Introduction
Education is not only
considered essential for the pleasant development of one’s personality but also
for the growth and progression of the country as a whole. Elementary education
is regarded as the foundation for the growth of not only the individuals but
for the welfare of the entire nation. Elementary education in India is regarded
as the foundation of compulsory schooling that is considered essential for the
individuals. It is preceded by pre-school or nursery education and is followed
by secondary education.
The central
government and the state government and other organizations have been involved
in the progression of elementary education. The 86th constitutional amendment
act, 2002 made education the fundamental right of the children within the age
group of six to fourteen years. Right of Children to Free and Compulsory
Education Act, 2009 was recognized. It stated all children should be made
provision of free education up to the age of fourteen years. The role of
education within the framework of social and economic progress is well
recognized. Universalisation of Elementary Education (UEE) has opened up
opportunities for the individuals leading to their social and economic
development. Education is not only meant to improve the knowledge of the
individuals but also in enhancing the overall quality of life.
Purpose (Aims) of
Elementary Mathematics Education
The primary grades
are often considered the most important years of a child’s school career. In
grades K–5, students acquire content knowledge that they will use as the
foundation for the rest of their education. As with all of the core subjects,
there are certain objectives for mathematics that should be addressed in the
primary grades.
The primary grades
are often considered the most important years of a child’s school career. In
grades K–5, students acquire content knowledge that they will use as the
foundation for the rest of their education. As with all of the core subjects,
there are certain objectives for mathematics that should be addressed in the
primary grades.
Mathematics
introduces children to concepts, skills and thinking strategies that are
essential in everyday life and support learning across the curriculum. It helps
children make sense of the numbers, patterns and shapes they see in the world
around them, offers ways of handling data in an increasingly digital world and
makes a crucial contribution to their development as successful learners.
Children delight in using mathematics to solve a problem, especially when it
leads them to an unexpected discovery or new connections. As their confidence
grows, they look for patterns, use logical reasoning, suggest solutions and try
out different approaches to problems. Mathematics offers children a powerful
way of communicating. They learn to explore and explain their ideas using
symbols, diagrams and spoken and written language. They start to discover how
mathematics has developed over time and contributes to our economy, society and
culture. Studying mathematics stimulates curiosity, fosters creativity and
equips children with the skills they need in life beyond school.
One of the skills
needed to be successful in the 21st century, elementary math is becoming an
increasingly important part of early childhood education.
According to the
National Association for the Education of Young Children, the math skills that
students learn at a young age build a foundation for future learning endeavors
and can be a good indicator of whether or not young people will be able to meet
and overcome new challenges as they mature.
"Mastery of
early math skills predicts not only future math achievement, it also predicts
future reading achievement," Greg Duncan, a researcher at Northwestern
University, said in a statement.
The following are
some of the important aims of Mathematics education at elementary level.
·
Acquire understanding of numbers and numeration
·
Develop ability to perform the four basic
operations
·
Develop skills in measurement, approximation and
estimation
·
Develop spatial concepts and ability to use them
·
Acquire the techniques of collecting,
representing and interpreting data
·
Develop positive attitudes towards Mathematic
and make good use of leisure time
·
Develop techniques of investigation and problem
solving strategies
·
Recognize that mathematics permeates the world
around us appreciate the usefulness, power and beauty of mathematics
·
Enjoy mathematics and develop patience and
persistence when solving problems
·
Understand and be able to use the language,
symbols and notation of mathematics
·
Develop a mathematical curiosity and use inductive
and deductive reasoning when solving problems
·
Become confident in using mathematics to analyse
and solve problems both in school and in real-life situations
·
Develop the knowledge, skills and attitudes
necessary to pursue further studies in mathematics
·
Develop abstract, logical and critical thinking
and the ability to reflect critically upon their work and the work of others
·
To develop in the child rational and scientific
attitude towards life
Objectives
Aims of teaching mathematics are genially scope
whereas objectives of the subject are specific goals leading ultimately to the
general aims of the subject. The objectives of teaching mathematics in school
can be described as under:
i.
Knowledge Objectives
Through mathematics, the pupil acquires the knowledge
of the following:
·
The child learns the mathematical language, for
example, mathematical symbols, formulae figures, diagrams, definitions etc.
·
The child understands and uses mathematical
concepts like concept of area, volume, number, direction etc.
·
The child learns the fundamental mathematical
ideas, processes, rules and relationships.
·
The child understands the historical background
of various topics and contribution of mathematicians.
·
The child understands the significance and use
of the units of measurement
ii.
Skill Objectives
Mathematics develops the following skills:
·
The child learns to express thoughts clearly and
accurately.
·
The child learns to perform calculations orally.
·
The child develops the ability to organise and
interpret the given data
·
The child learns to reach accurate conclusions
by accurate and logic reasoning.
·
The child learns to analyse problems and
discover fundamental relationships.
·
The child develops speed and accuracy in solving
problems.
·
The child develops the skill to draw accurate
geometrical figures,
·
The child develops the ability to use
mathematical apparatuses an tools skilfully
iii.
Appreciation Objectives
The child learns to appreciate:
·
The contribution of mathematics to the
development of various subjects and occupations.
·
The role played by mathematics in modern life.
·
The mathematical type of thought which serves as
model for scientific thinking in other fields.
·
The rigour and power of mathematical processes
and accrue of results.
·
The cultural value of mathematics.
·
The value of mathematics as leisure time activity.
iv.
Attitude Objectives
Mathematics helps in the development of following
attitudes:
·
The child develops the attitude of
systematically pursuing a task to completion.
·
The child develops heuristic attitude. He tries
to make independent discoveries.
·
The child develops the habit of logical
reasoning.
·
The child is brief and precise in expressing
statements and results,
·
The child develops the habit of verification.
·
The child develops power concentration and
independent thinking. (vii) He develops habit of self-reliance.
Curriculum of Elementary Mathematics Education
The word curriculum is derived from the Latin word
‘currere’ which means ‘run’. Thus curriculum means a course to be run for
reaching a certain goal or destination. Thus the traditional definition of
curriculum is a course of study or training leading to a product or education.
Teaching learning process does not operate in a vacuum. Certain planned
experiences have to be provided in a school so that optimum human development
according to the needs of a particular country is possible. Thus the term
curriculum in recent years has come to mean all the planned activities and
experiences available to the students under the direction of the school.
The primary grades are often considered the most
important years of a child’s school career. In grades K–5, students acquire
content knowledge that they will use as the foundation for the rest of their
education. As with all of the core subjects, there are certain objectives for
mathematics that should be addressed in the primary grades.
Numbers and Basic Functions
At the most basic level, mathematics involves
counting, recognizing numbers and performing simple operations like addition
and subtraction. In the primary grades, students should be provided with
numerous opportunities to master these skills. Upon exiting primary school,
children should be comfortable with writing and identifying numbers, rote
counting forward and backward and comparing numbers and quantities. Primary
students should have knowledge of number facts and families. They should also
be able to add, subtract, multiply and divide numbers.
Measurement and Estimation
In the primary grades, students should be taught about the measurement of length, weight and capacity. Children should be introduced to the comparative language, such as "shorter," "heavier" and
"longer," and should be able to apply these concepts when considering
different objects and units of measurement. Primary students should also learn
about money and time and be able to measure time in terms of hours, days,
months and years. In addition to measurement, children should be taught about the estimation of quantities and capacities.
Geometry
Learning about shapes, symmetry, position and
direction is a key objective for primary school mathematics. Students should be
exposed to two- and three-dimensional shapes and be able to identify, name and
draw them. Children in the primary grades should also have an understanding of
line and rotational symmetry, as well as the manipulation of objects in space.
Additional spatial concepts of position, like "above,"
"under," "next to" and "beyond," should be
addressed in primary mathematics education as well.
Data Collection and Interpretation
Being able to collect, organize and interpret data is
an important skill that is taught in the primary grades. Students should be
given opportunities to answer questions through sorting and organizing data by
using graphs, charts, tables and Venn diagrams. They should also learn to
compare objects and data based on given criteria.
Critical Thinking and Problem Solving
Mathematics should be used to develop critical
thinking and problem solving. Presented with a problem or situation, primary
students should be able to identify the proper strategies needed to come to
conclusions and carry out calculations. Throughout primary school, students
should progress from using concrete objects and written calculations to
carrying out operations mentally. Children in primary school should also have
the ability to identify and continue patterns, provide examples and
non-examples of mathematical statements, and form and test hypotheses.
Appreciation and Uses
One of the most important aims of students at this
level is to develop a positive attitude toward mathematics. Students should
understand and appreciate the functionality of mathematics. In addition to
valuing math, primary school students need to be taught how to use mathematics
in their everyday lives. They should be exposed to all the uses of mathematics,
from counting out change or telling time to use angles in architecture or
art.
Methodology
In the changing
scenario, students’ involvement in classroom, teaching method and teachers’
commitment are the factors that affect the teaching learning process. The
traditional aspects of educational outcomes are based on pure (factual) subject
knowledge and its memorization. On the other hand, modern aspects of
educational outcomes concentrate on the progressive side of knowledge and its
applications with individual involvement (Dogan, 2011)
According to the
National Council of Teachers of Mathematics (NCTM) there are five process
strands in teaching mathematics at any level.
i.
Problem Solving
ü
Build new mathematical knowledge through problem
solving
ü
Solve problems that arise in mathematics and in
other contexts
ü
Apply and adapt a variety of appropriate
strategies to solve problems
ü
Monitor and reflect on the process of
mathematical problem solving
ii.
Reasoning and Proof
ü
Recognize reasoning and proof as fundamental
aspects of mathematics
ü
Make and investigate mathematical conjectures
ü
Develop and evaluate mathematical arguments and
proofs
ü
Select and use various types of reasoning and
methods of proof
iii.
Communication
ü
Organize and consolidate their mathematical
thinking through communication
ü
Communicate their mathematical thinking
coherently and clearly to peers, teachers, and others
ü
Analyse and evaluate the mathematical thinking
and strategies of others;
ü
Use the language of mathematics to express mathematical
ideas precisely.
iv.
Connections
ü
Recognize and use connections among mathematical
ideas
ü
Understand how mathematical ideas interconnect
and build on one another to produce a coherent whole
ü
Recognize and apply mathematics in contexts outside
of mathematics
v.
Representation
ü
Create and use representations to organize,
record, and communicate mathematical ideas
ü
Select, apply, and translate among mathematical
representations to solve problems
ü
Use representations to model and interpret physical,
social, and mathematical phenomena
The elementary mathematics teachers must handle the mathematics teaching
in the following ways.
i.
Content
The teacher must
demonstrate and apply knowledge of major mathematics concepts, algorithms,
procedures, applications in varied contexts, and connections within and among
mathematical domains (Number and Operations, Algebra, Geometry and Measurement,
and Statistics and Probability)
ii.
Mathematical Practices
The elementary
mathematics teachers must handle mathematics teaching in the following
ways.
ü Use problem solving
to develop conceptual understanding, make sense of a wide variety of problems
and persevere in solving them, apply and adapt a variety of strategies in
solving problems confronted within the field of mathematics and other contexts,
and formulate and test conjectures in order to frame generalizations.
ü Reason abstractly,
reflectively, and quantitatively with attention to units, constructing viable
arguments and proofs, and critiquing the reasoning of others; represent and
model generalizations using mathematics; recognize structure and express
regularity in patterns of mathematical reasoning; use multiple representations
to model and describe mathematics; and utilize appropriate mathematical
vocabulary and symbols to communicate mathematical ideas to others.
ü Formulate, represent,
analyse, and interpret mathematical models derived from real-world contexts or
mathematical problems.
ü Organize mathematical
thinking and use the language of mathematics to express ideas precisely, both
orally and in writing to multiple audiences.
ü Demonstrate the
interconnectedness of mathematical ideas and how they build on one another and
recognize and apply mathematical connections among mathematical ideas and
across various content areas and real-world contexts.
ü Model how the
development of mathematical understanding within and among mathematical domains
intersects with the mathematical practices of problem solving, reasoning,
communicating, connecting, and representing.
iii.
Content Pedagogy
The elementary
mathematics teachers must handle mathematics teaching in the following
ways.
ü Apply knowledge of
curriculum standards for elementary mathematics and their
relationship to student learning
within and across mathematical domains in teaching elementary students and
coaching/mentoring elementary classroom teachers.
ü Analyze and consider
research in planning for and leading students and the teachers they coach/mentor
in rich mathematical learning experiences.
ü Plan and assist
others in planning lessons and units that incorporate a variety of strategies, differentiated
instruction for diverse populations, and mathematics-specific and instructional
technologies in building all students’ conceptual understanding and procedural proficiency.
ü Provide students and
teachers with opportunities to communicate about mathematics and make
connections among mathematics, other content areas, everyday life, and the workplace.
ü Implement and promote
techniques related to student engagement and communication including selecting
high quality tasks, guiding mathematical discussions, identifying key mathematical
ideas, identifying and addressing student misconceptions, and employing a range
of questioning strategies.
ü 3f) Plan, select,
implement, interpret, and assist teachers in using formative and summative assessments
to inform instruction by reflecting on mathematical proficiencies essential for
all students.
ü Monitor students’
progress and assist others, including family members, administrators and other
stakeholders, in making instructional decisions and in measuring and
interpreting students’ mathematical understanding and ability using formative
and summative assessments.
iv.
Mathematical Learning Environment
The elementary
mathematics teachers must handle the mathematics teaching in the following
ways.
ü Exhibit knowledge of
child, pre-adolescent, and adult learning, development, and behaviour and
demonstrate and promote a positive disposition toward mathematical processes
and learning.
ü Plan, create, and
coach/mentor teachers in creating developmentally appropriate, sequential, and
challenging learning opportunities grounded in mathematics education research
in which students are actively engaged in building new knowledge from prior
knowledge and experiences.
ü Incorporate knowledge
of individual differences and the cultural and language diversity that exists
within classrooms and include and assist teachers in embracing culturally
relevant perspectives as a means to motivate and engage students.
ü Demonstrate and
encourage equitable and ethical treatment of and high expectations for all
students.
ü Apply mathematical
content and pedagogical knowledge in the selection, use, and promotion of
instructional tools such as manipulatives and physical models, drawings,
virtual environments, presentation tools, and mathematics-specific technologies
(e.g., graphing tools and interactive geometry software); and make and nurture
sound decisions about when such tools enhance teaching and learning,
recognizing both the insights to be gained and possible limitations of such tools.
v.
Impact on Student Learning
The elementary
mathematics teachers must handle the mathematics teaching in the following
ways.
ü Verify that
elementary students demonstrate conceptual understanding; procedural fluency;
the ability to formulate, represent, and solve problems; logical reasoning and
continuous reflection on that reasoning; productive disposition toward
mathematics; and the application of mathematics in a variety of contexts within
major mathematical domains.
ü Engage students and
coach/mentor teachers in using developmentally appropriate mathematical
activities and investigations that require active engagement and include
mathematics-specific technology in building new knowledge.
ü Collect, organize, analyse,
and reflect on diagnostic, formative, and summative assessment evidence and
determine the extent to which students’ mathematical proficiencies have
increased as a result of their instruction or their efforts in
coaching/mentoring teachers.
Approaches in Teaching Elementary Mathematics
Use Everyday Objects
we already have everything we need to begin teaching math to our child.
Buttons, pennies, money, books, fruit, soup cans, trees, cars — we can count
the objects you have available. Math is easy to teach when we look at all of
the physical objects we can count, add, subtract, and multiply.
Everyday objects also help us teach our child that objects don't have to
be identical to be important in math. Counting apples is a great math lesson,
but counting apples, oranges, and watermelons together expand the thought
process. The child is connecting counting with various objects, instead of
running through a routine numbers game of 1, 2, 3.
Use Dramatizations
Invite children to pretend to be in
a ball (sphere) or box (rectangular prism), feeling the faces, edges, and
corners and to dramatize simple arithmetic problems such as: Three frogs jumped
in the pond, then one more, how many are there in all?
Use Children's Bodies
Suggest that children show how many feet, mouths, and so on they have.
When asked to show their "three arms," they respond loudly in
protest, and then tell the adult how many they do have and show
("prove") it. Then invite children to show numbers with fingers,
starting with the familiar, "How old are you?" to showing numbers you
say, to showing numbers in different ways (for example, five as three on one
hand and two on the other).
Use Children's Play
Engage children in block play that allows them to do mathematics in
numerous ways, including sorting, seriating, creating symmetric designs and
buildings, making patterns, and so forth. Then introduce a game of Dinosaur
Shop. Suggest that children pretend to buy and sell toy dinosaurs or other
small objects, learning counting, arithmetic, and money concepts.
Use Children's Toys
Encourage children to use "scenes" and toys to act out
situations such as three cars on the road, or, later in the year, two monkeys
in the trees and two on the ground.
Use Children's Stories
Share books with children that address mathematics but are also good stories.
Later, help children see mathematics in any book. In Blueberries for Sal, by
Robert McCloskey (Penguin, 1993), children can copy "kuplink, kuplank,
kuplunk!" and later tell you the number as you slowly drop up to four
counters into a coffee can.
Use Children's Natural Creativity
Children's ideas about
mathematics should be discussed with all children. Here's a "mathematical
conversation" between two boys, each 6 years of age: "Think of the
biggest number you can. Now add five. Then, imagine if you had that many
cupcakes." " Wow, that's five more than the biggest number you could
come up with!"
Use Children's Problem-solving Abilities
Ask children to describe how they
would figure out problems such as getting just enough scissors for their table
or how many snacks they would need if a guest were joining the group. Encourage
them to use their own fingers or manipulatives or whatever else might be handy
for problem solving.
Use a Variety of Strategies
Bring mathematics everywhere you go in your classroom, from counting
children at morning meeting to setting the table, to asking children to clean
up a given number or shape of items. Also, use a research-based curriculum to
incorporate a sequenced series of learning activities into your program.
Use Technology
Try digital cameras to record children's mathematical work, in their
play and in planned activities, and then use the photographs to aid discussions
and reflections with children, curriculum planning, and communication with
parents. Use computers wisely to mathematize situations and provide
individualized instruction.
Use Assessments to Measure Children's Mathematics Learning
Use observations, discussions with children, and small-group activities
to learn about children's mathematical thinking and to make informed decisions
about what each child might be able to learn from future experiences. Also try
computer assessments. Use programs that assess children automatically.
Assessment
Assessment is the
means by which we determine what students know and can do. It tells teachers,
students, parents, and policymakers something about what students have learned:
the mathematical terms they recognize and can use, the procedures they can carry
out, the kind of mathematical thinking they do, the concepts they understand,
and the problems they can formulate and solve. It provides information that can
be used to award grades, to evaluate a curriculum, or to decide whether to
review fractions. Assessment can help convince the public and educators that
change is needed in the short run and that the efforts to change mathematics
education is worthwhile in the long run.
Assessment basically
consists of two types, formative and summative. The formative is ongoing
process and the later is pencil-paper test. Whatever the type of assessment we
take, there are certain principles to follow.
Principles for
Assessing Mathematics Learning
i.
The Content Principle
Assessment should
reflect the mathematics that is most important for students to learn. Any
assessment of mathematics learning should first and foremost be anchored in
important mathematical content. It should reflect topics and applications that
are critical to a full understanding of mathematics as it is used in today's
world and in students' later lives, whether in the workplace or in later
studies. Assessments should reflect processes that are required for doing
mathematics: reasoning, problem solving, communication, and connecting ideas.
Consensus has been achieved within the discipline of mathematics and among
organizations representing mathematics educators and teachers on what
constitutes important mathematics. Although such consensus is a necessary starting
point, it is important to obtain public acceptance of these ideas and to
preserve local flexibility to determine how agreed-upon standards are reflected
in assessments as well as in curricula.
Assessment makes
sense only if it is in harmony with the broad goals of mathematics education
reform.
ii.
The Learning Principle
Assessment should
enhance mathematics learning and support good instructional practice. Although
assessments can be undertaken for various purposes and used in many ways, proponents
of standards-based assessment reform have argued for the use of assessments
that contribute very directly to student learning. The rationale is that
challenging students to be creative and to formulate and solve problems will
not ring true if all students see are quizzes, tests, and examinations that
dwell on routine knowledge and skill. Consciously or unconsciously, students
use assessments they are given to determine what others consider to be
significant.
iii.
The Equity Principle
Assessment should
support every student's opportunity to learn important mathematics. The equity
principle aims to ensure that assessments are designed to give every student a
fair chance to demonstrate his or her best work and are used to provide every
student with access to challenging mathematics. Equity requires careful
attention to the many ways in which understanding of mathematics can be
demonstrated and the many factors that may colour judgments of mathematical
competence from a particular collection of assessment tasks.
Equity also requires
attention to how assessment results are used. Often assessments have been used
inappropriately to filter students out of the educational opportunity. They might
be used instead to empower students: to provide students the flexibility needed
to do their best work, to provide concrete examples of good work so that
students will know what to aim for in learning, and to elevate the students'
and others' expectations of what can be achieved.
Equity also requires
that policies regarding use of assessment results make clear the schools'
obligations to educate students to the level of new content and performance
standards.
Purpose of Classroom
Assessment
One of the first
things to consider when planning for assessment is its purpose. Who will use
the results? For what will they use them?
Assessment is used
to:
ü
inform and guide teaching and learning
A good classroom
assessment plan gathers evidence of student learning that informs teachers'
instructional decisions. It provides teachers with information about what
students know and can do. To plan effective instruction, teachers also need to
know what the student misunderstands and where the misconceptions lie. In
addition to helping teachers formulate the next teaching steps, a good classroom
assessment plan provides a road map for students. Students should, at all
times, have access to the assessment so they can use it to inform and guide
their learning.
ü
help students set learning goals
Students need
frequent opportunities to reflect on where their learning is at and what needs
to be done to achieve their learning goals. When students are actively involved
in assessing their own next learning steps and creating goals to accomplish
them, they make major advances in directing their learning and what they
understand about themselves as learners.
ü
assign report card grades
Grades provide
parents, employers, other schools, governments, post-secondary institutions and
others with summary information about student learning.
ü
motivate students
Research (Davies
2004; Stiggins et al. 2004) has shown that students will be motivated and
confident learners when they experience progress and achievement, rather than
the failure and defeat associated with being compared to more successful peers.
The Assessment
Process
ü
Effective classroom assessment in mathematics:
ü
addresses specific outcomes in the program of
studies
ü
shares intended outcomes and assessment criteria
with students prior to the assessment activity
ü
assesses before, during and after instruction
ü
employs a variety of assessment strategies to
provide evidence of student learning
ü
provides frequent and descriptive feedback to
students
ü
ensures students can describe their progress and
achievement and articulate what comes next in their learning
ü
informs teachers and provides insight that can
be used to modify instruction.
The assessment
process starts with planning based on the program of studies learning outcomes
and involves assessing, evaluating and communicating student learning, as shown
in the following diagram.
What to Assess
The process of
assessment in mathematics includes the following dimensions of
mathematical
learning:
ü
concepts and procedures
ü
mathematical reasoning
ü
dispositions towards mathematics
ü
using mathematical knowledge and techniques to
solve problems
ü
communication
Types of Assessment
Pre-assessment or
diagnostic assessment
Before creating the
instruction, it’s necessary to know for what kind of students you’re creating
the instruction. Your goal is to get to know your student’s strengths,
weaknesses and the skills and knowledge they possess before taking the
instruction. Based on the data you’ve collected, you can create your
instruction.
Formative assessment
– Assessment for Learning
Formative assessment
is used in the first attempt of developing instruction. The goal is to monitor
student learning to provide feedback. It helps to identify the first gaps in
your instruction. Based on this feedback you’ll know what to focus on for
further expansion for your instruction.
Assessment for
learning is ongoing assessment that allows teachers to monitor students on a
day-to-day basis and modify their teaching based on what the students need to
be successful. This assessment provides students with the timely, specific
feedback that they need to make adjustments to their learning.
After teaching a
lesson, we need to determine whether the lesson was accessible to all students
while still challenging to the more capable; what the students learned and
still need to know; how we can improve the lesson to make it more effective;
and, if necessary, what other lesson we might offer as a better alternative.
This continual evaluation of instructional choices is at the heart of improving
our teaching practice.
Summative assessment
– Assessment of Learning
Summative assessment
is aimed at assessing the extent to which the most important outcomes at the
end of the instruction have been reached. But it measures more: the
effectiveness of learning, reactions on the instruction and the benefits on a
long-term base. The long-term benefits can be determined by following students
who attend your course, or test. You are able to see whether and how they use
the learned knowledge, skills and attitudes.
Assessment as
Learning
Assessment as
learning develops and supports students' metacognitive skills. This form of
assessment is crucial in helping students become lifelong learners. As students
engage in peer and self-assessment, they learn to make sense of information,
relate it to prior knowledge and use it for new learning. Students develop a
sense of ownership and efficacy when they use teacher, peer and self-assessment
feedback to make adjustments, improvements and changes to what they understand.
Confirmative
assessment
When your instruction
has been implemented in your classroom, it’s still necessary to take
assessment. Your goal with confirmative assessments is to find out if the
instruction is still a success after a year, for example, and if the way you're
teaching is still on point. You could say that a confirmative assessment is the extensive form of a summative assessment.
Norm-referenced
assessment
This compares a
student’s performance against an average norm. This could be the average
national norm for the subject History, for example. Another example is when the
teacher compares the average grade of his or her students against the average
grade of the entire school.
Criterion-referenced
assessment
It measures student’s
performances against a fixed set of predetermined criteria or learning
standards. It checks what students are expected to know and be able to do at a
specific stage of their education. Criterion-referenced tests are used to
evaluate a specific body of knowledge or skill set, it’s a test to evaluate the curriculum taught in a course.
Ipsative assessment
It measures the
performance of a student against previous performances from that student. With
this method you’re trying to improve yourself by comparing previous results.
You’re not comparing yourself against other students, which may be not so good
for your self-confidence.
Tools and Techniques
for Assessment
As students work with
mathematical tasks, they should be able to:
ü
explain, interpret and justify what they know in
their own ways, not just present what others have said about the topic
ü
make and explore connections that are not
immediately obvious
ü
speak to their peers about the personal
strategies they have used to arrive at their solutions
ü
provide evidence of their learning based on
explicit criteria
ü
create new ways to express ideas, insights and
feelings; e.g., making models or representations as they devise various ways to
solve a problem, justifying their solutions, creating simulations, working with
what they understand in new situations or contexts
ü
take action when they recognize that their
understanding of issues, problems and ideas could be improved.
Teachers can use a
variety of assessment tools and strategies to assess student performance. Some
of these strategies and tools include:
Anecdotal Notes
Anecdotal notes are used
to record specific observations of individual student behaviours, skills and
attitudes as they relate to the outcomes in the program of studies. Such notes
provide cumulative information on student learning and direction for further
instruction. Anecdotal notes are often written as the result of ongoing
observations during the lessons but may also be written in response to a
product or performance the student has completed. They are brief, objective and
focused on specific outcomes. Notes taken during or immediately following an
activity are generally the most accurate. Anecdotal notes for a particular
student can be periodically shared with that student or be shared at the
student’s request. They can also be shared with students and parents at parent–teacher–student
conferences.
The purpose of
anecdotal notes is to:
ü
provide information regarding a student's
development over a period of time
ü
provide ongoing records about individual
instructional needs
ü
capture observations of significant behaviours
that might otherwise be lost
ü
provide ongoing documentation of learning that
may be shared with students, parents and teachers.
Tips for Establishing
and Maintaining Anecdotal Notes
ü
Keep a binder with a separate page for each
student. Record observations using a clipboard and sticky notes. Write the date
and the student’s name on each sticky note. Following the note taking, place
individual sticky notes on the page reserved for that student in the binder.
ü
Keep a binder with dividers for each student and
blank pages to jot down notes. The pages may be divided into three columns:
Date, Observation and Action Plan.
ü
Keep a class list in the front of the binder and
check off each student's name as anecdotal notes are added to their section of
the binder. This provides a quick reference of the students you have observed
and how frequently you have observed them.
ü
Keep notes brief and focused (usually no more
than a few sentences or phrases).
ü
Note the context and any comments or questions
for follow-up.
ü
Keep comments objective. Make specific comments
about student strengths, especially after several observations have been
recorded and a pattern has been observed.
ü
Record as the observations are being made, or as
soon after as possible, so recollections will be accurate.
ü
Record comments regularly, if possible.
ü
Record at different times and during different
activities to develop a balanced profile of student mathematics learning.
ü
Review records frequently to ensure that notes
are being made on each student regularly and summarize information related to
trends in students' learning.
ü
Share anecdotal notes with students and parents
at conferences.
Observation
Checklists
Observing students as
they solve problems, model skills to others, think aloud during a sequence of
activities or interact with peers in different learning situations provides
insight into student learning and growth. The teacher finds out under what
conditions success is most likely, what individual students do when they
encounter difficulty, how interaction with others affects their learning and
concentration, and what students need to learn next. Observations maybe
informal or highly structured, and incidental or scheduled over different
periods of time in different learning contexts.
Observation checklists
allow teachers to record information quickly about how students perform in
relation to specific outcomes from the program of studies. Observation
checklists, written in a yes/no format can be used to assist in observing
student performance relative to specific criteria. They may be directed toward
observations of an individual or group. These tools can also include spaces for
brief comments, which provide additional information not captured in the
checklist.
Before you use an
observation checklist, ensure students understand what information will be
gathered and how it will be used. Ensure checklists are dated to provide a
record of observations over a period of time.
Tips for Using
Observation Checklists
ü
Determine specific outcomes to observe and assess.
ü
Decide what to look for. Write down criteria or
evidence that indicates the student is demonstrating the outcome.
ü
Ensure students know and understand what the
criteria are.
ü
Target your observation by selecting four to
five students per class and one or two specific outcomes to observe.
ü
Develop a data gathering system, such as a
clipboard for anecdotal notes, a checklist or rubric, or a video or audio
recorder.
ü
Collect observations over a number of classes
during a reporting period and look for patterns of performance.
ü
Date all observations.
ü
Share observations with students, both
individually and in a group. Make the observations specific and describe how
this demonstrates or promotes thinking and learning. For example; "Eric,
you contributed several ideas to your group's Top Ten list. You really helped
your group finish their task within the time limit."
ü
Use the information gathered from observation to
enhance or modify future instruction.
Conversations
Learning
conversations are particularly effective in helping students make connections.
There are a number of ways to keep track of learning conversations. For
example:
ü
Record the learning conversations by using a
digital recording device. Either the teacher or students can download the
recording and use audio editing software to identify the most salient parts of
the conversation and add them to their portfolios.
ü
Record the learning conversations by video.
Either the teacher or students can create video recording and use video
editing software to identify the most salient parts of the conversation and add
them to their portfolios.
ü
Record their emerging understandings, working
theories, solutions and reflections through a classroom Web site; e.g.,
chatroom, blogs, wiki. Students can then continue their conversations outside
of school, build on each other's ideas, and have a rich record of how their
knowledge was built and how deep understanding emerged through open
conversation.
Portfolios
A portfolio is a
purposeful collection of student work samples, student self-assessments and
goal statements that reflect student progress. Students generally choose the
work samples to place in the portfolio, but the teacher may also recommend that
specific work samples be included. Portfolios are powerful tools that allow
students to see their academic progress from grade to grade.
The physical
structure of a portfolio refers to the actual arrangement of the work samples,
which can be organized according to chronology, subject area, style or goal
area. The conceptual structure refers to the teacher's goals for student
learning. For example, the teacher may have students complete a self-assessment
on a work sample and then set a goal for future learning. The work sample
self‑assessment and the goal sheet may be added to the portfolio
Work samples from all
curricular areas can be selected and placed in a portfolio. These can include
stories, tests and reflections about work samples.
Effective portfolios:
ü
are updated regularly to keep them as current
and complete as possible
ü
help students examine their progress
ü
help students develop a positive self-concept as
learners
ü
are shared with parents or guardians
ü
are a planned, organized collection of
student-selected work
ü
tell detailed stories about a variety of student
outcomes that would otherwise be difficult to document
ü
include self-assessments that describe the
student as both a learner and an individual
ü
serve as a guide for future learning by
illustrating a student's present level of achievement
ü
include a selection of items that are the representative of curriculum outcomes, and what the student knows and can do
ü
include the criteria against which the student
work was evaluated
ü
support the assessment, evaluation and
communication of student learning
ü
document learning in a variety of ways—process,
product, growth and achievement
ü
include a variety of works—audio recordings,
video recordings, photographs, graphic organizers, first drafts, journals and
assignments that feature work from all of the multiple intelligences.
An essential
requirement of portfolios is that students include written reflections that
explain why each sample was selected. The power of the portfolio is derived
from the descriptions, reactions and metacognitive reflections that help
students achieve their goals. Conferencing with parents, peers and/or teachers
helps synthesize learning and celebrate successes. Some students become adept
at writing descriptions and personal reflections of their work without any
prompts. For students who have difficulty deciding what to write, sentence
starters might be useful.
Question and Answer
Questioning serves as
assessment when it is related to outcomes. Teachers use questioning (usually
oral) to discover what students know and can do. Strategies for effective
question and answer assessment include:
ü
Apply a wait time or 'no hands-up rule' to
provide students with time to think after a question before they are called
upon randomly to respond.
ü
Ask a variety of questions, including open-ended
questions and those that require more than a right or wrong answer.
ü
Use Bloom's Taxonomy when developing questions
to promote higher-order thinking.
ü
Teachers can record the results of question and
answers in anecdotal notes or include them as part of their planning to improve
student learning.
Checklists, Rating
Scales and Rubrics
Checklists, rating
scales and rubrics are tools that state specific criteria and allow teachers
and students to gather information and to make judgments about what students
know and can do in relation to the outcomes. They offer systematic ways of
collecting data about specific behaviours, knowledge and skills.
The quality of
information acquired through the use of checklists, rating scales and rubrics
is highly dependent on the quality of the descriptors chosen for assessment.
Their benefit is also dependent on students’ direct involvement in the
assessment and understanding of the feedback provided.
The purpose of
checklists, rating scales and rubrics is to:
ü
provide tools for systematic recording of
observations
ü
provide tools for self-assessment
ü
provide samples of criteria for students prior
to collecting and evaluating data on their work
ü
record the development of specific skills,
strategies, attitudes, and behaviours necessary for demonstrating learning
ü
clarify students' instructional needs by
presenting a record of current accomplishments.
Tips for Developing
Checklists, Rating Scales and Rubrics
ü
Use checklists, rating scales and rubrics in
relation to outcomes and standards.
ü
Use simple formats that can be understood by
students and that will communicate information about student learning to
parents.
ü
Ensure that the characteristics and descriptors
listed are clear, specific and observable.
ü
Encourage students to assist with constructing
appropriate criteria. For example, what are the descriptors that demonstrate
levels of performance in problem solving?
ü
Ensure that checklists, rating scales and
rubrics are dated to track progress over time.
ü
Leave space to record anecdotal notes or
comments.
ü
Use generic templates that become familiar to
students and to which various descriptors can be added quickly, depending on
the outcome(s) being assessed.
ü
Provide guidance to students to use and create
their own checklists, rating scales, and rubrics for self-assessment purposes
and as guidelines for goal setting.
Checklists usually
offer a yes/no format in relation to student demonstration of specific
criteria. This is similar to a light switch; the light is either on or off.
They may be used to record observations of an individual, a group or a whole
class.
Rating Scales allow
teachers to indicate the degree or frequency of the behaviours, skills, and
strategies displayed by the learner. To continue the light switch analogy, a
rating scale is like a dimmer switch that provides for a range of performance
levels. Rating scales state the criteria and provide three or four response
selections to describe the quality or frequency of student work.
Teachers can use
rating scales to record observations and students can use them as self-assessment
tools. Teaching students to use descriptive words, such as always, usually,
sometimes and never helps them pinpoint specific strengths and needs. Rating
scales also give students information for setting goals and improving
performance. On a rating scale, the descriptive word is more important than the
related number. The more precise and descriptive the words for each scale
point, the more reliable the tool.
Effective rating
scales use descriptors with clearly understood measures, such as frequency.
Scales that rely on subjective descriptors of quality, such as fair, good or
excellent, are less effective because the single adjective does not contain
enough information on what criteria are indicated at each of these points on
the scale.
References
1. A Vision of
Mathematics Assessment. National Research Council. 1993. Measuring What Counts:
A Conceptual Guide for Mathematics Assessment. Washington, DC: The National
Academies Press. doi: 10.17226/2235.×
