I’m working with a group of Year 7 mathematics teachers doing Lesson Study for the first time. The teachers chose to do a lesson study for what they believe to be the most difficult topic in this year level – integers. Part of my preparation as facilitator is to draw a map of what I know about teaching the topic. The map is more than a concept map because it includes not just related big ideas or concepts but also how these are taught and learned. Hence, I call this pedagogical content knowledge map (PCK map).
The PCK map I present here is a product of my own readings and my own experiences of teaching the topic. This means that it may not be the same as other teachers especially the ‘teaching part’ of the map, the ones in orange colors. For example, experience and research results back my claim that the number line is a very good way of representing the set of integers but not in teaching operations. Click here for my post about this. Notice that I gave emphasis on students knowing when a negative, a positive or a zero result rather than the rules for operation. I believe that without this, a conceptual understanding of the operation involving integers will be weak. Also, experience has taught me that although integers are numbers, the teaching of it must be algebraic. The instructions should be so designed so that students are learning algebraic thinking as well. I have noted this in the PCK map.
The map is not yet complete. I intend to include descriptions of effective activities and students’ learning trajectory of the concept after my research with the teachers. Please feel free to give your comments and share experiences for teaching integers that I could look into in my study.
The Program for International Student Assessment (PISA) of the OECD describes mathematical literacy as:
“an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen” (OECD,1999).
Mathematical literacy therefore involves more than executing mathematical procedures and possessions of basic knowledge that would allow a citizen to get by. Mathematical literacy is mathematical knowledge, methods, and processes applied in various contexts in insightful and reflective ways. According to de Lange, mathematical literacy is the overarching literacy that includes numeracy, quantitative literacy and spatial literacy. Each of these type of literacy empowers the individual in making sense of and understanding aspects of the world and his/her experiences.
De Lange’s tree structure of mathematical literacy.Spatial literacy empowers an individual to understand the three-dimensional world in which he/she lives and move. This necessitates understanding of properties of objects, the relative positions of objects and its effect on one’s visual perception, the creation of all kinds of three-dimensional paths and routes, navigational practices, etc. Numeracy is the ability to handle numbers and data in order to evaluate statements regarding problems and situations that needs mental processing and estimating real-world context. Quantitative literacy expands numeracy to include use of mathematics in dealing with change, quantitative relationships and uncertainties. Click here for deLange’s paper on this topic.
Implications to curriculum and instruction
To identify and understand the role that mathematics plays in the world is to be literate about mathematics and its applications. This means that individuals need to have an understanding of its core concepts, tools of inquiry, methods and structure.
To be able use mathematics in ways that meet the needs of one’s life as a constructive, concerned, and reflective citizen necessitates learning mathematics that is not isolated from the students’ experiences.
To be able to use mathematics to make well-founded judgment demands learning experiences that would engage students in problem solving and investigation as these would equip them to use mathematics to represent, communicate, and reason, to make decisions and to participate creatively and productively in the functioning of society.
These show that mathematical literacy requires learning mathematical concepts and principles that would be applicable to the individual and society’s life and activities; equip individuals the necessary skills in using mathematics to reason and make decisions; enable individuals to get a sense of the nature and power of the discipline in order to understand its role in the world.
To teach mathematical literacy, curriculum and instruction should therefore include these 3 R’s:
Relevant mathematical concepts, principles and procedures
Real-life context which can be investigated and modeled mathematically
