Our INK-12 research questions are the following:
How can technology that allows students to both draw and use representational tools support learning mathematics?
Our experience in grades two through four has shown that our technology helped students, particularly struggling students, to increase their understanding of multiplication and division. Our tools provided specific starting points for students unsure of how to approach a problem. The tools also enabled students to create their own visual representations to aid in problem solving and explanation of thinking. Drawing plays a key role in the creation of the representations, providing an important bridge between the specific and concrete objects in math problems and the abstractions that underlie mathematical reasoning.
The visual representations created by drawing and interacting with our tools enabled some students to grasp concepts that were obscured when they relied on rote algorithms. The representations also enabled students to see the relationship between a visual representation and the algorithms.
Students who were accustomed to filling out worksheets containing pre-created representations were able to learn to create their own representations, a process that requires a deeper understanding of a problem than does using pre-created representations. They also demonstrated preferences for particular kinds of representations, which enabled them to show more agency in their learning than previously had been possible with their paper-based curriculum.
Students in our field test classrooms were enthusiastically engaged in authentic mathematical reasoning; some participated more actively and energetically than they usually do in math class, according to their teachers.
We believe that experiences such as these can have long-term consequences for students, potentially changing their mathematical disposition and their desire to continue to study mathematics as their education proceeds.
How can technology that enables a teacher to view and share student work with her class support students’ learning?
Teachers found that the tablets made their students’ thinking visible, especially via our playback functionality, which enables a teacher to view the process a student uses to complete work. Teachers had a record of all student work and could review it in real time and after class to see what their students did and did not know. Sorting functionality enabled a teacher to choose student work examples that demonstrated the use of different approaches.
Using the technology, a teacher could select and project more than one example of student work for the class to discuss. The ease with which a teacher could show multiple examples of student work put the focus on thinking and reasoning, and the teacher’s ability to highlight and annotate projected work pages supported students’ learning.
The technology played an important role in supporting class discussions. Two critical factors were (1) the ability for students to submit work and see it displayed enabled students to get and give immediate feedback, and (2) the teacher’s ability to annotate displays using a tablet pen supported conversation that helped students learn to explain their thinking.
What role can machine "understanding" of student work play in facilitating the teaching and learning of multiplication and division?
We have created an infrastructure for machine analysis of visual representations that are created by students using a combination of our digital tools and freehand drawing. While we have not yet completely investigated the possibilities and limits of this system, we are convinced that it has significant potential as a technique that can support teachers in making better use of the work their students do in class, beyond marking it "right" or "wrong."
As we continue to develop and investigate machine analysis tools, a teacher will be able not only to view student work in real time, but also will get summary information about her students, including an overview of the problem-solving strategies students are using. This overview is based on our software's ability to analyze not only a particular visual representation, but also the process a student used to create the representation.
Selected Classroom Trials & Research Findings
We have conducted classrooms trials in 12 schools in five Boston-area school districts, working with 17 teachers and 800+ students learning math and science. The teachers partnered with us to help us design and test our software, working with us both in and outside their classrooms. We observed their classrooms, co-teaching with them at times, and interviewed both teachers and students. Classroom trials ran from two days to five weeks. Each daily session lasted between 50 minutes and several hours. After each trial, we continued to refine the software and focus on answers to our research questions based on the results of the trial. A subset of the trials are described below. (Click on a trial title to see details.)