## Learning By Design, the New Media Literacies, and Math

An experienced teacher might reasonably ask: Why do we need to teach linear algebra differently?

A common approach in grade 9 to teaching linear equations is to use basic everyday examples. A teacher can explain to students that if one goes to the store and wants to buy 4 candy bars, and each candy bar costs $2.50, how much money will that cost? Most students will pick up that 4 bars times $2.50 per bar equals $10. But what if one wants to buy 6 bars or 7? A teacher can then work with students to help them turn this into a linear equation: 2.5x = y where x is the number of bars one wants to buy and y is the amount of money it will cost. Most students get this concept. Then one can graph the equation as well so students can see the linear relationship. And then a teacher can introduce the y = mx + b concept. Unit accomplished.

This progression makes rational sense. So why are we proposing to change things?

There are a lot of reasons and there are whole sets of pedagogical ideas supported by research that point to problems with the traditional methods and suggest how we might design more meaningful approaches to the learning of mathematics. But if we had to boil all of that down to just one essential critique, it is that most students don’t find the approach described above very engaging. They can understand the logic of the procedure as described, but this experience of learning it is not transformative. It does not offer a solution to an authentic problem they have ever encountered in real life. There is little opportunity here for active and critical learning or the possibility of participating, even peripherally, in an authentic, evolving discourse. Yes, one likely needs to know something about algebra to become a mathematician, a business person, an architect, or any related field involving mathematics, but this knowledge, as presented above, is not situated within a context that suggests it is either meaningful or useful.

Our effort here, to re-imagine a grade 9 math class, centres on a whole cluster of exciting educational research and theory. We want to keep this document practical and accessible, but in the interest of transparency, we feel we should summarize a few of the core concepts.

The idea of multimodal literacy has emerged with the internet age (The New London Group, 1996). The thinking here positions linguistic literacy as just one of the modes of literacy needed to communicate effectively in our information age. The New London Group (1996) lists five distinct modes of literacy: linguistic, visual, gestural, spatial, and audio, as well as the various combinations of these types, referred to as multimodal literacy. Anna Sfard, building on these ideas, repositions children’s emerging thinking in mathematics as an effort to communicate (Caspi & Sfard, 2012). From this perspective, thinking and communicating develop together. As students learn to articulate ideas, their thinking sharpens. Recognizing communication as multimodal, students’ gestures, as they try to explain an emerging concept in math class, or their efforts to visualize a concept through a model or diagram, all become vitally important components of the process of learning. Mathematics is a particular multimodal discourse within an engaged community of practitioners who value abstract ideas.

Specifically, school-level algebra is understood as a meta-arithmetic discourse relating to the generalizing of procedures and the solving of equations. It exists both as a formalized mathematical discourse and also arises as a natural, informal discourse among school-aged learners (Caspi & Sfard, 2012). How to nurture these emerging conversations among students and gently guide them towards an engaged appropriation of the formalized discourse becomes our goal as teachers.

Situating mathematical activity in the context of larger, real-world conversations is a core strategy that grows out of the multiliteracy approach (The New London Group, 1996). In terms of linear algebra, we find those contexts in product pricing models, sports statistics, travel planning, art-creation and other common activities as well as through a meta-discussion about mathematicians, their real-world activities, and their histories. In the special context of British Columbia, we also recognize that these are important tools for First Nations communities concerned with understanding data in diverse contexts from resource management to history.

Another body of work that is central to our course design relates to the new media literacies. Jenkins, Purushotma, Weigel, Clinton & Robison (2009) describe these as the 11 skills needed to fully participate in our emerging digital culture: play, simulation, judgement, transmedia navigation, collective intelligence, appropriation, play, negotiation, performance, visualization, and multitasking. To this list, we are adding a meta-cognitive awareness of the creative process as described by Resnick (2007) that includes imagining, creating, playing, sharing, and reflecting. These are the 12 salient skills of our time. The gamut of multimodal literacies reconverge as we employ these skills in our social lives and at work. The younger generation knows this already. Many already possess these skills, in varying degrees, though they likely did not learn them at school. If we want math class to be relevant and engaging for students, these are the competencies most seminal for engagement and development. These are the ways many youth of today communicate.

In these skill areas, we see a gap between students who move confidently through the digital world and those who have not had opportunities to develop these needed competencies. In our vision, math class can help close this gap by scaffolding meaningful and relevant learning experiences for all students in the context of an exploration of fundamental concepts in mathematics.

Reflecting back on our original description of the typical approach to teaching linear algebra, as we reimagine this course, we intentionally slow down the progression, exploring real-world data using technological tools for collecting data and for modeling and simulating real-world situations. Through this activity and in the related class discussions, algebra emerges, naturally, as a way of generalizing about relationships and as a technique to facilitate calculations. Slowly the conversations shifts from a natural, invented language into an appropriation of the formalized discourse. Teaching practice transformed. Unit accomplished.

An experienced teacher might reasonably ask: Why do we need to teach linear algebra differently?

A common approach in grade 9 to teaching linear equations is to use basic everyday examples. A teacher can explain to students that if one goes to the store and wants to buy 4 candy bars, and each candy bar costs $2.50, how much money will that cost? Most students will pick up that 4 bars times $2.50 per bar equals $10. But what if one wants to buy 6 bars or 7? A teacher can then work with students to help them turn this into a linear equation: 2.5x = y where x is the number of bars one wants to buy and y is the amount of money it will cost. Most students get this concept. Then one can graph the equation as well so students can see the linear relationship. And then a teacher can introduce the y = mx + b concept. Unit accomplished.

This progression makes rational sense. So why are we proposing to change things?

There are a lot of reasons and there are whole sets of pedagogical ideas supported by research that point to problems with the traditional methods and suggest how we might design more meaningful approaches to the learning of mathematics. But if we had to boil all of that down to just one essential critique, it is that most students don’t find the approach described above very engaging. They can understand the logic of the procedure as described, but this experience of learning it is not transformative. It does not offer a solution to an authentic problem they have ever encountered in real life. There is little opportunity here for active and critical learning or the possibility of participating, even peripherally, in an authentic, evolving discourse. Yes, one likely needs to know something about algebra to become a mathematician, a business person, an architect, or any related field involving mathematics, but this knowledge, as presented above, is not situated within a context that suggests it is either meaningful or useful.

Our effort here, to re-imagine a grade 9 math class, centres on a whole cluster of exciting educational research and theory. We want to keep this document practical and accessible, but in the interest of transparency, we feel we should summarize a few of the core concepts.

The idea of multimodal literacy has emerged with the internet age (The New London Group, 1996). The thinking here positions linguistic literacy as just one of the modes of literacy needed to communicate effectively in our information age. The New London Group (1996) lists five distinct modes of literacy: linguistic, visual, gestural, spatial, and audio, as well as the various combinations of these types, referred to as multimodal literacy. Anna Sfard, building on these ideas, repositions children’s emerging thinking in mathematics as an effort to communicate (Caspi & Sfard, 2012). From this perspective, thinking and communicating develop together. As students learn to articulate ideas, their thinking sharpens. Recognizing communication as multimodal, students’ gestures, as they try to explain an emerging concept in math class, or their efforts to visualize a concept through a model or diagram, all become vitally important components of the process of learning. Mathematics is a particular multimodal discourse within an engaged community of practitioners who value abstract ideas.

Specifically, school-level algebra is understood as a meta-arithmetic discourse relating to the generalizing of procedures and the solving of equations. It exists both as a formalized mathematical discourse and also arises as a natural, informal discourse among school-aged learners (Caspi & Sfard, 2012). How to nurture these emerging conversations among students and gently guide them towards an engaged appropriation of the formalized discourse becomes our goal as teachers.

Situating mathematical activity in the context of larger, real-world conversations is a core strategy that grows out of the multiliteracy approach (The New London Group, 1996). In terms of linear algebra, we find those contexts in product pricing models, sports statistics, travel planning, art-creation and other common activities as well as through a meta-discussion about mathematicians, their real-world activities, and their histories. In the special context of British Columbia, we also recognize that these are important tools for First Nations communities concerned with understanding data in diverse contexts from resource management to history.

Another body of work that is central to our course design relates to the new media literacies. Jenkins, Purushotma, Weigel, Clinton & Robison (2009) describe these as the 11 skills needed to fully participate in our emerging digital culture: play, simulation, judgement, transmedia navigation, collective intelligence, appropriation, play, negotiation, performance, visualization, and multitasking. To this list, we are adding a meta-cognitive awareness of the creative process as described by Resnick (2007) that includes imagining, creating, playing, sharing, and reflecting. These are the 12 salient skills of our time. The gamut of multimodal literacies reconverge as we employ these skills in our social lives and at work. The younger generation knows this already. Many already possess these skills, in varying degrees, though they likely did not learn them at school. If we want math class to be relevant and engaging for students, these are the competencies most seminal for engagement and development. These are the ways many youth of today communicate.

In these skill areas, we see a gap between students who move confidently through the digital world and those who have not had opportunities to develop these needed competencies. In our vision, math class can help close this gap by scaffolding meaningful and relevant learning experiences for all students in the context of an exploration of fundamental concepts in mathematics.

Reflecting back on our original description of the typical approach to teaching linear algebra, as we reimagine this course, we intentionally slow down the progression, exploring real-world data using technological tools for collecting data and for modeling and simulating real-world situations. Through this activity and in the related class discussions, algebra emerges, naturally, as a way of generalizing about relationships and as a technique to facilitate calculations. Slowly the conversations shifts from a natural, invented language into an appropriation of the formalized discourse. Teaching practice transformed. Unit accomplished.