MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY

PHAM THI KIM CHAU

DESIGNNING AND USING LEARNING SITUATIONS
TO SUPPORT THE ASSESSMMENT OF CALCULATION
CAPACITY OF SENIOR PRIMARY SCHOOL GRADERS
THROUGH EXPERIENTIAL ACTIVITIES

Major: Theories and Methodologies in teaching Maths
Code: 9.14.01.11

SUMMARY OF DOCTORAL DISSERTATION
ON EDUCATIONAL SCIENCE

NGHE AN - 2019


This dissertation was ompleted in: VINH UNIVERSITY

Scientific supervisors:
1. Prof. PhD. DAO TAM
2. PhD. PHAM XUAN CHUNG

Reviewer 1: Assoc. Prof. PhD. Nguyen Thi Kim Thoa
Reviewer 2: Assoc. Prof. PhD. Tran Ngọc Lan
Reviewer 3: PhD. Thai Huy Vinh

The dissertation was successfully defended at the Boards of Examiners
of Univeristy level, in Vinh University, No 182, Le Duan Street, Vinh city, Nghe An

province
at……., on………..2019

This dissertation is available at:
1. National Library of Vietnam
2. The Information Center – Library of Nguyen Thuc Hao, Vinh University


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INTRODUCTION
1. Motivation of the study
Primary school is the foundation level, students inherit preschool calculation
operations such as: Counting, measuring, estimating, surveying, predicting, ... Students'
thinking which bears experience, perception often base on activities with visual objects or
models. Therefore, in teaching as well as in assessment, it is necessary to create
opportunities for students to take part in and experience. The 2018 general education
program for Math emphasizes the need to create opportunities for students to experience.
Math with its outstanding advantages, create many opportunities to develop students’
calculation capacity.
Vietnam is implementing its process of education reform. The formation and
development of capacity as the ultimate goal of education. Talking about capacity, we touch
upon all elements of the teaching process, in which assessment should be considered as a
teaching activity. Circular 22/2016 / TT-BGDĐT states: The assessment must be for the
student's progress or for students’ study improvement and for learning activities.
Assessment not only allows teachers to judge whether students meet the learning
requirements, but also helps students form and develop their capacity; help students realize
the mistakes from which they adjust their learning activities and teachers have a
foundation

to adjust teaching activities. The general education program in Maths


emphasizes that assessing students' ability must be done by evidence that demonstrates the
results achieved in the process of students’ performing actions. The directive points of
view are the guideline for the study of approaches to disclosure of students' abilities in the
process of calculation operations
Currently, coordination between students’ performance assessment and classroom
teaching activities through student behavioral demonstration is an urgent issue.
Furthermore, the problem of learning situation design has been interested by many
scientists, but the design of learning situations supporting the assessment of calculation
capacity of senior primary school graders through calculation activities has not been
thoroughly studied. For the above reasons, we choose the research topic of the thesis:
“Designing and using learning situations to support the assessment of calculation capacity
of senior primary school graders through experiential activities”.
.


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* Literature review
Regarding to calculation capacity, in the UK, the term "numeracy" is used.
According to Crowther Report (1959), numeracy means broad scientific knowledge. By
1976, numeracy was understood as the ability to use skills with numbers and concepts in a
practical context (Callaghan, 1987). Cockcroft (1982) defines numeracy as the necessary
skills and arrangements of ordinary people in work and daily life. These definitions focus on
skills with numbers, measurement, data processing, applying math skills to solve problems
in a specific context.
In the US, Mathematical literacy of PISA is nearly synonymous with numeracy,
emphasizing mathematical connections in many situations. Cambridge's 2017 elementary
mathematics program attached great importance to developing students’ calculation skills in
term of four aspects : listening-observing-dong and sharing and four supporting elements:
Audio learning-Visual learning-Interactive learning-Shared learning facilities.

In Russia, V.A. Kruchetxki considers that calculation capacity is the ability to
perform fast and accurate calculations, usually mental arithmetic. In Australia, in National
Report on Schooling in Australia (1997), the authors stated: Numeracy is an effective use of
mathematics to meet the general needs of daily life, in paid work, and in community life and
citizens. Numeracy in practice: Teaching, learning and using mathematics emphasized on
the effective teaching skills such as interdisciplinary, integrated teaching, in which
mathematics has a more proactive role than other subjects.
In Vietnam, calculation capacity used to be understood as human understanding and
confidence when using numbers and calculations. Pham Van Hoan (1981) concluded that
the calculation capacity was the quick and accurate performance of calculation, even in
mental arithmetic. Recently, the Ministry of Education and Training listed calculation
capacity as tool capacity in 8 core capacity groups. The 2018 General Education Program
the calculation capacity is seen as the specific capacity of students of general education
level, with mathematical ability being the most focused demonstration of calculation
capacity.
In terms of capacity assessment, Wolf (2001) argues that competency assessment is
based on the description of specific output products, so clearly that teachers, students and
stakeholders can visualize relative relativity. Relevant and accurate student achievement
after the learning process. The PISA international assessment program not only evaluates
knowledge and skills in the General Education program, but also focuses on 04 areas


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(understanding math, reading comprehension, scientific understanding, problem solving).
The issue of assessing the capacity of primary school students was also studied in TIMSS
(2011) through a system of questions and ways to quantify the capacity of primary students.
In Vietnam, recognizing the importance of capacity assessment, the Ministry of
Education and Training issued Circular 30 of 2014, Circular 22 of 2016 on assessing
primary students which emphasizes "Assessing primary students through activities of
observing, monitoring, exchanging, examining and commenting on the learning process of

students", "..through observing the students’ experiential activities", "assessment as a
learning method ". The capacity assessment has attracted many interested scientists such as:
Nguyen Cong Khanh (2014) in Renewal of examination and assessment of students
according to capacity approach. Nguyen Duc Minh (2014) in Guide for teachers to assess
the capacity of senior primary school graders. Nguyen Khai Hoan (2015) in Assessment of
students according to competency approach. Nguyen Thi Lan Phuong (2016) in The
capacity-led program and learners’ capacity assessment.
In general, the above studies only focus on students’ capacity assessment issues. The
problem of using learning situations to support assessing calculation capacity of senior
primary school graders has not been studied in a comprehensive way, so the research topic
“Designing and using learning situations to support the assessment of calculation capacity
of senior primary school graders through experiential activities” is an urgent issue.
2. The purpose of the study
The dissertation studies how to design learning situations and demonstrates that
designed learning situations can support assessing the calculation capacity of senior primary
school graders through experiential activities. Accordingly, the dissertation proposes
orientations to improve students' calculation capacity in the teaching process, contributing
to improving the effectiveness of mathematics teaching in primary school.
3. The subject and scope of the study
- Subject of the study: Learning situations support to the assessment of students’
calculation capacity through the demonstration of senior primary school graders during the
experiential process.
- Scope of the study: maths in senior primary school classes, students’ experience on
learning situation in class, assessment of the teaching process.
4. The task of the study


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- Studying relevant theoretical basis to propose concepts, components of calculation
capacity of the elementary school students and corresponding expressions; Concepts,

methods and tools for assessing the calculation capacity of senior primary school graders
through experiential activities.
- Researching the current situation of designing and using learning situations to
support the assessing the calculation capacity of senior primary school graders through
experiential activities, students’ mistakes in calculation activities .
- Proposing the process of designing learning situations, the process of testing the
designed learning situations, the process of using learning situations to support the
assessment of calculation capacity, the orientations to improve senior primary school
graders in the teaching process.
5. Methodologies of study
- Theoretical research method: Studying documents and situation of domestic and
international research on calculation capacity, designed learning situations, experiential
activities and capacity assessment to form the theoretical rationale of the dissertation. A
priority analysis and prediction of students' calculation plans will be done based on
mathematic knowledge and psychology.
- Case study: In chapter 1, we chose some 5th graders by random to examine the
mistakes of the senior primary school graders in the calculation activity, selected some
grade 4 investigate the difficulties and mistakes in calculation by senior primaru school
graders, 4-5 primary teachers to survey the current situation of designing and usinglearning
situations in assessing the calculation capacity of senior primary school graders. In chapter
2, we chose a few groups of students in grade 4 (or grade 5) if the content of the learning
situation is related to grade 4 (or grade 5) to examine the students’ demonstration in
performing the calculations during experience process. Similarly, in chapter 3, for each
learning situation, we chose 2 students by ramdom to identify each student's calculation
capacity. The results of this survey are the basis for us to identify the situation, the
feasibility of the learning situations and the calculation capacity of students.
- Observation - investigation: Use teacher survey forms to collect the situation of designing
and using learning situations in assessing students' calculationg capacity. Observe students'
computational experiential activities in learning situations that combine the use of student
surveys to collect their mistakes in the calculation. That is an important practical basis in the



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thesis, in adjusting learning situations, in supporting the assessment of calculation capacity
and in the orientation of improving the calculation capacity of students.
- Educational experiment: Organizing for students the calculation experience on the
designed learning situation, combining post-analysis to determine whether the learning
situation is appropriate and how to adjust fit, test modified learning situations and continue
to adjust if not appropriate (in chapter 2). When the learning situation is feasible, we
conduct experiments to evaluate students' computational competence (in chapter 3).
Because our assessment objective is not to classify students but to improve students'
computational capacity, we do not stop at assessment results, for students who are not
qualified to complete Study situation, we help children to successfully solve the learning
situation. The organization for students to experience in chapter 3 just to collect evidence to
determine the calculation capacity of the students to demonstrate the learning situation
designed to support the assessment of the students' calculation capacity, support help
improve students' calculaition capacity.
6. Scientific hyperthesis
If determining the calculation capacity components of the elementary school students
and their respective expressions, designing learning situations and appropriate assessment
tools, it is possible to support the assessment of ability. Student's calculation force.
Contribute to improving students' calculation capacity and improving the effectiveness of
elementary mathematics teaching.
7. Contributions of the dissertation
7.1. Theorietical contributions
- The system of theoretical bases of experience, calculation activities and capacity
to calculate, evaluate and evaluate calculaiton capacity, learning situations. Proposing
concepts and components of calculation capacity of the primary school students and
corresponding expressions, characteristics of the learning situation; Concepts, methods
and tools for calculating calculation capacity of elementary school students through

experiential activities.
7.2. Practical contributions
- Setting up the process of designing learning situations for students to
experience calculations.
- Setting up the process of testing the designed learning situations.


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- Setting up the process of using learning situations to support the assessment of
calculation capacity of senior primary school graders through experiential activities.
- Proposing orientations to improve the calculation capacity of senior primary school
graders in the teaching process.
8. Theorietical poits to be defended
- Designed learning situations offer opportunities not only for senior primary school
grdaers to reveal their calculation activities but also for teachers to investigate the students'
calculation activities, thereby serving as supporting tools to the assessment of calculation
capacity as well as measures to improve the calculation capacity of students.
- Learning situations create opportunities not only for senior primary school graders
to experience calculation but also for teachers to test the feasibility of learning situations.
9. Organization of the dissertation
In addition to the introduction, conclusions, references, appendices, the main content
of the thesis consists of three chapters:
Chapter 1: Theoretical and practical basis
Chapter 2: Designing and experimenting learning situations to support the
assessment of calculation capacity of senior primary school graders through
experiential activities
Chapter 3:

Using learning situations to support the assessmnent of calculation


capacity of senior primary school graders through experiential activities

Chapter 1
THEORETICAL AND PRACTICAL BASIS
1.1. Experiential activities of senior primary school graders
1.1.1

Conceptions on experiential activities of senior primary school graders

There have been many kinds of perpectives and classification of experiential
activities; however, the experiential activities of senior priamry school graders with two
kinds in this dissertation.
- Experience for changing strange phenomenon to familiar ones to students: For
unfamiliar situations, students can not immediately apply calculations, formulas, rules and
procedures for calculations but it requires that the problem must be changed either under


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understandable of processed forms so that the calculated contents can be revealed, that is to
turn the unfamiliarities to familiarities.
- Experience for connecting maths with the reality: Because the dissertation’s
approaches are in connection with learning situations in class, it is necessary to connect
mathematics with the reality by organizing such in-class activities as cake division, game
participation, ... or to create realistic simulation through hypothetical situations.
1.1.2

Learning by experience

Students’ participation in experiential activities, meaning that they are directly
involved in the activity, results in creating new experiences and abilities. The level of

experience shown on the participants’ self-improvement is reflected in the change from
narrow knowledge to broad understandings, from unable application to successful
application, from mechanical application to flexible application, from flexible application to
creative application.
1.2 Calculation activities of senior primary school graders
1.2.1 Conceptions on calculation activities of senior primary school graders
The calculation activity of senior primary school graders can be understood as the
activities the students do to change the strange maths problems to familiar ones, thereby
using the existing knowledge to solve the learning situation. Thus, the calculation activities
may include: using calculations, formulas, rules and processes; using math tools; using
thinking manipulations; using math language and mathematical models.
1.2.2

In class and out of class calculation activities of senior primary

school graders
Students’ experiential calulation activities are organized activities, initially
implementing external (speaking, writing, doing, creating) forms through interaction and
communication, then transforming into the internal forms (analysis, synthesis,
generalization, ...), which gradually turns into their capacities
1.2.3 Knowledge enhancing calculation activities of senior primary school graders
Knowledge plays the role in orienting, adjusting students’ calculation activities.
Therefore, weak knowledge will lead to difficulties or easy occurence of errors in
calculation.
1.2.3.1. Knowledge regarding to calculation methods
1.2.3.2. Knowledge regarding to dialectical materialism in phylosophy


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1.2.3.3. Knowledge regarding to the capacity of knowledge association and

mobilization
1.3 Calculation capacity of senior primary school graders
1.3.1. Conceptions on calculation capacity of senior primary school graders
Calculation capacity of senior primary school graders is their capacity to process
information, the relationship of quantity in solving maths problems in primary schools
1.3.2. Components of calculation capacity of senior primary school graders
1.3.2.1. Foundations to propose components calculation capacity of senior primary
school graders
In order to propose the calculation capacity components of senior primary school
graders, we rely on existing research on the components of calculation capacities, the
objectives and tasks designed in textbooks for senior primary school graders, the contentos
maths textbooks, the cognitive characteristics of senior primary school graders in math
learning, the reality of calculation capacities of senior primary school graders, and their
calculation activity.
1.3.2.2. Proposing the components in the calculation capacity of senior primary
school graders
Basing on the aboved-mentioned backgrouns, we propose components in calculation
capacity of senior primary school graders as follow:
a) Capacity to use calculations, formulas, rules and processes: The capacity to use
calculations, formulas, rules, processes in calculation activities in elementary schools can be
defined as the ability to identify and recall these capacities correctly and apply them directly
in calculations on a specific context.
b) Capacity to use maths tools: The capacty to use math tools in calculation
activities in elementary schools can be defined as the capacity to connect the contents of the
given maths problems with mathematical tools effectively in their direct calculation on a
specific context.
c) Capacity to use thinking operations: The capacity to use thinking operations in
calculation activities in elementary school can be defined as the capacity to perform
thinking and thinking styles to mobilize knowledge effectively in calculation activities
on a specific context.



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d) Capacity to use math language: The capacity to use math language in elementary
school math is reflected in the ability to understand, convert, express math language
(speaking, writing) to effectively change the maths problem in calculation activities on a
specific context.
e) Math modeling capacity: The capacity to model mathematics in calculation
activities in elementary school is reflected in the ability to understand, establish,
convert, explain and reflect mathematical models effectively in calculation activities
on a specific context.
1.3.3. The demonstrations of calculation capacityof senior primary school
graders
1.3.3.1. Fundations to propose the demonstrations of calculation capacity of senior
primary school graders
In order to propose the demonstrations of calculation capacity of senior primary
school graders, we base on the calculation behavior by the senior primary school graders
and the existing studies on the demonstrations of students' calculation capacity.
1.3.3.2. Proposing the demonstration of calculation capacity of senior primary
school graders
Basing on the above-mentioned foundations, we propose the demonstrations of
calculation capacity of senior primary school graders as follow:
a) Demonstrations of capacity to use calculations, formulas, rules, processes:
Implement four arithmetic operations smoothly; manipulate calculations, formulas, rules
and processes in familiar situations, apply calculations, formulas, rules, processes in
slightly different situations, apply calculations, formulas, rules and processes in
unfamiliar situations.
b) Demonstrations of capacity to use math tools: Use math tools as what has been
introduced and practice in familiar situations, use math tools in a slightly different situation,
use math tools in unfamiliar situations.

c) Demonstrations of capacity to use thinking manipulations: Present the correct
order of performing calculations in expressions, check the calculations, results and
calculation process, use simple inferences, able to monitor and evaluate a series of available
arguments, use existing knowledge to identify factors for calculation in the current situation.
adjust thinking process to another direction if the current calculation approach fails, develop


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a series of arguments in the calculation process. use thinking manipulations to find a
calculation plan in an unfamiliar situation, predict, propose appropriate hypothesis for the
calculation process. transform the calculated contents to change the unfimiliarities to
familiarities, summarize, reduce the calculation rules to put the cumbersome calculation on
simple calculation, consider the problem in many respects; do not be content with a
calculated way, reach for a fast, unique calculation, and breakthrough leap in reasoning,
point out the evidence to justify the correctness of the problem, evaluate the calculation
methods for the best option, generalize the results of calculations.
d) Demonstrations of capacity to use math language: Say or write the names of
learned maths, interprete calculations and results, dentify familiar symbols and forms in
known situations, explain familiar standard representations, switch between representations,
explain the process and calculation results; withdrawn relationships or statements;
mathematical expressions. associate, coordinate between different representations for
calculation (diagrams, tables, shapes, letters, symbols), understand and dealing with
languages and forms of expression in unfamiliar situations, explain the process and
calculation results in many ways; complex relationships or newly drawn statements. switch
between different types of performances to facilitate calculation and convert language easily
from unfamiliariy to familiarity. use performances to explain the correctness of the problem.
Combining representations to create ways of calculating problems, convert math language
to natural language to express the meaning of math knowledge in the reality.
e) Demonstrations of capacity to perform mathematical modeling: Identify familiar
mathematical models in similar situations, knowing how to create mathematical modeling

for slightly different situations or unfamiliar situations, convert, perform both forwarding
and conversing interpretation for mathematical models and situations, demonstrate and
evaluate the solution in a real context to consider the feasibility of the established model,
change to model if the calculation is not appropriate.
1.4. Learning situations to support the assessment of calculation capacity of
senior primary school graders through experiential activities
1.4.1. Conceptions on learning situations to support the assessment of calculation
capacity of senior primary school graders through experiential activities
Inheriting the conceptions of teaching situations, we consider learning situations that
support the assesment of senior primary school graders in experiential activities as the


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presentation or simulation of events, setting up learning activities to offer an unresolved
problem, requiring students to experience calculation activities to solve it. The learning
situations are in two forms:
- Pure mathematical situation: is a situation that solves mathematical problem and
relates only to requirements related to mathematical knowledge. The pure mathematical
situation can be divided into two types: The situations needing no visual representation and
the situations that need support of visual representation.
- Practical situation: A situation in which its subjects contains elements of
practical contents, in which at least one question / request / task exists and requires
students to experience calculation activities for solution. The practical factors here are
practical contexts, including: In interdisciplinary learning, personal or school life
activities, entertainment activities or social participation in the society.
1.4.2. The characteristics of learning situations to support the calculation capacity
of senior primary school graders through experiential activites
We know that not all calculation tasks can be fully simulated in a situation. However,
to encourage students to express the demontration of calculation capacity, the learning
situation is required to ensure the following characteristics: The learning situation must

contain requirements for students to experience; elements of calculation capacity, an
existence of conflict, motivated activities, and finally the subjects of activity.
1.4.3. The use of learning situations to support the assessment of calculation
capacity of senior primary school graders in experiential activities.
Depending on the purpose of use, the learning situation has its own functions. If the
sheet is used order to collect students' demonstrations of calculation capacity, it is necessary
to print out the contents of the study situation on A4 paper and give them to the students.
We call it a learning situation card and students write down their calculation on the blank
space. At that time, the learning situation sheet is considered as an assessment tool. If the
sheet is used to encourage students to demonstrate their calculation behavior, then the
learning situations can be viewed as an opportunity for students to experience and reveal
calculation capacity. If it is used to improve students' caculation capacity, the learning
situations can be considered a teaching solution. The dissertation is concerned with the
students’ experience on learning situations and emphasizes the teachers’ experience in
designing and testing learning situations.


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1.4.4. Assistance sheets
Each student's ability is different, so facing with the same problem, some may be
creative independent in solving while others will need help. Instead of instructions, we
designed an assistance sheet to assist students in experiential calculation activities. This is
also the foundation for us to propose the orientation to improve students' calculation
capacity in Chapter 3. The content of the assistance sheet contains suggestions and
orientations for students to do calculations
1.5. Assessing calculation capacity of senior primary school graders through
experiential activites
1.5.1. Conceptions on assessing the calculation capacity of senior primary school
graders through experiential activities
On inheriting the research on assessing students’ capacity, we believe that assessing

the calculation capacity of senior primary school graders through experiential activities are
the process of making judgments, drawing conclusions or judgments about the level of
students’ calculation; explaining the progress of students' calculaiton capacity. In particular,
the judgments, conclusions and judgments are based on analyzing the information
systematically collected in students’ behavioral expression in the process of experiential
calculation activitiies in learning situations.
1.5.2. Methods to assess the calculation capacity of senior primary school graders
through experiential activities
Calculation capacity is revealed through the students' behavorial expression during
experiential activities while behavior, attitudes, ... are difficult to be quantified and
quantitative assessment methods are difficult to access. On the other hand, due to the
characteristics of elementary students, which are often unplanned and unrestrained,
observing and explaining student behavior is often easier and more accurate. Therefore, we
conduct qualitative assessment by observing and studying students’ learning products.
1.5.3. Tools for assessing the calculation capacity of senior primary school graders
through experiential activities
Behavorial demonstrations are evidence to assess students' calculation capacity,
which are accessed by many different tools. The dissertation is concerned with scales
(assessment of each component of calculation capacity, table determining calculating


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capacity levels) and assessment tools (observation sheets, video clips / photos, learning
situation sheet, assistance sheet)
1.5.3.1. The scale to assess the calculation capacity of senior primary school graders
through experiential activities
- Level 1: Students cannot recall mathematical objects, concepts, properties, models
related to calculations in the current situation; or they can recall them but they do not work
in the current situation.
- Level 2 (Remembrance, recreattion): Students can recall, apply objects, concepts,

properties, mathematical models related to calculations in the current situation.
- Level 3 (Connection, integration): Students can link information for calculation
applicable in solving simple problems; create connections in different representations; read,
explain math languages and their relationships. The demonstration at this level are built on
the level 2 of calculation by bringing the calculation into a context that is not completely
familiar but still has an almost familiar structure.
- Level 4 (Reflection): Students can identify calculation content in situations; use
math knowledge to solve situational problems; do mathematical analyzing and reasoning.
These demonstrations relate to the ability to find calculation plans and perform calculations
in unfamiliar situations. It includes elements that reflect on processes needed or used to
solve problems.
1.5.3.2. Video clips and photos
Students’s expression behavior is diverse, complicated, and it can be timely
observed or it is not activated during the observation process. Therefore, the observer
needs to arrange video cameras / cameras to save the information, so as to test and
evaluate it and review it when it is needed.
1.6. The current situation of designing and using learning situations to support
the assessment of calculation capacity of senior primary school students through
experiential activities
1.6.1. The current situation of teachers’ awareness on the conceptions of
calculation capacity by senior primary school students
Many teachers have been vague not only of the pconceptions of calculation capacity by
senior primary school graders but also determining the students’ calculation activities.


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1.6.2. The current situation of teachers’ awareness on the conceptions of
calculation capacity by senior primary school graders
Most of the teachers agree that the student’s demonstration on calculation capacity
remained too low.

1.6.3. The current situation of innovation in assessing the calculation capacity of
senior primary school graders
1.6.4. The current situation of using methods and tools to assess the calculation
capacity of senior primary school graders through learning situations
1.6.5. The reality of designing and using learning situations to support the
assessment of calculation capacity by senior primary school graders through experiential
activities
Because teachers have never used learning situations to assist the assessment of
students’ calculation capacity they didn’t give any suggestions on the process of designing,
experimenting and using of learning situations to assist assessment.
1.6.6. The current situation of calculation capacity of senior primary school
graders in their calculation activities
Many teachers admitted that students’ calculation capacity was not high enough
1.6.6.1. Some difficulties and mistakes in applying calculation operations, formulas,
rules and processes
1.6.6.2. Some difficulties and mistakes in using calculation tools
1.6.6.3. Some difficulties and mistakes in thinking and speculating activities
1.6.6.4. Some difficulties and mistakes in using mathematical language
1.6.6.5. Some difficulties and mistakes in methematical modelling
1.6.6.6. Some other difficulties and mistakes
1.7. Conclusion for Chapter 1
From the data analysis has revealed that there is not clear about the awareness of
teachers on the demonstrations of calculation capacity. Assessments in primary schools
usually are based on test scores which are a little regard with the evaluation of students'
experience in the learning situation. Therefore, there is necessity approach for teachers to
take not only change their perceptions but also in their implementation. The performance of
students’ calculation capacity needs to be described in details. It is needed to establish a
scale of calculation capacity along with appropriate criteria, set up an observation sheet of



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students' calculation behavior on each element of computational competence to ensure
consistency between observations of a student or between students. On the other hand,
students often encounter difficulties in making calculations. These mistakes should be
illustrated in the learning situations to determine the student's calculation capacity and
improve the student's calculation capacity.
To respond to the requirement, chapter 1 has set of views on experience activities,
calculation capacity, and ability assessment, learning cases including analysis, and some
comments.

Thus, the thesis proposes types of computing experience, concepts, and

computational competence components and corresponding demonstrations of senior
students at elementary school; concepts and characteristics of learning situations to support
evaluating calculation capacity through experiential activities; calculation methods and
evaluation tools. The introduction of calculation competency is also the basis for designing
the observation sheet of students' calculation behavior. The introduction of specific learning
situations to create opportunities for students to interact, experience, and disclose
calculation activities; supporting helping sheet to encourage students to overcome mistakes
and orientations for calculation; video clips to record the process of students 'calculating
experience, as a demonstration of evaluating students' calculation capacity. This issue will
be further illustrated in chapter 2 and chapter 3. The results of the research of chapter 1
which have published in articles in magazines and conferences: Hanoi University of
Education Journal [ 2], Journal of Educational Science [1], Education Journal [4], Vietnam
Journal of Education [5], International Workshop at Vinh University [3].

Chapter 2
DESIGNING AND EXPERIMENTING LEARNING SITUATIONS
TO SUPPORT THE ASSESSMENT OF CALCULATION CAPACITY OF
SENIOR PRIMARY SCHOOL GRADERS THROUGH EXPERIENTIAL ACTIVITIES


In this chapter, we designed learning situations. With each designed learning
situation, based on the knowledge of mathematics and psychology, we analyze and analyze
the students' calculation plans. However, the designed products can be subjective, to ensure
that learning situations encourage students to reveal calculation activities that we conduct
through internal validation by nesting. Students experience the learning situation and collect


16
evidence, analyze the posterity to determine the appropriate learning situation and adjust
accordingly; continue testing modified learning situations on other student groups and the
process is repeated until the learning situation is feasible. Testing tools are learning
situations and assistance sheet Test subjects are groups of students in grades 4 and 5. The
observer is the author of the dissertation. Some students helped with filming the process of
students’ calculation experiential activities on learning situations.
2.1. The process of designing learning situations to support the assessment of
calculation capacity of senior praimary school graders through experiential activities
2.1.1. Proposal of process
Inheriting the research on the process of designing learning situations, we consider
the process of designing learning situations to assist the assessment of calculation capacity
of senior primary school graders the following diagram:
Identifying the objectives of assessment
Identifying the contents of assessment

Identifying the
format of
learning
situations

Identifying the

context of experience
in learning situation

Identifying
suitable
information, facts

Identifying
questions/requireme
nts to express the
problem

Designing learning situations

Diagram 2.1: The process of designing learning situations
a) Identifying the objectives of assessment: The problem questions :What is it to be
assessed? which are the elements to be assessed?
b) Identifying the contents of assessment: It is the behavoral demonstrations of
students’ calculation capacity.
c) Identifying the format of learning situations: whether they are pure mathematical
situations or visual representation situations or practical situations.
d) Identifying the context of experience in the learning situation: In primary school,
the context of experience in learning situations includes internal mathematical context
(mathematical rules, mathematical relationships, ...) and real context (daily life, studying
life or social life in the community).


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e) Identifying suitable information and facts: A fact is necessary information
containing relationships and calculation contents to solve in learning situations. The facts

can be expressed in words, tables, models, diagrams, pictures, ...
f) Identifying the questions / request to express the problem: Learning situations to
assess students’calculation capacity are required to have an open ending in the form of a
question / request for students to experience calculation in different ways. The contents of
the assessment and the evidence identified will be the foundation for the reference assessor
to design appropriate questions and requests.
g) Writing into a learning situation: Teachers sketch a structured learning situation:
Introduction (description of the context of events), the content (describing the evolution of
events), the questions / requests contain issues to be addressed. In addition, teachers need to
conduct learning situations to ensure mathematical accuracy.
2.1.2. Designing learning situations to support the assessment of calculation
capacity of senior primary school graders through experiential activities
2.1.2.1. Designing the learning situations on fraction model
2.1.2.2. Designing the learning situations on counting triangles
2.1.2.3. Designing the learning situations on the area of triagles
2.1.2.4. Designing the learning situations on cake division
2.1.2.5. Designing the learning situations on cloth selling
2.2. The process of experimenting learning situations to investigate the
calculation capacity of senior primary school graders through experiential activities
2.2.1. Foundations to suggest the process
In order to suggest the process of experienting the learning situations, we base on the
existing research on assessment, model of lesson study, methods of internal validation.
2.2.2. Proposal of the process
Based on the above-mentioned foundation, we consider the process of learning
situation experience to investigate the calculation activities of senior primary school graders
through experiential ctivities in the following diagram:


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Selecting the designed learning situations

Conducting priori analysis
Identifying evidence to be collected
Organizing calculation activities for students on learning situations
with the help of assistance sheet and collecting evidence

Analyzing results after experiment
Attaining evidence

Unattaining
evidence

Adjusting
learning
situations/
assistance
sheet

Asserting the values of learning situations
Diagram 2.2. The process of experimenting learning situations
a) Choose a designed learning situation: Choose a designed learning situation for
experiment.
b) A priori analysis: It is necessary to anticipate the calculation activities in solving
learning situations, which help orient us in observation. It is also necessary to anticipate the
students 'mistakes related to the learning situation to propose assistance sheets to ensure
students' progress and encourage students to expose their demonstrations to be investigated.
c) Identifying the evidence to be collected: Evidence in the scope of experience on
the learning situation includes the students’ calculation behavior expressed in the discussion
process, on the product of the learning situation card, assisting sheets, drafts through
speaking, writing, calculation doing.
d) Organizing calculation activities on learning situations and assistance sheets for

students to experience and collect evidence: We organize learning situations and assistance
sheets for students to experience. While the student experience their calculation activities
we observe, record video clips, take photos as evidence and review them when it is needed.
When the given learning situations are completed, we collect the learning sheets and
assistance sheets.
e) Post-analysis: Basing on students' demonstrations during the calculation
experience on the learning situation, we consider whether each calculation activity in a
priori analysis has revealed and has not yet revealed, students' difficulties and causes,
thereby orienting ways to adjust learning situations or assisting sheets to encourage students
to reveal their calculation demonstrations to be investigated.


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f) Affirming the learning situations: If the evidence ensures that the calculation of
calculation activities such as a priori or a minimum ensures the disclosure of the required
requirements of calculaiton capacity, the study situation is feasible. . Conversely, it is
necessary to adjust to a more appropriate learning situation.
Requirements for calculation capacity: Requirements for calculation capacity at the
end of primary school (most concentrated demonstration of calculation capacity) in
mathematis programs in the 2018 General Education Program focus on familiar calculation
activities; Student's calculation demonstration in simple, familiar or similar situations. They
correspond to level 2 in the scale of calculation capacity. We consider them as requirements
that need to be met in calculation capacity of the senior primary school graders.
g) Adjusting learning situations / assistance sheet: If students do the
calculationseriously but still have not disclosed the requirements of calculation capacity, the
learning situation is too difficult to encourage students to show off their demonstrations then
we should reduce the difficulty of the learning situation or adjust the assistance sheets. On
the other hand, if most students reveal demonstrations showed in the priori analysis, it
means that the learning situation is too easy, it is necessary to increase the difficulty of the
learning situation. If the expression of the learning situation and the assistance sheers have

not showed thestudents unaware of the task of calculation, then it is necessary to adjust
them for more appropriateness.
h) Organizing activities for student to experience in the learning situation and
assistance sheets helped adjustment and collection of evidence: We tested the
learning situation and adjusted assistance sheets on other different students and the
process is repeated.
2.2.3. Experimenting a number of specific learning situations to investigate the
calculation activities of senior primary school graders through experiential activities
2.2.3.1. Experimenting the learning situation on the fraction models
2.2.3.2. Experimenting the learning situation of counting triangles
2.2.3.3. Experimenting the situaltion on the questions calculating the area of
triangles
2.2.3.4. Experimenting the situation on cake division
2.2.3.5. Experimenting the learning situation of selling cloth
2.3. Conclusion on Chapter 2


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Learning situations provide opportunities for students to interact, experience and
disclose calculation behaviors. The student's mathematical expressions are a practical basis
for us to adjust the learning situation and helping sheets to ensure that students are exposed
to expected calculation activities. In chapter 2, we have proposed the process of designing
and testing learning cases, learning situations and feasible helping sheets will be depicted.
The modified learning situations will be used to organize the students' calculation
experience in supporting the calculation capacity in chapter 3, the modified helping sheets
will be the basis for proposing orientations to develop students' calculation competence in
the teaching process. The research results of chapter 2 which have published in articles in
the Journal of Education [6], Journal of Science Vinh University [7], Journal of Hanoi
University of Education [2], Journal of Educational Science [1].
Chương 3

USING LEARNING SITUATIONS TO SUPPORT THE ASSESSMENT OF
CALCULATION CAPACITY OF SENIOR PRIMARY SCHOOL GRADERS
THROUGH EXPERIENTIAL ACTIVITIES
In this chapter, we conduct experiments using the learning situation designed in
Chapter 2 to assist in assessing students' calculation capacity through the proposed
assessment toolkit. For each learning situation, we illustrate on two students because we
conduct case studies in qualitative assessment, based on the behavior of students' expression
so that we analyze students’ difficulties and mistakes and capacity.
- If students were successfull in solving the learning situations, we measured their
performance when students completed the learning situation, because the individual's ability
is reflected in specific tasks. We explained the progress of students' calculation capacity
based on how students overcame difficulties and mistakes in the process of calculations on
learning situations. That is the foundation for us to propose orientations to improve students'
calculation capacity in teacher’s designed learning situation activities.
- If the student were not qualified to complete the learning situations, we measured
their calculation capacity

when they asked for a assistance sheet. The objective of

evaluation in the thesis is not to classify but to support the progress of students, we do not
stop at the assessment results but help to relay students to complete learning situations. We
explained the progress of students' calculation capacity based on their ability to overcome
difficulties and mistakes and accumulate experience. The suggestions in the assistance sheet


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supporting the progress of students 'calculation capacity, which is the foundation for us to
propose an orientation to improve students' calculation capacity in teaching activities.
3.1. The process of using learning situations to support the assessment of
calculation capacity of senior primary school graders through experimential activities

Identifying the objectives of assessment
Identifying evidence to be collected
Identifying the methods and tools of assessment
Organizing learning situations for students to experience
calculation activities and collecting evidence
Identifying the level of calculation capacity of students
Explaining the students’ headway in calculation capacity

Diagram 3.1 : The process od using learning situations to support the assessment of
calculation capacity senior primary school graders
a) Identifying the assessment objectives: To identify the assesment objectives, it is
necessary to answer the questions: What is the assessment for? Which element will be
assessed?
b) Identify the evidence to be collected: The evidence in the experience of learning
situations of individual students include the calculation behavior shown on the product of
learning situation cards, assistance sheet, drafts through writing, calculating, creating
activities.
c) Identifying methods and assessment tools: Qualitative assessment is done by
observing and studying students' learning products. The assessment tools are the learning
situation sheets, the assistance sheets, and the observation sheets.
d) Organizing the learning situation for students to experience calculation activites
and collecting evidence: We delivered the learning situation sheets to students and
instructed the students how to use them. The process of students' experience on calculation
on the learning situation were also the process that we collected evidence by observing,
taking photos, recording video clips. When students completed the learning situations, we


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collected the students’ learning situation sheets, assistance sheets and considered them as
students' calculation products.

e) Identifying the level of students’ calculation capacity: First we associated each
evidence obtained with the students’ calculated behavior in the observation sheets to determine
the level of achievement in each element of calculation capacity. We then combined the results
of each element in accordance with the level of students’ calculation capacit.
f) Explaining the progress of students’ calculation capacity: We explained the progress
of students' calculation capacity primarily based on self-reference because we focused on the
students’ values and expectations rather than external standards. We based on the students’
performing tasks to identify their efforts to overcome the difficulties and mistakes made by
students in the calculation process to successfully solve the learning situation.
3.2. Using specific learning situations to support the assessment of calculation
capacity of senior primary school graders
3.2.1. Using learning situations on fraction models
3.2.2. Using learning situations on counting triangles
3.2.3. Using learning situations on calculating the area of triangles
3.2.4. Using learning situations on cake division
3.2.5. Using learning situations on cloth selling
3.3. Orientating the improvement of the students’ calculation capacity of senior
primary graders through experiential activities on learning situations
3.3.1. Orienting the improvement of the students’ calculation capacity of senior
primary school graders through teacher’ designing activities on learning situations
3.3.1.1. Designing learning situations in accordance with the proposed process
3.3.1.2. Orientating calculation for students through teacher’ designing activities on
learning situations
a) For pure mathematical situations: Designing learning situations should be related
to the students’ knowledge, calculation activities in accordance with generalization rules,
visual representation elements, students’ mistakes.
b) For practical situations: To design practical situations, if derived from the
practical context, teachers need to model their mathematics; If it comes from within math,
teachers need to put them in a real situation by completing them. Need to strengthen the
design of hypothetical situations, problems with practical content attached to students' lives.



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3.3.2. Orientating the

improvement of calculation capacity of senior primary

school graders in teaching activities
3.3.2.1. Orientation 1: Helping students understand the meanings and nature of
knowledge
Learning activities should be organized for students to follow the path of discovery,
exposing students in a situation where such knowledge arises. It is required that guidance on
calculations, formulas, rules and procedures should be carefully made. It is necessary to
train students in calculating and solving learning situations for the formation of skills. The
students’ difficulties and difficulties should be used for analysis, for solutions and
experience accumulation..
3.3.2.2. Orientation 2: Improving the students’ calculation capacity through
calculation activities in accordance with generalization rules
Students should practice preparing a table of facts in the learning situations. The
figures in the table will be a thinking fulcrum for students to predict the relationship
between the elements that make up the rules.
3.3.2.3. Orientation 3: Improving the students’ calculation capacity through
calculation activities on practical learning situations
It is necessary to create opportunities for students to experience calculation on a
variety of practical situations so that students accumulate experience and avoid similar
mistakes. Students should be encouraged to create new situations by association,
specialization, and generalization.
3.3.2.4. Orientation 4: Create opportunities for students to practice regular
calculation activities
It is necessary to train students to become familiar with the various types of learning

situations in the learning process, in discovering new knowledge and practicing practice. In
order to reinforce and expand knowledge for students, they have formed computational
skills, contributing to improving their calculation capacity.
3.4. Conclusion on Chapter 3
Chapter 3 has proposed the process of using a case of study to evaluate students'
calculation capacity with illustrative examples. The students’ calculation capacity will
determine based on this procedure, experience the learning situation, and helping sheets
improve the student's calculation capacity, the proposal evaluation toolkit is feasible