%) do not teach at a multi-track school. The number of teachers who work in Title I schools (n = 121; 47%) is almost equal to the number of those who do not (n = 138; 53%). In addition to responding to questions about multi-track configurations and Title I status, teachers were asked to rate their school climate in terms of the opportunities and support they receive to (a) implement important instructional strategies, and (b) participate in professional learning communities for teachers. Overall, the largest percentages of teachers indicate that they are implementing instructional practices in the classroom that benefit students (see Figure 10). Specifically, 58% of teachers indicate that they analyze student work to better understand the needs of their students at least two times a month. A slightly smaller percentage of teachers (45%) analyze student work at least two times a month as a means of evaluating their own teaching practices. The majority of teachers (56%) reported that at least two times per month they identify instructional strategies that might help address common student needs. Additionally, the majority of teachers (54%) stated that they identify unit objectives and plan unit instruction more than twice a month. Figure 10. BreakThrough Math teachers’ instructional practices

2% 0% 2%7%

16% 16%22% 19%

24% 27%32%

19%27% 26%

21% 24%31% 31%

23%31%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Analyze student work tounderstand student needs

Utilize instructionalstrategies to target student

needs

Analyze student work toevaluate teaching practices

Identify unit objectives andplan instruction for the unit

Not at all Less than once per month Once per month 2-3 times per month Weekly

Teachers responded to four questions that related to their perceptions of their school culture and level of support to improve student achievement (see Figure 11). A large percentage of teachers (77%) believed that the teachers in their school are involved “quite a bit” or “very much” in planning strategies for improving student achievement. The majority of teachers (69%) agreed that their school was “quite a bit” or “very much” engaged in an effort to improve student achievement by focusing on improving teaching in the classroom. Most teachers (61%) characterized the level of support they received from administration as “quite a bit” or “very much.” Additionally, when asked to consider the level of support for applying different instructional approaches in the classroom, 68% of teachers responded that their school supports them “quite a bit” or “very much.”


- 11 -

Figure 11. BreakThrough Math participants’ perception of school culture

0% 0% 1% 2%2%7% 6%

10%

21% 23% 24% 27%

44% 47% 47%42%

33%

22% 21% 19%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Teachers at your school areinvolved in planning

strategies for improvingstudent achievement

School wide effort toimprove student

achievement focus onimproving teaching in the

classroom

Your school supportsteachers in trying differentinstructional approaches in

the classroom

Teachers at your school feelsupported by administration

to do their job

Not at all A little bit Somewhat Quite a bit Very much

Almost all of the teachers (96%) participated in workgroup meetings. Of these teachers, 43% agreed that these meetings enhanced their teaching in the classroom “quite a bit” or “very much.” A mere 3% of teachers said that workgroup meetings did not enhance their teaching in the classroom, and 54% agreed that it enhanced their teaching “a little bit” or “somewhat.”

- 12 -

END-OF-COURSE SURVEY RESULTS A total of 327 teachers enrolled in the BreakThrough Math course during the 2007-2008 school year. Of these teachers, 76% (n = 249) completed the End-of-Course survey before it closed in July 2008. Frequency reports for every item on the End-of-Course Survey can be found in Appendix B. The majority of teachers (n = 145, 58.2%) achieved “highly qualified” status prior to taking the course. As a result of taking the course, 27% of the teachers (n = 67) not previously “highly qualified” report that they now achieved “highly qualified” status. The largest percentage of teachers (64%) indicated that they enrolled in either the Problem Solving, Fraction Concepts, or Concepts of Measurement course (see Figure 12). Most teachers (81%) indicated that their course was “Blended,” a combination of online and face-to-face meetings. The remaining teachers (19%) indicated that the course they took was purely online. The following sections report the aggregated results from the 249 teacher responses to the End-of-Course survey. The results are organized by topic areas assessed by the survey: (a) teachers’ perceptions of BreakThrough Math courses, (b) school climate for sharing and sustaining knowledge from BreakThrough Math courses, and (c) application of BreakThrough Math courses to the classroom. Figure 12. Participant enrollment in BreakThrough Math courses

26%

5% 6% 3%

17% 19%

7%2%

9%1% 5%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Perspect

ives o

n Proble

m Solving

Numbers &

Num

eratio

n

Mult

i-Digit W

hole

Number

Compu

tation

Solving Equ

ation

s

Fractio

n Conc

epts

Concep

ts of

Measur

emen

t

Geometr

ic Shap

es & Soli

ds

Ratio &

Proportio

n

Equali

ty & In

equal

ity

Ration

al Num

bers &

Ope

ration

s

Statist

ics, D

ata A

nalys

is, & Proba

bility

Teacher Perceptions of BreakThrough Math Courses In the End-of-Course survey, teachers responded to a variety of questions to assess their perception of the quality of the course and their gains in content knowledge as a result of taking the course. As detailed below, most teachers reported positive ratings of the course and indicated personal gains in content knowledge.

Quality of the course The majority of teachers (72%) indicated that they completed a previous math professional development course that focused on improving the teaching of math, but that this was their first BreakThrough Math


- 13 -

course. Almost all of the teachers (91%) who had taken a previous course that focused on improving math instruction agreed that the BreakThrough Math course was better than (22%) or as good as (69%) their previous course. The BreakThrough Math course either met or exceeded most teachers’ expectations (see Figure 13). In addition, the teachers provided high ratings for the facilitator’s preparedness. Almost all teachers (86%) indicated that the facilitator was either exceptionally prepared (44%) or quite prepared (42%) to facilitate the course and could not have done anything more to facilitate learning. Overall, teachers rated the quality of the BreakThrough Math course highly. The majority of teachers (80%) indicated that they thought the BreakThrough Math course was either above average or excellent (see Figure 14). Additionally, 91% of teachers responded that they would probably or definitely recommend a BreakThrough Math course to others (see Figure 15). Figure 13. Degree to which the BreakThrough Math course met participants’ expectations

1% 2%

16%

56%

25%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Not at all A little Somewhat Met most of myexpectations

Exceeded myexpectations

- 14 -

Figure 14. Participant rating of the BreakThrough Math course quality

3%

18%

51%

29%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Below average About average Above average Excellent

Figure 15. Participant rating of the degree to which they would recommend a BreakThrough Math course to another teacher

0%3% 5%

34%

57%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Definitely not Probably not Unsure Probably Definitely

Perc

ent o

f Tea

cher

s


- 15 -

Teachers responded to questions about what they believed to be the three major strengths of the BreakThrough Math course. They also provided feedback about three areas of the BreakThrough Math course that could use improvement. Additionally, teachers selected what they believed was unique about the BreakThrough Math course.

Based on open-ended responses about major strengths of the program that were gathered from previous courses, teachers were given a set of 12 options from which to select the three areas that they believed to be major strengths of the course. The options included: (a) Discussions and/or feedback from other teachers (e.g., forums); (b) Convenience of online portion (e.g., ability to work from home, follow own schedule); (c) Learning more about teaching (e.g., rich problems, effective questions, Big Ideas); (d) Video lesson analysis; (e) Learn more about math content/concepts; (f) Facilitator; (g) Useful strategies, (h) Organization of the course; (i) Useful, relevant readings; (j) Practice writing a lesson plan; (k) None. Additionally, an “other” category was included for those teachers who wished to write in a major strength that was not among the options listed. The largest percentage of teachers selected the following as the major strengths of the BreakThrough Math course: The convenience of the online portion (17%), Discussions (15%), and Pedagogical learning (15%. See Figure 16). Figure 16. Aggregated reported strengths of the BreakThrough Math course

17% 15% 15%12% 11% 8% 6% 4% 3% 1% 4% 4%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Conve

nienc

e

Discuss

ions

Peda

gogic

al Lear

ning

Video A

nalys

is

Math

Lear

ning

Strate

gies

Reading

s

Organiz

ation

Facil

itator

Lesson

Plan

None

Other

A similar strategy was used with the Areas of Improvement question. Teachers were given a set of options from which to select three areas of the BreakThrough Math course that needed improvement. Options for this question were derived from previous open-ended responses to this question. The eight options included: (a) More face-to-face meetings, (b) More discussion, (c) Less busy work, (d) Clarification of expectations, (e) Eliminate technology problems, (f) Reduce workload, (g) None, and (h) Other. A large percentage of teachers (14%) selected “Other” and wrote in additional suggestions for improvements. These suggestions created four new categories: (a) organization of the course, (b)

- 16 -

feedback from the facilitator, (c) repetitive or unnecessary content, (d) and forum features. The largest percentage of teachers (51%) selected “None” for the areas that could use improvement. The remaining teachers selected the following top three areas for improvement: (a) Eliminate technology problems, (b) Reduce workload, and (c) Organization of the course (see Figure 17). Figure 17. Aggregated reported areas for improvement for the BreakThrough Math course

51%

9% 9% 8% 6% 4% 4% 3% 2% 1% 1% 1%0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

None

Elimina

te Tech

nolog

y Prob

lems

Reduc

e Work

load

Organiz

ation

Less B

usy W

ork

Clarifi

cation

of C

ourse

Expec

tation

s

More

Face-to

-Face

Meet

ings

Feed

back

Conten

t

More

Disc

ussio

ns

Forum

Featu

resOthe

r

Teachers responded to the question, “What was unique about this course compared with other professional development experiences (other than BreakThrough Math courses) that you have had?” Teachers were encouraged to select all of the features that they found to be unique. The set of features included: (a) Convenience of an online course, (b) Discussions and/or feedback from other teachers, (c) Video lesson analysis, (d) Relevance of content to classroom instruction, and (e) Other. The “Other” category included a text box for teachers to record their responses. The largest percentage of teachers (55%) selected “Convenience of an online course” as being the most unique feature of the BreakThrough Math course (see Figure 18).


- 17 -

Figure 18. Participant perceptions of unique features of the BreakThrough Math course*

55%

28% 28%

35%

6%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Convienience of anonline course

Discussions withteachers

Video lessonanalysis

Relevance ofcontent to classroom

instruction

Other

* Note: Percentages do not sum to 100% because categories are not mutually exclusive.

Gains in content knowledge The majority of teachers (89%) who completed a BreakThrough Math course agreed that they gained at least some new content knowledge as a result of completing the BreakThrough Math course (see Figure 19). The extent to which knowledge gains were made as a result of the course was strongly and positively correlated with teachers’ perceptions of the course, as measured by Spearman’s rho2 (see Table 1). Specifically, those who rated their math content knowledge gains higher tended to rate the BreakThrough Math course facilitator as more prepared. They also rated the course as higher quality and as meeting more of their expectations.

2 Spearman’s rho was used in lieu of the Pearson’s correlation coefficient because the data for these questions is ranked scale data. Spearman’s rho is a measure of association ranging from -1 to +1, with values closer to 0 indicating a weaker to no relationship between 2 variables (rho = 0 means no relationship). Positive values mean higher values on one variable are associated with higher values on the other, while negative values mean higher values on one variable are associated with lower values on the other.

- 18 -

Figure 19. Participants’ perceived gains of content knowledge after completed BreakThrough Math course

5% 6%

26%31% 32%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

B T M a th c o u rse re su lte d in g a in s in o w n c o n te n t k n o w le d g e

Not at all A little Some Quite a bit Very much

Table 1. Correlations with participants’ perceptions of knowledge gain

Mean Standard Deviation Correlation

Preparedness of the facilitator 4.28 0.78 .29* Course met expectations 4.02 0.76 .39* Quality of the course 4.06 0.75 .46* *indicates that correlation is statistically significant at the p<.01 level.

School Climate for Sharing and Sustaining Gains from this Professional Development A series of questions on the End-of-Course survey assessed teachers’ perceptions of their school climate. These questions focused on the degree to which (a) the course aligned with curriculum or learning goals, (b) the teacher was engaged in a learning network, and (c) the teacher shared knowledge from the course with other teachers. Teachers were just as likely to enroll in the BreakThrough Math course with other teachers from their school (47%) as they were to enroll independently (53%). Of the 47% of teachers who enrolled in the course with other teachers from their school, 73% indicated that they had opportunities to discuss the course and/or network with other teachers during the BreakThrough Math course. Most teachers (85%) agreed that the BreakThrough Math course was “quite a bit” or “very much” aligned with their school curriculum. Similarly, 84% of teachers agreed that the BreakThrough Math course was “quite a bit” or “very much” aligned to their students learning needs. The majority of teachers (79%) indicated that were engaged in a learning network during the course, where they spent at least some time working with their peers during the BreakThrough Math course (see Figure 20).


- 19 -

Figure 20. Participant perceptions of the BreakThrough (BT) Math course meeting professional development standards

9

0

0

13

3

4

29

12

12

38

50

45

12

35

39

0 10 20 30 40 50 60

Engaged in a learningnetwork

BT Math coursealigned withcurriculum

BT Math coursealigned to students

learning needs

Percent of Teachers

Very muchQuite a bitSomeA littleNot at all

As a way to assess the sustainability of the BreakThrough Math course, teachers responded to the following question: “At your school, is there a system in place for sharing what you’ve learned in this professional development course with your fellow teachers?” Almost equal percentages of teachers indicated that there was (48%) or was not (52%) a system in place to share what was learned in the BreakThrough Math course. The teachers who indicated that there was a system in place for sharing what they learned were probed to indicate what the sharing system included. Teachers selected as many of the following six options that applied: (a) written summary to teachers (e.g., newsletter), (b) recommend this to my school as part of our professional development, (c) face-to-face presentation to teacher workgroups, (d) casual conversations with fellow teachers, (e) share course materials with fellow teachers, and (f) other. The largest percentage of teachers (54%) indicated that they planned on telling other teachers about what they learned in the BreakThrough Math course through casual conversations (see Figure 21).

- 20 -

Figure 21. Ways BreakThrough Math participants share what was learned in BreakThrough Math course with teachers*

5%

21% 18%

54%

37%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Written summary(e.g., newsletter) to

teachers

Recommend this tomy school as partof our professional

development

Face-to-facepresentation to

teacherworkgroups

Casualconversations with

fellow teachers

Share coursematerials with

fellow teachers

* Note: Percentages do not sum to 100% because the categories are not mutually exclusive.

Application to the Classroom Teachers reported the degree to which the BreakThrough Math course impacted their teaching and student learning. In addition to a quantitative rating, teachers were also given the opportunity to elaborate on their rating and explain how the course impacted teaching and student learning. They were also asked to describe any “aha!” moments they experienced during the course. Following are the quantitative and qualitative findings on how the course impacted teaching and student learning.

Impact on Teaching and Student Learning Teachers were asked to indicate whether the BreakThrough Math course impacted their teaching, if they had been able to apply course teachings to their classroom instruction, and whether they thought the instruction had an impact on student learning. Most teachers (n = 114, 77%) agreed that the BreakThrough Math course resulted or will result in a change in classroom practice (Figure 22). It is important to note—especially when considering the questions about opportunities to apply course teachings and impact student learning—that most of the teachers took the BreakThrough Math courses in the summer. Thus, the majority (n = 174, 75%) had not taught the math content covered in the course by the time they took the survey (see Figure 23).


- 21 -

Figure 22. Percentage of participants who changed or anticipate changing their teaching because of the BreakThrough Math course

Yes77%

No23%

Figure 23. Percentage of participants who taught the content covered in the BreakThrough Math course

Yes25%

No75%

All teachers rated, on a scale of 1 to 10, the degree to which the BreakThrough Math course changed classroom practices to enhance student learning. Even though the majority of the teachers did not have the opportunity to teach the content from the BreakThrough Math course, it is assumed that the teachers are anticipating the impact on their classroom practice for the upcoming school year. The mean rating was 6.8. The distribution of the ratings is detailed below in Figure 24. If teachers indicated that they taught the content covered in the BreakThrough Math course, then they were asked to rate if they noticed any changes in student learning that they attribute to the course. Almost all of teachers (95%) indicated that they noticed changes in student learning. Most teachers commented on the types of changes that they noticed; however, half (49%) of the changes were focused on changes in teacher practice rather than changes in student learning (51%). Included in Table 2 are several excerpts that highlight the types of comments teachers made regarding changes in student learning and changes in their own practice that will ultimately impact student learning.

- 22 -

Figure 24. Extent to which the BreakThrough Math course changed classroom practices to enhance student learning

1%4% 3% 4%

11%15%

24% 25%

5%9%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Not atall- 1

2 3 4 5 6 7 8 9 A lot- 10


- 23 -

Table 2. BreakThrough Math teachers’ comments about changes in teaching and student learning Changes in strategies used to teach math “Learned many strategies and my students have taken upon themselves to write books (several) as to how they LOVE

math.” “I allow more manipulatives to be used and I also allow students to struggle with problems much more than I used to. I have also tried to allow more time and more often for students to share their strategies with each other…”

Changes in questioning techniques and interactions with the students “Questioning and Big Ideas mostly but I liked the realization about making connections. We use this in reading and I

never thought about doing this on a mathematical level. Better attitudes and more enthusiasm.” “My questions have become more focused and clear. My students are more comfortable answering questions.” “I have worked on improving my questioning to keep the cognitive demand high. My improved questioning has improved my students’ intrinsic motivation.” “They will take ‘ownership’ of their learning when they have to think for themselves. I have noticed that my students aren’t as afraid to explain the wrong answer. Because we all learn as a class together.”

Change in teaching specific math content “When studying music note values in my music class, we compared groups of note values and used the process of

equality vs. inequality when creating balance between the measures. The break through moment in my music room was when a fifth grade student finally realized more clearly musical note values through the equality vs. inequality lesson. When he told his math teacher, ‘Now this, I get!’, I was so thrilled for him and this important breakthrough!” “I now teach area and volume using the idea that area is based on 2 dimensions (squared) and volume is based on 3 dimensions (cubed) and I used cut up graph paper (parallelograms) to let my students explore the idea of the moving and additive principals which is visual and hands on. Responses to area and volume have been much more accurate.”

Observed changes in students’ learning and/or approach to math “Remarkable - I've taught this unit before and have never taught it so well - my students did marvelously on every

assessment during our fractions unit - much better than ever before. Students had a MUCH better grasp of our fractions unit than ever before.” “Students have become more confident problem solvers. I think students have made more connections, and have deepened their understanding of mathematical concepts.” “Students are picking up some of the content with more ease. Students are more eager to express and describe their mathematical thinking.” “My students were much more excited about math; no matter what level they are.”

Changes in Instructional Practices Described by Teachers To assess what the teachers learned in the BreakThrough Math courses, teachers were asked to reflect on changes in their thinking as a result of the key concepts that were emphasized in all of the BreakThrough Math courses. The key concepts included: effective questioning, cognitive demand of problems, anticipating student responses, lesson planning, and emphasizing core mathematical concepts. Additionally, in a separate question, teachers indicated if they had any “aha!” moments in the course. Following are the results from teachers’ reflections on their change in thinking due to the BreakThrough Math course, as well as teachers’ proclaimed “aha!” moments.

- 24 -

When questioned about the impact of learning more about each key concept, the largest percentage of teachers reported a change in thinking and/or teaching (Figure 25). The key concepts that were prominent in impacting teacher thinking and/or practices included effective questioning (82% of teachers reported a change), cognitive demand (71% of teachers reported a change), and anticipating responses (73% of teachers reported a change). Examples of the types of changes that teachers reported are in Table 3. Figure 25. Percentage of participants who indicated changes in thinking and/or teaching for the key concept areas of all BreakThrough Math courses

82%

71% 73%

62% 65%

18%

29% 27%

38% 35%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

EffectiveQuestioning

Cognitive Demand AnticipatingResponses

Lesson Planning Emphasizing CoreMathematical

Concepts

Change in Classroom Practice No Change in Classroom Practice


- 25 -

Table 3. Participants’ examples of changes in thinking and/or teaching about the key concepts in the BreakThrough Math course Effective Questioning “I want to ask more open-ended questions and use the word ‘why’ a lot. I will focus more on my questioning this year to

check for student understanding.” “I will ask more why questions and let the students think about their answers, so that I do not lead them to what I want.” “I realized that I have to change my questioning if I am going to have my students gain a deeper understanding of math. I also know that it will take time and that I will need to be patient. One step at a time.”

Cognitive Demand “I am more aware of the types of problems that I need to provide for my students that allow them to discover a more deep

understanding of a concept instead of just a formula or procedure.” “Realizing that I need to do a better job of incorporating a variety of ways to solve problems, I now feel encouraged that all students can tackle cognitively demanding problems.” “I need to reevaluate my questions on test and make them more in depth and thinking questions.”

Anticipating Responses “I definitely will work on longer response time which I hope will encourage more thinking.”

“I have learned that I need to take more time to in my lesson planning for anticipating my students’ responses to my questioning.” “Anticipating responses is an area which I tend to overlook due to time constraints. I need to stop and reflect more during my lesson planning in order to better guide my students.”

Lesson Planning “I will just be putting more time into thinking through the lesson first and having more thoughtful, meaningful

introductions and assessments.” “Using the LessonLab approach to lesson planning I have come to realize takes time, but time that is worthwhile, for not only does it serve as a plan for activities and a hypothesis of how students will respond, but it also plans for how a teacher will respond to anticipated questions asked by the student. Responding appropriately to the anticipated questions help students to overcome their misconceptions of a given problem or task.” “The sample lesson plans from the course gave a different perspective on lesson planning. One was including the questions you are planning to ask the students, and also including their anticipated responses”

Emphasizing Core Mathematical Concepts “The biggest idea I took from this was the idea that we need to get away from teaching a ‘mile wide and an inch deep.’

We need to give kids a much deeper understanding of the core concepts.” “The importance of spending time on the core concepts versus giving all standards equal time.” “I realize that implementing the BIG IDEAS will not only help my students become better thinkers, but also better in mathematical processes and concepts.”

The majority of the teachers (76%) indicated that they had an “aha!” moment in the course, which suggests that they had new learning in their course. Most responses focused on either new math content knowledge (43%) or new strategies for teaching math concepts (57%). Table 4 illustrates a sample of teachers described “aha!” moments.

- 26 -

Table 4. Participants’ descriptions of "aha!" moments during the BreakThrough Math course Aha moment focused on new math content knowledge “Different algorithms for the computations. I was only familiar with the standard procedures.”

“I forgot that I use the = sign to mean an operation rather than a process of solving and balancing equations. Many students use the = sign to show just an answer.” “I hadn't considered how easy it would be to explain mixed numbers and improper fractions using a number line instead of shaded parts.” “I was surprised at how many ways there were to look at fractions. You get yourself in a certain way of thinking and it is hard to change some things.” “In one of the articles that I read it was explained in detail why we can use cross multiplication. Although this is the technique that I used throughout my math career, it finally made sense why I can do this and how it related to what we were learning.” “Learning the differences in odds and probability and a clear explanation of theoretical and experimental probability were the major ‘ahas!’ in this course.”

Aha moment focused on new strategies for teaching math concepts “I think my biggest ‘aha’ was realizing how often I want to jump in and rescue my students from struggle, rather than

encouraging them to figure things out on their own (even if they fail at first). Also, I'm now more in tune to anticipating student responses to a lesson and having good questions prepared to meet their responses (in order to facilitate greater learning vs. telling them the answer).” “I've realized through taking this course that there are better ways for me to ask questions. Students need to be actively involved in their learning and I need to facilitate that, not tell them what to do or guide them to the ‘correct answer’.” “In the lesson plans, teachers need to anticipate student responses and PLAN actual questions to help facilitate the students’ learning. I think this is WONDERFUL strategy. The whole issue of good questions really made me say aha!” “It suddenly occurred to me that I need to change the way I respond to student answers. I need to ask for explanations of thinking from all students, not telegraphing to them whether I think their answer is right or wrong.” “Just realizing that anticipating student responses and effective questioning really can help me develop more meaningful lessons. That some problems that seem to demand a higher level of cognitive effort really are not rich problems at all.”


- 27 -

TEACHERS’ CONFIDENCE IN MATHEMATICS RESULTS In both the preliminary Teacher Background and School Climate Survey and the End-of-Course survey, teachers were presented with items assessing three domains: (a) confidence in math, (b) beliefs about correct answers in math, and (c) beliefs about math as operations. Two-hundred and twenty teachers completed both pre- and post-course survey items (77% of the total number enrolled). These items were taken from a measure of teachers’ beliefs and practices related to mathematics instruction, developed and tested by Stipek, Givvin, Salmon, & MacGyvers (2001)3. The following items comprise each of these three domains and are rated on a scale ranging from 1 (“strongly disagree”) to 6 (“strongly agree”): Confidence in math

I feel confident that I understand the math material I teach. I think of myself as being good at math. When I teach math I often find it difficult to interpret students’ wrong answers. I'm good at communicating math content to students. When my answer to a math problem doesn't match someone else’s, I usually assume that my

answer is wrong. I'm not competent enough in math to teach it beyond the middle school grades.

Beliefs about correct answers

It doesn’t matter whether students get the right answer as long as they understand the math concepts inherent in a problem.

To assess students’ math understanding it is important to observe them while they are working and to listen to their math conversations.

Students who aren’t getting the right answers need to practice on more problems. In mathematics, answers are either right or wrong. Students who produce correct answers have a good understanding of the mathematical concepts. Children’s reasoning in their mathematical problem solving is more important to assess than

whether they solve problems correctly. Beliefs about math as operations

Mathematics involves mostly facts and procedures that have to be learned. The best way to understand math is to do a lot of problems. There is usually only one way to solve a math problem. In mathematics you can be creative and discover things on your own. Studying a few problems in depth is a good way to learn mathematical concepts.

After combining the items for each domain into three subscales, scores on each baseline subscale (obtained from the Teacher Background survey completed pre-course) were compared to scores on the post-course subscales using a paired samples t-test. The test produces a t-statistic that measures whether the difference score (post-course minus pre-course scores) is statistically significantly different from 0. A significant t-test indicates that statistically detectable change occurred between two time points. 3 Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to

mathematics instruction. Teaching and Teacher Education, 17(1) 213-226.

- 28 -

Results of the paired t-tests indicate that there was a statistically detectable difference between baseline and post-course on measures of beliefs about correct answers and beliefs about operations (see Table 5). Notably, beliefs were slightly lower post-course, possibly due to improved math content knowledge. Although the differences from pre- to post-course are statistically significant for teachers beliefs about correct answers and beliefs about operations, this is not very meaningful because the differences are very small — a quarter of a point on a six-point scale. Conversely, there were no detectable changes in confidence in math subscale. Table 5. Paired samples t-test results for pre-course to post-course domain comparison 95% Confidence

Interval of the Difference

Mean

Difference Std.

Deviation

Std. Error Mean Lower Upper t df

Sig. (2-tailed)

Confidence in math 0.09 1.04 0.07 -0.05 0.23 1.33 217 .19

Beliefs about correct answers 0.23 0.68 0.05 0.13 0.32 4.87 217 .00

Beliefs about math as operations

0.34 0.72 0.05 0.24 0.43 6.88 217 .00

Note: Std Dev = Standard Deviation, a measure of variability. For changes to be statistically detectable, “Sig” (right-most column) should be ≤ .05. To assess whether changes occurred in single items, each item was assessed for change from pre- to post-course. For several items, the change in score from pre- to post-course was in the desired direction, indicating that, on average, teachers made a detectable move toward the measured belief in the expected direction for each of these items (see Table 6). For example, for the item “The best way to understand math is to do a lot of problems,” the desired direction is to disagree with this statement, indicating a belief that mathematical understanding requires more than simple repetition or rote learning. An important caveat to these analyses is that it is likely quite clear for many of the teachers which direction on the scale is most desirable when responding to the items. Therefore, the improvement — although statistically detectable—may not be very meaningful. Changes ranged from 0.19 to 0.54 points on the six-point scale. The largest average change (.54 points from pre- to post-course) was for the item: “In mathematics you can be creative and discover things on your own.”


- 29 -

Table 6. Paired samples t-test results for pre-course to post-course individual item comparison 95% Confidence

Interval of the Difference

Mean

Difference Std.

Deviation

Std. Error Mean Lower Upper t df

Sig. (2-tailed)

Confidence in math

When I teach math I often find it difficult to interpret students’ wrong answers.

0.19 1.32 0.09 0.02 0.37 2.17 216 .03

When my answer to a math problem doesn't math someone else’s, I usually assume that my answer is wrong.

0.25 1.59 0.11 0.04 0.46 2.30 217 .02

Beliefs about correct answers

To assess students’ math understanding it is important to observe them while they are working and listen to their math conversations.

0.38 0.89 0.06 0.26 0.50 6.26 216 .00

Students who aren’t getting the right answers need to practice on more problems.

0.33 1.26 0.09 0.16 0.50 3.88 217 .00

In mathematics, answers are right or wrong. 0.30 1.14 0.08 0.15 0.45 3.88 217 .00

Students who produce correct answers have a good understanding of the mathematical concepts.

0.23 1.35 0.09 0.05 0.41 2.56 217 .01

Children’s reasoning in their mathematical problem solving is more important to assess than whether they solve problems correctly.

0.29 1.11 0.08 0.14 0.44 3.83 217 .00

Beliefs about math as operations

In mathematics you can be creative and discover things on your own.

0.54 1.12 0.08 0.39 0.69 7.10 217 .00

Studying a few problems in depth is a good way to learn mathematical concepts

0.45 1.29 0.09 0.28 0.63 5.22 217 .00

Mathematics involves mostly facts and procedures that have to be learned.

0.35 1.23 0.08 0.19 0.52 4.23 217 .00

The best way to understand math is to do a lot of problems.

0.20 1.01 0.07 0.06 0.33 2.89 217 .00

Note: Std Dev = Standard Deviation, a measure of variability. For changes to be statistically detectable, “Sig” (right-most column) should be ≤ .05.

- 30 -

CHANGE IN PEDAGOGICAL CONTENT KNOWLEDGE RESULTS Several measures of pedagogical content knowledge (PCK) were assessed in all of the BreakThrough Math courses at the beginning and end of the course. The tasks at the end of the course measure the same conceptual content as the tasks at the beginning of the course, with minor alterations in the details of the problem from pre- to post-course (see Figure 26). All courses included open-ended measures of PCK developed specifically for this evaluation (see Appendix C for Pre- and Post Course Thinking Tasks). Three of the courses (Fraction Concepts, Multi-Digit Whole Number Computation, and Numbers & Numeration) also included a series of closed-ended questions developed by Ball and colleagues (e.g., Hill, Schilling, & Ball, 2004) to measure PCK. Items developed by Ball and colleagues are advantageous because these items have benefited from years of development, revisions, and reliability and validity testing. However, because the content measured in the Ball et al. items are not directly aligned with the content for all of the BreakThrough Math courses, we designed tasks to embed in the courses that are intended to consistently assess PCK across all of the BreakThrough Math courses. The wording of these custom developed tasks and questions were derived from Ma’s (1999) discussion of assessing mathematical PCK4. At the time of this report, reliability and validity assessments for these tasks have not been completed and therefore caution must be used when interpreting results. In this report the open-ended PCK tasks developed for this evaluation are referred to as the pre- and post-course thinking tasks. The closed-ended measure developed by Ball and colleagues will be referred to as the further and final tasks. Figure 26. Pre- and Post-Course Thinking Tasks (examples from the Fraction Concepts course)

Pre-Course Thinking Task Post-Course Thinking Task

Task questions given both pre- and post-course: 1. What does a student need to know or be able to do solve the problem? That is, what are the basic mathematical concepts being assessed? 2. Give at least two different ways that a student could solve this problem, and briefly describe each (2-3 sentences each) so it is clear what each solution would look like.

4 Ma, L. (1999). Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates.

Which of the figures shows fourths? Explain your thinking.

Which of the figures shows fourths? Explain your thinking


- 31 -

3. Describe the most common errors students make or misunderstandings they have when learning the mathematical concepts that underlie this task.

Pre- and Post-Course Thinking Tasks Depending on the course, teachers responded to two to four tasks that measured their PCK for the BreakThrough Math course in which they were enrolled. The tasks were comprised of at least one problem that students might encounter in their math textbook (see Figure 26 above) and at least one problem that included an incorrect student solution (see Figure 27). Following each task, teachers responded to a series of three open-ended questions about the task. For the tasks that did not have a student solution, teachers responded to open-ended questions that focused on three aspects of PCK, including: (a) What does a student need to know or be able to do to solve the problem (PCK aspect: concepts to solve problem); (b) Give at least two different ways that a student could solve this problem, and briefly describe each so it is clear what each solution would look like (PCK aspect: solutions); (c) Describe the most common errors students make or misunderstandings they have when learning the mathematical concepts that underlie this task (PCK aspect: common student errors). For tasks that focused on problems with an incorrect student solution, the teachers responded to the following open-ended questions that also focused on various aspects of PCK: (a) What does a student need to know or be able to do to solve the problem (PCK aspect: concepts to solve student work problem); (b) Explain if the student correctly solve the problem; (c) Looking at the students work and trying to get “into their head”, describe the reasoning the student used to solve the problem (PCK aspect: student reasoning). Figure 27. Example of a Pre-Course Thinking Task with student work (from Concepts of Measurement course)

Pre-Course Thinking Task Post-Course Thinking Task

Task questions given both pre- and post-course: 1. What does a student need to know or be able to do to solve the problem? That is, what are the basic mathematical concepts being assessed? 2. Did Lateshia/Henry correctly solve the problem? Please explain your answer. 3. Looking at Lateshia’s/Henry’s work and trying to “get into her/his head,” describe the

- 32 -

reasoning she/he used to solve the problem. Teachers’ open-ended responses to the task questions were scored by a set of nine mathematics education experts who have an in-depth knowledge of the BreakThrough Math courses. Teachers’ responses to task questions resulted in a score between zero and four for each question. A score of 0 indicated “No understanding,” a score of 1 indicated “Minimal understanding,” a score of 2 indicated “Adequate understanding,” a score of 3 indicated “Good understanding,” and a score of 4 indicated “Excellent understanding.” Scorers did not know if the responses they scored were from the pre- or post-course tasks. Pairs of scorers worked independently to score all of the teacher responses to a particular course. Any differences in scores were resolved through discussion between the two scorers. As described previously, teachers responded to two to four tasks depending on the BreakThrough Math course in which they were enrolled. For each task, teachers responded to the same three questions. Teachers’ scores from similar questions across tasks were averaged together to create a single score for each question type. There was one score for concepts to solve problem, one score for different solutions, one score for common student errors, one score for concepts to solve student work problem, and one score for student reasoning. The same methodology was used to derive post-course scores. A total of 327 teachers enrolled in the BreakThrough Math course during the 2007-2008 academic year. Of these teachers, 72% (n = 234) completed both the pre- and post-course thinking tasks for at least one of the two to four tasks. Following are the results from an aggregate analysis of all courses by PCK aspect. Paired sample t-tests were conducted to determine if there were differences in teachers’ responses between pre- and post-test. A significant t-test indicates that a statistically detectable change occurred between two time points.

Concepts students need to understand to solve problems In every BreakThrough Math course, teachers responded to at least one question before and after the course in which they described what a student would need to know or understand in order to solve the problem given in the task. Across all courses and tasks, 169 teachers responded to this type of question in both pre- and post-course tasks. The mean pre-course score was 1.67 and the mean post-course score was 1.90 (see Figure 28). The difference between pre- and post-course scores was statistically significant; t(168) = -4.35, p < .001.


- 33 -

Figure 28. Mean pre- and post-course scores for questions assessing pedagogical content knowledge about concepts to solve problems

None Minimal Adequate Good Excellent None Minimal Adequate Good Excellent

Different solutions methods to solve a problem In every BreakThrough Math course, teachers responded to at least one question before and after the course in which they described two different solution methods that a student could use to solve the problem in the task. Across all courses and tasks, 167 teachers responded to this type of question in both the pre- and post-course tasks. The mean pre-course score was 2.22 and the mean post-course score was 2.28 (see Figure 29). The difference between pre- and post-course scores was not statistically significant, although the difference in scores indicates a positive trend in that the post-course scores were higher than the pre-course scores.

- 34 -

Figure 29. Mean pre- and post-course scores for questions assessing pedagogical content knowledge about different solutions

None Minimal Adequate Good Excellent None Minimal Adequate Good Excellent

Most common errors students make while solving a specific problem In every BreakThrough Math course, teachers responded to at least one question before and after the course in which they described the most common errors or misunderstandings that a student might have while solving the problem given in the task. Across all courses and tasks, 144 teachers responded to this type of question in both the pre- and post-course tasks. The mean pre-course score was 1.66 and the mean post-course score was 1.81 (see Figure 30). The difference between pre- and post-course scores was statistically significant; t(143) = -2.39, p < .05.


- 35 -

Figure 30. Mean pre- and post-course scores for questions assessing pedagogical content knowledge about common student errors

None Minimal Adequate Good Excellent


Concepts that a student used to solve a problem In every BreakThrough Math course, teachers responded to at least one question before and after the course in which they reviewed a problem that contained student work to solve the problem. One of the questions for this task was for the teacher to describe the most common errors or misunderstandings that the student might have when solving the problem in the task. Across all courses and tasks, 210 teachers responded to this type of question in both the pre- and post-course tasks. The mean pre-course was 1.69 and the mean post-course score was 1.82 (see Figure 31). The difference between pre- and post-course scores was not statistically significant.

- 36 -

Figure 31. Mean pre- and post-course scores for questions assessing pedagogical content knowledge about concepts to solve student work problem



Student reasoning used to solve a problem In every BreakThrough Math course, teachers responded to at least one question before and after the course in which they reviewed an example of student work to solve a particular problem. The final question for this task asked the teacher to describe the type of reasoning that the student might use while solving the problem in the task. Across all courses and tasks, 209 teachers responded to this type of question in both the pre- and post-course tasks. The mean pre-course score was 2.12 and the mean post-course score was 2.51 (see Figure 32). The difference between pre- and post-course scores was statistically significant; t(208) = -4.85, p < .001.


- 37 -

Figure 32. Mean pre- and post-course scores for questions assessing pedagogical content knowledge about student reasoning



Further and Final Tasks As mentioned in the beginning of this section, three of the BreakThrough Math courses (Fraction Concepts, Multi-Digit Whole Number Computation, and Numbers & Numeration) included a series of closed-ended questions developed by Ball and colleagues (e.g., Hill, Schilling, & Ball, 2004). At the time of this report, the Fraction Concept course had responses from 20 teachers at both pre- and post-course by which to assess change. For Numbers & Numeration, only 15 teachers completed the items both pre- and post-course. For Multi-Digit Whole Number Computation, only nine teachers completed both sets of items. The following summarizes the aggregated results and the results by courses. The results by courses must be taken with caution due to the very low number of teacher responses from pre- to post-course.

Aggregate Results In three of the courses (Fraction Concepts, Numbers & Numeration, and Multi-Digit Whole Number Computation) there were 10 to 16 closed-ended items that assessed teachers’ pedagogical content knowledge. Due to the range in the total number of items in each course, the raw score was not used to compare teachers’ scores across different courses; rather, the proportion of correct answers was used. On average, teachers provided correct responses for 62% of the pre-course items and 65% of the post-course items. The gain in the proportion of correct responses from pre- to post-course is not statistically significant, although the trend is approaching the correct direction (i.e., higher post-course scores in comparison to pre-course scores). Figure 33 depicts the distribution of teachers’ proportionally correct scores at pre- and post-course.

- 38 -

Figure 33. Proportion of correct answers at pre- and post-course

Fraction Concepts Ten items assessed pedagogical content knowledge (see Appendix D for Further and Final Thinking items). Each item was worth 1 point, for a maximum of 10 points. Teachers’ average scores were 5.6 (56% correct) on the pre-course items and 5.5 (55% correct) on the post-course items. The distributions of scores ranged from 2 to 9 on the pre-course items and from 2 to 8 on the post-course items (see Figure 34). A paired t-test was used to assess the difference between the pre- and post-course total scores. There was no statistically significant (i.e., detectable) difference between the two. Yet, when assessed item-by-item rather than by total score, it appears that, as a group, teachers made improvements in most items, while doing worse with others (see Table 7). Figure 34. Distribution of teachers’ scores for Further and Final Thinking Tasks in Fraction Concepts


- 39 -

Table 7. Changes in Further and Final Thinking Task by item (Fraction Concepts)

Item

Proportion Correct PRE-

COURSE

Proportion Correct POST-

COURSE Proportionate Gains/Loss

% of Individual Teachers

Improving 1 0.6 0.7 0.1 15% 2 0.7 0.6 -0.1 10% 3 0.0 0.2 0.2 20% 4 0.3 0.5 0.2 30% 5 1.0 1.0 0.0 0% 6 0.4 0.4 0.1 15% 7 0.3 0.3 0.0 5% 8 0.8 0.9 0.1 15% 9 0.9 1.0 0.0 10%

10 0.7 0.0 -0.7 0% There were only two items (#2 and #10) on which teachers scored higher pre-course compared to post-course. On item #5, all of the teachers provided correct answers both pre- and post-course. Aside from items #5 and #10, at least 10% or more of the teachers had an incorrect answer on the pre-course item and a correct answer on the post-course item (see far right column in Table 7 above).

Numbers & Numeration Seventeen items assessed pedagogical content knowledge; however one item (#9) was dropped because it was redundant (see Appendix D for Further and Final Thinking items). Each item was worth 1 point, for a maximum of 16 points. Teachers’ average scores were 11.27 (70% correct) on the pre-course items and 11.87 (74% correct) on the post-course items. The distributions of scores ranged from 9 to 13 on the pre-course items and from 7 to 16 on the post-course items (see Figure 35). A paired t-test was used to assess the difference between the pre- and post-course total scores. There was no statistically significant (i.e., detectable) difference between the two. Yet, when assessed item-by-item rather than by total score, it appears that as a group, teachers made improvements in most items, while doing worse with others (see Table 8).

- 40 -

Figure 35. Distribution of teachers’ scores for Further and Final Thinking Tasks in Numbers & Numeration

Table 8. Changes in Further and Final Thinking Task by item (Numbers & Numeration)

Item


COURSE




Improving 1 0.27 0.33 0.07 27% 2 0.67 0.67 0.00 20% 3 1.00 0.87 -0.13 0% 4 0.93 1.00 0.07 7% 5 0.80 0.73 -0.07 7% 6 0.73 0.73 0.00 13% 7 0.87 0.93 0.07 13% 8 0.53 0.67 0.13 27% 9 dropped dropped dropped dropped

10 0.13 0.40 0.27 27% 11 0.73 0.87 0.13 13% 12 0.07 0.20 0.13 20% 13 1.00 1.00 0.00 0% 14 0.93 0.93 0.00 7% 15 0.87 0.80 -0.07 0% 16 0.87 0.73 -0.13 7% 17 0.87 1.00 0.13 13%

There were four items (#3, #5, #15, and #16) on which teachers scored higher pre-course compared to post-course. On item #13, all of the teachers provided correct answers both pre- and post-course. On all


- 41 -

other items, at least 7% or more of the teachers had an incorrect answer on the pre-course item and a correct answer on the post-course item (see far right column in Table 8 above).

Multi-Digit Whole Number Computation Thirteen items assessed pedagogical content knowledge (see Appendix D for Further and Final Thinking items). Each item was worth 1 point, for a maximum of 13 points. Teachers’ average scores were 7.78 (60% correct) on the pre-course items and 9.11 (70% correct) on the post-course items. The distributions of scores ranged from 5 to 12 on the pre-course items and from 7 to 13 on the post-course items (see Figure 36). A paired t-test was used to assess the difference between the pre- and post-course total scores. There was a statistically significant (i.e. detectable) difference between the two; t(8)= -2.83, p < .05. Teachers had higher scores in the Further and Final Thinking Tasks after completing a BreakThrough Math course focused on the same content as the tasks. Yet, when assessed item-by-item rather than by total score, it appears that, as a group, teachers made improvements in most items, while doing worse with others (see Table 9). Figure 36. Distribution of teachers’ scores for Further and Final Thinking Tasks in Multi-Digit Whole Number Computation

- 42 -

Table 9. Changes in Further and Final Thinking Task by item (Multi-Digit Whole Number Computation)

Item


COURSE




Improving 1 0.56 1.00 0.44 44% 2 0.44 0.56 0.11 11% 3 0.89 0.89 0.00 0% 4 0.89 0.78 -0.11 11% 5 0.78 0.89 0.11 22% 6 0.67 0.56 -0.11 11% 7 0.78 0.78 0.00 11% 8 0.67 0.89 0.22 22% 9 0.33 0.33 0.00 22%

10 0.56 0.78 0.22 22% 11 0.78 0.89 0.11 11% 12 0.33 0.56 0.22 33% 13 0.11 0.22 0.11 11%

There are only two items (#4 and #6) on which teachers scored higher pre-course compared to post-course. On all other items, at least 11% or more of the teachers had an incorrect answer on the pre-course item and a correct answer on the post-course item (see far right column in Table 9 above).


- 43 -

SUMMARY & RECOMMENDATIONS This evaluation report summarizes data that have been collected through the BreakThrough Math professional development courses offered through the Ohio Department of Education. The goal of offering these courses is to help teachers improve their math pedagogical content knowledge as a means by which to improve the teaching of math. The link between math pedagogical content knowledge and student achievement has been documented extensively in the literature (e.g., Ma, 1994). It is expected that if these courses can enhance math pedagogical content knowledge and those gains can be sustained, this professional development will contribute to improving instructional quality. It is understood that a limitation of these courses is that they are 18 hours in length and do not include a component by which to sustain gains made by taking these courses. At Pearson, current development is underway to “marry” the content specificity of the BreakThrough Math courses (now known as “Enhancing Instruction in Mathematics”) with strategies for sustaining the learning that teachers attain from these courses. Some of the teachers surveyed in this report indicate having a system in place at their school by which they can share what they’ve learned in these courses. However, until those systems are structured specifically to address sustainability of these courses, one should not assume that teachers will be able to implement changes suggested by these courses in any systematic, sustainable way. The data reported here are all collected from the teachers before, during, and after they take one or more BreakThrough Math courses. Measures addressing changes in instructional practices and student learning are all based on teachers’ self-reports, not objective assessments such as classroom observations or student assessments. These objective evaluation data are not feasible to collect for this project for several reasons, namely that the teachers generally take these courses in the summer. As such, because school is not in session, teachers cannot administer student assessments or make classroom observations. Thus we can only ask teachers about the perceived or expected changes they attribute to taking the BreakThrough Math professional development course(s). Results, therefore, should be regarded with this caveat in mind. Measures of pedagogical content knowledge have been designed specifically for these courses in the form of pre- and post-course tasks. To provide additional information and to serve as “back up” data, items developed by Deborah Ball and colleagues at the University of Michigan (Hill, Schiller, & Ball, 2004) to assess math pedagogical content knowledge have been embedded in three of the courses: Fraction Concepts, Numbers & Numeration, and Multi-Digit Whole Number Computation. However, it appears to be difficult to obtain teacher cooperation in completing these items. We have asked the facilitators to mandate that item completion is required for course completion; however, the small amount of responses (47% to 64% of the teachers per course) indicates that teachers are still not completing these items. If we can gather more data from the teachers using Hill et al.’s (2004) items, we can correlate these responses to those of the Pre- and Post-Course Thinking tasks developed specifically for these courses. Hill et al.’s items were selected based on similarity to the course content. However, because these items were developed independently of the courses, they do not align perfectly with the BreakThrough Math content. In contrast, the Pre- and Post-Course Thinking tasks were designed by math content experts in tandem with BreakThrough Math developers and facilitators to ensure that the item content aligned

- 44 -

closely with the course content. Therefore we do not expect a high correlation between the Pre- and Post-Course Thinking tasks and Hill et al.’s (2004) items. A moderately small to moderate correlation would be expected, and when sufficient data exist, this will be tested. With these limitations in mind, evaluation results suggest that, in general, the teachers who take these BreakThrough Math courses enjoy the courses and regard them as providing useful information that can be applied in the classroom. Although some teachers initially may have been skeptical about the online format for these courses (only a small minority are taught face-to-face), a large portion of the teachers indicate that the online delivery was actually advantageous as far as flexibility and convenience. Additionally, features such as forums (which provided a way to interact with peers) and video lesson analysis were highly regarded. Only a small minority rated these courses as low in quality and failing to meet expectations. Teacher comments about “aha!” moments or new insights focused on new math content knowledge and new ways to teach math concepts in the classroom, both of which are key components of math pedagogical content knowledge. Most of the teachers believe that they have made or will make changes to their teaching, and that they see or expect to see changes in student learning as a result of having taken these courses. Again, these results are based on teachers’ perceptions, not objective observations.


- 45 -

APPENDIX A. FREQUENCY REPORT FOR TEACHER BACKGROUND AND SCHOOL CLIMATE SURVEY

- 46 -

Meet highly qualified status

Frequency Percent Valid Percent Cumulative Percent

Yes 233 86.9 87.3 87.3

No 34 12.7 12.7 100.0

Valid

Total 267 99.6 100.0

Missing System 1 .4

Total 268 100.0

Taking course to meet highly qualified status


Yes 19 7.1 38.0 38.0

No 31 11.6 62.0 100.0

Valid

Total 50 18.7 100.0

Missing System 218 81.3

Total 268 100.0

Achieve highly qualified status after the course


Yes 7 2.6 14.0 14.0

No 43 16.0 86.0 100.0

Valid

Total 50 18.7 100.0


Total 268 100.0


- 47 -

Years teaching


Less than 1 year 5 1.9 1.9 1.9

1 to 2 years 8 3.0 3.1 5.0

More than 2, less than 5

years 34 12.7 13.0 17.9

More than 5, less than

10 years 76 28.4 29.0 46.9


20 years 96 35.8 36.6 83.6


30 37 13.8 14.1 97.7

More than 30 6 2.2 2.3 100.0

Valid

Total 262 97.8 100.0


Total 268 100.0

- 48 -

Years in current school


Less than 1 year 29 10.8 11.2 11.2

1 to 2 years 24 9.0 9.3 20.5


years 50 18.7 19.4 39.9


10 years 88 32.8 34.1 74.0


20 years 48 17.9 18.6 92.6


30 18 6.7 7.0 99.6

More than 30 1 .4 .4 100.0

Valid

Total 258 96.3 100.0


Total 268 100.0

Years teaching in current grade level


Less than 1 year 22 8.2 8.5 8.5

1 to 2 years 28 10.4 10.8 19.3


years 55 20.5 21.2 40.5


10 years 81 30.2 31.3 71.8


20 years 54 20.1 20.8 92.7


30 19 7.1 7.3 100.0

Valid

Total 259 96.6 100.0


Total 268 100.0


- 49 -

Years teaching math


Less than 1 year 13 4.9 5.0 5.0

1 to 2 years 14 5.2 5.4 10.3


years 48 17.9 18.4 28.7


10 years 87 32.5 33.3 62.1


20 years 70 26.1 26.8 88.9


30 25 9.3 9.6 98.5

More than 30 4 1.5 1.5 100.0

Valid

Total 261 97.4 100.0


Total 268 100.0

Elementary Education Credential


Not selected 29 10.8 14.4 14.4

Selected 172 64.2 85.6 100.0

Valid

Total 201 75.0 100.0


Total 268 100.0

- 50 -

Mathematics Credential


Not selected 43 16.0 39.8 39.8

Selected 65 24.3 60.2 100.0

Valid

Total 108 40.3 100.0


Total 268 100.0

Special Ed Credential


Not selected 64 23.9 63.4 63.4

Selected 37 13.8 36.6 100.0

Valid

Total 101 37.7 100.0


Total 268 100.0

Grade Taught

Selected

Count

K teacher 32

1st teacher 32

2nd teacher 47

3rd teacher 40

4th teacher 48

5th teacher 52

6th teacher 34

7th teacher 23

8th teacher 11

9th teacher 39


- 51 -

Number of grades taught


0 9 3.4 3.4 3.4

1 221 82.5 82.5 85.8

2 12 4.5 4.5 90.3

3 11 4.1 4.1 94.4

4 2 .7 .7 95.1

5 7 2.6 2.6 97.8

6 5 1.9 1.9 99.6

7 1 .4 .4 100.0

Valid

Total 268 100.0 100.0

College major


Did not major or minor

in math 212 79.1 79.4 79.4

Minored in math 22 8.2 8.2 87.6

Majored in math 33 12.3 12.4 100.0

Valid

Total 267 99.6 100.0

Missing System 1 .4

Total 268 100.0

- 52 -

Last course/training in math


In the past year 141 52.6 53.0 53.0

1-2 years ago 34 12.7 12.8 65.8

2-5 years ago 36 13.4 13.5 79.3

Over 5 years ago 55 20.5 20.7 100.0

Valid

Total 266 99.3 100.0

Missing System 2 .7

Total 268 100.0

Highest degree


Bachelor 61 22.8 22.8 22.8

Master 206 76.9 77.2 100.0

Valid

Total 267 99.6 100.0

Missing System 1 .4

Total 268 100.0


- 53 -

Field for highest degree


Other 48 17.9 18.0 18.0

Elementary Ed 111 41.4 41.6 59.6

Curriculum &

Instruction 21 7.8 7.9 67.4

Math Education 28 10.4 10.5 77.9

Special Ed 23 8.6 8.6 86.5

Reading 25 9.3 9.4 95.9

Administrative 11 4.1 4.1 100.0

Valid

Total 267 99.6 100.0

Missing System 1 .4

Total 268 100.0

Multi-track school


Yes 63 23.5 24.7 24.7

No 192 71.6 75.3 100.0

Valid

Total 255 95.1 100.0


Total 268 100.0

Title I school


Yes 121 45.1 46.7 46.7

No 138 51.5 53.3 100.0

Valid

Total 259 96.6 100.0


Total 268 100.0

- 54 -

Analyze student work to better understand student needs


Not at all 4 1.5 1.5 1.5

Less than once per

month 42 15.7 16.2 17.8

Once per month 61 22.8 23.6 41.3

2-3 times per month 71 26.5 27.4 68.7

Weekly 81 30.2 31.3 100.0

Valid

Total 259 96.6 100.0


Total 268 100.0

Identify instructional strategies that might help address common student needs


Less than once per

month 42 15.7 16.1 16.1

Once per month 70 26.1 26.8 42.9

2-3 times per month 67 25.0 25.7 68.6

Weekly 82 30.6 31.4 100.0

Valid

Total 261 97.4 100.0


Total 268 100.0


- 55 -

Analyze student work to evaluate teaching practices in the classroom


Not at all 5 1.9 1.9 1.9

Less than once per

month 57 21.3 22.0 23.9

Once per month 82 30.6 31.7 55.6

2-3 times per month 55 20.5 21.2 76.8

Weekly 60 22.4 23.2 100.0

Valid

Total 259 96.6 100.0


Total 268 100.0

Identify unit objectives and plan instruction for the unit


Not at all 18 6.7 6.9 6.9

Less than once per

month 50 18.7 19.2 26.2

Once per month 49 18.3 18.8 45.0

2-3 times per month 63 23.5 24.2 69.2

Weekly 80 29.9 30.8 100.0

Valid

Total 260 97.0 100.0


Total 268 100.0

- 56 -

Teachers at your school are involved in planning strategies for improving student

achievement


A little bit 6 2.2 2.3 2.3

Somewhat 54 20.1 20.7 23.0

Quite a bit 116 43.3 44.4 67.4

Very much so 85 31.7 32.6 100.0

Valid

Total 261 97.4 100.0


Total 268 100.0

School wide effort to improve student achievement focus on improving teaching in

the classroom


Not at all 1 .4 .4 .4

A little bit 18 6.7 7.0 7.4

Somewhat 60 22.4 23.3 30.6

Quite a bit 121 45.1 46.9 77.5

Very much so 58 21.6 22.5 100.0

Valid

Total 258 96.3 100.0


Total 268 100.0


- 57 -

Your school supports teachers in trying different instructional approaches in the

classroom


Not at all 3 1.1 1.1 1.1

A little bit 16 6.0 6.1 7.3

Somewhat 64 23.9 24.4 31.7

Quite a bit 124 46.3 47.3 79.0

Very much so 55 20.5 21.0 100.0

Valid

Total 262 97.8 100.0


Total 268 100.0

Teachers at your school feel supported by administration to do their job


Not at all 5 1.9 1.9 1.9

A little bit 25 9.3 9.5 11.5

Somewhat 72 26.9 27.5 38.9

Quite a bit 111 41.4 42.4 81.3

Very much so 49 18.3 18.7 100.0

Valid

Total 262 97.8 100.0


Total 268 100.0

- 58 -

Teacher workgroup meetings enhance your teaching in the classroom


Not at all 7 2.6 2.7 2.7

A little bit 33 12.3 12.7 15.4

Somewhat 102 38.1 39.2 54.6

Quite a bit 78 29.1 30.0 84.6

Very much so 30 11.2 11.5 96.2

Not Applicable 10 3.7 3.8 100.0

Valid

Total 260 97.0 100.0


Total 268 100.0

Ongoing programs at school that focus on improving teaching in the

classroom


Yes 166 61.9 64.1 64.1

No 93 34.7 35.9 100.0

Valid

Total 259 96.6 100.0


Total 268 100.0


- 59 -

Extent than the ongoing program is effective in improving pedagogy


Not at all 2 .7 .9 .9

A little 15 5.6 7.0 7.9

Somewhat 81 30.2 37.9 45.8

Quite a bit 65 24.3 30.4 76.2

Very much so 15 5.6 7.0 83.2

Not applicable 36 13.4 16.8 100.0

Valid

Total 214 79.9 100.0


Total 268 100.0

PD courses that focus on improving math teaching


None 44 16.4 16.6 16.6

1 38 14.2 14.3 30.9

2-4 109 40.7 41.1 72.1

More than 10 16 6.0 6.0 78.1

5-10 58 21.6 21.9 100.0

Valid

Total 265 98.9 100.0


Total 268 100.0

- 60 -

Personal satisfaction with teaching math


Not at all satisfied 2 .7 .8 .8

A little satisfied 11 4.1 4.2 4.9

Average 74 27.6 28.1 33.1

Good 121 45.1 46.0 79.1

Excellent 55 20.5 20.9 100.0

Valid

Total 263 98.1 100.0


Total 268 100.0

Level of assistance school provides to become a more effective teacher


None 7 2.6 2.7 2.7

A little 22 8.2 8.4 11.1

Some 112 41.8 42.9 54.0

Quite a bit 90 33.6 34.5 88.5

A lot 30 11.2 11.5 100.0

Valid

Total 261 97.4 100.0


Total 268 100.0


- 61 -

Level of assistance admin provides to become a more effective teacher


None 11 4.1 4.2 4.2

A little 36 13.4 13.9 18.1

Some 92 34.3 35.5 53.7

Quite a bit 97 36.2 37.5 91.1

A lot 23 8.6 8.9 100.0

Valid

Total 259 96.6 100.0


Total 268 100.0

- 62 -

APPENDIX B. FREQUENCY REPORT FOR END-OF-COURSE SURVEY


- 63 -

The funding agency requires info about "highly qualified" teachers. As a result

of taking this course, do you meet highly qualified status?


1 145 58.2 58.9 58.9

2 34 13.7 13.8 72.8

3 67 26.9 27.2 100.0

Valid

Total 246 98.8 100.0


Total 249 100.0

Which BreakThrough Math content did this course cover? (SELECT ONLY

ONE):


PS 63 25.3 26.0 26.0

NN 13 5.2 5.4 31.4

MD 14 5.6 5.8 37.2

SE 7 2.8 2.9 40.1

FC 41 16.5 16.9 57.0

CM 45 18.1 18.6 75.6

GSS 18 7.2 7.4 83.1

RP 5 2.0 2.1 85.1

EI 22 8.8 9.1 94.2

RNO 3 1.2 1.2 95.5

SDP 11 4.4 4.5 100.0

Valid

Total 242 97.2 100.0


Total 249 100.0

- 64 -

How was the course delivered?:


Online only 44 17.7 18.2 18.2

Face-to-Face 2 .8 .8 19.0

Blended 196 78.7 81.0 100.0

Valid

Total 242 97.2 100.0


Total 249 100.0

Have you ever enrolled in a similar course (i.e. focused on improving the teaching

of math) that was NOT another BreakThrough Math course?


No 68 27.3 28.3 28.3

Yes 172 69.1 71.7 100.0

Valid

Total 240 96.4 100.0


Total 249 100.0

How did this current course (the one you just completed) compare?


N/A 41 16.5 38.0 38.0

Worse 6 2.4 5.6 43.5

About the same 46 18.5 42.6 86.1

Better 15 6.0 13.9 100.0

Valid

Total 108 43.4 100.0


Total 249 100.0


- 65 -

To what extent did this course meet your expectations?


Not at all 2 .8 .8 .8

A little 5 2.0 2.1 2.9

Somewhat 39 15.7 16.4 19.3

Met most of my

expectations 133 53.4 55.9 75.2

Exceeded my

expectations 59 23.7 24.8 100.0

Valid

Total 238 95.6 100.0


Total 249 100.0

How prepared was the facilitator to facilitate the course?


Not at all 1 .4 .4 .4

A little 5 2.0 2.1 2.5

Somewhat 26 10.4 11.0 13.6

Quite prepared 100 40.2 42.4 55.9

Exceptionally prepared 104 41.8 44.1 100.0

Valid

Total 236 94.8 100.0


Total 249 100.0

- 66 -

Was there anything the facilitator could have done to make your learning more

meaningful and effective?


Yes 44 17.7 19.0 19.0

No 188 75.5 81.0 100.0

Valid

Total 232 93.2 100.0


Total 249 100.0

Did you have any AHA moments in the course?


Yes 178 71.5 75.4 75.4

No 58 23.3 24.6 100.0

Valid

Total 236 94.8 100.0


Total 249 100.0

AHA moment coded


New math content

knowledge 52 20.9 42.6 42.6

New strategies for

teaching math concepts 70 28.1 57.4 100.0

Valid

Total 122 49.0 100.0


Total 249 100.0


- 67 -

Strength #1 of the course


Discussions 34 13.7 14.5 14.5

Convenience of online

course 57 22.9 24.4 38.9

Learning more about

teaching 45 18.1 19.2 58.1

Video analysis 27 10.8 11.5 69.7

Learning more about

math 26 10.4 11.1 80.8

Facilitator 8 3.2 3.4 84.2

Useful strategies 14 5.6 6.0 90.2

Organization of the

course 8 3.2 3.4 93.6

Useful, relevant

readings 6 2.4 2.6 96.2

Practice writing lesson

plans 1 .4 .4 96.6

None 4 1.6 1.7 98.3

Other 4 1.6 1.7 100.0

Valid

Total 234 94.0 100.0


Total 249 100.0

- 68 -



Discussions 37 14.9 17.0 17.0


course 35 14.1 16.1 33.0

Learning more about

teaching 31 12.4 14.2 47.2


Learning more about

math 19 7.6 8.7 67.9



Organization of the

course 9 3.6 4.1 81.7

Useful, relevant

readings 20 8.0 9.2 90.8


plans 6 2.4 2.8 93.6

None 7 2.8 3.2 96.8

Other 7 2.8 3.2 100.0

Valid

Total 218 87.6 100.0


Total 249 100.0


- 69 -



Discussions 28 11.2 14.1 14.1


course 21 8.4 10.6 24.7

Learning more about

teaching 23 9.2 11.6 36.4


Learning more about

math 25 10.0 12.6 61.1



Organization of the

course 6 2.4 3.0 78.3

Useful, relevant

readings 10 4.0 5.1 83.3


plans 2 .8 1.0 84.3

None 15 6.0 7.6 91.9

Other 16 6.4 8.1 100.0

Valid

Total 198 79.5 100.0


Total 249 100.0

- 70 -

Improvement #1 of the course


More face-to-face

meetings 6 2.4 2.6 2.6

More discussions 1 .4 .4 3.0

Less busy work 21 8.4 9.0 12.0

Clarification of course

expectations 14 5.6 6.0 17.9

Eliminate technology

problems 26 10.4 11.1 29.1

Reduce workload 30 12.0 12.8 41.9

None 88 35.3 37.6 79.5

Other 4 1.6 1.7 81.2

Feedback 9 3.6 3.8 85.0

Organization 23 9.2 9.8 94.9

Content 5 2.0 2.1 97.0

Forumn Features 7 2.8 3.0 100.0

Valid

Total 234 94.0 100.0


Total 249 100.0


- 71 -



More face-to-face

meetings 5 2.0 3.0 3.0

More discussions 6 2.4 3.6 6.6





problems 15 6.0 9.0 24.7


None 85 34.1 51.2 86.1

Other 1 .4 .6 86.7

Feedback 5 2.0 3.0 89.8


Content 4 1.6 2.4 100.0

Valid

Total 166 66.7 100.0


Total 249 100.0

- 72 -



More face-to-face

meetings 10 4.0 6.5 6.5

More discussions 1 .4 .6 7.1





problems 10 4.0 6.5 18.1


None 112 45.0 72.3 93.5

Other 1 .4 .6 94.2


Content 2 .8 1.3 100.0

Valid

Total 155 62.2 100.0


Total 249 100.0

Did you think the video analysis was valuable?


Yes 225 90.4 95.7 95.7

No 10 4.0 4.3 100.0

Valid

Total 235 94.4 100.0


Total 249 100.0


- 73 -

Did other teachers from your school enroll in the course?


Yes 110 44.2 47.0 47.0

No 124 49.8 53.0 100.0

Valid

Total 234 94.0 100.0


Total 249 100.0

Did you have any opportunities to discuss the course and/or network with other

teachers during the BT math course?


Yes 76 30.5 73.1 73.1

No 28 11.2 26.9 100.0

Valid

Total 104 41.8 100.0


Total 249 100.0

Engaged in a learning network


Not at all 20 8.0 8.5 8.5

A little 30 12.0 12.8 21.4

Some 67 26.9 28.6 50.0

Quite a bit 89 35.7 38.0 88.0

Very much so 28 11.2 12.0 100.0

Valid

Total 234 94.0 100.0


Total 249 100.0

- 74 -

BT math course aligned with curriculum


Not at all 1 .4 .4 .4

A little 6 2.4 2.6 3.0

Some 27 10.8 11.6 14.6

Quite a bit 117 47.0 50.2 64.8

Very much so 82 32.9 35.2 100.0

Valid

Total 233 93.6 100.0


Total 249 100.0

BT math course aligned to students learning needs


Not at all 1 .4 .4 .4

A little 9 3.6 3.9 4.3

Some 27 10.8 11.7 16.0

Quite a bit 105 42.2 45.5 61.5

Very much so 89 35.7 38.5 100.0

Valid

Total 231 92.8 100.0


Total 249 100.0


- 75 -

BT math course resulted in gains in own content knowledge.


Not at all 12 4.8 5.1 5.1

A little 13 5.2 5.5 10.6

Some 62 24.9 26.3 36.9

Quite a bit 74 29.7 31.4 68.2

Very much so 75 30.1 31.8 100.0

Valid

Total 236 94.8 100.0


Total 249 100.0

Effective questing coded


Change described 203 81.5 81.5 81.5

None 46 18.5 18.5 100.0

Valid

Total 249 100.0 100.0

Cognitive demand of problems coded



None 72 28.9 28.9 100.0

Valid

Total 249 100.0 100.0

Anticipating student responses coded



None 66 26.5 26.5 100.0

Valid

Total 249 100.0 100.0

- 76 -

Lesson planning coded



None 95 38.2 38.2 100.0

Valid

Total 249 100.0 100.0

Emphasizing core concepts coded



None 88 35.3 35.3 100.0

Valid

Total 249 100.0 100.0

Have you taught the mathematics content covered in the course while taking the

course?


Yes 59 23.7 25.3 25.3

No 174 69.9 74.7 100.0

Valid

Total 233 93.6 100.0


Total 249 100.0


- 77 -

Have you noted any changes in student learning in your classroom that you

would attribute to having taken this course?


Yes 56 22.5 94.9 94.9

No 3 1.2 5.1 100.0

Valid

Total 59 23.7 100.0


Total 249 100.0

Types of changes due to BT Math course


Change in student

behavior 20 8.0 44.4 44.4

Change in strategies to

teach math 7 2.8 15.6 60.0

Change in interactions

with students 11 4.4 24.4 84.4

Change in specific math

content 7 2.8 15.6 100.0

Valid

Total 45 18.1 100.0


Total 249 100.0

Have you made any changes in your teaching that you would attribute to this

course?


Yes 114 45.8 76.5 76.5

No 35 14.1 23.5 100.0

Valid

Total 149 59.8 100.0


Total 249 100.0

- 78 -

APPENDIX C. PRE- AND POST COURSE THINKING TASKS


- 79 -

Concepts of Measurement PRE COURSE THINKING POST COURSE THINKING

PRE COURSE THINKING POST COURSE THINKING

- 80 -



- 81 -

Equality and Inequality PRE COURSE THINKING POST COURSE THINKING


- 82 -




- 83 -

Fraction Concepts PRE COURSE THINKING POST COURSE THINKING


- 84 -




- 85 -

Multi-Digit Whole Number Computation PRE COURSE THINKING POST COURSE THINKING


- 86 -



- 87 -

Numbers & Numeration PRE COURSE THINKING POST COURSE THINKING



- 88 -



- 89 -

Perspectives on Problem Solving (K-5) PRE COURSE THINKING POST COURSE THINKING


- 90 -

Perspectives on Problem Solving (6-8)




- 91 -

Rational Numbers & Operations PRE COURSE THINKING POST COURSE THINKING


- 92 -




- 93 -

Solving Equations PRE COURSE THINKING POST COURSE THINKING

The cost to rent a bike is $10.00 for the first hour and an extra $1.00 for every additional hour. If Angie rents a bike for 8 hours, how much does she have to pay?

To make 1 milkshake, you need 2 cups of strawberries and 3 cups of milk. How many cups of strawberries does Mike need to make 4 milkshakes?


The teacher is covering up one number in the equation below. Answer the questions that follow.

a. What number is underneath the teacher’s hand? _____ b. How did you figure it out?

The teacher is covering up one number in the equation below. Answer the questions that follow. 17 – 6 ( ) = 3 a. What number is underneath the teacher’s hand? _____ b. How did you figure it out?


Mimi is 4 times as old as Tom. Tom is 4 years old. a. Write an expression that represents Mimi’s

age b. How old is Mimi? Explain how you figured it

out

The area of a rectangle is 132cm2. The width is 11cm and the length is 1 more than the width.

a. write an expression that represents the length of the rectangle

b. What is the length? Explain how you figured it out.

- 94 -


The problem: 6x + 10 = 2x - 4 David’s work appears below.

6x + 10 = 2x – 4 (6x + 10)/2 = (2x – 4)/2

3x + 5 = x – 2 3x + 5 - 5 = x – 2 - 5

3x = x – 7 3x - x = x – 7 – x

2x = - 7 x = - 3.5


- 95 -

Statistics, Data Analysis, and Probability PRE COURSE THINKING POST COURSE THINKING

What are the odds of drawing an ace from a standard deck of cards? Explain your thinking.

What are the odds of drawing a diamond from a standard deck of cards? Explain your thinking.


- 96 -

PRE COURSE THINKING POST COURSE THINKING The problem:

Suppose you want to find out whether students will eat one slice of pizza or more than one slice of pizza for lunch on a given day. Since it would take too much time to ask all of the students at your school, you will only survey 35% of them. How would you select the students to participate in the survey? Why is this a good data collection method?

Javier’s answer:

Since there are 1000 students at my middle school, it means that I have to survey 35% or 350 of them. I am going to use a convenience sample: There are 350 8th graders and they all eat lunch at 12:30. It will be easy to collect information from this group and that is why I have selected them.

The problem: Suppose you want to find out whether students would rather eat hot dogs or tacos for lunch on a given day. Since it would take too much time to ask all of the students at your school, you will only survey 33% of them. How would you select the students to participate in the survey? Why is this a good data collection method?

Vikram’s answer:

Since there are 1000 students at my middle school, it means that I have to survey 33% or 330 of them. I am going to use a convenience sample: There are 330 6th graders and they all eat lunch at noon. It will be easy to collect information from this group and that is why I have selected them.


- 97 -

APPENDIX D. FURTHER AND FINAL THINKING TASKS

- 98 -

Fraction Concepts 1. Takeem’s teacher asks him to make a drawing to compare 3/4 and 5/6. He draws the following:

and claims that ¾ and 5/6 are the same amount. What is the most likely explanation for Takeem’s answer? (Mark ONE answer.)

Takeem is noticing that each figure leaves one square unshaded. Takeem has not yet learned the procedure for finding common denominators. Takeem is adding 2 to both the numerator and denominator of 3/4 and he sees that equals 5/6 All of the above are equally likely. I’m not sure.

2. Mr. Stephenson’s students have had trouble comparing fractions. He was puzzled one day when many of them thought that 10/10 was less than 4/4. What is the most likely reason that they thought this? (Mark ONE answer.)

They don’t know how to find a common denominator for 10 and 4. They don't know where 10/10 goes on the number line so they cannot compare it with 4/4. They know that tenths are smaller than fourths. These answers are all equally likely. I’m not sure.


- 99 -

3. At a professional development workshop, teachers were learning about different ways to represent multiplication of fractions problems. The leader also helped them to become aware of examples that do not represent multiplication of fractions appropriately. Which model below cannot be used to show that 1/12 x 2/3 = 1 (Mark ONE answer).

I’m not sure.

4. Mr. Grimes asked his students what they knew about finding the lowest common denominator(LCD) in order to add fractions. His students came up with a variety of ideas. Which of the following is true? (Mark ONE answer.)

You cannot add fractions without first finding the LCD. If you use the LCD, you will never have to simplify your answer. You can always find the LCD by multiplying the two denominators. None of these is true. I’m not sure.

- 100 -

5. Mr. Thompson’s class was working on finding the least common multiple (LCM) of pairs of numbers. He asked students to find the LCM of 12 and 18 and heard the following answers: 1, 3, 6, and 36. Next he gave them the problem: Find the LCM of 9 and 15. What would be the likely answer given to this problem by a student who answered “6” to the previous problem? (Mark ONE answer.)

1 3 6 45 I’m not sure.

6. Mrs. Wise wants to include some word problems on her fractions quiz. Could she use the following word problem as word

problem for 1/2 - 1/3?

I have 1/2 of a pizza left. My brother comes in and eats 1/3 of my leftover pizza. How much pizza is left?

Yes No I'm not sure

7. Mr. Lewis asked his students to divide 6/8 by 1/2. Charlie said, “I have an easy method, Mr. Lewis. I just divide numerators and denominators. I get 6/4 which is correct.” Mr. Lewis was not surprised by this as he had seen students do this before. What did he know? (Mark ONE answer.)

He knew that Charlie’s METHOD was wrong, even though he happened to get the right answer for this problem. He knew that Charlie’s ANSWER was actually wrong. He knew that Charlie’s METHOD was right, but that for many numbers this would produce a messy ANSWER. He knew that Charlie’s METHOD only works for some fractions. I’m not sure.


- 101 -

8. An item on a quiz requires students to draw a picture to show 1/6. Students draw the following:

Which of the following would you accept as correct responses? (Mark ONE answer.)

I and II only. I and IV only. I,II, and III only. All are correct responses. I’m not sure.

9. Mr. Lee asked his students to compare 5/9 to 3/7. Which of the following should he accept as a correct explanation? (Mark ONE answer.)

5/9 is greater than 3/7 because 5 is greater than 3. They are equal because each is missing four pieces from the whole. They are equal because adding two to the numerator in 3/7 and two to the denominator in 3/7 produces 5/9. 3/7 is greater because the pieces will be bigger. 5/9 is greater because it is more than one-half, while 3/7 is less than one-half. I'm not sure.

10. Mr. Webb gave his students the following problem:

Which would you rather have: 6/10 of a dollar or 4/5 of a dollar?

Explain your reasons for choosing your answer.

He was surprised to discover that most of his students thought that 6/10 was worth more than 4/5. One student wrote: "If I had 6/10, I would have 2 more than 4/5. I would choose 6/10 so I could have more money." What most likely explains the student error? (Mark ONE answer.)

The student's answer was based on her knowing that 10 is twice as big as 5. The student was only considering the size of the pieces. The student made an error while trying to get a common denominator. The student was only considering the number of pieces. I'm not sure.

- 102 -

Numbers & Numeration

1. Mr. Schilling asked his second graders to compare pairs of numbers to see which is greater. Several students go this pair wrong:

53 > 205

What is the least likely reason for their error?

They only compared the first digit in each number. They saw that 3 was larger than 0. The number line in their class only goes up to 100. They thought "205" meant twenty-five. I'm not sure.

2. Ms. Belle's students are learning about even and odd numbers. She gives them a worksheet with a list of numbers and asks them to identify which numbers are even and which are odd (below).

19

8

10

87

Many students get stuck on the number 87. What is the most likely reason for their difficulty?

They are not comfortable with such large numbers. They see that the 8 is even and the 7 is odd and cannot decide. They don't know how to split 87 into two groups. They are not using manipulatives. I'm not sure.

3. Sometimes it is difficult to understand what students are doing with manipulatives. Mrs. Bradley saw one of her students counting out three blocks, then counting out seven blocks, then counting all the blocks from one to ten and she tried to figure out what the student was doing. What problem might this student be solving (select ONE answer)?

3 + 7 = 10 10 - 7 = 3 Show 10 blocks Jane has 3 blocks and Jon has 7. How many more blocks does Jon have? I'm not sure.


- 103 -

4. Mrs. Akellar was reading a short story that had the following passage in it:

Ten frogs sat on the side of a riverbank. Three frogs hopped in the water.

Mrs. Akellar asked her students to represent this passage using number sentences, and a student volunteered, "10 - 3 = 7." Mrs. Akellar continued to read:

Along came the rain, and three more frogs jumped in the water.

What number sentence should Mrs. Akellar's class choose to represent this next part of the story (mark ONE answer)?

10 - 3 = 7 10 - 4 = 6 7 - 3 = 4 7 - 4 = 3 I'm not sure.

Ms. Wilson's class is working in groups to decompose 391 into hundreds, tens, ones, and tenths. As she walks around, she sees groups have arrived at very different answers. Which of the following ways to represent 391 should she accept as correct? Please mark Yes, No, or I'm not sure for each choice below.

*Note: this question was dropped from the analyses 10. Suppose you give the following problem to a child who does not yet know how to multiply:

Miguel has 3 packs of gum. There are 5 sticks of gum in each pack. How many sticks of gum does Miguel have?

Which of the following is the student least likely to do? 3 +3 + 3 + 3 + 3 = 15 5 + 5 + 5 = 15 3 + 5 = 8 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 The student would be equally likely to do any of these. I'm not sure

Yes No I'm not sure5. 3 hundreds + 90 tens + 1 one 6. 2 hundreds + 19 tens + 1 one 7. 1 hundred + 119 tens + 1 one 8. 3 hundreds + 9 tens + 10 tenths 9. I’m not sure*

- 104 -

Mr. Rowan has never taught third grade before,and he is wondering what to expect at the beginning of the year. Because he teachers in a school that enrolls many new students from outside of the state, he cannot rely on the state framework to provide him the guidance he needs. Which of the following would most third graders be likely to be able to do, and which would they be unlikely to be able to do? Please mark likely to be able to do, unlikely to be able to do, or I'm not sure for each.

15. Ms. Wright is writing a test for her students, and she wants to put the easiest problems first and the most difficult problems last. Which of the following problems would be the most difficult for students?

Julie has 4 balloons. Jeff gives her some more. Now she has 12 ballloons. How many balloons did Jeff give her? Julie has 4 balloons. Jeff gives her 8 more. How many balloons does Julie have now? Julie has some balloons. Jeff gives her 8 more balloons. Now Julie has 12 balloons. How many balloons did Julie have initially? They are all equally difficult. I'm not sure.

16. You are working individually with Bonny, and you ask her to count out 23 checkers, which she does successfully. You then ask her to show you how many checkers are represented by the 3 in 23, and she counts out 3 checkers. Then you ask her to show you how many checkers are represented by the 2 in 23, and she counts out 2 checkers. What problem is Bonny having here?

Bonny doesn't know how large 23 is. Bonny thinks that 2 and 20 are the same. Bonny doesn't understand the meaning of the places in the numeral 23. All of the above.

17. Mrs. M assigned the following problem to her students:

Use base ten blocks to show the number 16.

What response are students least likely to give?

One tens rod, six cubes One cube, six cubes 7 cubes in a pile 16 cubes in a pile

Likely to be able to do

Unlikely to be able to do

I'm not sure

11. Know that if you want to add up a list of numbers (e.g., 2+8+7+5+3), you can group them together and then add the groups to get the total (e.g., [(2+8) + (7+3) + 4]) 12. Produce a definition of even numbers. 13. Write five hundred twenty-six as 526. 14. Multiply a two-digit number by a two-digit number.


- 105 -

Multi-Digit Whole Number Computation 1. Imagine that you are working with your class on subtracting large numbers. Among your students' papers, you notice that some have displayed their work in the following ways:

Which of these students is using a method that could be used to subtract any two whole numbers?

A only B only A and B B and C A, B, and C I'm not sure

- 106 -

Imagine that you are working with your class on multiplying large numbers. Among your students' papers, you notice that some have displayed their work in the following ways:

Which of these students is using a method that could be used to multiply any two whole numbers?

5. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students focused on particular difficulties that they are having with adding columns of numbers. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:

Which have the same kind of error?

I and II I and III II and III I, II, and III

Method would work for all whole numbers

Method would NOT work for all whole numbers I'm not sure

2. Student A 3. Student B 4. Student C


- 107 -

6. As Mr. Callahan was reviewing his students' work from the day's lesson on multiplication, he noticed that Todd had invented an algorithm that was different from the one taught in class. Todd's work looked like this:

What is Todd doing here?

Todd is regrouping ("carrying") tens and ones, but his work does not record the regrouping. Todd is using the traditional multiplication algorithm, but working from left to right. Todd has developed a method for keeping track of place value in the answer that is different from the conventional algorithm. Todd is not doing anything systematic. He just got lucky- what he has done here will not work in most cases.

When learning about multi-digit subtraction, Mrs. Jackson's class encounters the following problem:

A student answers "209" by completing the problem on the board as shown above. Mrs. Jackson wants to ensure that other students understand why the procedure works, and asks another student to explain this answer. Which explanation(s) should she feel comfortable accepting as evidence this student understands why the procedure works?

Yes No I'm not sure

7. You can't take 7 from 6, so you cross out the 0 and make it a 9, and the 6 becomes a 16, and then cross out the 3 and it becomes a 2. Then I take away. 16 take away 9 is 7, 9 take away 9 is 0, and you just have 2.

8. I regrouped 306 to be 2 hundreds, 9 tens, and 16 ones. That's the same amount as 306. Then I could do the problem. I took away 7 from 16 and took away 9 from 9.

9. I borrowed from the tens place to make the 6 a 16. But since it was a 0, I had to borrow again from the hundreds place, making the 3 a 2. Then I just subtracted.

- 108 -

10. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students around particular difficulties that they are having with subtracting from large whole numbers. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:


I and II I and III II and III I, II, and III

11. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students around particular difficulties that they are having with multiplication computation. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:


I and II II and III I and III I, II, and III I'm not sure


- 109 -

12. Mrs. Carpenter asked her students to add one to 7,399. Which of the following answers is LEAST likely for a student to give?

7300 7400 7500 8400 I'm not sure

13. As Mrs. Boyle was teaching subtraction one day, she noticed a few students subtracted in the following way:

What were these students most likely doing?

The students "subtracted up," by taking 3 away from 8, and then tried to compensate for this mistake. The students compensated by subtracting 30 from 63, then dealt with the 8 and the 3 in a second step. The students made a mistake with the standard procedure, crossing out the 2 rather than the 6. The students added 10 to both 63 and 28, then subtracted. I'm not sure

Date post:	04-Aug-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

K-9 Math Teachers Implementation of BreakThrough...

Documents