Mahmoud Abdulwahed

Mathematics Teaching Interventions For at-Risk non-Specialist Students,

a Cybernetics Point of View

• Analysis of why they are at risk?– Mathematics Perception of Non-Specialist– Math Weakness Factors– Cognitive Psychology Background– Pedagogical Background

• Designing Interventions:– Mathematical Model of a Pedagogical Model– Engineering Approach in Lecturing Design– Analysis and Design Implications

• Implementation:– Interventions Implementation– Potential Technology Assisted Intervention Tools

• Evaluating Interventions

• Conclusions

Overview

Mathematics Perception of Non-Specialist

• Math is Boring.• Math is Too Difficult.• Don’t Enjoy/Like it.• Not Useful in Life!.• Prefer Other Courses• Not Needed for Future Degree

or Career.

A recent large scale study on A-Level students

questioning why they don't choose continuing math study,

many reasons where revealed (Brown et al 2008):

Math Weakness Factors

• Cognitive Reasons.• Pedagogical Reasons.• Affective Reasons.• Cultural Reasons.

Hence, for better design of teaching interventions, those factors must be investigated and taken into consideration.

Cognitive Psychology Background

The information processing model of Human Memory (Slavin 2006)

• Information should be ATTENDED to be transferred from the senses to the short term memory, otherwise they are forgotten (sensory register stores for 2 seconds only).

• Short term memory capacity is limited to five to nine bits at once, storing time is about 30 seconds. But, short term memory can retrieve information from the long term memory.

• The probability of transferring the information from the short term memory to the long term memory increases with the time they last in the former, hence, REPITION is important for achieving this goal.

• Implications to educational design:– Stimulating the students towards the taught subject.– Avoid cognitive overload when giving too much information at once.– Increasing the exercises and the reflection activities to make the students thinkmore and more about the learned material leading to storing data for long term.

External Stimulus

External Stimulus

Sensory Register

Sensory Register

Initial Processing

Initial Processing

Short Term Memory

Short Term Memory

Long Term Memory

Long Term Memory

Repetition

RetrievalRehearsal and Coding

Information

Forgotten Forgotten

Short Term Memory

Cognitive Psychology Background

• Procedural memory stores algorithmic procedures.• Episodic memory stores visual, auditorial, and kinetic

information.• Semantic memory stores concepts, facts, and general

information in a connected network of ideas and relationships called schemata.

• Implications to educational design:– Activating the episodic memory through the use of

visual and aduiotorial learning material, simulations, and teamwork.

– Activate reflection of information taught couple of weeks before to fix it again in the long term memory.

– Linking taught information with previous knowledge of the student and make it relevant.

External Stimulus

External Stimulus Sensory

Register

Sensory Register Initial

Processing

Initial Processing Short Term

Memory

Short Term Memory

Long Term Memory

Long Term Memory

Forgotten Forgotten

Repetition

RetrievalRehearsal and Coding

Information

Episodic Memory

Semantic Memory

ProceduralMemory

Long Term Memory

Pedagogical Background, Constructivism• Student centred approach in constructing knowledge through assimilation and

accommodation (Piaget 1977).

• Emphasize on meaningful learning, relevance to reality and students knowledge.

• Emphasize on social learning and social interactions (Vygotsky 1978).

• Strong emphasize on learning through experience (Dewey 1938, Kolb 1984).

Abstract Conceptualization

Concrete Experience

ReflectiveObservation

Active Experimentation

Constructivist Design Model, Kolb’s Experiential Learning Model• Kolb Model is well received in

the pedagogical literature, Kolb’s book (Kolb 1984) is cited 6000+ times.

• Optimal learning occurs when a balance between CE, RO, AC, and AE stages is given.

• Traditional teaching gave higher attention to AC.

• Enhanced learning is reported with Kolb’s Cycle design (Abdulwahed et al 2008 [3])

Mathematical Model of a Pedagogical Model

Kolb Cycle with Normal vs. at Risk Student

Output

-

+Input

Feedback/Measurement

Low Pass Filter

Mapping Kolb’s Cycle into Mathematical Model.

-

Rf Y

• Mathematical modelling of Kolb’s cycle to understand the learning dynamics when it is admitted as pedagogical model of designing educational interventions.

• Kolb’s model seems to be effective in closing the gap between normal and at risk students, mainly due to the feedback loop (associated with RO stage) and the AE stage.

• More details about the modelling process can be found in (Abdulwahed et al 2008 [1])

Modelling Kolb’s Model !

Motivation and Findings:

Module outcomes for normal vs. at risk students

Engineering Approach in Lecturing Design

Prerequisites

Final Module

OutcomeLecture 1

Lecture 2 Lecture N

Block diagram of closed loop lecturing model for at risk-students.

Lecture 2 Objectives

Lecture N Objectives

Lecture 1 Objectives

1

1 1 1 0

2 2 2

2

12 12 12

12

10 0

1 0 01

1 0 0 1 0

0 1 11

0 0

ax x r x

x x ra

x x r

a

Closed Loop Lecturing Mathematical Model (Abdulwahed et al 2008 [2])

• Break the course into self contained simpler modules (Breaking Complexity).

• Continuous assessment and feedback loop around each module.

• Set up clear objectives for each module.• Design leads to reduce the gap between

normal and at risk students.

Designing Lecturing:

Design Outlines:

Analysis and Design Implications• Very important role of continuous assessment, in

particular formative assessment.• Very important role of reflective activities and feedback.• Very important role of experiential activities.• Introducing concrete experiences at the beginning of any

new thing learned to left the students attention, reduce anxiety, and filter the taught information from external noise.

• Incorporating repetitive activities in different formats and on timely distant spaces for activating long-term information retention.

• Social learning should play essential role.• Importance of incorporating visual learning materials to

give mathematical abstracts visual meaning, towards Mathematical Object Oriented Thinking (MOOT).

• Make the taught mathematics relevant to the students study/degree through relevant real world examples.

• Try to break affective and cultural dams in face of math learning.

Interventions Implementation

• Implementations would follow the previous guidelines, probably different models will be suggested depending on the subject.

• Effective use of technology in implementing interventions.

• Embedding cognitive tools such as concept maps, brain storming, and mind maps in teaching and learning mathematics.

• Relying on both proximal and cyber social learning, effective use of web 2.0 tools for the later is needed. Moving towards Learning 2.0.

• The particular use of technology is for implementing the Kolb’s model through facilitating the following:– Measuring learning outcomes.– Feedback providing.– Building multiple reflection spaces.– Building multiple active experimentation spaces.

Potential Technology Assisted Intervention Tools

• Voting systems.• Mobile phones.• Internet mathematical resources (there is very huge already

designed material in the web).• Learning objects.• Computer simulation softwares such Matlab, Maple, and

LabView.• Web 2.0 tools such as wikies, facebook, twitter, youtube,

skype, blogs, etc.• Open Course Ware (OCW) sites such as MIT OCW.• Computer assessment tools such as Maple TA.• Peer assessment tools such as webPA.• Virtual Learning Environments VLEs.• 3D virtual worlds, in particular SL with Moodle through Sloodle.• Virtual conferencing and classrooms such as DimDim.• Internet whiteboards.• Gaming!

Hybrid Evaluation Methods

Evaluating Interventions

• Questionnaires.• Pre and Post Tests.• Statistical Hypothesis testing.• Investigating the use of cognitive

assessment tools if any is suitable.

Initial Suggested Tools:XX

XX

Treatment

Pre Post

Experimental Group

Equivalentgroups

Different Outcome?

Control Group

YY

YtYt

Conclusions!