Current projects

Flexible knowledge of rational numbers with multiple graphical representations in ITS (more)

Sequencing learning with multiple representations of rational numbers

Affordances of the problem context when learning with multiple graphical representations of fractions (more)

Sequencing learning with multiple representations of rational numbers

Status: creating fraction tutors

We are currently planning a new study on learning with multiple graphical representations in the domain of fractions in order to investigate how learning with multiple representations should be sequenced to effectively support students' conceptual understanding. In order to benefit from multiple representations, learners must achive some level of fluency in interpreting and manipulating the individual representations, and must also engage in sense-making activities across the representations, to relate them and abstract underlying concepts.
The question arises how tasks involving different representations should be sequenced so that both these aspects of robust learning will be realized. In particular, how frequently should student switch between representations?

The proposed research will be carried out with a think-aloud lab study followed by an in vivo experiment.
The purpose of the think-aloud study is to investigate empirically (1) which representations are commensurate with which problem types, and (2) how the individual representations complement each other in terms of supporting comprehension of fractions. A small number of students will work on paper-and-pencil tasks involving the graphical representations targeted in the proposed research, and will be asked to express their thoughts while solving these problems.
Following the think-aloud study, we will carry out a 3-condition in vivo study. In each of the three conditions, the students will work with a series of CTAT-built fractions tutors. These tutors will cover an introduction into different representational models, fractions as division, equivalent fractions, and fraction addition. All conditions "inherit" key features from the most successful condition in that experiment: multiple graphical representations of fractions combined with prompts to self-explain how each graphical representation relates to the standard symbolic fraction notion. Compared to our earlier experiment, we decided to bring down the number of graphical representations from 5 to 3, namely, numberline, pie, and set representations.
The three conditions differ in how frequently students switch between representations:

1. Switch frequently - students will switch representations after every other tutor problem
2. Switch with moderate frequency - students will switch representations after every 5 tutor problems
3. Switch infrequently - within each unit of the tutor curriculum, the representations are used in a "blocked" manner: students first work with one representation, then with the next, then with the third
4. Switch with gradually-increasing frequency - students start out within each unit by switching representations with moderate frequency (as in the second condition), but the frequency increases gradually until at the end of each unit it reaches the same frequency as the first condition

Each condition will use (in iterating fashion) the three graphical representations of fractions described above: area models (circle), number line, and set model. The order in which the representations are presented to each student will be random ized, so that the results of the experiment will not be specific to any particular order of presenting the representations . In this experiment, the tutor curriculum will not yet contain any connection-making activities. That is, students will not yet see different graphical representations simultaneously on the screen, although they will be faced with different graphical representations across problems. It is important to note, however, that in Experiment 1, as in our prior experiment, many activities in the tutors will combine a graphical representation and symbolic notation of fractions (as opposed to two graphical representations), as illustrated by the example discussed above.