Category: Teaching mathematics

proportion
Posted in Teaching mathematics

What is proportional reasoning? Does cross multiplication help learn it?

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proportionProportional reasoning is a capstone of children’s elementary school arithmetic and a cornerstone of all that is to follow (Lesh & Post, 1988). But for some reason, the teaching of elementary school mathematics topics either become an end in itself or has become more a preparation for learning algebra. Proportional reasoning is not being given its due attention. Solving proportion problems has become an exercise of applying routine procedure than an opportunity to engage students in proportional reasoning.

What is proportional reasoning? Why is it important?

Proportional reasoning is a benchmark in students’ mathematical development (De Bock, Van Dooren, Janssens, & Verschaffel, 2002). It is considered a milestone in students’ cognitive development. It involves:

  1. reasoning about the holistic relationship between two rational expressions such as rates, ratios, quotients, and fractions;
  2. synthesis of the various complements of these expressions;
  3. an ability to infer the equality or inequality of pairs or series of such expressions;
  4. the ability to generate successfully missing components regardless of the numerical aspects of the problem situation; and
  5. involves both qualitative and quantitative methods of thought and is very much concerned with prediction and inference.

Proportional reasoning involves a sense of co-variation and of multiple comparisons. In this sense it is a ‘subset’ of algebraic thinking which also give emphasis on structure and thinking in terms of relationship.

What is cross multiplication? Does it promote proportional reasoning?

Cross multiplication is a procedure for solving proportion of the type A/B = x/D. It solves this equation by this process: A*D = x*B. This algorithm is not intuitive. It is not something that one will ‘naturally generate”. Studies have consistently shown that only very few students understand it although many can carry out the procedure. I know many teachers simply tell the students how to do cross multiplication and use specific values to verify that it works without explaining why the algorithm is such.

Many mathematics textbooks and lessons are organized in such a way that students are taught to do cross multiplication before asking them to do problems involving proportion. This practice deprives the students from understanding the idea of proportion and developing their proportional thinking skills. Research studies recommend to defer the introduction of cross multiplication until students have fully understood proportion and have had experiences in solving proportion problems using their knowledge of operation and their understanding of fraction, ratios, and proportion.

References and further readings:

  1. Number Concepts and Operations in the Middle Grades
  2. Proportional reasoning tasks and difficulties
  3. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-8
Posted in Curriculum Reform, Teaching mathematics

What is reasoning? How can we teach it?

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The  world does not give us complete information that’s why call on our power of reasoning to complete this information the best we can and produce new knowledge.  Mathematics is one of its most famous product.

What is reasoning? When do we learn it?

Reasoning is defined as  the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions and beliefs. We can make this definition more specific using Ol’eron’s:

“Reasoning is an ordered set of statements, which are purposefully linked, combined or opposed to each other respecting certain constraints that can be made explicit.” – Ol’eron (1977; 9)

Teachers’ knowledge of learning trajectory for reasoning is as important as their knowledge of students’ typical learning trajectory for specific content topics. In this post I will share a framework that I think will be useful for teachers in developing the reasoning skills of learners. I cannot anymore trace where I got this idea but I know it’s from a Japanese lesson study document I was reading last year. Reasoning is a skill highly emphasized in Japanese mathematics lessons. They have developed a framework for analyzing their students ‘reasoning trajectory’. This is applicable even in non-mathematics context. The framework even specifies the grade level to which a particular way of reasoning and arguing it is expected.

Levels of reasoning
  1. At the end of 2nd grade, students begin using expressions such as “because…” to describe their reasons and support their ideas.
  2. In 3rd grade, they begin comparing their own ideas with others and use expressions such as “my idea is similar to so-and-so’s idea…”
  3. In 4th grade, students use expression such as “for example…” and “because…,” more frequently Moreover, they begin to use hypothetical statements such as “if it is… then…”
  4. In 5th grade, they can become more sophisticated and make statements such as, “If it is … then it will be *, but if it is # then I think we can say @” by looking at different conditions.
  5. Finally, in 6th grade, students can start describing in ways such as, “It can be said when it is … but in this situation # is much better,” and starting to make decisions about how to choose a better idea.
In teaching mathematics, reasoning need not always be restricted to that of formal, logical or mathematical forms of reasoning. Words and phrases such as those listed above should be part of the students communication. It is therefore important to listen to the way students make their arguments or reason out in whole class and small-group discussion. If these are not yet part of the everyday communication of mathematics in our classes then its time for us to design the lesson that creates the environment where these kind of thinking and communicating is encouraged. Problem solving and mathematical investigation activities are great context where this can happen.

 

Posted in Teaching mathematics

What is the role of visualization in mathematics?

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Like abstraction and generalization which I described in my earlier posts here and here,visualization is central to the learning and understanding of mathematics. (Note that these processes are also natural human mental dispositions and so we can claim that doing mathematics is a natural human activity.)

Visualization used to be considered only for illustrating otherwise abstract ideas of mathematics but now visualization has become a key component of mathematical processes such as reasoning, problem solving, and even proving.

What is visualization?

Synthesizing the definitions of visualization offered by Zimmermann and Cunningham (1991, p. 3) and Hershkowitz (1989, Abraham Arcavi proposes the following definition:

Visualization is the ability, the process and the product of, creation, interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings– Abraham Arcavi, ESM, 2003

What are examples of use of visualization in mathematics?
  1. For communicating information, the graph is perhaps the most recognizable of the visual representations of mathematics.
  2. For proving, visual proofs are already accepted as legitimate proofs. For example, here’s a visual proof of the Pythagorean Theorem. Click here for source of movie. [iframe 350 500]
  3. Of course, visuals can also be used to challenge students to reason and explain in words and symbols. For example teachers can show the visual in #2 then ask the students what the visual is telling them about the relationships between the areas of the three squares and about the sides of right triangles. Students should be asked to support their claim.
  4. Visualization tasks also trains students mind to ‘think outside the box’. Click here for an example of a problem solving tasks which can be solved by visualizing possible arrangements. Patterning activity like Counting Hexagons are great activities not only for generating formulas and algebraic expressions but trains the mind to look for relationships, an important component in algebraic thinking.
  5. Because what we see usually depends on what we know, visuals can also be used as context for assessing students knowledge of mathematics. Click here for an example on how to assess understanding by asking students to construct test items.
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.