Negotiating the transition from high school to undergraduate

Negotiating the transition from
high school to undergraduate
mathematics: some reflections
by Zimbabwean students
Zakaria Ndemo
Bindura University of Science
Description of problem
• Students face challenges in learning
undergraduate mathematics.
• Discrepancies characterizing high to
university mathematics has been referred to
as the “shock” of the “new” inevitable (Clark
& Lovric, 2009)…...
• Transitional phase is characterized by hurdles:
change in social environment, mode of
delivery and multi-quantified symbolism
Description of problem
• Transitional hurdles has resulted in low uptake
of undergraduate mathematics.
• Students fail to grasp fundamental distinction
between analytic and synthetic
• Current study sought to unravel the nature of
transitional challenges among Zimbabwean
undergraduate mathematics students.
Theoretical considerations
Theoretical constructs explicating the nature of
transitional difficulties include:
• Notions of set-before and met-befores (Tall,
2004), Ausbel’s theory of learning
• Synthetic and analytic definitions (Selden &
Selden, 2003).
• Anthropological idea of the rite-of –passage
Theoretical considerations
• Set-befores are genetic structures we are born, 3
basic set-befores (recognition, language,
• Pertinent to the study was language dimension in
light of the multi-quantified symbolism of
undergraduate mathematics learning
• A met-before is a current mental facility shaped
by specific prior experience of the learner. There
are negative and positive met-befores
• Ausbel’s theory of learning
Synthetic and analytic definitions
• Synthetic definitions are everyday contextual
meanings e.g. accumulation point , process of
gathering together and increasing in amount
• Analytic definition- formal axiomatic meaning
that is complete e.g., accumulation point, or a
number  is an accumulation point of a subset A
of real numbers if given a real  > 0;
( should be conceived of as a small radius),
the interval ( – ,  + ) contains at least one
point of A distinct from 
Rite-of-passage framework
• separation phase, refers to period when students
are in high school and preparing for university.
• Liminal phase, is the time lapse between end of
high school and beginning of university
mathematics as well as time in between.
• Incorporation phase, covers first year university
study duration – a strong synthetic model
interferes with assimilation of new information
• Content taught at incorporation phase not
compatible with robust synthetic model
Beinstein’s theory of pedagogical
• Classification: refers to the strength of the
boundaries between discourses and group of
actors (Bernstein in Jablonka et al.,2012, p. 2)
• Strict separation between largely informal
intuitive discourse in high school and broadly
axiomatic discourse of the incorporation
• Undergraduate mathematics is a more
classified knowledge system.
Beinstein’s theory of pedagogical
• recognition rule refers to the need for students to
understand principles for distinguishing between
high school and undergraduate mathematics
learning contexts.
In this regard, students need to recognize the
speciality of the discourse at tertiary level, as a
necessary condition for their capacity at producing
what counts as legitimate mathematics in the
undergraduate learning contexts.
Research questions
• What are Zimbabwean undergraduate students’
perceptions of the nature of university
• What are the sources of difficulties faced by the
students during the transition from school to
university mathematics?
• To what extent has high school mathematics
prepared students for undergraduate
• The study involved mathematics majors and
mathematics education students from two
universities in Zimbabwe.
• Sampling was purposive: 29 CPD mathematics
education students, 40 mathematics majors:
16 first and 24 second year students.
• Qualitative data were gathered that consisted
of students’ written responses
Data analysis
• Students’ written responses were mapped onto the
theoretical constructs discussed (Varghese, 2009).
• However we do not claim that there was a one to one
correspondence between the constructs and students
expressions of their perceptions about the nature of
university mathematics and the kinds of transitional
• In recognition of the possibility of the emergency of
categories not covered by literature, summative
content analysis technique (Berg, 2009).
• Manifest elements blended with latent meanings
• The dominant student perception about the nature of
university mathematics is that it involves logical
reasoning (9 out of 36) for university A and (6 out of
16) for university B.
• It is worrisome to note that a significant of
informants ( 8 out of 36) from University A and (4 out
of 16) from university B and expressed that
mathematics learning involves doing calculations, a
view consistent with high school mathematics (Tall,
2008, p.5). This is indicative of a robust synthetic
model, negative met-befores which make negotiation
of transitional phase difficult
• “Mathematics requires reasoning and logic
while with other subjects its just
• “ … a lot of calculations are involved”
• “Mathematics involve calculations and there
exact solutions to a problem”
• Content related sources of difficulties were dominant among
participants from both universities. The results are very much
consistent with our findings from research question 1 where most
students perceived mathematics as a discipline involving
calculations, a view not compatible with university mathematics
(Tall, 2008).
• “… failing to understand limits.”
• “… the use of many variables”
• Once again pointing to a robust synthetic model where negative
met-befores from informal intuitive discourse interferes with
axiomatic formalism dominant at undergraduate level (poor
classification and recognition skills in Beinstein’s terms)
• Research question 3 was addressed by analyzing students’
responses to the item: Do you think school mathematics prepared
you well for university mathematics? Explain your answer
• most students expressed that high school mathematics has a link
with undergraduate mathematics.
• Very few students suggested that high school and undergraduate
mathematics are disjointed knowledge systems, pointing to weak
knowledge classification skills.
• Such students probably had not developed recognition rules to
distinguish between the highly intuitive school mathematics
involving numerous calculations and the highly specialized
axiomatic formalism in tertiary mathematic
Most of the material found in the first courses Linear maths 1 and
Calculus 1 are covered at A level except that some of the methods were
were being discarded in Linear Maths”
• “Most complex concepts we meet at th is level actually build from school
Responses classified as partial include:
• “for other courses school mathematics is used at the university but for
other courses completely new courses are learnt.”
Concluding remarks and looking ahead
Overall, students’ responses revealed:
• Prevalence of negative met-befores and
dominance of synthetic model of definitions.
• Inconsistencies lack classification and
recognition skills needed to negotiate the
transitional phase
• Delayed gratification
• Low self-attribution factor among

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