Dr Hang Kim Hoo

PrincipalNUS High School of Mathematics & Science

6 September 2010

Holistic Assessment of

Mathematics Learning – The NUS

High Experience

6 Years Integrated Programme

2-2-2 year curriculum structure

Modular System

Grade Point System

Core (Graded) - key modules

that students must pass

Elective (Graded) - modules that

builds on the Core to give a

deeper understanding and

greater exposure to the subject

Enrichement (Ungraded) –

modules that provide breath in

learning

NUS HIGH SCHOOL CURRICULUM

MATHEMATICS, COMPUTING & STATISTICS

Core Modules

The core modules of the mathematics curriculum are

designed to cover most topics covered in the GCE ‘O’

level, GCE ‘A’ level, AP Calculus AB & BC and AP Statistics.

Elective modules

Extension of topics covered in the core modules may be

offered as elective modules. These elective modules aim

to provide a deeper understanding and greater exposure

to the topics.

Research Programme

Provides students avenues for independent research and

project work.

MATHEMATICS CURRICULUM FRAMEWORK

MATHEMATICAL

PROBLEM

SOLVING

Attidudes

Pro

cesses

Metacognition

Concepts

Skills

Beliefs

Interest

Appreciation

Confidence

Perserverence

Numerical Calculation

Algebraic Manipulation

Spatial Visualisation

Data Analysis

Measurement

Use of Mathematical Tools

Estimation

Monitoring of one’s own thinking;

Self regulation of learning

Reasoning,

Communication &

Connections,

Thinking Skills & Heuristics;

Applications & Modelling

Numerical ;

Algebraic;

Geometrical;

Statistical;

Probabilistic;

Analytical

NUS High School Mathematics Curriculum – Some

Highlights The Department of Mathematics, Statistics &

Computing uses a variety of learning

approaches:

Interactive Learning – collaborative learning, peer teaching

Independent Learning – guided self-study, project

work

Integration & Connection – application in real-life

problems

Inquiry Learning – explorative & investigative work

Interdisciplinary Learning – teaching via problem solving, research

Enriched learning – mentorship, talks, seminars, etc

INSTRUCTIONAL STRATEGY & PEDAGOGY

USE OF TECHNOLOGY

Use of Technology (Graphing calculators TI89,

Autograph, Geometer’s Sketchpad, Matlab,

Maple and Mathematica, Statisticals Exploration

Tools) are or will be used regularly by students

and teachers:

To support learning through exploration and

discovery;

To enable learning through the management

and interpretation of data

To support realistic applications of

mathematics

INSTRUCTIONAL STRATEGY & PEDAGOGY

NUS High School Mathematics Curriculum – Some

Highlights Assessment of student learning - students are

required to produce a variety of products during their courses to display their understanding and learning. Some examples include: Graded class exercises

Quizzes or topical tests

Take back assignments

Individual projects / Group projects

Class participation & oral presentations

Mathematical Tasks/Journal

Written semestral examinations

MATHEMATICS, COMPUTING & STATISTICS

ASSESSMENT

Assessment of LearningDid students learn?

How much and how well did they

learn?

Assessment for LearningHow can we use feedback to help students learn better?

What are the learning gaps?

Assessment as LearningHow can students be developed

into self-directed learners?

What habits of mind are desirable?

Summative AssessmentAssignments, Quizes, Tests, Oral

Presentations,Examinations,

Problem solving tasks, Projects

Formative AssessmentAssignments, Quizes, Tests, Oral Presentations,Examinations,

Problem solving tasks, Projects

Self-Directed LearningSelf-assessment & regulation,

metacognitive reflection, open-

ended tasks, journals etc.

A S S E S S M E N T A S L E A R N I N G

A quadrilateral has a pair of

opposite sides parallel. We know

that two of its interior angles are

right angles. What is the shape of

this quadrilateral?

Demonstration – A Mathematics Problem

TAXONOMIES & HIERARCHICAL MODELS

Development in learning theories has surfaced

Taxonomies of Learning or Hierarchical Models of

Understanding. Some of these are:

BLOOM’S TAXANOMY OF LEARNING DOMAINS

• Cognitive Domain

• Affective Domain

• Psychomotor Domain

SOLO TAXANOMY

• Bigg’s structure of the observed learning outcome

VAN HIELE THEORY – Thinking in Geometry

TAXONOMIES & HIERARCHICAL MODELS

EASY DIFFICULT

CONCRETE

LOW LEVEL

SIMPLE

UNI-DIMENSIONAL

ABSTRACT

HIGH LEVEL

COMPLEX

MULTI-DIMENSIONAL

S C A F F O L D I N G

LEARNING OUTCOME LEARNING OUTCOMEEND

START

T A S K A N A L Y S I S

Demonstration – An Example from Mathematics

Suppose that a, b, c and d are positive real

numbers such that the polynomial

4 3 2 2 3 4f ( ) 4 6 4x x ax b x c x d

has four distinct positive roots.

Show that (i) c > d ; (ii) b > c (iii) a > b

T A S K A N A L Y S I S

Demonstration – An Example from Mathematics

In this question, you may assume that if k1, …, kn are distinct

positive real numbers, then

4 3 2 2 3 4f ( ) 4 6 4x x ax b x c x d has four distinct positive roots.

Suppose that a, b, c and d are positive real numbers such

that the polynomial

1

1 1

1 nn n

r r

r r

k kn

(i) By considering the relationship between the coefficients of f

and its roots, show that c > d.

(ii) By differentiating f, show that b > c.

(iii) Show that a > b.

A S S E S S M E N T OF L E A R N I N G

ASSESSMENT – Norm- & Criterion-Referenced Approach

Tests can be categorized into norm-referenced tests and

criterion-referenced tests.

They differ in their intended purposes, the way in which

content is selected, and the scoring process which

defines how the test results must be interpreted.

A norm-referenced test is used to classify students. They

are designed to highlight achievement differences

between and among students to produce a dependable

rank order of students across a continuum of

achievement from high achievers to low achievers

A S S E S S M E N T OF L E A R N I N G

ASSESSMENT – Norm- & Criterion-Referenced Approach

A criterion-referenced test reports how well students are

doing relative to a pre-determined performance level on

a specified set of educational goals or outcomes.

It is used wish to see how well students have learned the

knowledge and skills which they are expected to have

mastered. The information may be used to determine

how well the student is learning the desired curriculum.

Test content is an important factor in choosing between

an norm- and a criterion-referenced test.

MATHEMATICS MODULE

ASSESSMENT – An Illustration of a Criterion-Referenced Approach

Assignments

Quizzes

Topical Tests

Examination

Problem Solving

Grades

up to B

or B+Grades

up to A+

Rubrics, descriptors

for each grade type defined

R E L I A B I L I T Y & V A L I D I T Y

ReliabilityThe consistency of a measurement, or the degree to which

an assessment measures the same way each time it is used

under the same condition with the same students. In short,

it is the repeatability of your measurement. A measure is

considered reliable if a person's score on the same test

given twice is similar. It is important to note that reliability is

not measured, it is estimated.

ValidityThe strength of conclusions, inferences or propositions.

More formally, it is defined as the "best available

approximation to the truth or falsity of a given inference,

proposition or conclusion.“

There are 4 types of validity: (1) conclusion validity; (2)

internal validity; (3) construct validity; (4) external validity.

R E L I A B I L I T Y & V A L I D I T Y

ReliabilityThe consistency of a measurement, or the degree to which

an assessment measures the same way each time it is used

under the same condition with the same students. In short,

it is the repeatability of your measurement. A measure is

considered reliable if a person's score on the same test

given twice is similar. It is important to note that reliability is

not measured, it is estimated.

ValidityThe strength of conclusions, inferences or propositions.

More formally, it is defined as the "best available

approximation to the truth or falsity of a given inference,

proposition or conclusion.“

There are 4 types of validity: (1) conclusion validity; (2)

internal validity; (3) construct validity; (4) external validity.

Reliability estimates the consistency of a measurement, or more simply the degree to which an assessment measures the same way each time it is used in under the same conditions with the same students. Validity, on the other hand, involves the degree to which we are measuring what we are supposed to, more simply, the accuracy of our measurement. Validity is more important than reliability because if an instrument does not accurately measure what it is supposed to, there is no reason to use it even if it measures consistently (reliably).

M O V I N G A H E A D

START END

BLOOM’S

TAXONOMY

KNOWLEDGE

COMPREHENSION

APPLICATION

ANALYSIS

EVALUATION

SYNTHESIS

SCAFFOLDING

ASSESSMENT

OF FOR

LEARNING

AS LEARNING

PEDAGOGY

I HEAR & I FORGET,

I SEE & I REMEMBER,I DO & I UNDERSTAND,

I REFLECT & I IMPROVE

FORMATIVE

SUMMATIVE

M O V I N G A H E A D

TOTAL INSTRUCTIONAL DESIGN

Classroom

Learning

(Formal)

Beyond Formal (Classroom) Learning

Learning Theory Pedagogy Assessment

Delivery

I/II

B E N C H M A R K I N G & C O M P A R I S O N

NUS

NU

S

NUS

NU

S

Students are encouraged to take up NUS modules

NUS High School

Diploma

CAP

Modules Taken & Grades

Individual Teacher’s

Comments

Detailed student portfolio

NUS High Diploma

External Exams such as AP and SAT are optional

L E A R N I N G A T N U S H S

At NUS High School of Mathematics &

Science, we hope to enable teachers to

engage students actively in learning.

From guiding them to construct their own

understanding and create new

knowledge through exploration and

experimentation, it is hoped that they will

be enabled to provide creative solutions

to difficult real life problems and make

breakthrough inventions in the long term

C O N C L U S I O N S

ISSUES & FUTURE POSSIBILITIES

Assessment as a motivation tool Teaching to the test

Alternative/Authentic

Assessment Modes

Reliably valid & implemented

on a mass scale

Snapshot Synoptic Exams Album / Portfolio Assessment

While we can continue to make assessments more reliable and

valid, it is never an exact Science.Exploring the feasibility of implementing alternative assessment

modes on a mass scale reliably is a first step towards moving away

from using proxy measures of learning

Q U E S T I O N & A N S W E R

T H A N K Y O U