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
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