Associate University: EMA511 Math Pedagogies Review and Lesson Plan

This assignment critically reviews two mathematics pedagogies: learning trajectory and collaborative learning. The report analyzes their effectiveness, impact on student learning, and application in lesson planning. Part 1 focuses on a critical review of the selected pedagogies, considering factors like mathematical proficiency, problem-solving skills, and reasoning. Part 2 involves designing a lesson plan based on a hypothetical learning trajectory to support student's mathematical thinking, integrating elements like mathematical goals, developmental progression, and instructional strategies. The assignment emphasizes critical thinking, problem-solving, and the interpretation of results in real-world contexts, providing a comprehensive analysis of mathematics education approaches and practical lesson design.

1
Learning trajectory in the form of lesson plan
Student Name
Associate University
Tutor
Date

Trajectory 2
Trajectory
PART1: Critical Review of two Mathematical Pedagogies
Pedagogy that is also referred to as education, is a profession and science of teaching that
involves a range of techniques as a whole class and structured group work, guided learning and
individual activity. In simple words, (Wentzel, 2014, p. 6) Pedagogy is One of the important
subjects a teacher trainee has to learn to become an efficient Teacher by understanding the
psychological aspects of children in their learning process and the techniques /tools for
successful teaching.
On the other hand, Mathematical pedagogy refers to an approach to mathematics
education that includes a practical and rational commitment to liberation. To build the critical
mathematics pedagogy out of critical pedagogy, some focus on the empowerment of the learners
as experts and actors for change in their own world. It also requires that students and teachers use
mathematics to understand the relations of power, resource inequalities between different social
groups and explicit discrimination. (Alexander, 2013, p. 11) There is no set curriculum for
mathematics pedagogy, therefore it is designed to be responsive to the lives of students in their
local context. Because ineffective teaching is a waste of time. Some people don’t know anything
and don’t want to learn anything. Teaching them is extremely rewarding. Some people don’t
know anything but want to learn. Teaching them is relatively easy. Some people know
everything, don’t need to learn anything. These people are almost impossible to teach.
Out of the many types of mathematics pedagogy, this case will focus on two approaches
namely;
i. Learning trajectory

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ii. Collaborative learning approach
i. Learning trajectory
Learning trajectory is the overall thinking and learning of students in a mathematical domain,
these are aimed at developing the student’s mental processes. Therefore, these trajectories help
classify particular endpoints of an educational context. (Clavijo-Olarte & Sharkey, 2018, p. 8) It
describes the facts that helps the students get a desired endpoint.
Clement’s and Surama’s argument is that hypothetical learning trajectory contains objectives
that helps in boosting the critical thinking of the student. (Ilmonen, 2012, p. 4) According to their
formulation there are two or more interacting trajectories which are closely related or ordered
parts aimed at reaching the goal as a given below;
i. Teachers use experimented tasks to produce actions that will help progress into a
given direction. Teachers mostly use a designed through a mathematics panel.
ii. The students thinking and development usually go through a particular domain/levels
thereby leading toa desired goal.
There are three components of learning trajectories
i. Mathematical goal- this is the first part of learning trajectory. The main objective here
is to determine the children’s ability to think and generate the knowledge acquired to
solve equations in future. These ideas mostly come from the national mathematics
panel.
ii. Development progression this is the second part of learning trajectory that consist of
different tasks that are aimed at developing the student’s capability and thinking
about that particular topic therefore achieving the mathematical goal that is targeted.

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iii. Instructional strategies- this is the third part helps the students develop their thinking
since the tasks assigned to them help them develop critically thereby moving from
one level to the other.
To develop a particular mathematical understanding, children construct each level of thinking
and reasoning, sequentially if they are provided with appropriate teaching approaches and tasks.
2. Collaborative Approach
Collaborative learning is a method of teaching and learning in which students’ team
together to find answers to a problem that a teacher has given them. Collaborative learning is
where 2 or more people work together to learn. (Kiswanto & Suryana, 2017, p. 11) An example
of collaborative learning is when the teacher has students work together in groups to create a
project. Students learn from each other and use the resources of everyone in the group. Another
example of collaborative learning is when students form a study group. We use collaborative
learning in the real world all the time. Companies make use of the power of collaborative
learning by creating teams of workers with the same goal or by assigning mentors to new
workers. Think of collaborative learning as the opposite of independent learning where everyone
does the work alone and provides all needed resources alone. It motives the mathematics
teachers to embrace joint work in order to improve the student’s capabilities and understanding
in various areas e.g. measurement and percentage. Since it is believed that learning is based on of
social act, this helps the students share their knowledge thus helping weaker students acquire
more knowledge from the sharp students.
It helps the student to solve the problem/ algebraic equations and apply the knowledge acquired
to tackle similar situations suppose it arises. (Edgington, 2014, p. 6) Secondly it transforms the
quality of teaching and lecture to the students thereby providing a quality assurance to the

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students. Lastly collaborative teaching is usually used and viewed as a means of enhancing and
improving the teaching outcomes. This helps improve the general performance of the students.
Effective teams combine their strengths to overcome individual weakness, and continually cross-
train and coach each other so that those weaknesses are minimized. An effective team shares
equally any joint success, and works together to own and learn from any failures.
Techniques used in collaborative to make learning interesting
Students may ask for clarification of any words that they do not recognize, but they may not
be told how to interpret ambiguities. That is left up to them. The group is comprised of five
students so that it is harder to have a student simply choose to agree with a friend in order to rush
the exercise along. (López-Gopar, 2018, p. 10) Also, an odd number means that there will be a
tie breaker if the group is divided. It is better to use a statement that is ambiguous or
controversial in order to elicit a stronger response to it. Now, students must first write a sentence
in which they state that they agree with the statement or disagree with the statement and tell the
reader why they agree or disagree. This is followed by a sentence in which they must tell the
reader what in their experience of life has taught them to think the way they do about the
statement. Once the writing is done, students take turns, a minimum of fifteen and a maximum of
thirty seconds to tell the rest of their group what they wrote. Everyone else must listen without
comment until everyone has had his/her turn to speak. The group may now discuss the statement
with members telling how they interpreted the statement to arrive at their response. Next, the
group must decide if it can reach consensus on the statement, in other words, do they all agree
that they agree or disagree with the statement as it was presented. (Kiswanto & Suryana, 2017,
p. 04) Finally, each group will tell the rest of the class if it reached consensus or failed to reach

Trajectory 6
consensus on the statement and what the binding ideas were. There are several benefits of
collaborative learning, they include;
i. It sharpens the student’s memory, thereby enabling the student to solve the task
assigned to them within the shortest time possible.
ii. Encourages student’s creativity and techniques in approaching the questions assigned
to them.
iii. Collaborative learning is a powerful tool since it enhances the students thinking
capity therefore enabling the student participate better during discussion.
Most curriculum developers argue that collaborative learning is important to students its
assured that students with lesser ability gain from those with greater abilities.
PART 2: Lesson plan: Development of Learning Trajectory
In this section we are explaining the usefulness of mathematical learning trajectory as a
tool to support teacher’s attention to student’s mathematical thinking from various study findings
learning trajectory support teachers in choosing appropriate tasks and learning goals, and
anticipating students likely approaches and difficulties. ("Lesson Plan," 2012, p. 15) In particular
learning trajectory was used to attend to both short- and long-term learning goals, provided
information about a specific student strategies and misconceptions, and supported the teachers in
considering levels of sophistication among their students’ mathematical conceptions.
Some of the learning goals are employment that is, integrated competitive employment
and also independent living skills.

Trajectory 7
Independent living skills- Reading is the best independent living skills one should adopt.
There are a lot you can learn independently from books, newspapers, magazines. You can easily
learn from the experience of another person who wrote something. Sometimes a book is the
result of many years of life and its ups and downs. As humans we never have enough time to
experience and learn from those experiences, so we have to cling on somebody else’s thoughts,
advices, and lessons. Read every day and learn about life before it is too late. Here are what are
important independent living skills
a. The ability to dream and create clear goals and objectives for yourself and your life in
every aspect of your life
b. Being willing to go for your dreams and not get stopped by your negative beliefs or
theories
c. Knowing that you have what takes to get the results you want and trusting yourself
d. Knowing that you are accountable for your own life and what you have or don’t have
e. Understanding your own OS and how you think and act
f. Being able to observe and adjust your behavior to achieve your objectives
g. Being able to communicate clearly and effectively with others
h. Caring and honoring other people and their preferences
i. Being accountable for the promises you make to yourself and others in every aspect of
your life
j. Being centered and trust that if you do the right thing, you will always come on top.

Trajectory 8
Designing such a plan for a diverse group can be very challenging and there are always going
to be a loafer or two in the class. Do not let these discourage you. This is not meant to be an
insult in any way, it is merely an anagram to instruct. (Szymczyk, 2017, p. 9) The teacher should
try to Keep It Simple and Stupid. What this means is that you begin with simply put but
interesting explanations of the subject matter. Interjecting a little humor acts as an ice breaker
and alleviates nervousness on the part of the students as well as the teacher. If you want a
successful instruction plan, you have to keep it malleable. Don’t be afraid to backtrack to clarify
points of confusion. The idea is to successfully teach the material, not rush through it and get it
over with. The teacher should be respectful but firm. Don't call anyone out with the intent to
humiliate, but don't take any crap from them. Above all, if teachers themself do not show interest
in the subject matter, they shouldn’t expect the students to either. Passion is contagious. It will
visibly infect those who you can captivate and they will work harder to know more.
There are several levels of thinking in paths that help children develop higher levels of
critical thinking. These parts help the students improve the mathematical thinking. They include:
i. Mathematical goals
Mathematics not only improves critical thinking but even improves the students’ focus,
brainstorming, way of tackling the problems, application of acquired laws and many more. It's
really fun to do mathematics. You can try yourself. When you won't get correct answer, you try a
number of ways to solve it. Here the student won't notice but in reality, it improves the ability of
critical thinking. Students who lament abysmal mathematical skills. The ability to reason
mathematically can quickly rule out any number of seemingly creative ideas that when pressed to

Trajectory 9
their conclusions would ultimately fail. It can also lead one to see how certain ideas have the
potential to succeed and help to narrow the vision of how the idea will need to be pursued in
order to achieve that success. (Setiawan, 2019, p. 7) Right there, that is a huge piece of critical
thinking, being able to weed out the truth from fiction. This will also improve the teacher’s
critical thinking.
ii. Development of progression
It consists of levels that helps the teacher achieve mathematical goal. It provides a simple
label and description of different levels with an example of the students thinking and behavior.
This will help the teacher to know the ability of the students and put them in respective groups
according to their capability. Putting them in various groups will help the weaker students learn
from the sharp students therefore gaining a lot, since students learn much faster when they are
taught by fellow students than when the teacher is teaching them.
iii. Instructional tasks
The tasks are to help the learner/ student to acquire ideas and skills that are useful in
developing the children’s critical thinking. These tasks will eventually help the teachers to
promote the students from one level to the other. (Setiawan, 2019, p. 17) Some learning
trajectories involve smaller research with few guidelines, this is a major challenge to teachers.
Pick the right person. Picking the wrong person for a task is a major reason for failure. Match the
requirements of the task to the abilities of the person. Be sure that the person you delegate the
task to is capable of doing the assignment. Delegate effectively to the right person. This frees the
teacher from doing more things of higher value. Delegate the entire job. The more often you
assign responsibilities to the right people, the more competent and knowledgeable they become.

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Delegate with participation and discussion. Invite questions and be open to suggestions. You
need to delegate in such a way that students walk away feeling that they are capable of doing the
tasks with a lot of ease.
Aim Description Examples of activities and
weblinks
Lesson 1 Understanding So, first of all try to
learn the basics of
number and algebra,
measurement and
probability. practice
the questions which
cover those basic
concepts.
Concentrate on how to
approach the question
rather than the final
answer. The student
should not memorise
the formulae but try to
derive them.
Automatically the
student will
understand each and
Reinforcing significance
of working mathematics
with the content.

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everything that they
need to know.
Lesson 2 Fluency The teacher should
choose appropriate
units that students can
easily understand and
use necessary
instruments for
calculation.
Describing how the
content is explored and
developed.
Lesson 3 Problem solving In class, the teacher
should solve problems
(decimas, percentages
and measurements
using whole numbers),
and ensure the student
put the acquired
knowledge into the
solution working,
covering a myriad
variety of new
problems given in
class.
Providing the language to
build the development
aspects of learning
mathematics.
Lesson 4 Measuring Assign tasks to the
students and see their
Reflecting the content and
encompassing the

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ability capabilities to
handle calculations on
algebraic and
measurements.
proficiencies.
Conclusion
To develop a particular mathematical understanding, children construct each level of
thinking and reasoning, sequentially if they are provided with appropriate teaching approaches
and tasks. The collaborative techniques make learning interesting thereby making the students to
acquire more knowledge since the environment will be favorable to acquire new concepts.
Mathematical learning trajectories not only improve the teacher’s critical thinking but it’s also
beneficial to the students’ too. Mathematical learning trajectory is an important tool to both the
teacher and the student since it helps both to develop mentally.
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