# Matrices inquiry

# The prompt

Mathematical inquiry processes: Interpret and reason; identify connections; generate examples. Conceptual field of inquiry: Matrix addition and subtraction.

Brian Ridpath devised and developed the matrices prompt for an online environment. He teaches high school mathematics at the Nevada Virtual Academy. Here he reports on how he has adapted Inquiry Maths for his classes.

Online schooling is very challenging here in the US since many of our students are from the bottom half of schools. They leave traditional schooling for a place where they think they can be successful. I have been involved in online schooling for some time and have taught fairly traditionally up through last month. I have noticed over thirty years of teaching 8th grade through college courses that students of all ages give up quite easily in problem solving and fail to self-assess in a meaningful way.

This year I have been experimenting with various self-assessment approaches and have grown more discouraged than usual at the engagement rate of my students (as measured by who turns in their work and participation rates in class). It was then I happened on the surds prompt and I saw the Inquiry Maths website. I have since tried to create some prompts and see what happens in my classes.

## Virtual environment

We use a program called Blackboard Connect which has a virtual whiteboard. Much material can be shown through uploaded slides with students and teachers mostly interacting through a chat or microphone. There are whiteboard tools for everyone - writing, typing and drawing icons, etc. I can place random groups of students into separate rooms (usually two, three or four students) to work together and I can observe their work and/or visit the room to give guidance and hints. The online environment works a bit slower than it might in a regular classroom - students like to chat rather than use the mic. So timing is an issue. Not all students show up to class. Many watch the recording of class so I usually leave one group of students in the main room to be recorded. At present, I have 170 students of which about 50 show up to classes on a regular basis.

I have designed prompts for an Algebra 2 course that involve matrix operations. The first matrix prompt (above) involved addition and subtraction. The class was well received by students and we have carried out three more inquiries (on scalar multiplication and some problem solving with matrices) since. I am not an expert by any means, but the first inquiry was fun and students wanted more so I want to try and get one per week. A few weeks into this process, I am intrigued and hopeful for its use in a virtual environment.

# Engagement and creativity through online inquiry

Brian Ridpath reports on how the inquiry developed:

My hope for that day was that the students would try to find a way to combine the objects they were looking at and then eventually discover how to add and subtract a matrix. I started with the whole class looking at the prompt and asked them to write comments about what they noticed. Of course, many asked what they should do with all those numbers. I said nothing. But a few said they saw that 1 + 3 = 4 and 1 - 3 = -2. Immediately after this insight I gave the students three regulatory cards:

Work in a group to continue.

Ask questions for two minutes then work in a group.

Continue to work as whole class.

They voted roughly 20 - 1 to work in a small group (we have a polling tool that is easy to set up and have students choose). I am able to cruise through groups and ask questions or spur them on or just watch and head to a group that is struggling to start. I had six groups of four that day and four groups were able to come up with a reasonable conclusion while two were on their way at time stop. One of the groups had started labelling entries as A1 + B1 = C1 and so on, while another group had colour coded the pairs.

We came together and I had one group explain their results, and then I showed results from other groups especially the A1 notation and colour coding and asked them why they were doing that. A student-led discussion with my questions ensued and we worked from labelling and colours to spreadsheets until a student said we could always use numbers and write A(1,1) to refer to a specific number. Eventually, to wrap up, I summarised their main points for addition of matrices, the labelling of elements and the relationship between the two. I sent them on their way with a few practice problems at that point.

The students' self-assessment of the activity was positive and in general I have found more participation in class. They did well and I was completely surprised by their creativity and questions, which were much more to the point. They were definitely more willing to attempt things. I would like to extend the time the students think and work. I see that I have to train them to develop their sense of adventure and inquisitiveness further."

# Discovering the inverse of a matrix

Professor Merrilyn Goos (of the University of Queensland) carried out research into inquiry learning. She studied classes of upper secondary students, which she described as Communities of Inquiry. One inquiry involved the students discovering the algorithm for

finding the inverse of a 2 x 2 matrix. The inquiry features in the journal article Learning Mathematics in a Classroom Community of Inquiry, Journal for Research in Mathematics Education (2004).