Assessment in mathematics

Assessment in Junior High School
Mathematics
Dr. Carlo Magno
De La Salle University, Manila
1

Answer the following questions:
• What is assessment for you?
• When do you conduct assessment?
• How do you assess math skills?
2

Advance Organizer
• Standards in Math (Junior HS)
• Sources of Assessment Information
– Assessment literacy
– Reading Assessment results
– Assessment for Learning
• Formative Assessment
• Guidelines in Developing Items
3

Why do we need standards?
• To make sure that
everyone delivers
quality work
• To produce quality
students
• To deliver quality
programs
• Basis on what to assess
4

Mathematics Standards for Junior HS
• Algebra
– explore the concepts involving a quadratic
function and its graph and solve problems
involving quadratic functions and equations.
– solve equations involving rational expressions
– explore relationships of quantities that involve
variation and solve problems involving direct,
indirect and joint variation
5

– simplify expressions with rational exponents and
solve problems involving them.
– perform fundamental operations on expressions
involving radicals and solve problems involving
expressions and equations with radicals.
6

• Trigonometry
– explore the concept of trigonometric ratios and use
these to solve problems on angles of elevation and
depression and navigation.
– generate an arithmetic and a geometric sequence, find
the sums of the terms in the sequence and solve
problems involving these sequences.
– explore polynomial functions
7

Mathematics Competencies for Junior HS
• Geometry
– use the fundamental theorems of proportionality
– prove and use concepts on triangle similarity,
particularly on similarity of right triangles to solve
problems.
– prove and use theorems involving quadrilaterals.
– find parts of a circle and solve problems involving
the circle and its parts.
8

Mathematics Competencies for Junior HS
– explore geometric figures on the rectangular
coordinate plane.
• Statistics
– describe a set of data using measures of position.
– count occurrences of an event and arrangements
using the Fundamental Counting Principle,
Permutations and Combinations.
– find the probability of compound events.
9

DepEd Taxonomy
• real-life application
of understanding
• enduring big ideas,
principles, and
generalizations inherent to
the discipline
• cognitive
operations that
the student
performs
• content of the
curriculum, the
facts and
information that
the student
acquires
Knowledge Process
Product/PerformanceUnderstanding

Explain
Which of the following statements of the relationship
between market price and normal price is true?
a. Over a short period of time, market price varies directly
with changes in normal price.
b. Over a long period of time, market price tends to equal
normal price.
c. Market price is usually lower than normal price.
d. Over a long period of time, market price determines
normal price.

Translation from symbolic form to another form, or vice versa
Which of the graphs below best represent the supply situation
where a monopolist maintains a uniform price regardless of
the amounts which people buy?
A B C D
S
Price
Quantity
S
Price
Quantity
S
S
Price
Quantity
S S
Price
Quantity
S
Interpret

Apply
In the following items (4-8) you are to judge the effects of a particular policy on the
distribution of income. In each case assume that there are no other changes in
policy that would counteract the effect of the policy described in the item. Mark
the item:
A. If the policy described would tend to reduce the existing degree of inequality in
the distribution of income,
B. If the policy described would tend to increase the existing degree of inequality
in the distribution of income, or
C. If the policy described would have no effect, or an indeterminate effect, on the
distribution of income.
__ 4. Increasingly progressive income taxes.
__ 5. Confiscation of rent on unimproved
__ 6. Introduction of a national sales tax
__ 7. Increasing the personal exemptions from income taxes
__ 8. Distributing a subsidy to sharecroppers on southern farms

Have perspective
After reading the passage answer the following questions…
1. Where was Carol walking?
a. park
b. beach
c. mall
d. city hall
2. How did she feel on this walk?
a. envied
b. sad
c. relaxed
d. happy

Have perspective
3. Carol envied the people around her because they
_____________________.
a. were sad and lonely
b. love the city life
c. were laughing and joking
d. don’t like the city

Empathize
• Your new maid from the mountain destroyed
your very expensive Narra door and she used
it as firewood and cooked rice in your newly
landscaped garden. How should you react?
• A…
• B…
• C…
• D…

Sources of Assessment Information
• Assessment Results
– Classroom Assessment: Quarterly Test, Quizzes
– National Assessment: NAT Results (Grade 6)
17

Sources of Information on Student
Mastery
• Forms
– Formative
– Summative
• Types
– Paper and Pencil
– Alternative forms: Performance, authentic, Portfolio
• Approaches
– Assessment “of” learning
– Assessment “for” learning
18

Assessment Literacy
• (1) Assessment comes with a clear purpose
• (2) focusing on achievement targets
• (3) selecting proper assessment methods
• (4) sampling student achievement
19

Reading Assessment Results
Levels of Proficiency
20

Beginning Developing Approaching
Proficiency
Proficient Advanced
22

• Mathematics NAT
• 15 items
– Place value = 1
– Fraction = 2
– Measurement = 2
– Multiplication = 1
– Division = 2
– Lines = 1
– Addition = 3 (Problem solving)
– Ratio and proportion = 1
– Statistics = 2 (interpreting graphs)
24

Approach in Assessment
Assessment of
Learning
Assessment for
Learning
Assessment as
learning
25

ASSESSMENT OF LEARNING AND ASSESSMENT
FOR LEARNING
Effect of Previous Practices:
rank students on
achievement by graduation
New Expectation: Assure
competence in Math,
Reading, Writing, etc.
• Implications?
Assessment and grading
procedures should help
students succeed.
26

ASSESSMENT FOR LEARNING
• We need to close the gap between
standards and students competencies
• Risk: our society will be unable to
productively evolve in social and
economic sense.
• Assessment is a tool to ensure student
mastery of essential standards.
27

ASSESSMENT FOR LEARNING
• Mistaken beliefs about how to
use assessment to support
school improvement:
1.High-stakes tests are good for all
students because they motivate
learning
2.If I threaten to fail you, it will
cause you to try harder
3.If a little intimidation doesn’t
work, use a lot of intimidation
28

MISTAKEN BELIEFS
4. The way to maximize learning is to
maximize anxiety
5. It is the adults who use assessment
results to make the most important
instructional decision.
29

MISTAKEN BELIEFS
PROFOUND MISTAKE
Teachers and leaders don’t need to understand
sound assessment practices – the testing people
will take care of us.
COUNTER BELIEF
They do need to understand sound assessment
practices.
30

Assessment “for” Learning
• Assessment Crisis: The Absence Of
Assessment FOR Learning
• By Rick Stiggins
31

• School improvement requires:
– the articulation of higher achievement standards,
– the transformation of those expectations into
rigorous assessments, and
– the expectation of accountability on the part of
educators for student achievement, as reflected in
test scores.
32

• When we assess for learning, teachers use the
classroom assessment process and the
continuous flow of information about student
achievement that it provides in order to
advance, not merely check on, student
learning.
33

• understanding and articulating in advance of
teaching the achievement targets that their students
are to hit;
• informing their students about those learning goals,
in terms that students understand, from the very
beginning of the teaching and learning process;
• becoming assessment literate and thus able to
transform their expectations into assessment
exercises and scoring procedures that accurately
reflect student achievement;
34

• using classroom assessments to build students’
confidence in themselves as learners and help them
take responsibility for their own learning, so as to lay
a foundation for lifelong learning;
• translating classroom assessment results into
frequent descriptive feedback (versus judgmental
feedback) for students, providing them with specific
insights as to how to improve;
35

• continuously adjusting instruction based on the
results of classroom assessments;
• engaging students in regular self-assessment, with
standards held constant so that students can watch
themselves grow over time and thus feel in charge of
their own success; and
• actively involving students in communicating with
their teacher and their families about their
achievement status and improvement.
36

Formative Assessment
• Need not be graded as summative
assessments (end-of-unit exams or
quarterlies, for example) are.
• They serve as practice for students
• They check for understanding along the way
and guide teacher decision making about
future instruction;
• they also provide feedback to students so they
can improve their performance
37

• For assessments to be accurate, teachers need
multiple measures of student understanding.
• Teachers need evidence gathered over time in
different ways to evaluate how effective the
teaching and learning process has been.
• Tomlinson and McTighe (2006) suggest that when
teachers gather a "photo album" rather than a
"snapshot" of our students, we can differentiate
instruction based on a more accurate evaluation
of our students' learning needs.
38

• 1. Student friendly targets from the beginning
• 2. Models of strong and weak work
• 3. Continuous descriptive feedback
• 4. Teach self-assessment and goal setting
• 5. Teach one facet at a time.
• 6. Teach focused revision.
• 7. Teach self-reflection to track growth
39

• Group Assessment - allows you to quickly
identify problems or misconceptions, which
you can address immediately.
• Individual assessment - Provide some
feedback to the learner, perhaps in the form
of a brief comment or, at the very least, a
check, check-plus or check-minus, with a brief
verbal explanation about what each symbol
indicates
40

• Summaries and Reflections .Students stop and reflect, make sense of
what they have heard or read, derive personal meaning from their
learning experiences, and/or increase their metacognitive skills. These
require that students use content-specific language.
• Lists, Charts, and Graphic Organizers Students will organize information,
make connections, and note relationships through the use of various
graphic organizers.
• Visual Representations of Information Students will use both words and
pictures to make connections and increase memory, facilitating retrieval of
information later on. This "dual coding" helps teachers address classroom
diversity, preferences in learning style, and different ways of "knowing."
• Collaborative Activities Students have the opportunity to move and/or
communicate with others as they develop and demonstrate their
understanding of concepts.
41

• Formative Assessment can be an integral part
of instruction (Guskey, 2007):
• (1) use assessments as sources of information
for both students and teachers,
• (2) follow assessments with high-quality
corrective instruction, and
• (3) give students second chances to
demonstrate success
42

• By varying the type of assessment you use over the
week, you can get a more accurate picture of what
students know and understand, obtaining a "multiple-
measure assessment ‘window' into student
understanding" (Ainsworth & Viegut, 2006).
• Using at least one formative assessment daily enables
you to evaluate and assess the quality of the learning
that is taking place in your classroom and answer
these driving questions: How is this student evolving as
a learner? What can I do to assist this learner on his
path to mastery?
43

• Response to Intervention (RTI) model
– Tier 1 interventions include monitoring at-risk
students within the general education classroom,
ensuring that each student has access to a high-
quality education that is matched to his or her needs.
– RTI focuses on improving academic achievement by
using scientifically based instructional practices.
– Use alternative assessment which utilizes quality
interventions matched to student needs, coupled with
formative evaluation to obtain data over time to make
critical educational decisions.
44

Techniques in Writing Items
45

Multiple Choice
1. Rene Descartes is
a. a famous Italian.
b. important in mathematics.
c. known for his analytical geometry.
d. the author of many books.
•It is recommended that the stem be a direct question.
•The stem should pose a clear, define, explicit, and singular
problem.
Why is the item faulty?

Multiple Choice
IMPROVED: With which one of the
mathematics field is Rene Descartes
associated?
a. Analytical geometry
b. Differential calculus
c. Discrete mathematics
d. Computational mathematics

Multiple Choice
2. Milk can be pasteurized at home by
a. heating it to a temperature of 130o
b. Heating it to a temperature of 145o
c. Heating it to a temperature of 160o
d. Heating it to a temperature of 175o
•Include in the stem any words that might otherwise be repeated
in each response.

Multiple Choice
IMPROVED: The minimum temperature that
can be used to pasteurize milk at home is:
a. 130o
b. 145o
c. 160o
d. 175o

Multiple Choice
3. Although the experimental research, particularly that by
Hansmocker must be considered equivocal and assumptions
viewed as too restrictive, most testing experts would
recommend as the easiest method of significantly improving
paper-and-pencil achievement test reliability to
a. increase the size of the group being tested.
b. increase the differential weighting of items.
c. increase the objective of scoring.
d. increase the number of items.
e. increase the amount of testing time.
Items should be stated simply and understandably, excluding
all nonfunctional words from stem and alternatives.

Multiple Choice
IMPROVED: Assume a 10-item, 10-minute paper-
and-pencil multiple choice achievement test has a
reliability of .40. The easiest way of increasing the
reliability to .80 would be to increased
a. group size
b. scoring objectivity
c. differential item scoring weights
d. the number of items
e. testing time

Multiple Choice
4. None of the following is a prime number
except
a. 17
b. 64
c. 96
d. 98
•Avoid negatively stated items

Multiple Choice
IMPROVED: Which of the following is a prime
number?
a. 17
b. 64
c. 96
d. 98

Multiple Choice
5. Who is the primary proponent of the Field
theory in solving equations?
a. Ernst Steinitz
b. Manny Paquiao
c. Lea Salonga
d. Mark Twain
•If possible the alternatives should be presented in some logical,
numerical, or systematic order.
•Response alternatives should be mutually exclusive.

Multiple Choice
IMPROVED: Who is the primary proponent of
the Field theory in solving equations?
a. Ernst Steinitz
b. Richard Dedekind
c. Leopold Kronecker
d. Heinrich M. Weber

Multiple Choice
6. Which of the following statements makes
clear the meaning of the word “electron”?
a. An electronic tool
b. Neutral particles
c. Negative particles
d. A voting machine
e. The nuclei of atoms
•Make all responses plausible and attractive to the less
knowledgeable and skillful student.

Multiple Choice
IMPROVED: Which of the following phrases is
a description of an “electron”?
a. Neutral particle
b. Negative particle
c. Neutralized proton
d. Radiated particle
e. Atom nucleus

Multiple Choice
7. What is the area of a right triangle whose
sides adjacent to the right angle are 4 inches
long respectively?
a. 7
b. 12
c. 25
d. None of the above
•The response alternative “None of the above” should be used
with caution, if at all.

Multiple Choice
IMPROVED: What is the area of a right triangle
whose sides adjacent to the right angle are 4
inches and 3 inches respectively?
a. 6 sq. inches
b. 7 sq. inches
c. 12 sq. inches
d. 25 sq. inches
e. None of the above

Multiple Choice
8. If the sum of 70 and 60 is 130, then the sum of
700 and 600 is
a. was greater than 1500.
b. to be a whole number.
c. was less than 900.
d. to be approximately 1300.
Make options grammatically parallel to each other and consistent
with the stem.

Multiple Choice
IMPROVED: If the sum of 70 and 60 is 130, then the
sum of 700 and 600 is…
a. 1300.
b. 1400.
c. 1500.
d. 1600.

Multiple Choice
9. The “standard error of estimate’ refer to
a. the objectivity of scoring.
b. the percentage of reduced error variance.
c. an absolute amount of possible error.
d. the amount of error in estimating criterion
scores.
Avoid such irrelevant cues as “common elements” and “pat
verbal associations.”

Multiple Choice
IMPROVED: The “standard error of estimate”
is most directly related to which of the
following test characteristic?
a. Objectivity
b. Reliability
c. Validity
d. Usability
e. Specificity

Multiple Choice
10. What is the number that raises a base to a
certain power?
a. exponent
b. logarithm
c. tangent
d. cosecant
In testing for understanding of a term or concept, it is generally
preferable to present the term in the stem and alternative
definitions in the options.

Multiple Choice
IMPROVED: Which of the following statements
is the best description of a logarithm?

Reference
Magno, C., & Ouano, J. (2010). Designing
Written Assessment for student learning.
Manila: Phoenix.

• Ability to Recognize the Relevance of
Information
67

• Ability to Recognize Warranted and
Unwarranted Generalizations
68

• Ability to Recognize Inferences
69

• Ability to Interpret Experimental Findings
70

• Ability to Apply Principles
71

• Ability to Recognize Assumptions
72

Reading comprehension
• Bem (1975) has argued that androgynous people are
“better off” than their sex-typed counterparts
because they are not constrained by rigid sex-role
concepts and are freer to respond to a wider variety
of situations. Seeking to test this hypothesis, Bem
exposed masculine, feminine, and androgynous men
and women to situations that called for independence
(a masculine attribute) or nurturance (a feminine
attribute). The test for masculine independence
assessed the subject’s willingness to resist social
pressure by refusing to agree with peers who gave
bogus judgments when rating cartoons for funniness
(for example, several peers might say that a very
funny cartoon was hilarious). Nurturance or feminine
expressiveness, was measured by observing the
behavior of the subject when left alone for ten
minutes with a 5-month old baby. The result
confirmed Bem’s hypothesis. Both the masculine sex-
typed and the androgynous subjects were more
independent (less conforming) on the ‘independence”
test than feminine sex-typed individuals.
Furthermore, both the feminine and the androgynous
subjects were more “nurturant” than the masculine
sex-typed individuals when interacting with the baby.
Thus, the androgynous subjects were quite flexible,
they performed as masculine subjects did on the
“feminine” task.
73
35. What is the independent variable in the
study?
a. Situations calling for independence and
nurturance
b. Situation to make the sex type react
c. Situations to make the androgynous be
flexible
d. Situations like sex type, androgynous and
sex role concepts
36. What are the levels of the IV?
a. masculine attribute and feminine attribute
b. rating cartoons and taking care of a baby
c. independence and nurturance
d. flexibility and rigidity

Interpreting Diagrams
Instruction. Study the following illustrations and answer the following
questions.
Figure 1
74
Pretest Posttest
101. Which group received the treatment?
a. group A b. group B
b. c. none of the above
102. Why did group B remain stable across the
experiment?
a. there is an Extraneous Variable
b. There was no treatment
c. ceiling effect occured
103. What is the problem during the pretest phase of
the experiment?
a. the two groups are nonequivalent
b. the groups are competing with each other
c. the treatment took place immediately
Group B
Group A

Activity 1
• Groupwork
• Form 6 groups and each group is assigned
with one facet of understanding
• Write 4 multiple choice items for each facet of
understanding
• Group presentation will follow
• Working time is only 25 minutes

Activity 1
• Group 1: Evaluating functions
• Group 2: Identifying linear functions
• Group 3: Identifying Quadratic functions
• Group 4: Finding the Properties of a circle.
• Group 5: Evaluating Trigonometric functions
76

Insights on Assessment
1. After the activities, I realized that my test …
2. After the activity, now I know that …
3. After the activity, every time I construct test,
I will …

Assessment in mathematics

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