Tag: habits of mind

Posted in Teaching mathematics

The Four Freedoms in the Classroom

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You will find that by providing the following freedoms in your classroom an improved learning environment will be created.

The Freedom to Make Mistakes

Help your students to approach the acquisition of knowledge with confidence. We all learn  through our mistakes. Listen to and observe your students and encourage them to explain or demonstrate why they THINK what they do. Support them whenever they genuinely participate in the learning process. If your class is afraid to make mistakes they will never reach their potential.

The Freedom to Ask Questions

Remember that the questions students ask not only help us to assess where they are, but assist us to evaluate our own ability to foster learning. A student, having made an honest effort, must be encouraged to seek help. (There is no value in each of us re-inventing the wheel!). The strategy we adopt then should depend upon the student and the question but should never make the child feel that the question should never have been asked.
classroom quote

The Freedom to Think for Oneself

Encourage your class to reach their own solutions. Do not stifle thought by providing polished algorithms before allowing each student the opportunity of experiencing the rewarding satisfaction of achieving a solution, unaided. Once, we know that we can achieve, we may also appreciate seeing how others reached the same goal. SET THE CHILDREN FREE TO THINK.

The Freedom to Choose their Own Method of Solution

Allow each student to select his own path and you will be helping her to realize the importance of thinking about the subject rather than trying to remember.

These freedoms help develop students skills and habits of mind.

Posted in Teaching mathematics

Making generalizations in mathematics

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Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students’ mental disposition or  habits of mind in learning and dealing with mathematics is one of the important goals of mathematics education.  Making generalizations is a skill, vital in the functioning of society. It is one of the reasons why mathematics is in the curriculum. Learning mathematics (if taught properly) is the best context for developing the skill of making generalizations.

What is generalization?

There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of “finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction (click here to read my post about abstraction). The second meaning includes extension (empirical or mathematical) of existing concept  or a mathematical invention. Perhaps the most famous example of the latter is the invention of non-euclidean geometry. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

For further discussion on these meanings, read Michael Mitchelmore paper The role of abstraction and generalization in the development of mathematical knowledge. For discussion about the importance of generalization and some example of giving emphasis to it in teaching algebra, the book Approaches to Algebra – Perspectives for Research and Teaching is highly recommended. There is a chapter about making generalizations and with sample tasks that help promote this attitude.

Sample lessons

Mathematical investigations and open-ended problem solving tasks are ways of engaging students in making generalizations. The following posts describes lessons of this type:

  1. Sorting number expressions
  2. Lesson study: Teaching subtraction of integers
  3. Math investigation lesson: polygons and algebraic expressions
  4. Polygons and teaching operations on algebraic expressions

Of course it is not just the type of tasks or the design of the lesson but also the classroom environment that will help promote making generalization and make it part of classroom culture. Students will need a classroom environment that allows them time for exploration and reinvention. They will need an environment where a questioning attitude is promoted: “Does that always work?” ,”How do I know it works”? They will need an environment that accords respect for their ideas, simple or differing they may be.

Posted in Algebra

Teaching algebraic expressions – Counting smileys

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This is an introductory lesson for teaching the concept of variable and algebraic expressions through problem solving. The problem solving task combines numerical, geometric, and algebraic thinking.  The figure below shows the standard version of the task. Of course some easier versions would ask for the 5th figure, then perhaps 10th figure, then the 100th figure, and then finally for the nth figure. This actually depends on the mathematical maturity of the students.

An alternative version which I strongly encourage that teachers should try is to simply show first the diagrams only (see below).

Study the figures from left to right. How is it growing? Can you think of systematic ways of counting the number of smileys for a particular “Y” that belongs to the group? This way it will be the students who will think of which quantity (maybe the number of smileys in the trunk of the Y or the position of the figure) they could represent with n.The students are also given chance to study the figures, what is common among them, and how they are related to one another. These are important mathematical thinking experiences. They teach the students to be analytical and to be always on the lookout for patterns and relationships. These are important mathematical habits of mind.

Here are possible ways of counting the number of smileys: The n represents the figure number or the number of smiley at the trunk.

1. Comparing the smileys at the trunk and those at the branches.

In this solution, the smileys at the branches is one less than those at the trunk. But there are two branches so to count the number of smileys, add the smileys at the trunk which is n to those at the two branches, each with (n-1) smileys. Hence, the algebraic expression representing the number of smileys at the nth figure is n+2(n-1).

2. Identifying the common feature of the Y’s.

The Y’s have a smiley at the center and has three branches with equal number of smileys. In Fig 1, there are no smiley. In Fig 2, there is one smiley at each branch. In fact in a particular figure, the number of smileys at the branches is (n-1), where n is the figure number. Hence the algebraic expression representing the number of smileys is 1+ 3(n-1).

3. Completing the Y’s.

This is one of my favorite strategy for counting and for solving problems about area. This kind of thinking of completing something into a figure that makes calculation easier and then removing what were added is applicable to many problems in mathematics. By adding one smiley at each of the branches, the number of smileys becomes equal to that at the trunk. If n represents the smileys at the trunk (it could also be the figure number) then the algebraic representation for counting the number of smilesy needed to build the Y figure with n smiley at each branches and trunk is 3n-2, 2 being the number of smileys added.

4. Who says you’re stuck with Y”s?

This is why I love mathematics. It makes you think outside the box. The task is to count smileys. It didn’t say you can not change or transform the figure. So in this solution the smileys are arrange into an array. With a rectangular array (note that two smileys were added to make a rectangle), it would be easy to count the smileys. The base is kept at 3 smileys and the height corresponds to the figure number. Hence the algebraic expression is (3xn)-2 or 3n-2.

The solutions show different visualization of the diagram, different but equivalent algebraic expressions, and all yielding the same solution. Of course there are other solutions like making a table of values but if the objective is to give meaning to algebraic symbols, operations, and processes, it’s best to use the visuals.

A more challenging activity involved Counting Hexagons. Click the link if you want to try it with your class.

Posted in Curriculum Reform

What is mathematical literacy?

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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: