1. Dispositions, Motivations and Aspirations
Investigators:
Julian Williams (PI), Laura Black, Pauline Davis, Birgit Pepin and Geoff Wake
Research Associates: Valerie Farnsworth, Paul Hernandez-Martinez, Maria Pampaka
Associate Research Fellow: Diane Harris
Associate Research Students: Kamila Jooganah and Irene Kleanthous
Research Statistician: Graeme Hutcheson
Administrator: Tim Millar

2. The Study of Mathematics
England allows participation in mathematics education by post-16 students to be optional which has the following consequences:
the vast majority of students elect to opt out of studying mathematics
approximately 50% of students in England might be considered capable of studying mathematics beyond the age of 16
just over 10% of students in England choose to take the first year of a course in A-level mathematics in preparation for study in HE.
participation in maths post-16 is compulsory elsewhere in the world
each year there are around 300,000 England students who disengage from mathematics at 16.
All of which brings issues of choice-making and identity formation sharply into focus.

3. Transition
The transition period for students seems crucial as they move into advanced study of mathematics.
Deeper understanding of mathematics seems particularly important as we seek to develop a ‘workforce’ that is prepared to ensure effective participation in the high-tech world of the 21st century.
Achieving this outcome requires high standards of student performance but before this can be achieved, students must be disposed towards and engaged with mathematics.
Point 2 – important in the UK and also Europe

4. Methodology Institution (college) Classroom culture Aspirations
Longitudinal series of interviews Learning outcome measures Surveys
Case study
Lesson observations & videos
Teacher survey instrument
Interviews
Institution
(college)
Classroom culture
Teacher
Student
Background & experiences
Aspirations
Programme
(1st year of A Level)

5. The two longitudinal studies

6. Measuring Dispositions during the two projects
Disposition to go into HE
Disposition to finish chosen course in HE
Disposition to study more mathematics (Maths disposition)
Background Variables: Course, Gender, Ethnicity, Language of first choice, EMA, LPN, First generation into HE
Background Variables:
Course (Math, Science/Engineering, Other) Gender, Ethnicity, Language of first choice, Country of origin, First generation into HE

7. Aims of the two Projects
TLRP: To understand how cultures of learning and teaching can support learners in ways that help them widen and extend participation in mathematically demanding courses in Further and Higher Education (F&HE)
AS Mathematics Vs AS Use of Mathematics
TransMaths: To understand how 6th Form and Further Education (pre-university) students can acquire a mathematical disposition and identity that supports their engagement with mathematics in 6fFE and in Higher Education (HE)
Focus on Mathematically demanding courses in HE (‘control’ : non mathematically demanding, e.g. Medicine and Education)
In other words, the project sample included many students who had what might be considered low grades for progression to advanced courses:
typically a grade C or B as opposed to a grade A or A* in the qualification that marks the end of compulsory study of mathematics i.e. GCSE.

8. The TLRP Findings
Students’ dispositions to go into HE overall are increasing during their AS year and this continues until the middle of their A2s.
Students’ dispositions to go into HE overall are increasing during their AS year and this continues until the
middle of their A2s.
Logits
"Logit" is a contraction of "Log-Odds Unit" (pronounced "low-jit"). It is no more obscure a measurement unit of an underlying and invisible variable than an "Ampère" is of invisible electric current. The essential ingredient of Amps and logits is that they be additive.
Real apples are not additive. One Apple + One Apple = Two Apples. But Two Apples are twice as much as One Apple only when the Two Apples are perfectly identical. Real apples are not perfectly identical. When we say One Amp + One Amp = Two Amps, we say "all Amps are identical," wherever they appear on the Ammeter. Logits form an equal interval linear scale, just like Amps. When any pair of logit measurements have been made with respect to the same origin on the same scale, the difference between them is obtained merely by subtraction and is also in Logits. This is how Amps work.
Like an Ammeter, the logit scale is unaffected by variations in the distribution of measures that have been previously made, or by which items (resistances) may have been used to construct and calibrate the scale. The logit scale can be made entirely independent of the particular group of items that happen to be included in a test this time, or the particular samplings of persons that happen to have been used to calibrate these items.
We construct a logit scale in the same way that we construct an Amp scale. We deduce a theory that produces equal interval, linear measures and derive a method for applying that theory. In the case of qualitative ordered observations (right/wrong, present/absent, none/some/all), the necessary and sufficient theory is the Rasch model, and the method of application is numerous administrations of similar agents (test items) to relevant objects (persons).
The theory is
This is a "linear" model because all elements can be represented as fixed positions along one straight line. In games of chance, the (Probability of Success)/(Probability of Failure) is called the "odds of success".
"Loge[(Probability of Success)/(Probability of Failure)]" is called log-odds. The units of measurement constructed by this theory are called "log-odds units" or "logits".
Not all numbers represent equal interval scales, no matter how equally spaced their values appear. Rank orders are counted with equally spacing, but rank order numbers do not specify whether the distance between 1 and 2 is equal, greater or less than the distance between 2 and 3.
How do we know that logits are equal interval? By observing that when data fit the theory, the specification that a one logit positive difference between any person and any item anywhere on the scale always has the same stochastic consequence.
When data fit, the interval specification of the theory is realized in the data. For these data, the interval scale is established. The implication is that for similar data the scale will continue. This implication, however, is always tested when fit is analyzed for each new application, just as wise use of an Ammeter requires that it too be continually checked. When in the new application, the fit criteria are met, then the linear scale continues - the logit unit is maintained.
Benjamin D. Wright
Logits? Wright BD. … Rasch Measurement Transactions, 1993, 1993, 7:2 p.288

9. Students’ dispositions to study more mathematics

10. Students’ dispositions to study more mathematics - TLRP
Students’ dispositions to go study more mathematics decrease over time.
AS Trad students reported statistically significant higher disposition at all times. Even though the drop is consistent for the two groups during the AS year, what is interesting is the UoM students seem to report higher disposition during their A2 year, and the difference between the two groups is not statistically significant at this stage. (Note: we should bear in mind the missing data at this stage and the possible implications for interpretation of the results).
Gender differences are not statistically significant at any given DP. However the interesting point which could be made from the figure below (right) is the steeper drop of female students mathematics dispositions, when compared to the male ones (for whom the change is not very noticeable between DP2 and DP3).

11. The TransMaths Findings
The disposition of students to finish their chosen course does not change very much over time: there is a very weak decreasing trend but this varies when different subgroups of students are compared.
As far as students’ disposition to study more mathematics in the future, it seems that the trend is again declining, however not as steeply as found in the TLRP project.
The graphs are on the next slide

12. Students’ dispositions to complete their HE course
TransMaths measure 1: Disposition to finish chosen course in HE
The disposition of students to finish their chosen course is not changing very much over time: there is a very weak decreasing trend but this varies when different subgroups of students are compared.
There are some differences based on students’ attended course. Maths and Science and engineering students (i.e. STEM) are scoring overall lower on this measure compared to students of ‘other’ courses (mainly social sciences and medicine). The difference however is getting smaller when students are on their second year.
Another point to be made is the slight decrease of the disposition of ‘other’ students and the slight increase of the maths students to complete their chosen course, over time.
The figure on the right hand side shows how female students have at all times, higher disposition to complete their chosen course than male students. We should note however that maybe the two figures show confounding relationships, since the mathematics and engineering courses are dominated by male students.

13. Students’ dispositions to study more mathematics
TransMaths measure 2: Disposition to study more mathematics
As far as students’ disposition to study more mathematics in the future, it seems that the trend is again declining, however not as steep as found in our previous study.
As shown, students of social sciences (and medicine) overall score significantly lower than STEM students in this measure. However, their scores are stable. Maths and engineering students overall score much higher: during DP1 maths students score significantly higher than engineering students as well. However, the disposition of the engineering (and science) students remains stable over time, whereas there is a significant drop in maths’ students disposition during their first year at university (as shown between DP1 and DP2).
Gender differences
Overall, male students score higher at all data points (and the difference is statistically significant). There is also a declining trend, observed for the female students.

14. Effect of pedagogy on math dispositions - TLRP
Positive effect: Math Disposition at DP1, ‘Mathematical demand of other subjects’
Negative effect: pedagogy

15. Dispositions, motivations, aspirations
Learners’ dispositions to study mathematics are in steady decline through the two year period of ‘advanced’ study, but this decline is exacerbated by ‘transmissionist pedagogy’.
Different classroom experiences relate to distinct mathematical identities.
A leading identity, e.g. becoming an engineer, can be important in shaping a student's motives for mathematical activity.
Pampaka et al., forthcoming
Williams , J. , L. Black , P. Hernandez-Martinez , P. Davis , M. Pampaka , and G. Wake Repertoires of aspiration, narratives of identity, and cultural models of mathematics in practice . In Social interactions in multicultural settings , M. Cesar and K. Kumpulainen , 39 – 69 . Rotterdam : Sense.
Black, L., Davis, P., Hernandez-Martinez, P., Pampaka, M., Wake, G. and Williams, J Developing a ‘leading identity’: The relationship between students’ mathematical identities and their career and higher education aspirations. Educational Studies in Mathematics , 73(1): 55–72.

16. Bringing it all together….
(e.g. Pedagogy, m dispositions, transitional experience etc)
Pearson Correlations
UK results
Pedagogy at Uni
Pre-University Pedagogy
Math Support at University (DP5)
Non significant
-0.19 (p<0.05)
Transitional Feelings (DP5)
-0.20 (p<0.001)
Disposition to Finish Course_DP5
-0.12 (p<0.05)
Math confidence (DP5)
-0.17 (p<0.001)
MHE disposition (DP5)
-0.19 (p<0.001)
Those used to Transmissionist pedagogy at school had a somewhat lower perception of maths support in transition, less Maths confidence and lower disposition for further maths study.
Transmissionism at uni is also associated with some negative perceptions of transition and disposition for further study...