ALTERNATIVE ASSESSMENT
IN THE MATHEMATICS
CLASSROOM
Prepared by: Maricris B. Bugasan
Objective:
At the end of the chapter, the students are
expected to:
a.
define what is alternative assessment;
b.
identify some alternative assessment tools,
and;
c.
appreciate the importance of these
assessment tools
Table of Contents
01
Using Alternative Assessment in Mathematics Classroom
Traditional Form Of Assessment
vs.
Carefully Designed Assessments
02
Alternative Assessment
03
Assessment Practices in Mathematics Classroom
04
Suggested Alternative Assessment Practices
Overview
Overview
Assessment
 is an integral part of the teaching and learning process.
 regarded as dynamic.
Educational researchers and teachers
 have suggested and practiced a wide range of
alternative ways of assessing pupils‘ learning to prevail
over the shortfalls of the traditional paper-and-pencil
test since the mid-1980s.
Overview
In the last ten years
 alternative assessment has also gained
increasing attention from educational policy
makers, administrators, researchers, and teachers,
particularly since the early 2000s
Overview
In Singapore,
 the PERI report indicates that there is a need to
shift assessment practices away from an overemphasis on assessment of learning as an endoutcome;
 especially at the lower primary levels and;
 shape mind sets to view “assessment as an
integral part of ongoing processes to support
learning”. (MOE, 2009, p. 30).
Overview
Mathematics teachers
 frequently hear concerns about implementing new
assessment practices in the classrooms;
 understand the value of alternative assessment in
assessing some instructional objectives more validly
than the traditional mathematics test but this
information may or may not be translated well into
classroom practices.
Overview
Basic education teachers in mathematics;
 may not feel confident about designing,
implementing or judging pupils‘ work on alternative
assessment tasks.
•
need to equip them with a set of newer alternative
assessment practices to be integrated into their
classroom instruction.
Overview
This need is congruent with the PERI recommendations
about assessments in that teachers:



must be aware of a range of pupils‘ ability and
learning styles;
be fair to all pupils and free from bias;
as well as delineate and communicate
assessment standards to pupils and parents
Lesson 1.
Using Alternative Assessment
in Mathematics Classroom
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
Educational and assessment policies come and go
but the main aim of assessment will continue to inform
teaching and learning.
Assessment is expected to be among the most
contentious issues in the mathematics classroom.
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
Over the last five years in Singapore, the mathematics
curriculum was revised to place emphasis on:


reasoning, communications and connections;
applications and modelling in addition to heuristics
and;

thinking skills as processes that encompass the
mathematics curriculum.
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
As a result of this revision, there is a need to examine
the:

instructional approach;

types of problems used and;

assessment approach in the primary mathematics
classroom.
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
The Ministry of Education (MOE) in Singapore
announced in July 2010 that by year 2013,


there will be no formal examinations for all primary
one pupils (Davie, 2010);
instead of taking formal examinations, they will go
through bite-size formal assessment.
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
These bite- sized assessments inform the pupil, teacher
and parents about the:

pupil‘s areas of strength and areas to work on in
his or her overall development
In other words, assessment will be continual, rather than
limited to the mid-year and year-end examinations.
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
With all these changes in the lower primary mathematics
classroom;

teachers must use assessment tools such as rubrics to
assess and provide pupils with richer feedback on their
development in both academic and non-academic
areas (MOE, 2009).
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
This is aligned with the established statement on the
concept of assessment in mathematics that was given in
the National Council of Teachers of Mathematics (NCTM)
Assessment Standards for School Mathematics, which
defined assessment as:
“the process of gathering evidence about a student‘s
knowledge of, ability to use, and disposition toward
mathematics and of making inferences from that
evidence for a variety of purposes” (NCTM, 1995)
Lesson 1. Using Alternative Assessment in
Mathematics Classroom
One of the PERI recommendations is:

to balance the acquisition of knowledge with the
development of skills and values through increased
use of engaging pedagogies, more holistic assessment
to support pupils‘ learning and development, and a
stronger emphasis on non-academic aspects within
the curriculum (Fu, 2010).
ALTERNATIVE ASSESSMENT
Traditional Form Of Assessment
vs.
Carefully Designed Assessments
Traditional Form Of Assessment vs. Carefully
Designed Assessments
Carefully Designed
Assessments
Traditional Form of
Assessment


used in all levels of
education, that is, in
elementary, secondary and
even in higher education
often fails to assess deeper
forms of learning.




not only evaluate what students
have learned but:
can motivate students in their
approach to learning;
helping them develop thinking
and problem-solving skills, and;
allowing them to assess their own
understanding of the course
content.
ALTERNATIVE ASSESSMENT
OR
AUTHENTIC ASSESSMENT
Alternative Assessment Or Authentic Assessment
An alternative to standard tests and exams,
provide a:
 true evaluation of what the student has learned;
 going beyond acquired knowledge to focus on what
the student has actually learned by looking at their
application of this knowledge.
Alternative Assessment Or Authentic Assessment
Alternative forms of assessment
 can allow you to see what student can and cannot
do, versus what they do and do not know

tend to evaluate applied proficiency rather than
measuring knowledge, allowing for problem solving
and reflection, rather than merely providing facts as
answers to specific questions.
Authentic Or Alternative Assignments
Alternative or authentic assignments


should involve written and performative measures
so that students can develop meaningful and
applicable skills, and advance their knowledge of
the “how over that of the what”.
meant to help develop disciplinary behaviors in
students, making new connections between existing
skills (Rousseau, 2018).
Assessment Practices in
Mathematics Classroom
Assessment Practices in Mathematics Classroom
As assessment is a critical
part of the teaching and
learning process in
classrooms.
Mathematics teachers need to
keep abreast of new
developments in assessment
and be equipped with the
necessary knowledge and
skills in implementing various
assessment practices.
Assessment Practices in Mathematics Classroom
Traditionally, mathematics teachers:
● have relied on paper-and-pencil tests to assess pupils‘
mathematics learning.
● to reproduce memorised procedures without thinking
about why the processes work and what the answer
means
● does not provide a comprehensive measure of pupils‘
ability.
Assessment Practices in Mathematics Classroom
Traditional assessment techniques also:
● make it difficult to develop inferences about pupils‘
learning that may be helpful in designing new
approaches to improve pupils‘ learning.
Webb (1992) claims that tests are important quantitative
assessment tools, but they do not constitute the totality
of assessment.
Assessment Practices in Mathematics Classroom
Thus in this age of
accountability, teachers
need:
• more varied information about their pupils'
mathematical understanding and
competence.
• to use effective classroom assessments that
measure what pupils are learning and how
they are learning it.
• use the assessment information to plan and
adjust instructional processes.
In other words, classroom assessments should be integrated with the instructional
process for mathematics teachers to understand and strengthen pupils‘ learning.
Assessment Practices in Mathematics Classroom
These are the
key principles
of:
Assessment
FOR learning:
or formative
assessment as
encouraged by
many assessment
experts.
a process by which
assessment information is
used and interpreted by
teachers to adjust their
teaching strategies for
improvement in subsequent
learning and teaching.
Assessment
OF learning:
commonly associated with
national and school
semester examination, is
assessment which gathers
evidence for summative
judgement of pupils‘
performance.
Assessment Practices in Mathematics Classroom
Alternative
Assessment
practices
provide
a
more
comprehensive picture of the pupil
and provide more authentic
information
than
traditional
assessment
practices
which
provide limited information about
pupils‘ understanding.
may be appropriate for the
assessment of non-academic
domains because some of these
domains cannot be assessed using
pencil-and-paper tests.
Assessment Practices in Mathematics Classroom
There are two related
reasons for
implementing
alternative
assessment in
classroom practice.
1. Reveals at a very
concrete level what
the curricular
objectives are.
2. Offers teachers
examples of how
they can activate
their pupils‘ thinking
and learning.
Assessment Practices in Mathematics Classroom
Other terms used for alternative
assessment are:
performance
assessment, practical
assessment, or
authentic
assessment.
Possible alternative
assessment practices include:
practical tests, oral
presentations, journal
writing and openended tasks.
One common aspect of these
methods of assessment is that
they represent “alternatives” to
the more traditional paper-andpencil test formats found in so
many classrooms.
Suggested Alternative
Assessment Practices
Suggested Alternative Assessment
Practices
1.
2.
3.
4.
Practical Tests
Oral Presentations
Journal Writing
Open-ended tasks
Note: All these alternative assessment practices, which
can be easily implemented in the lower primary
mathematics classrooms, are described below.
Suggested Alternative Assessment Practices
1.
●
●
●
●
Practical Test
include the provisions of short-term learning goals
enhanced motivation as well as instant and explicit
feedback (Clarke, 1992).
pupils are expected to use manipulatives, materials and
instruments to deduce mathematical principles.
an effective assessment tool and has a high degree of
assessment validity as the skills are assessed in practice in
the manner in which they have been learnt through the
use of hands-on experiences.
Suggested Alternative Assessment Practices
More examples of practical test items include:
1. Putting objects on the balance to find out which object is
heavier than the tube of glue.
2. Using paper clips to measure the length of the mathematics
textbook on the table.
3. Using the measuring tape to measure the length, breadth and
height of the box on the table.
4. Providing three bags of marbles, A, B and C and kitchen scale
to find out the mass of the bags. Pupils arrange the bags in
order of mass, starting from the lightest to the heaviest.
5. Providing two containers, A and B, filled with water to compare
volume by pouring the water into the two beakers to find out
which container holds more water.
Suggested Alternative Assessment Practices
2. Oral Presentations
● enable pupils to give solutions verbally and the process of
interaction between a teacher and pupils facilitates sharing
of thoughts and clarification of understanding
●
create an opportunity for the teacher to listen to what the
pupils are saying about their thinking about mathematics,
how they communicate mathematically and their
understanding of mathematics using their own words.
Suggested Alternative Assessment Practices
2. Oral Presentations
In addition, according to the Communication Standard for
Grades 6-8, teachers using oral presentation tasks must
provide opportunity for pupils to:

think through questions and problems;
 express their ideas;
 demonstrate and explain what they have learnt;
 justify their opinions;
 and reflect on their understanding and on the ideas of
others (NCTM, 2000).
Suggested Alternative Assessment Practices
2. Oral Presentations
There are two main benefits of using oral presentations.
● First, the teachers can gather information about their
pupils‘ learning of mathematics and use this information
to direct instructional process.
● The pupils‘ can develop communication skills.
● Teachers need to be aware that opportunity for pupils
to be involved in active and meaningful verbal
communication is a critical process for their learning
and knowledge acquisition (Fan and Yeo, 2007).
Suggested Alternative Assessment Practices
2. Oral Presentations
●
●
They also need to give pupils necessary guidance
including clarity of expectations, especially at the early
stage, and in particular, create an encouraging
classroom environment for pupils to engage themselves
in such communication.
As lower primary school pupils are still young, it is
necessary to structure the oral presentation tasks so
that the pupils and teachers are engaged.
Suggested Alternative Assessment Practices
2. Oral Presentations
Examples of oral presentation tasks include:
1. Pupils‘ previous writing tasks on their learning reflection or
perceptions.
2. Pupils‘ solutions to non-routine problems.
3. Pupils‘ previous writing tasks about learning of mathematical
concepts.
4. A chosen idea that is pre-agreed before discussion.
5. To share their problem-solving behaviours.
6. To share results or findings of a learning journey through
mathematics trial.
7. To present pair or group work activity.
Suggested Alternative Assessment Practices
3. Journal Writing
●
●
offers pupils opportunities to reflect on their
learning by writing about their thoughts and
feelings about the mathematics they are learning.
can be a valuable technique to further develop
and enhance pupils‘ mathematical thinking and
communication skills in mathematics
Suggested Alternative Assessment Practices
3. Journal Writing
●
●
●
can also assist pupils to learn how to communicate
mathematically when they try to explain what they have
learnt
help them to clarify their own understanding (Stempien
and Borasi, 1985).
provide opportunities for pupils to self assess what they
have learned
Suggested Alternative Assessment Practices
3. Journal Writing
The following are some examples of mathematics
journal prompts that might help pupils start their
journal writing.
1. The things I want to know about in mathematics are …
2. I knew I was right when......
3. I wish I knew more about......
4. Were you frustrated with this problem? Why or why not?
5. What decisions had to be made when solving this
problem?
6. Is mathematics your favourite subject? Why or why not?.
Suggested Alternative Assessment Practices
3. Journal
Writing
The following examples of mathematical journal prompts assist
pupils to revisit important mathematical concepts and
processes:
1. Explain in your own words what addition means.
2. Explain what is most important to understand about fractions.
3. What would happen if you missed a step in solving a problem?
Why?
4. How many times did you try to solve the problem? How did you
finally solve it?
5. The thing you have to remember with this kind of problem is........
6. What method did you use to solve this problem and why?
Suggested Alternative Assessment Practices
3. Journal Writing
● In addition, Waywood (1992) proposed three forms of
journal writing: recount, summary, and dialogue.
i.
ii.
iii.
recount approach - pupils write what they have
observed in their lessons
summary approach - pupils review what they have
learnt during their lessons
dialogue approach - pupils elaborate on what they
have learnt.
Suggested Alternative Assessment Practices
3. Journal Writing
●
●
In the use of mathematics journals, pupils are
required to express their understanding through
drawing, mathematical formulae or words.
It is thus useful to evaluate pupils‘ understanding
and create opportunity for teachers to provide
feedback to the pupils through journal writing.
Suggested Alternative Assessment Practices
3. Journal Writing
●
●
journal writing as a formative assessment (Miller,
1992).
This form of journal writing should not be graded,
otherwise the pupils may pretend that they
understand everything in the mathematics lesson.
Suggested Alternative Assessment Practices
4. Open-ended Tasks
●
●
elicit a range of responses from pupils including a
chance for pupils to show all that they know about
the relevant content.
provide pupils with the opportunity to communicate
their understanding in depth.
Suggested Alternative Assessment Practices
4. Open-ended Tasks
●
●
Open-ended items ―have more than one answer
and/or can be solved in a variety of way (Moon and
Schulman 1995, p. 25).
In addition to producing a solution, pupils must also
explain their solution process and justify their
answer.
Suggested Alternative Assessment Practices
4. Open-ended Tasks
● a task that is open for pupils‘ process and solution is a
way of motivating pupils‘ high quality thinking (De
Lange,1995)
●
open-ended tasks are exemplars of good questions in
that they advance significantly beyond the surface.
(Sullivan and Lilburn, 2005)
Suggested Alternative Assessment Practices
4. Open-ended Tasks
● require pupils to think more intensely and to provide a
solution which involves more than remembering a fact
or repeating a skill
●
●
offer opportunities for pupils to reveal their
mathematical thinking, reasoning processes as well as
problem-solving and communication skills.
an attempt to make assessment more of an open
process that will benefit both teachers and pupils.
Suggested Alternative Assessment Practices
4. Open-ended Tasks
Consider the following open-ended tasks for the lower primary
levels:
1. Write down five whole numbers between 178 and 202.
2. List five 3-digit numbers that have digit 6 in the tens place.
3. List two sets of five numbers that have a sum of 100.
4. Draw a shape where the sum of all the sides is 36 cm
5. Gilbert and Hazel have 40 postcards. After Gilbert gives a few postcards to
Hazel, how many postcards does Hazel have? Explain your answer.
6. Draw a triangle. Write a number in the centre of the triangle. Write three
numbers in the corners of the triangle that add up to the number in the
centre. Now challenge yourself by choosing greater numbers. Draw and
write as many triangles and numbers as you can.
Suggested Alternative Assessment Practices
4. Open-ended Tasks
●
●
emphasize the importance of thoroughly
understanding concepts and carefully
communicating mathematics, rather than shortanswer items that ask pupils to simply practice rote
memorization of mathematical facts.
One criterion for a good open-ended task is that it
will elicit responses that are amenable to partial
credit according to some established rubric.
Suggested Alternative Assessment Practices
4. Open-ended Tasks
Scoring rubrics with a greater emphasis on making more holistic
judgments on pupil‘s work, with less emphasis on counting up rights
and wrongs.
Rubrics are brief outlines that describe the content and quality of
work needed to achieve a specific grade in an open ended task and
enable the teacher assessing the piece of work to determine the
evidence of pupils‘ understanding and mathematical communication.
Four Alternative Assessment
exemplify how
teachers and pupils
could benefit from
implementing
alternative
assessment in the
lower primary
mathematics
classroom
highlight the
different learning
experiences that
pupils will gain
when they work on
diverse types of
tasks.
are just the first
steps towards
making the use of
alternative
assessment in the
classroom a
meaningful one
where emphasis is
on the process
(reasoning and
thinking) rather
than the product
(final answer).
GENERALIZATION
“All students can learn
and succeed, but not
all on the same day,
in the same way”.