Threshold Concepts in QuantsDr Richard DiamondEnquiry Presentation

18th May 2011

Postgraduate Certificate in Learning and Teaching in Higher Education

University College London

Enquiry Aims and Activities

• Disciplinary application of the threshold concepts approach to analyse quants (statistics and finance)

• Identification of threshold concepts and effective learning pathways to navigate the webs of concepts

• Analysis of VLE data (grades and time spent)

• Analysis of student feedback (CATs and informal)

• Reflection on and improvement of techniques for teaching, engagement and support

Threshold Concepts Approach Literature Review

Key References

Cousin (2006), Davies and Mangan (2007), Meyer and Land (2003, 2005), Land et al. (2005) and Meyer et al. (2008)

What Is Threshold Concepts Approach About?

• Recognising and focusing on central concepts in contemporary ‘stuffed’ curriculum

• Linking thinking and practicing through re-working of prior naive or premature understandings (common sense vs. science)

• Dealing with the troublesomeness of knowledge that stems from (a) opening up concepts deemed understood and (b) uncertainty about future understanding

Surface Learning Issues in Economics

• Students acquire formal knowledge of a discipline but seem unable to use it when making sense of experience in work or their everyday lives

• Students struggle with underpinning theory and resort to verbatim learning of isolated aspects of the subject that they are unable to juxtapose

Davies and Mangan (2007:18)

Initial Findings from Assessment Data

Design and Sampling

• Three subsequently delivered modules of 14, 51 and 37 participants• Curriculum made ‘simpler’ from one run to next• Same textbook (Barrow 2009) and MathXL questions• Scores had a range of 1 to 100, presenting categorial and finely ranked data

Grades DistributionHighly discretised data bins give approximate Normal

Where is the mean?10% bins provide more difference between students

Distribution is Multi-ModalFitting to the Normal still leaves a problem with bins

Surface Learners

Deep Learners

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100

More

Freq

uenc

y

Bin

Histogram

Distribution is Bi-ModalReflects a transition through troublesome knowledge

Surface Learners

Deep Learners

Amount of time spent indicates problem, not result

Group One28 Sections

Group Two20 sections

Initial Findings from Significant Correlations I

• Total time spent by students had no significant correlation with their results. Time spent per section had negative correlation with the section’s score

• In order to score high overall, the students needed to score high for all chapters (grading emphasised particular chapters)

Initial Findings from Significant Correlations II

• Scores in Chapters 1-3 have significant correlation with subsequent results (procedural thresholds being learned, especially in Chapter I)

• Normal Distribution - Hypothesis Testing - Regression Modelling showed the strong triangle relationship. They are threshold concepts and procedures of Statistics

Threshold Concepts for Knowledge Integration

Example of a conceptual mapNotice multiple dimensions that link concepts

Webs of Threshold Concepts

• Two kinds of maps: local and grand

• Learners should have some basic and threshold concept knowledge before being guided through the mapping

• Maps are drawn in front of and together with learners and questions get answered

Technique Improvement:What has been tried in class

Visualisaiton

• Visual simulations for complicated matters. Examples: Type I and Type II errors in justice system. Thousands of weight combinations to show the Efficient Frontier of a financial portfolio

• Sketches of probability distributions. Example: finding any area under the Normal Distribution curve

Improvisation

• Improvising with different meaningful data sets on the go: downloading data from the Internet, applying a statistical technique then soliciting interpretations and acknowledging or providing the most sensible.

• Example: sorting hamburgers and coffee drinks in percentiles for the amount of fat

Experiments with Assessment

• Giving an integrating document such as a final exam early, during the first workshop

• Exam design: calculation question plus several interpretation questions (could be multiple choice)

• Questions on interpretation aim to check the deeper understanding

Exam Design Pendulum in Quantitative Disciplines

• From solely multiple choice questions to sections of several related questions that require calculation and interpretation

• This was the case in how I developed my own exams (over three years) and exams done by colleagues elsewhere

This only scratched a surface of insight which threshold

concepts approach can offer

Technical Slides

Types of Conceptual ChangeDiscipline: Business Statistics

Type of Change Type of Transformation Examples in Business Statistics

Basic

The concepts that have relevance to everyday

experience.

Calculation skills and ability to identify a

statistical measure

Central tendency and dispersion

Mean vs. median

Standard Deviation

Probability (sources, ways of definition)

Discipline

Acquisition of theoretical perspective--ability to

see the world as a statistician.

Basic concepts are related to the outside world

using discipline threshold concepts (e.g., mean

and standard deviation can be plugged in to

probability distribution in order to describe a

specific situation).

Probability Distribution

Continuous vs. discrete

Hypothesis Testing. Significance

Correlation

Regression

Time Series Data

Index

Procedural

(Same description as in quantitative finance, a

more intense area)

An understanding and mastery of the subject’s

modelling procedures that enable the

construction of discipline-specific models,

arguments and ways of practising.

“Mathematical Transformation” - “Magic of

working through a proof”

Essential mathematical notation (summation with indexes)

Operation with equations. Polynomials

Operations with percentages

Meaning of greek letters

Normal Distribution Tables

Ability to visualise a distribution (use a sketch)

(there could well be other, lower level procedural threshold not covered)