Posted in Curriculum Reform

What is mathematical literacy?

Defining mathematical literacy

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
  • Rich mathematical tasks that fosters conceptual understanding and development of skills and habits of mind

Check out these great books on mathematical literacy:

Posted in Curriculum Reform, Mathematics education

What is mathematical investigation?

Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended.

I first heard about math investigations in 1990 when I attended a postgraduate course in Australia.  I love it right away and it has since become one of my favorite mathematical activity for my students who were so proud of themselves when they finished their first investigation.

Problem solving is a convergent activity. It has definite goal – the solution of the problem. Mathematical investigation on the other hand is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. The exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations. It is these thinking processes which enable an individual to learn more mathematics, apply mathematics in other discipline and in everyday situation and to solve mathematical (and non-mathematical) problems.

Teaching through mathematical investigation allows  for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. The ultimate aim of mathematical investigation is develop students’ mathematical habits of mind.

Although  students may do the same mathematical investigation, it is not expected that all of them will consider the same problem from a particular starting point.  The “open-endedness” of many investigation also means that students may not completely cover the entire situation. However, at least for a student’s own satisfaction, the achievement of some specific results for an investigation is desirable. What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:

  • systematic exploration of the given situation
  • formulating problems and conjectures
  • attempting to provide mathematical justifications for the conjectures.

In this kind of activity and teaching, students are given more opportunity to direct their own learning experiences. Note that a problem solving task can be turned into an investigation task by extending the problem by varying for example one of the conditions. To know more about problem solving and how they differ with math investigation read my post on Exercises, Problem Solving and Math Investigation.

Some parents and even teachers complain that students are not learning mathematics in this kind of activity. Indeed they won’t if the teacher will not discuss the results of the investigation, highlight and correct the misconceptions, synthesize students’ findings and help students make connection among the math concepts covered in the investigation. This goes without saying that teachers should try the investigation first before giving it to the students.

I think mathematical investigation is constructivist teaching at its finest. For a sample lesson, read Polygons and algebraic expressions.

The book below offers investigation “start-up” for college students.

Posted in Algebra, Curriculum Reform

Algebraic thinking in algebra

Algebraic thinking is an approach to thinking about quantitative situations in general and relational manner. This kind of thinking is optimized by a considerable understanding of the objects of algebra, a disposition to think in generality, and engagement in high-level tasks which provide contexts for applying and investigating mathematics and the real-world.

big ideas in algebra
Ingredients in Algebraic Thinking
Objects of Algebra

The objects are the content of algebra which I classify into three overlapping categories. The first category and the most basic are those for representing changing and unchanging quantities and relationships. These include the idea of variables, numbers, graphs, equations, matrices, etc. The second category are ideas for working with unknown quantities which involve solving equations and inequalities under which are linear equations and inequalities in one variable, systems of linear equations and inequalities, exponential equations, quadratic, trigonometric equations, etc. The third and last category involves the ideas for investigating relationships between changing quantities which include directly and inversely proportional relationships; relationships with constant rate of change; relationships with changing rate of change; relationships involving exponential growth and decay; periodic relationships, etc.

Thinking dispositions

Knowledge of algebraic content do not necessarily translate in algebraic thinking. Computational fluency in simplifying, transforming, and generating expression for example, while important, do not necessarily involve a person in algebraic thinking if one is doing it for its own sake. Thinking processes that contribute to the development of algebraic thinking are those that require purposeful representations of quantities and relationships, multiple interpretations of representations, finding structures, and generalization of patterns, operations and procedures. These should become part of students’ thinking disposition.

High level tasks

The higher-order tasks in mathematics  include problem solving, mathematical investigations (sometimes referred to also as open-ended problem solving tasks), and modeling.

Posted in Curriculum Reform, Mathematics education

Understanding by Design from WikiPilipinas

I think the following entry from WikiPilipinas needs revising. “Learning of facts”? Check also the last statement.

“Teaching for understanding” is the main tenet of UbD. In this framework, course design, teacher and student attitudes, and the classroom learning environment are factors not just in the learning of facts but also in the attainment of an “understanding” of those facts, such as the application of these facts in the context of the real world or the development of an individual’s insight regarding these facts. This understanding is reached through the formulation of a “big idea”– a central idea that holds all the facts together and makes these connected facts worth knowing. After getting to the “big idea,” students can proceed to an “understanding” or to answer an “essential question” beyond the lessons taught.

One of my initial concerns about UbD in my previous post is about not checking first if the bandwagon we jumped in to will run in our roads although  I received a comment that said the DepEd did pilot it and are confident that it can. The results of the pilot I believe are not for public consumption. We just have to believe their word for it. But with this post at WikiPilipinas, I don’t know if it is clear to us what the wagon is.  Here’s the next paragraph:

Through a coherent curriculum design and distinctions between “big ideas” and “essential questions,” the students should be able to describe the goals and performance requirements of the class. To facilitate student understanding, teachers must explain the “big ideas” and “essential questions” as well as the requirements and evaluative criteria at the start of the class. The classroom environment should also encourage students to work hard to understand the “big ideas” by having an atmosphere of respect for every student idea, including concrete manifestations such as displaying excellent examples of student work.

But I love the description of traditional method of constructing the curricula in the following paragraph. Very honest. But I can’t agree about the analogy with Polya’s.

The UbD concept of “teaching for understanding” is best exemplified by the concept of backward design, wherein curricula are based on a desired result–an “understanding” or a “big idea”–rather than the traditional method of constructing the curricula, focusing on the “facts” and hoping that an “understanding” will follow. Backward design as a problem-solving strategy can even be traced back to the ancient Greeks. In his book “How to Solve It” (1945), the Hungarian mathematician George Polya noted that the Greeks used the strategy of “thinking backward” by knowing what you want as a solution in order to solve a problem.

If I remember right, G. Polya wrote “look back” as the last step for solving a problem. It means you reflect on your solution and answer in relation to the problem. But wait, there is a problem solving strategy called “working backwards” which is probably what is meant here but as an analogy to backward design? Uhmmm …

Oh, by the way, “backward design” is a problem solving strategy?

Not that I’m happy we’re adapting Understanding by Design but who cares if I’m happy with it or not. There isn’t anything I can do in that department but just to help now to make sure we make the most of it. It is is a multimillion peso project. That’s our taxes. The one in WikiPilipinas is by far the only resource in the net for UbD Philippines. If you happen to know other related sites, please share.

Here’s one research about UbD in Singapore. Here’s my other UbD related post