This is one of assignments given when I was in Realistic Mathematics Education course. We have to describe what steps that the articles show.
The Hypothetical Learning Trajectory (HLT)
By : Dwi Afrini Risma (F112661)
Simon used a brilliant metaphor to describe how a teacher needs to make planning before teaching, i.e. journey metaphor. Based on this metaphor, in order to reach the learning goal, teacher has to make a learning trajectory in which give a guideline for every learning step made. This trajectory is not only describes how the learning process started but also how to shift students progress in thinking from model of to model for. An actual teaching experience in the classroom perhaps will differ from the design because of the students’ thinking and big idea. When teacher experience this situation, they have to revise the design and may implant the new design. This cyclic process is introduced by Simon (1995) with the term hypothetical learning trajectory (HLT).
In Koeno Gravemeijer article, Measurement and Flexible Arithmetic can be begun by situating the classroom teaching experiment on measurement and arithmetic in context of the research that proceeded. In this learning, the goal is to helps students develop a framework of number relations, which would enable them to construe flexible computation strategies for addition and subtraction up to 100. In the design, we use ruler as a model of iterating some measurement unit and the empty number line as a model for of mathematical reasoning. The trajectory of this activity can be seen in the following scheme.
The scheme in figure 1 designed following every students progress in thinking, in which the learning process begin with constructing context-based meaning of measuring and end with reasoning with a flexible ruler.
According to Gravemeijer (2004), teacher has to investigate whether the thinking of students actually evolves as conjecture, and he or she has to revise or adjust the learning trajectory on the basis of his or her finding (Gravemeijer, 2004). Thus, Cobb noted that design researcher not only envisions as a sequence of instruction but also forms conjectures about the potential mathematical argumentation and evolving tool use that might accompany the realization of the sequence (Gravemeijer, Bowers, & Stephan, 2003). The following scheme (Figure 2) is a manner in global structure of a hypotetical learning trajectory.
Pacing to structure linear distance:
In the first step we construct students understanding towards context-based meaning of measuring. Based on this article, students have to start with King’s foot as a standardized of measuring unit. This activity will be considered as important starting point to students to think that they need a standard in measurement since each person has different length of foot. In the other hand, it is also become meaningful and realistic for them since measuring by foot is familiar to them.
Measuring with different iterable unit:
In this activity, students are allowed to use a different iterable unit: unifix cubes in a smurf scenario in which it is as representative of smurfs food cans. This activity is important because it would allow students deal with larger numbers easily.
Can we use a small number for measuring a large quantity?
In this activity, students are confronted with a dilemma in which the smurfs decide to only use small number of food cans instead of 50 cans. This activity is important to develop students’ sense of counting by ten.
Counting by ten and adjusting by adding or subtracting ones
As continuance of the previous activity, this activity can help student reason with tens and ones to structure the number up to 100.
Measuring paper strip of ten units of one cube then followed by reasoning using the strip
The idea of this activity is to introduce a measurement strip that made of a paper strip on which 10 units of ten were iterated, each subdivided into 10 units of one cube. We hope that eventually a number on the measurement strip show the students about the distance extended from beginning of measurement strip to the line to which this number belongs.
Using the strip to reason about comparing, incrementing, and decrementing lengths of object that are not physically present.
The goal of this activity is to develop students’ arithmetical solution strategies. The focus is on indentifying and comparing the length and the height of an object numerically.
Reasoning about magnitudes using measurement stick
The goal of this activity is to let them get used to a measurements stick in which the number has been erased. Therefore, students can read the sign in stick even though the stick has no number on it. In addition, they can use it to give reason about the magnitudes.
Introduction to empty number line
The important of this activity is by using this tool; teacher can support students’ effort to reason with a tool that was not decremented with specific unit
Reasoning with empty number line
In this stage, we hope students can use the same solution strategy and explanation when reasoning. In other word, students can relate another same problem with different context by using a same strategy, in which they use empty number line as their tool on reasoning.
Gravemeijer, K. (2004). Creating opportunities for students to reinvent mathematics. ICMI 2004 (pp. 1-17). State College, PA, USA: ICMI.
Gravemeijer, K., Bowers, J., & Stephan, M. (2003). A Hypotatical Learning Trajectory on Measurement and Flexible Arithemetic. Journal for Research in Mathematics Education 12, pp. 56-64.