Teaching mathematics for life skills and for the examination: Possibilities for reconciliation

David MtetwaDepartment of Science and Mathematics

Education, University of Zimbabwe

Some contextual observations

Curricula docs emphasize thrust on teaching and learning for life skills and mental capabilities [developmental orientation]

Classroom instructional action focuses on exam [reward system]

Student learning action focuses on the exam [currency] Parental reaction is centred on the exam [accountability] Employer fraternity appear to require both [evidence] Researchers: rhetoric on life skills, pragmatic on exam

[academic game]

Purpose

Draw attention to an apparent tension [exam – life skills] Put on the table some issues related to the

tension Encourage reflection on it while exploring

possibilities for resolving tension Develop common practitioner understanding Raise fresh questions

Some issues and implications

Validity/reliability of exams Same exam produces an A student who struggles

and one who does not: questioning Zimsec ?? Zimsec exam an omnibus instrument (serves

multiple purposes, one size fits all [placement, certification, aptitude, achievement]

No single common but different validitiesTeacher tests focus on predicting Zimsec exam

Practicality considerationsTeaching for life skills requires too much time and

resources Often teachers deal with pupils (uninterested); not

scholars (motivated) Equal opportunity aim (ability range too wide) Assessing for deep understanding costly (time,

resources, expertise) Special capabilities on the teacher are assumed

(yet often lack)

Searching for middle ground: A proposal

Acknowledge ruling constraints (time, resources, work & learn environment, special teacher capacities that may lack)

Aim for attaining both teaching objectives (exam and life skills)Notion of preparation and incubation [Poincare &

Hadamard in Tall] Notion of gestalt [Gestalt Psychologists] [Kohler] Notion of PCK [Shulman]

FLOW OF INSTRUCTION

Introductory

Extension and exploration

Development

consolidatory

Progression of activity: 3 phases

• Investigations• Mathematical language• Defining elements• Implications

• Motivation and appreciating phenomenon• Qualitative descriptions• Reflecting (on phenomenon)

interestmeaningfulness

Progression of activity cont• Investigations

• Mathematical language• Defining elements• Implications

• Problem solving• Interpreting/meaning

• Practice• Reflection (within discipline)

Formal theory and Language

•Big picture within topic and discipline.•Pass examinations.•Applications in and outside mathematics.•Life skills enhanced.

Linking theory, phase, activity

Invest in preparation and incubation (introducing phase) appreciating motivation for the mathematical objects

being introduced [interest]reflecting on qualitative description of the objects [local

meaning]Focus on development of main principle and language

(gestalt) [elaborating phase] investigate and describe properties of objects formulate definitions of objects and explore implications

[formal theory]

Frame and administer detail filling mathematical tasks [concluding phase] (PCK an asset) engage in problem-solving around the definitions

and their implications (meaningful rather than drill practice) [consolidating understanding, proficiency]

reflect on objects again and attempt to appreciate their place in the topic and the subject (overview, global big-picture meaning)

Remarks Many teachers invest more in the 3rd phase and far less in the first

2 phases; or begin by stating definitions Many begin and end with the 3rd phase, typically called exercises

or practice (largely routine and computational; little strategic thinking)

Practice, if done in class, is usually ‘busy work’, and if at home it is for ‘accountability’

If practice involves problem solving at all, it is usually the type involving recognition and recalling only, little productive thinking

The key to reconciliation between teaching for exam and for life skills lies in the idea of preparation-incubation, gestalt, and PCK

Lesson structure: 3 phases

Genesis of Object(recognition or creation)

Exploring characteristics (operational bevaviour)

Developing object(structure and applications)

An example of lesson flow on vectors

Genesis of a vector Consider encounters with phenomena (from

maths/everyday life) requiring consideration of a new idea/object

Focus on the agency and explaining power of the idea/object (may still be nameless)

Make need for such an idea/object compelling and apparent

Examine defining characteristics (when orientation and amount of movement are important) searching for hidden treasure striker pulling a shot boxer throwing a punch disease carrier in an epidemic locating a point relative to another on a plane/space

focus on consequences of missing one or both parameters

Developing idea/object combination of the 2 qualities important,

deserves a name [vector]needs representation (conventions and contexts

of use: geometrical, algebraic, other?) mathematical properties (intuitive) [geometrical –

practical meaningfulness; algebraic – consistency with other familiar algebraic objects and operations, other?]

formal definition formulation [existence/creation issue]; meaning to be established before definition

implications [theoretical, e.g., theorems, and generation of other objects such as momentum, vector spaces, etc]

implications [applications, e.g., as tools for problem solving as in engineering, aviation, etc.]

Thank You