Hypothetical
Learning
Trajectory
Regular Course on Differentiated Instruction in Education
for Senior High School Mathematics Teachers
Yogyakarta, June 18 2022
Wisnuningtyas Wirani
💼
Training Specialist
(SEAMEO QITEP in Mathematics)
✉
tyaswirani@staff.qitepinmath.org
Aim of the Session
1. able to predict what is going to happen in the
classroom with a certain mathematical activity
2. able to respond to students’ idea, thinking, answer
(students’ responses)
3. able to write and use the HLT for developing,
observing, and reflecting lesson
Agenda




HLT: What and why
How to use HLT
Activity and Discussion
Q&A
Survey Result
Have heard of HLT
LP developed in a year
Yes
17
3-5 times
More than 5 times
8
No
Twice at most
13
7
Make your own LP
23
3
1
Yes
No
HLT: What Why
Survey Result
What do you
know about HLT
• I dont know very well, but that 3 of my friend topic for their
dissertation
• Hypothetical learning trajectory is a theoretical model for the
design of mathematics instruction
• Hypothetical learning trjectory is the steps in learning that
engage the students in learning mathematics
• The hypothetical learning trajectory consists of three aspects:
learning objectives, learning activities, and learning hypotheses
• HLT is essential when we design the learning process. By
predicting student's response or ways of thinking, teachers could
prepare suitable reaction. HLT could be developed based on
student's prior knowledge combined with teacher's experience in
the last few years.
• Hypothrtical learning trajectory is lintasan belajar.
• HLT means how to plan lesson based on students' learning
obstacle.
HLT: What Why
DISCLAIMER!!!
It might be a new term for you. However, you are already familiar with it.
HYPOTHETICAL
LEARNING
TRAJECTORY
predictions
of the learning
HLT: What Why
Survey Result
Optimising LP
Some think that they have optimised the LP
• I think for some lesson plan, yes I did optimize it. I developed it my self, I imagine what
will happen in my class and design the plan based on students conditions. Then I do the
learning activities according to the plan. Some times I improvise if the condition is out of
control.
• I think yes, because we make the lesson plan every day, so we know what student need
• Yes, because i plan it for student learning.
• I always try to optimize the use of lesson plan for learning. In the next academic year, I
remodify the lessson plan with the experiences of the previous academic year.
• Yes...i have optimised the use of lesson plan for the learning make my teaching is more
focused
• Yes, the use of lesson plan make our teaching have clear goal
• I think I have, by following every step that I have included in my lesson plan. But, I don't
Know why are there still some that I feel have not been achieved from the learning
objectives.
HLT: What Why
Optimising LP
Some think that they have not optimised the LP
• Not yet, because ussualy the reality doesn't suit the plan, changeable depend on the
situation.
• I think that I have made use of the lesson plan. but not in an optimized way yet. A lot of
unexpected situation happening in class that we can not completely fully follow the lesson
plan have been made, instead I improve it to adapt with the situation.
• No, because it is flexible based on the learning needs, style, and interests of the students.
• No, i dont. Sometimes, learning activities that occur are not as planned.
• No, I need more
toisdevelop.
More
time and materials.
HLT
used
to
bridge
our
• No sometimes there are some problems that make me have to change or modify the
lesson plan.expectation (goal) of the
• No, because of the format used in my school doesn't give enough space for teachers to
learning to the reality
develop practical lesson plan.
• I don't think it's fully optimal because I'm still improving the quality of my lesson plans to
make it more optimal later during the lesson. I feel that the lesson plans are optimal
when we can implement all the lesson plans that we make into lessons according to the
target but in reality the content of the lesson plans is sometimes only an initial design and
during teaching there is still something added from the current conditions.
• Maybe not. Sometimes depends on the students ability, if students unable to master the
skill, have to change the planning
HLT: What Why
Survey Result
Elements in LP
• Aims/objectives/learning outcomes/learning indicators
• Arrangement of environment
• Activities/task/problem
• Assessment/reflection
• Strategies
• Curriculum standard/basic competence
• Duration
• Learning media/materials and tools
• Subject/grade of students
learning goals
Components
HLT
(Simon, 1995:136)
learning activities
hypothetical learning
process: predictions of
students’ responses
and the teacher’s
responses
Compared to LP
Students’ Responses
Linear framework (left) and hypothetical learning trajectory (right)
(Fosnot & Dolk, 2001)
How to Use HLT
Observing
the lesson
Hypothesis
of teacher
knowledge
Improving
instructions
Activity and Discussion
Given an equilateral triangle with
a circle inside and an equilateral
triangle.
Determine the exact ratio of the
areas of the two triangles.
Think and work on the
problem for 10 minutes
Math activity for grade 10
Goal: using geometric properties to solve problems
Component 1: Goal
Component 2: Activity
Component 3:
Hypothetical learning
process
Students’ Responses
Has difficulty getting started
Teacher’s Responses
• What do you know about the angles or lines in the diagram? What do
you need to find out?
• Can you add some helpful labels to the lengths?
• Can you add any helpful construction lines to your diagram? What do
you know about them?
• Can you find relationships between the lengths from what you know
about geometry?
Students’ Responses
Teacher’s Responses
Works out the ratio by measuring the
dimensions of the triangles
• What are the advantages/disadvantages of your method?
• Are your measurements accurate enough? How do you know?
Does not explain the method clearly
• Would someone unfamiliar with your type of solution easily understand
your work?
• How do you know these triangles are similar/congruent?
• It may help to label points and lengths in the diagram.
For example: The student does not explain
why triangles are similar.
Or: The student does not explain why
triangles are congruent.
Has problems recalling standard ratios
The student makes an error using the ratios
for a 30°, 60°, 90° triangle (1, 3 , 2).
Uses perception alone to calculate the ratio
For example: The student rotates the small
triangle about the center of the circle,
assuming the diagram alone is enough to
show the ratio of
areas is 4:1.
• What do you know about cos 30° and sin 30°? How can you use this
information?
• How might the Pythagorean Theorem help you?
What math can you use to justify your answer?
Students’ Responses
Makes a technical error
Teacher’s Responses
Check to see if you have made any algebraic errors.
For example: The student makes an error
manipulating an equation.
Uses ratios of lengths rather than ratios of
areas
• What is the formula for the area of the circle?
• How can you use it to find the ratio of the areas of the circles?
For example: When finding the ratio of the
areas of the two circles, the student finds the
ratio of the radii, rather than the squares of
the radii.
Produces correct solutions
Can you solve the problem using a different method? Which method do you
prefer? Why?
How this session related to DI?
Refer to Mr. Yoga’s session (this morning)
Content
Strategies,
scaffoldings
Process
Products
Discussion, doodle on GeoGebra
classroom, assignments
Students’ characteristics,
needs, feelings
Affect
Learning
environment
Socio norms
(Tomlinson & Eidson, 2003)
You may initially plan the whole journey or only part of it.
You set out sailing according to your plan. However, you
must constantly adjust because of the conditions that you
encounter. You continue to acquire knowledge about
sailing, about the current conditions, and about the areas
that you wish to visit. You change your plans with respect
to the order of your destinations. You modify the length
and nature of your visits as a result of interactions with
people along the way. You add destinations that prior to
the trip were unknown to you. The path that you travel is
your [actual] trajectory. The path that you anticipate at
any point is your “hypothetical trajectory”.
(Simon, 1995, p. 136–137)
References
Fosnot, C. T., & Dolk, M. (2001). Constructing multiplication and division: Young mathematicians at
work, (Vol. 2).
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal
for research in mathematics education, 114-145.
Tomlinson, C. A., & Eidson, C. C. (2003). Differentiation in practice: A resource guide for differentiating
curriculum, grades 5-9. ASCD.
Thank you
qitepinmath.org