Heather Gramberg Carmody

“For many students studying mathematics in school, the beliefs or feelings that they carry away about the subject are at least as important as the knowledge they learn ofthe subject”

(Philipp, 2007, p. 257)

The challenge for STEM education is not simply one of proficiency but one of interest as well (President’s Council of Advisors on Science and Technology, 2010a)

Condition of STEM 2013 (ACT, 2014) showed that almost half of graduating high school seniors in 2013 expressed an interest in STEM but many of those students with an interest have no intention to pursue careers in those fields

81% of 4th graders have positive attitudes towards mathematics, but only 35% of 8th graders have those same responses (U.S. Department of Education, 2003)

“Troubling sign” of high-achieving students moving away from STEM (PCAST, 2010b)

Motivation & affect among key factors in preventing talent loss (Hong & Milgram, 2008)

Krutetskii (1976): formalized perceptions, rapid and broad generalizations, flexibility in thinking, drive for clarity and rationality of solutions

NCTM Task Force (1995): function of ability, motivation, belief and experience or opportunity

Sriraman (2008): ability to abstract, ability to master logical thinking, capacity for fast-paced learning among other characteristics

Over 40 years of work with gifted mathematics students (Benbow, 2005)

Students have exceptional reasoning and can quickly learn material at young ages (Brody & Stanley, 2005)

Above grade-level testing and flexible programing for younger students to enroll in advanced courses (Brody & Stanley, 2005; Assouline & Lupkowski-Shoplik, 2011)

Acknowledged by foundational theorists: Vygotsky (Mahn and John-Steiner, 2002 ; Wertsch, 1985) Bloom (Krathwohl, Bloom, & Masia, 1964) Simon and Binet (Binet and Simon, 1916)

In mathematics publications NCTM 2011 yearbook Decades of research (Aiken, 1976; Leder & Forgasz, 2006; McLeod, 1989) National reports (National Mathematics Advisory Panel, 2008; NRC, 2001)

Included in gifted education: Renzulli (2005) Operation Houndstooth Sternberg (2005) Wisdom – includes intelligence, motivation, and affect

Variety of definitions, can encompass temperament, moods, beliefs, emotions, attitudes, interests, values and more (Anderson & Bourke, 2000; Corno et al., 2002; DeBellis & Goldin, 2006)

McLeod’s (1994) definition: emotions, attitudes and beliefs“increasing affective involvement, decreasing cognitive

involvement, increasing intensity, and decreasing stability” (McLeod, 1989, p. 246)

Overlap between the two constructs and need for a more integrated model (Linnenbrink, 2006)

Empirical study showed close relationships between affect and motivation (Schweinle, Meyer & Turner, 2006)

Gifted students - gender differences in motivation, confidence and interest (Preckel et al., 2008, TerwillingerTitus, 1995)

Students with a mastery goal orientation tend to:have strong levels of self-efficacy (Middleton et al., 2004; Patrick

et al., 2006)have larger gains in mathematics (Keys et al., 2012) Use positive academic coping strategies (Friedel et al., 2007)

Students with a performance-avoidance goal tend to avoid seeking help, use less effective strategies and deal with negative emotions (Patrick et al., 2006)

Self-efficacy and math performance or achievement have a reciprocal relationship, higher levels in one linked to higher levels of the other (Pajares & Graham, 1999; Patrick et al., 2006; Usher, 2009)

High levels of self-efficacy connects to the selection of more difficult math and science courses (Betz & Hackett, 1983; Hackett & Betz, 1989; Stevens et al., 2004)

Strong, often negative emotions from high school students (Hoyles, 1982)

Strong negative emotions attached to mathematics, but positive attitudes too (Stodolsky et al., 1991)

Significant presence of math anxiety (Betz, 1978)Correlation between high levels of anxiety and a

decrease in the selection of advanced mathematics courses (Hembree, 1990; Reyes, 1984)

Early studies found significant differences due to gender (Fennema & Sherman, 1976)

Collegiate female students more likely than male students to report higher levels of math anxiety (Betz, 1978) and lower self-efficacy (Betz & Hackett, 1983)

High school female students had higher levels of anxiety but not lower performance (Hembree, 1990)

No significant differences (Hackett & Betz, 1989; Shoenfeld, 1989)

Pajares and Graham (1999) found no gender differences in self-efficacy beliefs

Few qualitative differences in boys’ and girls’ comments about their own competence (Usher, 2009)

Self-concept beliefs results mixed when compared to peersGifted students’ perceptions decline from ages 9 to 13 (Shi, Li, &

Zhang, 2008)Gifted students more confident and higher self-concept (Chan,

1996; McCoach & Siegle, 2000; Pajares, 1996)Underachieving gifted students have low self-efficacy and high

anxiety (Reis & McCoach, 2000)Links between self-efficacy, mastery goal orientation and

academic success (Chan, 2008; Malpass et al., 1999)Gender differences favoring boys (Junge & Dretzke, 1995;

Preckel et al., 2008; Terwilliger & Titus, 1995)

Rich description of the ways adolescents in a private school setting characterize their own affect and motivation

Examination of possible group differences in affect and motivation responses between genders, and levels of acceleration

How do adolescents, those in grade level classes and those who are accelerated, at a college preparatory school describe their own motivation and affect in mathematics?

What differences exist between the way girls and boys characterize their motivation and affect in mathematics?

What differences exist in the characterization of motivation from students on grade level, those accelerated one year, and those accelerated two or more years? If differences exist are they similar between boys and girls on grade level, those accelerated one year, and those accelerated two or more years?

Qualitative-dominant concurrent design, quan+QUAL(Johnson & Christensen, 2008)

Qualitative: phenomenological goal to described the lived experience of participants (van Manen, 1990)

Quantitative: cross-sectional design with descriptive statistics, MANOVA, and DFA in analysis

Purposive sampling from adolescents in a private schoolAll students grades six through twelve invited to

participate (approximately 630 students)Grade Total Number of

StudentsNumber of

Participating Students

Response Rate

6th 65 56 86%7th 73 39 53%8th 77 66 86%9th 98 71 72%

10th 101 65 64%11th 103 56 55%12th 108 41 38%

Total sample size 625 394 63%

Eight open-ended questions about students’ emotions, attitudes and motivation in mathematics

Questions one through four from study of honors students’ perceptions of high school (Siegle, Rubenstein, & Mitchell, 2014)

Questions five through seven from study of middle students self-efficacy in mathematics (Usher, 2009)

Final question unique to this study

Fennema-Sherman Mathematics Attitude Scales

Patterns of Adaptive Learning Scales

Scale Number of Items Internal Consistency Reliability (this study)

Usefulness of Mathematics 12 0.94

Mathematics Anxiety 12 0.95

Scale Number of Items Internal Consistency Reliability (this study)

Mastery Goal Orientation 5 0.86

Performance – Approach 5 0.83

Performance – Avoid 4 0.86

Academic Efficacy 5 0.87

Permission obtained from administration

All parents of students grades 6-12 received study description and notice of passive consent

Participating students anonymously completed surveys during math class in May 2014 through QualtricsMiddle school is 1:1 with iPadsHigh school is Bring Your Own Device (BYOD) with supplemental

carts as well

Open then axial coding (Strauss & Corbin, 1998) to look for emerging themes and trends

Use of Nvivo 10 and 11 throughout analysis

Reflective journaling, memos, and other techniques to record analysis

Meeting with “critical friend” to check bias and help with code clarification

IV’s: gender, level of acceleration, grade as co-variate

DV’s: usefulness of mathematics, anxiety, mastery orientation, performance approach, performance avoid, self-efficacy

Descriptive statistics and correlations

Multivariate analysis of variance (MANOVA) for correlated data to guard against Type I Error

Discriminant function analysis (DFA) to see which dependent variables best predicted group membership

128 comments about value of resilience, 18 with lack of “Teachers should start to teach the students that failing

isn't the worst thing to happen to them and also how to get back up if one does fail” (8th grade girl, accelerated one year, id#45)

206 comments suggesting high challenge leads to high motivation; 96 comments suggesting the opposite

“Math generally comes easier to me than some others and I want to find where my breaking point is and I want to continue doing harder work” (9th grade girl, accelerated two years, id#200)

171 comments about time outside of class“I asked her a question, she explained it, and I

understood it. It became easier for me, then, to go and see her whenever I had issues” (11th grade girl, not accelerated, id#148).

120 comments of support (contrasted with 40 negative teacher descriptions)

“[my teacher] told me that no matter how much I struggled in math he was never going to give up on me….I never gave up and neither did he.” (10th grade boy, not accelerated, id#284)

393 comments relating to future use of mathematics

“Mathematics class is valuable to me because it allows me to better understand the world around me.” (11th grade boy, accelerated 2 years, id#10)

250 comments linking enjoyment and motivation

“[Enjoyment of a course] is a major, possibly the most important, aspect in affecting my motivation.” (11th grade boy, accelerated 2 years, id#10)

Variable M SD

Usefulness 4.00 .77

Anxiety 2.70 .93

Mastery Goal Orientation 3.45 .82

Performance Approach Goal Orientation 3.56 .85

Performance Avoid Goal Orientation 2.84 .87

Self-efficacy 3.69 .81

Belief that success was possible: evident in qualitative comments and self-efficacy as the second highest scale for nearly all subgroups.

Usefulness of mathematics was common reason for enjoyment and motivation. Usefulness scale was the highest rated for all subgroups.

Multiple goal orientations possible. Qualitative suggest mastery orientation yet nearly all subgroups had higher performance approach orientation in quantitative.

More frequent comments from girls (38% vs. 33%) about anxiety and a different progression of anxiety described during tests.

Girls described seeking help from others more frequently than boys

Male students (n = 183)

Female students (n = 211)

M SD M SD

Usefulness 4.09 .80 3.93 .71

Anxiety 2.51 .90 2.83 .94

Mastery Orientation 3.56 .84 3.36 .81

Performance Approach Orientation

3.63 .85 3.50 .87

Performance Avoid Orientation 2.82 .88 2.85 .88

Self-efficacy 3.88 .77 3.57 .80

There were not statistically significant main effects for gender (Wilks’ λ = .975, 𝐹𝐹6,382= 1.631, 𝑝𝑝 = .137).

The interaction effect of gender and acceleration was not statistically significant (Wilks’ λ = .975, 𝐹𝐹12,764= .812, 𝑝𝑝 = .638, partial 𝜂𝜂2 = .013).

The quantitative results show no difference between responses of male and female students

Both groups had high self-efficacy scores. Girls seeking help from others, does not seem to mean a lack of self-efficacy.

Girls wrote more often of anxiety in the open-ended questions, and yet that scale was the lowest of the six for them.

The results from the MANOVA analysis of the quantitative data support the conclusion that the groups had similar experiences.

There was a significant association between gender and level of acceleration 𝜒𝜒2 2 = 23.15, 𝑝𝑝 < .001.

There are more boys than expected at the highest level of acceleration and fewer girls.

Students on grade level wrote more frequently of teachers in all subtopics.

Most accelerated students wrote of enjoying mathematics for its own sake

Different word choice for negative emotions: Grade level students used stronger words like “angry” “mad”, “hate” but accelerated students wrote more of “frustration” or a sense of obligation

Students on grade level

(n = 134)

Students accelerated

one year(n= 200)

Students accelerated two or more

years(n = 60)

M SD M SD M SDUsefulness 3.80 .81 4.04 .73 4.37 .55Anxiety 2.84 .93 2.75 .94 2.10 .74Mastery Orientation 3.38 .83 3.43 .82 3.72 .80Performance Approach 3.49 .92 3.61 .84 3.54 .78Performance Avoid 2.96 .90 2.82 .88 2.60 .77Self-efficacy 3.50 .85 3.77 .79 4.05 .60

Main effects for acceleration (Wilks’ λ = .885, 𝐹𝐹12,764= 4.014, 𝑝𝑝 < .0001, partial 𝜂𝜂2 = .059).

Effect sizes indicated that 5.9% of the total variation was accounted for by acceleration.

Usefulness, anxiety, and mastery orientation were important variables in distinguishing groups

Differences in emotions. Grade level students used stronger word choice for negative

emotions. Moving up in acceleration the students had lower levels of anxiety. This

scale was also significant in distinguishing between the three groups.

Role of challenge in learning. Grade level students responded positively to high challenge less often

than their peers. Mastery orientation in the DFA was one of the factors that

distinguished between groups. As the level of acceleration increased so did the mean scores for mastery orientation.

Qualitative comments considered with the quantitative data also support the idea of multiple orientations.

Extension of other research (Chan, 2008; Chouinard & Roy, 2008; Kaplan et al., 2002; Keys et al., 2012)

Perhaps the culture at Meridian Academy was competitive, not only for admittance, but also once enrolled.

Most significant difference between boys and girls was their placement in accelerated classes an unfortunate confirmation of existing research.

Differences from Preckel et al. (2008) with significant gender differences

Similar results to Pajares and Graham (1999) with non-significant gender effects and lower self-efficacy in general education students

Later study at same school showed some differences on engagement and motivation at different acceleration levels with particular mathematical task types (Pumphrey, 2018)

Teachers: Even when students were not asked about teachers, their importance and influence of is forefront on the minds of students. It is a strong indication for teachers that our role is meaningful and important.

Teachers: Consideration for practioner research.

Schools: Need attention to the lack of balance of boys and girls at the highest levels of acceleration

Researchers: Need for further research on placement balance at highest levels of acceleration. Supports multiple goal orientations, especially when both types of data are analyzed.

Extraneous variables such as: teacher factors, courses, and socioeconomic factors limits the conclusions

Timing of the data collection. Survey just before semester examsResponse rate for 12th grade students

Questions may not have been appropriate for 6th grade students

Replication research at Meridian AcademyOther causes for student variation: ethnicity, current

mathematics course, current teacher or socio-economic status

Intentional focus on teachersPossible extension of Academic Orientation ModelOther school settings:

Local private schoolsLocal public schools

In what ways do you see these ideas in your context?

What questions do you have about how affect and motivation may be at work in your school?

Contact info:Heather Carmody, Park Tudor Schoolhcarmody@parktudor.org