Maths Education: Exploring Learning Trajectories in Early Mathematics

This report delves into the concept of learning trajectories in mathematics education, emphasizing their significance in enhancing teaching and learning outcomes. It explores the implications of authentic teaching, assessment, and the development of learning tasks to improve student performance. The report examines the natural learning progressions of children in mathematics, highlighting the importance of understanding these developmental sequences for educators. It discusses the structure of numeracy programs, referencing the F–10 Australian Curriculum and the essential elements for successful learner development, including algebra, number sense, measurement, geometry, probability, and statistics. The report also analyzes the relationship between numeracy progression and the Australian Curriculum, emphasizing the need for consistent opportunities to recognize and apply mathematical skills. Furthermore, it provides insights into how the National Numeracy Learning Progression can be used at various levels to identify individual student performance and foster collaboration among teachers. The report explores the role of learning trajectories in connecting new ideas with prior knowledge and facilitating effective mathematical communication. It also addresses authentic learning, assessment, and their implications, including language skills and teaching methods. The report provides examples of learning activities and learning trajectory samples for counting, along with bonus examples of mathematical activities such as 'Shop and Add it Up,' 'Less than Greater or Equal,' and building objects. Overall, the report concludes that learning trajectories are crucial for improving teaching and learning in mathematics, providing a framework for curriculum development and real-world application.

Maths
Education - Mathematics

Abstract:-
Presently changes in the education system leads towards the many challenges. This study is
related to the implication of authentic teaching, assessment, and learning. Development in the
learning task improves student learning and helps to achieve higher standards. In this study, we
will consider the implication in the activities and learning trajectories related to mathematical
activities.
Introduction:-
For development, natural learning progressions followed by the Children. For example,
children first learn very slowly, in a specific manner such as walking, running, jumping, and
skipping with increased ability and speed. Similarly, student follow the same procedure for the
progressions of natural development in math learning. Children learn mathematics skills and
ideas in according to their mental ability. An educator needs to understand these Sequence's
activities and developmental progressions for the children's development. Educators can develop
enriched learning environments for mathematics so that the development of children can be
effectively and appropriately (Stone, 2019).
One question is arising that on what basis Numeracy program is structured for
Mathematics? F–10 Australian Curriculum is essential for the development of the learner.
According to ‘The National Numeracy Learning Progression’, three elements are necessary for
the successful development of the learner. These elements comprise of Algebra and Number
Sense, Measurement and geometry, Probability, and Statistics (Curriculum, 2020). For the
numeracy capability it is essential aspects for the present developmental sequences, in which
each element contains the sub-elements. There are a total of nine sub-elements in the algebra and
Number system, two in the Probability and statistics, and four in Geometry and measurement
(Woodcock, Sisco, & Eady, 2015).
What is the relation of the numeracy progression and Australian Curriculum? In the
Australian Curriculum, Numeracy skill is a kind of explicit teaching for Mathematics. A learner
needs opportunities consistently to recognize the Mathematics at outside the classroom. So,

numerous peoples can apply for the general mathematical skills in unfamiliar and familiar
situations. In the Curriculum, using mathematical skills enriches the study in the learning area,
which helps to develop a deeper understanding of numeracy. Mathematical ideas are essential for
the students so that they can interact with their lives (Hammond, & Flook, 2019).
How can the progression of numeracy be used? The NNLP (National Numeracy Learning
Progression) can be used in a school, at the individual or team teacher level. This Numeracy
Progression provides the maximum benefits in the student learning. It also relates to the
collaboration and professional learning between teachers in a whole-school. This progression is
used to identify the individual student’s numeracy performance within the sub-elements. In class,
student abilities can be varied. Each student cannot be fit in a specific progression level. Therefore,
a logical sequence uniformly provides the progression.
Teacher judgments about the learner numeracy capability are based on the learning
experiences. Oral explanations, Number talks, and written or any task from any area of learning
can provide evidence of the learner's capability related to numeracy. Teachers use progressions,
which lead to the development of learners during learning and teaching programs. Students can
increase robust knowledge in a transferable and adaptable manner so that learners can develop
new ideas. They can connect new ideas with previous ideas and how they can represent their
thoughts differently. A learner can describe their thinking mathematically in an effective manner
(Lambdin, 2009).
The Curriculum of the Australian is designed to help all young Australians so that
children can become creative, confident, and successful learners. Foundation presented the
developmental Sequence of learning of year - 10, in which learning Curriculum of Australia
describes to the parents, teachers, students, and others in the community about what is to be
taught. The learning quality of young people can be expected as they progress in school (Massie,
2019).
The Foundation 3-D designs - Year 10 of the Australian Curriculum identify the
importance of skills and disciplinary knowledge and understand the priorities of the cross-
curriculum. Understanding, skills, and Disciplinary knowledge are related to the Australian

Curriculum 8 learning areas such as Mathematics, English, Physical Education and Health,
Science, Social Sciences, and Humanities, The Technologies, Languages, and Arts. In each
learning subject or area, specific content describes the student what will learn. Achievement
standards describe the understanding of the depth and knowledge sophistication (Beams,
Manners, & Morgan, 2010).
Development of the child relates to the physical, language, cognitive development, math,
emotional-social, and learning characteristic. Traditionally precollege and vocational
mathematics separated in the schools. In the world of technology, the learner can connect
mathematics with real-life applications. Mathematics based on the workplace can be better
mathematics for everyone for every student. Trajectories learning consist of mainly three parts-1)
Goal of Mathematics 2) a path of development from which learners can reach the goal 3) set of
tasks or instruction activities. Each of these levels can help in the development of learners' higher
thinking.
The big idea of the mathematical learning trajectory is the make clusters of the skills and
concepts. This central idea relates to the thinking of the children, and there generates future
learning capability. This big idea comes from the several projects which are includes from the
National Math Panel and NCTM (National Council of Teachers of Mathematics) panels:
a) Establishment of the objective b) from where we have to start? C) Where we go next,
and how? D) How do we reach the goal?
Authentic learning is a type of learning design that connects with the students what they
taught in the school from the real word problems, issues, and applications. Learning experiences
in the real world also relates to the ambiguities and complexities of real life. Students learn
something new by' learning by doing approach.' Traditional forms of assessment, which includes
examinations and essays, have no particular application in the real world. The authentic
assessment helps the learner to investigate their learning skills(A, 2016).
The aim of any learning experience relates to the authenticity in every lesson, unit, and task
to ensure that students can develop the confidence and skills to solve the problem with its
learning abilities. Outside life, school life means to prepare the students in such a way that they

can make meaning of their learning. Learner can link their new knowledge with previous
education (Roberts, 2013).
Some implications are taken into account during the authenticity, learning, and assessment.
This implication includes language skills and teaching issues during the teaching phase and lack
of existing knowledge and experience of a teacher, which affects the learning activity of the
student. New innovating methods of teaching are also necessary for the development of learner's
so that they can create new ideas.
Example:- Online or Live Lectures:-In this information is provided to the student orally,
which is supported by the slides and into 4-6 minute blocks in short interaction, ask the student
to respond to the question related to the task. Ask a question to the student, and the student will
explain and summarize it. After asking the question, wait for 15-20 seconds before calling a
randomly selected student or for volunteers to respond. It is mandatory to show the visual clue to
the student after posing the question. After a student response, call another student to describe
the reaction of the first student. In case if the response of the first student is not entirely accurate,
then call the second student to respond to the answer of the first student.eg. What do you think in
his topic [name], would you agree? This learning activity is relevant for student progress towards
declarative knowledge and learning outcomes (Nctm, 2012).
Learning Trajectory Sample for Counting:-
Age Developmental Progression Instructional Tasks
1 year Pre-Counter- Non Verbal or
verbal counting
Chanter-Verbal Chants with
“sing-song” or identify words
Associate different number
with Quantities
Repeat varying context
counting series
2 year Reciter Verbal – Count,
Verbally not mandatory it
will be in Sequence
Repeat Counting sequence
and frequently experience
Student count with a
computer and will add an
apple in the basket with each

count
3 year Reciter – Count Verbally up-
to ten with some objects
Corresponder- Arrangement
of smaller and bigger words
Move and Count- For
example, children move with
each count from 1 to 10.
Touch head-on count 1,
Touch arm on count 2, Touch
ear on count 3
Children will click on food
items on every count with a
computer
4 year Counter-Accurately counts
objects in a line to 10 and
answers the “how many”
question with the last number
counted
Producer -Counts out objects
to 10. Recognizes that
counting for the situation is
relevant in which no. is
placed.
Cubes in the Box-Child will
count a small set of cubes and
put them into the box and
with close lid. Then tell the
child how many cubes are
hiding by you. After answer
by a child, write that number
and count all cubes with
children together.
Count motion in numeracy
that how many times children
jump, or clap.
Ice cream ice-cream - Learner
count items up to 10, putting
the cherry on an Ice cream to
match a target amount.
5 year Counter and Producer –
Counts the number from 1 to
Allow children to count up to
20 or beyond and tell children

10. Keep the object on a track
or in a different arrangement.
to play with coins. Make the
tower of coins with counting.
Ask the children to put
marbles in the box to match
the target (Clements, &
Sarama, 2010).
Other Bonus Example of Mathematics Activities (Leo, 2020):-
1. Shop and Add it UP: - When you are in the store with your child, then ask the children to
add different items together. E.g.:- How many fruits you have to buy? How many boxes of
apple are there? How many types of chocolates are there?
2. Less than Greater or Equal: - Make the chart of shapes Ask the children which two or
more shapes are equal or different.
Figure 1:- Chart of images (Worlds, 2020)
In the chart, sort the similar images in one column and unequal in the other column.

Figure 2:- Chart of equal or unequal images (Worlds, 2020)
Another example for the greater or less than is that make some cards in which 2 or more cards of
> Sign, some are < sign and some cards with the = sign. Play a game with these cards with a
particular number on a table. Ask the child to put the signed card between these placed numbers
on the table quickly. The track of how much time and how many counts he can get correct.

Figure 3:- Chart of word problem (Worlds, 2020)
Give the chart of the word problem to the children and see how fast he can solve the problems.
Figure 4:- Chart of word problem (Worlds, 2020)
Give the chart of word problem of the addition of a number to the children and ask him to sort it
into even or odd slots and see how fast he can solve the problems.
3. Build Objects: -- Use building toys or blocks to build towers or houses. After you make
Count Pieces up to 20 and then subtract or add pieces.
4. Take the Poll: - Create the graph of a family member and ask children a question about the
family member and ask a question to the children related to the pictures.
5. Order Up & Down:-Compare the sizes of objects and ask your child to arrange the object in
the smaller or more substantial order. Ask him to put all his toys in larger of shorter Sequence.
Analyze his time how fast he can do this activity (Ackerman, & Kloss, 2019).

Conclusion: - From the above study it is concluded that learning trajectory improves teaching
and learning in the area of mathematics. Curriculum is mandatory for the development of the
student. Authentic learning related to the learning trajectories. Some examples are taken of
different age students and how mathematics can be linked by real life. How student learn by the
process of ‘Learning by Doing’. Thus, learning trajectories can facilitate developmentally
appropriate teaching and learn for all children

References:-
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