Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
PROSPECTIVE TEACHERS USING GUIDED MATH TO
DIFFERENTIATE MATHEMATICS INSTRUCTION
Yvonne J. John, Ph. D
Centre for Education Programmes
The University of Trinidad and Tobago
Old Southern Main Road, Curepe
Trinidad and Tobago
Email: [email protected]
1-(868)-642-8888 Ext. 29193
Stephen Joseph, Ph. D
Avril Sampson, MSc
Centre for Education Programmes
The University of Trinidad and Tobago
ABSTRACT
In Trinidad and Tobago large numbers of children found in schools are not developing the
mathematical skills needed to achieve basic numeracy (Ministry of Education, Trinidad and
Tobago National Test, 2013). This study examined the effect of differentiated math instruction
training on prospective teachers’ ability to meet students’ needs. Prospective teachers placed in
primary schools of Trinidad and Tobago participated in a two-week, field-teaching practice
focusing on small group instruction in mathematics. The study used a convergent, mixed-method,
research design aimed at triangulating a single group pretest/posttest quasi-experiment with survey
and focus group responses, and reflections from the sample group of prospective teachers.
Findings of the study revealed that the ability of prospective teachers to meet students’ needs in
mathematics greatly improved with differentiated instruction training in guided mathematics
frameworks.
Keywords: prospective teachers, guided math
1
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
1. PROSPECTIVE TEACHER USING
GUIDED MATH FRAMEWORKS TO
DIFFERENTIATE MATHEMATICS
Mathematics is essential to everything that
children learn in school. Understanding how
children learn mathematics, apply what they
learn to solve problems, and how they plan and
assess that knowledge daily is fundamental to
meeting their needs. Creating a classroom
population of students eager and ready to apply
mathematical concepts to their everyday lives
presents teachers with many challenges and
choices. The purpose for conducting this study
stemmed from specific weaknesses analyzed in
student outcomes based on the Trinidad and
Tobago Primary School National Test (2013),
and observations of classroom practices during
practicum sessions. In Trinidad and Tobago, the
National Test is a standardized test administered
annually to primary school students in Standard
One and Three, in basic subjects of English
Language Arts and Mathematics, and Standard
Two and Four in Science and Social Studies
(The National Test, (n.d.). The major objectives
of this examination postulated by the (Ministry
Of Education, n.d.) are: (1) gathering
information which enables administrators at the
school, district and national levels to make
decisions, (2) identifying areas of the primary
school system that require further investigation,
(3) identifying national norms, (4) comparing
students’ performance by school and educational
districts and (5) tracking students’ progress
through school. The National Test, (2013) report
reveals that Standard Three students in 295 of
536 primary schools (55%) are not meeting
benchmarks in total mathematics (Table 1). Too
many children experience mathematical literacy
problems.
Table 1:Mathematics Performance on National
Test 2013
Status
No.
of
Schools
% Primary
schools
Failing = (< 50%)
students not meeting
benchmarks
Passing =
(≥50%)
students
meeting
benchmarks
Total
295
55%
241
45%
536
100%
in the primary schools of Trinidad and Tobago
continue to give whole group instruction in
mathematics with little or no differentiated
instruction provided, particularly for the at-risk
students. Overall, the statistics show that
students in some primary and special schools of
Trinidad and Tobago continue to fail in basic
areas of mathematics such as number/number
sense, measurement and money, geometry and
statistics (National Test, 2013). In response to
the student data and observation of methods
employed by teachers, the researchers decided to
conduct a study with prospective teachers
pursuing studies in special needs education in
differentiating math using formative assessment
data to group and match instruction to meet the
needs of students in the content areas of
mathematics.
2. LITERATURE REVIEW
Limited research is being done
concerning the role of prospective teachers’
ability to use a framework in mathematics to
meet the needs of at-risk students in Trinidad
and Tobago and the Caribbean. However, a
review of the literature for the use of guided
math has produced some informative articles.
Sammons (2006) postulates that teachers are
finding it difficult to use methods they have
used successfully in the past since these no
longer are working for the children that they
teach. The demands of the new curriculum
standards (MOE, 2014) require new ways of
teaching mathematics in schools. Sammons
states that it is important to develop a model that
offers students the opportunities to develop their
mathematical skills and be able to apply that
knowledge to function independently in the
world of mathematics. Sammons (2006)
suggests a model which comprises the following
components: a classroom environment of
numeracy, morning math warm-ups and
calendar
board
activities,
whole-class
instruction, guided math instruction with small
grops of students, math workshop, individual
conferencing and an ongoing system of
assessment.
These components, Sammons
proposes, will allow the teacher to support
student’s needs.
2.1. Assessment and analysis of data
Moreover, in spite the fact that so many of
our children are experiencing challenges in basic
mathematics skills, it was observed that teachers
2
National Council for Teachers of
Mathematics (NCTM), 2000) stressed the Assessment
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
Principle in the Principle and Standards for School
Mathematics. The two main ideas of the principle are
assessment to enhance students’ learning, and
assessment as a valuable tool for making instructional
decisions.
2.2. Making instructional decisions
On-going assessment provides teachers
with timely information about class and
individual needs, hence effective teachers are
constantly engaged in formative assessment.
Van de Walle, et al., (2014) point out that (The
Council of Chief State School Officers
(CCSSO), 2010; and NCTM, 2000) reiterate that
assessment is not separate from instruction, but
that it should include the critical mathematical
practice and processes that occur in effective
problem-based instructional approaches.
2.3 The classroom environment of numeracy
Numeracy skills are fundamental in
order for children to reach their maximum
potential.
The Ministry of Education of
Trinidad and Tobago has made raising
numeracy and literacy skills a priority across the
new primary schools’ curriculum. Sammons
states that the creation of a classroom environment
which supports numeracy enables students to
build on their acquired knowledge of numbers.
Van de Walle claims that the logistics, physical
environment and tone of the classroom must be
addresed to meet the needs of the students. He
further states, that in attending to the needs of the
students the seating arrangement, grouping
strategies and access to materials must be
considered.
2.4. Effective math instruction
What are the characteristics of effective
math instruction? Research has indicated that it
takes more than a good teacher, in fact, it involves
good teachers, a numeracy rich classroom
environment and a curriculum that has depth and
breadth (Protheroe, 2007). Trinidad and Tobago.
MOE curriculum guide points out that teachers
should emphasize numeracy across the
curriculum. In the classroom, that means
making consistent connection between Math and
real life situations. It also emphasizes a focus on
problem solving, communication, and
representation, critical thinking and reasoning.
Students should be exposed to daily
2
opportunities to read, write and speak Math
(MOE, 2013).
Many teachers are moving away from the
traditional, teacher-centered approach, however, it
is an excellent method for presenting strategies or
for making connections at the beginning of the
lessons and for ongoing review of mastered skills
(Sammons, 2007). (Murray & Jorgensen, 2007;
Small, 2009) state that though it is challenging to
plan a lesson that focuses on a target mathematical
concept or skill that is suitable for whole-group
instruction while meeting the needs of a variety of
students, teachers should consider parellel tasks and
open questions.
2.5. Principles
mathematics
and
standards
of
school
The Equity Principles and Standards for
School Mathematics states: “Excellence in
mathematics education requires equity --- high
expectations and strong support for all students.
All students, regardless of their personal
characteristics, backgrounds, or physical
challenges must have opportunities to study --and support to learn mathematics” (NCTM,
2000, p. 12). Teaching for equity is much more
than providing children with an equal
opportunity to learn mathematics, instead, it is
the ability to attain equal outcomes for all
students by being perceptive to their individual
needs (Van de Walle, 2014). Planning, teaching
and assessing students with exceptionalities,
including students who are gifted pose
challenges for many teachers. Baroody, (1987)
points to the fact that students who struggle with
math may continue to do so because they are
‘instructionally disabled’. The equity principle
points to the fact that there are students who
must be considered if teachers are to address and
maintain equal results – high expectations while
providing for individual differences (Van de
Walle, 2014). This can only be achieved through
small-group instruction, and frameworks such as
guided math.
2.6. Guided math
Guided math according to Newton,
(2013) allows teachers to address students at
their instructional level so that you can take them
to their potential. He reiterates that guided math
allows the teacher to support learning by
grouping students in small instructional groups
to teach them in their zone of proximal
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
development (Vygotsky, 1978). Newton (2013)
states that the goals of guided math are for
students to become proficient mathematicians
who have conceptual understanding, procedural
fluency,
strategic
competence,
adaptive
reasoning and mathematical confidence. Guided
math aims at getting students comfortable with
numbers, operations and mathematical concepts
so that they can work independently with new
and different contexts. Cobb County School,
(n.d.) describes guided math as an environment
in which some students are given the opportunity
to work independent of teacher guidance to
building student’s skills, concepts and strategies
through the use of teacher directed tasks and/or
Math Learning Centers. The teacher pre-selects
a group to observe and conference with for the
purpose of assessing student growth and
development, while noting areas where
additional support is needed. Ideally, Guided
Math should take place daily for at least 15
minutes, but as little as once per week, has a
significant benefit in building student selfreliance, independence and critical thinking
skills (Newton (2013).
a guideline in order to differentiate math
instruction, the consensus in the literature is that
the particulars of ‘how-to' deliver that
instruction should be left to the teacher, yet,
there are core skills, which make up any
differentiated math instruction methodology.
The researchers’ position in this study was to
give prospective teachers the core skills
necessary to differentiate math instruction, and
evaluate the success of an individual teacher's
math instruction on the basis of fulfilling
students' needs.
3. METHODOLOGY AND DESIGN
3.1. Purpose of the Study
The purpose of this study was to
investigate whether training in guided math
frameworks will increase the ability of
prospective teachers to better meet the needs of
students. These prospective teachers were
exposed to the primary school new curriculum
prior to practicum placement in nine primary
schools in Trinidad and Tobago.
2.6. Guided Math groups
3.2. Hypothesis and Research Questions
Students should be grouped by
instructional level, that is, the level at which
instruction is not too easy or difficult but “just
right” for students to work in their zone of
proximal development. Sammons recommends
that guided math groups should be
homogeneous, according to performance on a
variety of mathematical assessment, but should
also be flexible.
The hypothesis for this study was as
follows:
Prospective
teachers
receiving
systematic and explicit training using
differentiated mathematics instruction --- guided
math frameworks --- will be better able to meet
students’ needs in mathematics.
Additionally, the following three
research questions were investigated.
1. What are prospective teachers’ perceptions
about training in diagnostic math assessment
and analysis to meet the needs of students?
2. What are prospective teachers’ perceptions
about training in developmental math
pedagogy aimed at increasing their ability to
meet the needs of students?
3. To what extent do prospective teachers
understand that the instruction in guided
math frameworks will improve their ability
to meet the needs of students?
2.7. ‘How To’ Differentiate Mathematics
Instruction
The problem most teachers face when
differentiating instruction to meet the needs of
students in small-group teacher-led activity is
‘how to’ get everything done and increase
student achievement (Gibson, 2008). There is
no explicit guide to systematic and explicit
instruction in ‘how to’ deliver differentiated
instruction. In fact, it is reported that scientific
research has not provided procedural models to
differentiation, mainly because of the
uncertainty surrounding what differentiation is
and the limited research surrounding how to
implement it in classrooms.
Even though there is no current
standard step-by-step procedure to give teachers
3
3.3. Population and sample
The population for the study comprised
all prospective year-3 and year-4 teachers (inservice and pre-service) completing a Bachelor
of Education degree in Special Needs, Early
Childhood Care Education (ECCE) and Primary
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
Education at the University of Trinidad and
Tobago (UTT).
Participants for this study, 31
prospective teachers (29 females and two males),
were purposively selected from a larger sample
of 49 prospective teachers (46 females and three
males). These participants were registered for the
courses Teaching Mathematics II To Students
With Mild To Moderate Disabilities, and
Engaging In Classroom Practice/Enhancing And
Improving Classroom Practices for Semester 2.
They must also have successfully completed the
courses of Teaching Mathematics I To Students
With Mild To Moderate Disabilities and
Deepening the Field Teaching Experience for
Semester 1. Selection criteria for these
participants required that they participated in a
prior study on differentiating instruction varying
content, process and product to meet the needs of
their students based on students’ interest (Joseph
& John, 2014).
3.4. Research Site
3.5. Research design
The researchers set out to support or
refute the hypothesis, and answer the
aforementioned questions, by utilizing a
convergent,
mixed-method
design
that
triangulates teacher reflections and survey
responses with a single-group, pretest-posttest,
quasi-experiment.
3.6. Single-group,
experiment
pretest-posttest,
quasi-
The dependent variable in this study
was prospective teachers ability to meet student
needs in mathematics, operationalized by three
student
scores
in
prospective
teacher
performance
obtained
from
presentations/demonstration,
ongoing
preparation, and examined field teaching in
math, triangulated with data from prospective
teacher perceptions of themselves shown in
surveys and reflection notes.
This study was conducted in nine (9) of
537 primary schools in Trinidad and Tobago.
The nine primary school sites were assigned on a
quota-sampling basis and contained eight
‘inclusive’ schools (students with/without mild
to moderate exceptionalities) and one of the 16
special
schools.
The
assignment
was
representative of the population of failing (<50%
of the students meeting benchmark on the
Mathematics test of the National Test) and
passing schools (≥50% of the students meeting
benchmark on the total Mathematics test). The
site included nine (9) principals, thirty-one (31)
directing teachers, one hundred and twenty-four
(124) students in thirty-one (31) targeted primary
classes from Infants One to Standard Three
(Table 2).
3.6.1.
Table 2: School Assignment for sample group
Outstanding [4]
The
prospective
teacher
consistently demonstrates the
accomplishment of the criteria
and
surpasses
the
knowledge, skills, disposition
and/or performance skills of
an initial educator
Proficient [3]
The
prospective
teacher
adequately demonstrates the
accomplishment of the criteria
and meets the knowledge,
skills,
disposition
and/or
performance skills of an initial
educator
Developing [2]
The
prospective
demonstrates
Percentage of students
School 1
School 2
School 3
School 4
School 5
School 6
School 7
School 8
School 9
# P.T
Fail (< 50%)
Pass
(≥50%)
0
1
1
1
1
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
4
5
5
3
5
2
2
1
4
6
2
9
31
P.T = Prospective-Teachers
4
Total
Schools
presentation/demonstration
The lecturer in practice assessed
prospective teachers ability to choose, create,
modify and use appropriate resources to match
curriculum content being delivered in classrooms
based on the choice of a teaching strategy to be
demonstrated in front of peers. This assessment
was based on the ability to discuss how and why
the strategy was useful, and to provide
justification that it was the best decision in the
circumstances for meeting the needs of their
children. This assessment was completed using
the Demonstration of an Instructional Strategy
Rubric, adapted from UTT Practicum course
content, (2013), (Table 3).
Table 3: Rubic for demonstration/presentation
teacher
some
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
accomplishment
of
the
criteria and meets the
knowledge, skills, disposition
and/or performance skills of
an initial educator
Beginning [1]
The
prospective
teacher
demonstrates
limited
accomplishment
of
the
criteria. There is much room
for improvement on the
knowledge, skills, disposition
and/or performance skills of
an initial educator.
Not observed [0]
The prospective teacher did
not demonstrate the criteria.
OBSERVED
Relevance of the strategy
 to lesson/context in which it
is used
 to subject area
 to learners’ level and abilities
 theoretical
underpinnings:
Differentiated instruction
etc
Suitability of explanation re: use in
delivery
 Diction / clarity of speaker
 Content presented: name of
strategy, purpose, targeted
learners
characteristics;
construction; other uses etc.
 Time period for presentation
maximized
 Benefits and challenges
Creativity re design of the strategy

Originality of ideas used/
Innovation

Selection of materials used to
create resource

Repeatable design
Technical quality

Visual/tactile stimulation; size of
print, etc.

Durability (can be reused in
other lessons on other occasions,
other subject areas, etc.

Can be used for classroom
display (as a learning tool)

Adaptability: can be used for
teacher
demonstration/student
discovery and manipulation;
with/without teacher supervision.

Intricate details
Effectiveness/Overall impact

Aesthetically pleasing; general
visual impact

Comprehensiveness
of
presentation
3.6.2.
5
on-going preparation
0
1
2
3
4
Prospective teachers were expected to
be fully prepared each day while on field
teaching. All unit and lesson plans were
submitted to and approved by the lecturers in
practice before these lessons were taught. They
were kept in binders and made available for
scrutiny during field visits. The evaluation of this
assignment was based on specific criteria
identified in the on-going preparation rubric,
(Table 4).
Table 4: Ongoing Preparation Rubric [15%]
13 - 15%
Fully prepared for field-orientation and
field-teaching visits. Records of teaching
and
reflections well maintained in a
Practicum Folder and available at all
times.
9 - 12%
Mostly prepared for field-orientation and
field-teaching visits. Records of teaching
and reflections maintained in a Practicum
Folder, but partially incomplete.
5 - 8%
Mostly unprepared for field-orientation
and field-teaching visits. Records of
teaching
and reflections not readily available or
are mostly incomplete.
0—4%
Always unprepared for field-orientation
and field-teaching visits Records of
teaching
and reflections not available.
3.6.3. examined field teaching of math
Prospective teachers planned, delivered,
and were evaluated on lessons taught over the
two-week period. On field-teaching days, each
prospective teacher taught two (2) pre-planned
lessons, inclusive of one math lesson, by
him/herself while being observed by his/her
cooperating teacher or lecturer in practice daily.
The examination of this assignment was based
on specific criteria identified in the differentiated
instruction rubric (Table 5).
Table 5:Tool B4: Differentiated Instruction Rubric
(Chicago Public School n.d.)
KNOWLEDGE-CENTERED CLASSROOM0 1 2 3 4


The lesson is based on clear
objectives (based on
learning goals and content
standards) and all students
are supported to meet those
objectives
Students are presented with
tasks that are appropriately
challenging
(e.g.
from
questioning, student work)
and focused on achieving
Evidence
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
the learning objectives
LEARNER-CENTERED CLASSROOM
01234
student can engage in or
make sense of the content
(based on student interest,
need, learning style, etc.)
Evidence
 The teacher demonstrates an
understanding of individual
student
skills
and
characteristics.
 The classroom instruction and
classroom environment are
culturally sensitive.
PACING
PRODUCT

01234
 The teacher provides adequate
time to gauge student prior
knowledge.
 Students have sufficient time to
meet the lesson objectives
given different learner needs.
GROUPING






AND

6
Evidence
3.8. Intervention
01234
Evidence
01234
Evidence
Based on the research and best practices
encountered in the literature on teaching
mathematics, training was designed to
increase the knowledge and proficiency of
prospective teachers.
The
intervention
approach chosen combined scheduled training in
(a) instruction in designing and implementing
instruction in the domains of conceptual
understanding,
declarative
knowledge,
procedural knowledge and problem solving, and
(b) training in working with guided math
frameworks (Table 6).
The lesson plan reflects
potential struggles students
may have with content
The lesson is differentiated
on one or more of the
following variables: content,
product, process
The teacher has prioritized
what content students are
expected to learn (e.g.
essential
material
vs.
incidental facts/enrichment)
PROCESS

.
01234
CONTENT

The
independent
variable
was
differentiated instruction training in guided math
frameworks and instruction in designing and
implementing instruction in the domains of
conceptual
understanding,
declarative
knowledge, procedural knowledge and problem
solving. In the pretest no instruction was given to
prospective teachers in differentiating math
instruction. This means that in the pre-test stage,
systematic training was not conducted in guided
math frameworks or instruction in designing and
implementing instruction in the domains of
conceptual
understanding,
declarative
knowledge, procedural knowledge and problem
solving.
Evidence
The materials in the
classroom
environment
allow for all learner needs
(i.e. materials at different
levels, allow for multiple
learning styles.
The materials are accessible
by students
PLANNING
The
teacher
has
differentiated the method by
which students gain access
to the content (e.g. audio,
visuals,
guided
notes,
technological assistance)
The teacher has planned for
different activities by which
Evidence
3.7. Independent variable
The teacher utilizes varied
instructional
groupings
(whole group, small groups,
partners, heterogeneous, and
homogenous groups).
The teacher interacts with all
of the different groups during
the course of the lesson.
The
teacher
circulates
among the groups to ensure
students are learning and
provide feedback
PROACTIVE
INSTRUCTION

.
01234
MATERIALS
Evidence
01234
The teacher has planned
various
performance
indicators for students to
demonstrate evidence of
learning (tiered assessments,
choice
of
culminating
products, etc.)
3.9. Schedule
Table 6: Training schedule for prospective teachers
01234
Evidence
Co
W
1
Prac
C
Math
C
W
2
W
3
W
4
W
5
W
6
W
7
W
8
W
9
W
1
0
C
C
C
F
C
C
C
F
F
O
P
P
C
C
C
C
C
C
C
F
F
P
P
W = Week, C = Classroom sessions, FTP = Field
Practice, R/I = Reflection/Instruction, Co = Course
W
1
1
R
/I
C
W
1
2
R
/I
C
Teaching
3.10. Module 1: Assessment and analysis of
data
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
Hands-on training was conducted in the
assessment, and analysis of test data.
ESSENTIAL
QUESTIONS
What open-ended, grade-level appropriate questions will prompt exploration,
innovation, and critical thinking about the big ideas?
3.12. Module 3: Unit and lesson planning
This resource can be used to create a process for developing
lesson plans that outline essential elements of unit design—
curriculum standards, high quality instruction, and a
balanced assessment system. A unit consists of a coherent
series of lessons where concepts and/or skills advance and
deepen over time for all students.
(Adapted from Wisconsin Department of Public Instruction)
GENERAL
INFORMATION
Class: Mathematics
Unit Title : Transportation – How I
get arount
Length of Unit: 2
weeks
Sequence:
UNIT OVERVIEW
Addition: Solving real life problems involving addition (concrete, pictorial and
symbolic modes) (MOE, 1.1.3)
UNIT STANDARDS
Mathematics (Infant 2) Addition Standard [1.1.2], Subtraction [1.1.4]
7
Used
Consistently
throughout
School
Explicitly
Taught and
Assessed
Teacher
Language
addition
subtraction
DATA ANALYSIS &
PRIOR KNOWLEDGE
NEEDED TO SUPPORT
LEARNING
 What is it that students need to know and be able to do
prior to this unit? Add without regrouping, units, tens.
 How will you determine what students know and can do in
preparation for this unit? Pre-assessment
DISCIPLINARY
LITERACY
Table 7: Unit plan
Grade: Infant 2
Which general academic and domain-specific words deepen student
understanding? Decide which words will be included in teacher language,
included in assessment language, known and used by all students, explicitly
taught and assessed in context, and used consistently throughout the school.
Word
Training included systematic instruction
in preparing unit frameworks and lesson plans
that utilize guided math. Table 6 illustrates the
plan of training, which was conducted over a
twelve-week period and practicum placement
followed for a two-week period. All unit/lesson
plans for the Field Teaching Practice (FTP)
assignment prepared were handed in to the
lecturer in practice on week 7 (W7) initial draft
and week 8 (W8) final draft of the training plan.
The final draft of the unit was based on
approval/editing discussions between lecturers in
practice and prospective teachers.
For each of the two FTP weeks, each
prospective teacher designed one unit of work
for Math to be covered in five lessons – shared
math, and five guided math lessons per week,
one for each of the different instructional level
groups in the assigned class. A copy of a
guided/shared unit plan is included (Table 7).
UNIT CONCEPTS AND
ACADEMIC
VOCABULARY
Known and
Used by All
Students
Prospective teachers were given direct,
explicit instruction on how to adjust/manage the
learning environment, transitioning from whole
group instruction to small groups – some
independently run, others teacher-led.
What is/are your learning target/s? What does proficiency look like? How will
you communicate that to students? Students will…
Assessment
Language
3.11. Module 2: Management of the learning
environment
UNIT LEARNING
TARGET(S)
 Which texts will be used?
 Are those texts available in multiple modes?
 Do the texts include a range of print and digital text in
diverse media, lengths, and formats at students’ grade and
independent levels?
 Which texts will be created?
 Do the texts include a range of print and digital texts in
diverse media, lengths, and formats?
SEQUENCE OF
LESSON PLANS
 Sequence the lesson plans that will be taught within this
unit.
o
Adding groups of (concrete objects).
o
adding sets in
(pictorial/representational) form.
o
Adding using (abstract form).
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
INTRA- AND
INTERDISCIPLINARY
CONNECTIONS
 How does this unit connect to other units in the course of
study?
 How does this unit connect to other disciplines?
 How does this unit connect to (theme for level)?
ADDITIONAL
CONSIDERATIONS
 What opportunities do students have to
— be flexible,
— make choices,
— take initiative,
— interact with others,
— be accountable, and
— be a leader?
 How will you provide students with opportunities for
application of skills, student-directed inquiry, analysis,
evaluation, and/or reflection?
 How does technology support teaching and learning?
 Are there other staff members involved (e.g., team taught,
media specialists, guest speakers)?
 Will there be field trips or community projects related to
this unit?
Table 8: Example of a lesson plan
No. of Students: 18
Class: Infant 2
Theme: My Sense of Belonging
Duration: 20 – 25 minutes
Subject: Mathematics– Statistics – Picture Charts and
pictographs
Context: Items around the school or at home can be
collected, classified by a criteria and organized.
Outcomes: Given a plethora of real life items, students with
90% accuracy, will be able to demonstrate the ability to:
1. collect data
2. classify data
3. organize data
4. interpret data.
Vocabulary and Phrases:
Data
Collect data
Classify data
Organize data
Interpret data
Mini-Lesson: Teacher models collecting data from the
classroom environment.
Students are asked to look around the class/school for
items to collect, classify, organize by a specific criteria.
Connect to prior knowledge
Reinforce the vocabulary
Guided practice problems/Check for understanding
Student activities: Small groups
Students will collect, classify and organize their:

lunch kits according to color

pencils by length

foods by food groups vegetables, fruits,
Share:
Reinforce criteria for collecting, classifying and organizing
items.
Reinforce vocabulary
8
ASSESSMENTS
 How will you use benchmark and summative assessments
to elicit direct, observable evidence in order to monitor
and/or measure student learning and inform instruction?
 How will you use the results of your benchmark and
summative assessments to differentiate instruction?
 How will you use the results of your formative
assessments?
 How will you communicate student learning?
 How do students provide feedback about their learning?
 In what ways do students have multiple options to
demonstrate their learning?
 How will your assessments be culturally responsive?
Pre-assessments
Formative Assessments
Summative Assessments: Final Performance Tasks
4. RESULTS
4.1. Quantitative results
Training prospective teachers to use
assessment to determine the need areas of the
students that facilitate formulation of group
structures for small group instruction,
significantly improved their performance on
three dependent variables. A paired-sample t-test
compared systematic training for prospective
teachers in differentiating math instruction in
Semester II with no training conditions in
Semester I. For each independent variable, with
99% confidence, the null hypothesis (mean
differences were equal to zero) was rejected.
These findings support the researchers’
hypothesis that explicit training in guided math
frameworks and instruction in designing and
implementing instruction in the domains of
conceptual
understanding,
declarative
knowledge, procedural knowledge and problem
solving will positively affect prospective
teachers ability to meet students’ needs (Table 9).
Table 9: Paired t-test for prospective teachers scores
before – Semester 1, after – Semester II
Variables
T
P value
Mean
of
Difference
presentations/
demonstrations
Ongoing_preparation
fld_teach_guidedmath
-7.48
p = 0.00000004
-2.14
-2.86
p = 0.007964
-1.73
7.80
P< 0.00001
-9.87
4.2. Qualitative results
The quantitative data analysis was
triangulated with the qualitative data gathered
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
from a survey instrument and the prospective
teachers’ reflections. The results showed that
prospective teachers followed the procedural
guidelines given during training and they were
able to administer and use diagnostic math
assessment to determine students’ needs, form
semi-permanent
groups
by
homogenous
instructional levels, and carefully select matched
instructional materials to deliver guided math
lessons.
Prospective teachers perceived that they
met the needs of students in their growth of
number and operations, measurement and
money, statistics and geometry (Table 9). One
hundred percent (100%) of the prospective
teachers perceived that they met the needs of
their students while using guided math lessons in
the assigned content areas taught.
Table 10: How needs were met during Guided math
Perceived Needs Met
# Prospective teachers
reporting
Number and operations
Statistics
Measurement and Money
Geometry (Tessellations, 3D
& 2D shapes)
Fractions
14
4
5
4
5
Thirty percent (30%) of the prospectiveteachers experienced challenges of classroom
management while instructing students at the
teacher-led stations. Seventeen percent (17%)
experienced challenges of time management and
lack of resources, while three percent (3%)
experienced challenges in the pacing of their
shared and guided math lessons. Table 11
provides the original utterances of some of the
prospective teachers.
Table 10 shows the breakdown of
difficulties encountered during Shared math
identified by the sample of prospective teachers.
Nineteen percent (19%) identified Time
Management, thirteen percent (13%) Hooking or
Capturing Students’ Attention, 7% experienced
challenges from Disruptive Students, and having
students Stay On Task, and three percent
experienced challenges such as Lack of
Resources, and Managing Levels, respectively;
while forty-eight percent (48%) did not
experience challenges.
Table 11: Challenges encountered during Shared
Math
Challenges
Hooking students
Staying on Task
9
# of
prospective
teachers
4
2
Percent
13%
7%
Time Management
Managing Levels
Disruptive Students
Lack of Resources
No challenges to report
Total
5.
6
1
2
1
15
19%
3%
7%
3%
48%
31
100.
DISCUSSION
Conducting training and giving
continuous support to prospective teachers in
implementing differentiated instruction to meet
the needs of students during practicum positively
impacted
their
level
of
performance,
empowering them to better meet the needs of the
students that they taught. The majority of
prospective teachers in the study felt that they
were better able to meet the needs of the students
during their field teaching practice while
conducting their guided math sessions. They felt
that they were able to meet the needs of students
by helping students develop skills number and
operations e.g. in adding to ten. Some felt that it
was the most effective tool to improve a student's
basic math skills, and also to help the students
understand the cultural
and
historical
significance.
Prospective teachers were
observed during Guided Math instruction
directly and explicitly helping students to
establish fundamental skills necessary for
proficient
problem
solving,
strengthen
specifically identified weaknesses, improve
attention to detail, build fluency, and develop
vocabulary knowledge in mathematics.
Most prospective teachers perceived
that they had the ability to better meet students'
needs and they emphasized that they would
continue to use the information and experience
gained in this study, in future practicums.
However, some prospective teachers felt that
they needed strategies to manage the class during
Guided Math with small groups. Others felt that
they needed to pace the lessons more
appropriately, especially in shared math.
Though
managing
small-group
instruction has shown to be difficult for some 3%
of the practicing teachers, while at the same time
managing the independent groups in the
classroom, 48% perceived that their practicum
was successful as they met the needs of students
in various identified ways including managing
and organizing the environment while all
students were appropriately engaged.
Most prospective teachers enjoyed engaging
the students in shared and guided math groups,
since the activities blended with the new primary
school’s thematic, integrative curriculum (MOE,
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
2013). Many reported that it was easy to assign
activities for independently led stations since
they structured and organized the class so that
their expectations were communicated and
articulated to the children. During shared math
most prospective teachers conducted their
instruction in a spirited manner, while others
conducted it using a more structured approach.
Whichever approach they chose, it made learning
math enjoyable for their students. Included in
(Table 11) are the utterances of some prospective
teachers.
Table 11: Reflections on guided math (Excerpts)
Reflections from guided math
“I particularly liked guided math because I had the
opportunity to meet students needs – by targeting what
they needed. Many of my shy students participated in
their small groups and one could see the sense of pride
as they answered questions accurately.”
“ During my guided math groups, students were given
the opportunity to develop and use math strategies. My
students became contented with different types of
problems, and some of them were able to discuss
measurement of the items that they were assigned.”
“After conducting a mini-lesson in class, I took my
students (physically handicap) on a tour of the school to
find cylindrical shapes. The class was divided into small
groups, with four adults controlling a group each. My
students were able to discuss and talk about the various
items around the school that were cylindrical in shape”.
My guided and shared math session was a wonderful
experience for all.”
Most of the prospective teachers reported
that they provided their students with the
opportunities to actively participate in math
activities. Their students learned to focus on the
strategies for measurement and statistics, or
geometry. Students had opportunities to expand
and develop new math vocabulary. Some
prospective teachers reported that students were
given the opportunity to develop and use math
strategies that allow them to become comfortable
with different types of problems.
Prospective teachers reported that their
students benefited from guided math as:
– students who were struggling with some
of the math skills were given the
opportunity to develop their own
strategies for solving problems.
– it promoted self-confidence and the
understanding that there are multiple
ways to solve problems.
– it gave students the opportunity to teach
and support each other to clarify their
thinking as they share ideas.
– it promoted student articulation about a
problem.
10
–
–
it gave the teacher time to observe and
assess student needs.
it supported cooperation among peers.
5.1. Limitations
5.1.1. Access to the students at the respective
sites
The short time span of the Field Teaching
practice, as well as, the timing of the study
limited the results of the study. Prospective
teachers were on the field during the eleventh
and twelfth weeks of a thirteen-week school
term. Cooperating teachers were preparing their
classes for end of term tests, and some schools
were actively preparing for school and/or District
Sports Meetings. Therefore, prospective teachers
had little time to use a systematic approach to
adjusting the learning environment to suit the
teaching of differentiated lessons. While they
were effective in the diagnosis of math needs,
and to some extent in matching instruction to
needs, they perceived that they needed more
practice in managing the rest of the class during
Guided Math.
5.1.2. Longitudinal effects
Most experimental studies are spread over a
long time. In this case, the time from pretest to
posttest was ninety (90) days. The time available
to investigate the research problem and to
measure change within the sample was
constrained by the duration of the assignment.
The researchers were not able to control the
length of the practicum, since the decision for the
length of the study is subject to MOE timetable
and the University’s calendar. Shortness of time
during this practicum also impacted on
prospective teachers ability to appropriately
manage the learning environment. Students need
to learn skills to work independently during
guided math times. The study time was too short
for some prospective teachers to get their
students used to the rules of working
independently. However, future practicums in
the participants’ year-four, which will be longer,
will provide them with the opportunity to
implement rules and provide their students with
opportunities to work with a variety of
classmates, without teacher assistance.
5. 2. Future Research
Based on the results of this study, further
experimental research will be conducted with
Running head: INVESTIGATING GUIDED MATHEMATICS INSTRUCTION
emphasis on classroom management of
independent groups while the teacher conducts
instruction in small groups with guided math
lessons, and time management during shared
math. Further research that will focus on the
organization,
behavior
management
and
development of engaging activities for students
at the independent stations will continue to
empower prospective teachers.
Practitioners may benefit from future
research that tracks the same prospective
students in their fourth year of the Bachelor of
Education programme, as well as the
establishment of quality assurance measures to
ensure the training programme continues to be
administered with fidelity. Also, future research
should be done measuring the impact of
differentiated math instruction on student math
achievement.
6. REFERENCES
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thinking: A developmental framework for
preschool, primary, and special education
teachers. Teachers College Press.
Cobb County School District, (n.d.). Retrieved
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Chicago Public School (n.d.). Tool B4:
Differentiated
Instruction
Rubric.
Retrieved
http://www.chicagopublicschool.org.
Gibson, V. & Hasbrouck, J. (2008).
Differentiated Instruction: Grouping for
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Joseph, S.& John, Y. (2014). Practicum Experiences
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Leone, P., Wilson, M. & Mulcahy, C. (2010)
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Ministry of Education, the Republic of Trinidad
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11
https://drive.google.com/a/moe.edu.tt/fold
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from
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Building Each Student’s Mathematical
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like.
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instruction.
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