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 Combinatorics

What makes counting problems difficult

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I don’t have the numbers but I think not many will disagree with me that among those who like mathematics and find joy in the challenge of solving math problems, only very few will also like to solve combinatorial or counting problems. And if I may be allowed to push my observation a bit further, those who are more inclined to algebra are the ones who have more aversion to problems that asks them to count.

Indeed, combinatorial problems are odd. Unlike other mathematics problems, these type of problems cannot easily be categorized and solved with predictable algorithms. Each problem always seems to be a case in itself. Knowing all the formulas for different cases of permutations and combinations is not a guarantee that one will be able to solve these problems. For many, these are only useful if the problem already explicitly asks for number of permutations and combinations. And when indeed the problem says so, one still has to decipher other conditions and the assumptions embedded in the context. Knowing when to add or multiply is also difficult, if not the most difficult part. Knowing when one already double counted also requires depth of understanding not only of the formulas but also of the context of the problem. Of course for some these are also the very reason why they love to solve counting problems.

Take the case of this family of problems:

 

  1. How many three-digit numbers can you form from the digits, 5, 6, 7, 8, and 9? (This requires direct application of the permutation formula or better, the multiplication principle.)
  2. How many three-digit numbers are there in all? (That you will choose from 0 to 9 is implied only plus you ask yourself how about 027? Is it a 3-digit number of a 2-digit number?)
  3. How many numbers can be formed from 5, 6, 7, 8, and 9?
  4. How many three-digit numbers greater that 600 can be formed from the digits 5, 6, 7, 8, and 9? (Oops … so what formula is applicable now?)
  5. How many numbers greater than 3,000,000 can be formed by arrangements of the digits 1, 2, 2, 3, 6, 6, 6? (More Ooops…)
Having identified some of the difficulties inherent to counting problems, what teaching tip can I propose?
  1. Ask for different ways of solving the problem, slowly discouraging students’ reliance to listing. Thinking in terms of tree diagrams and empty boxes are great help.
  2. Defer introduction of formulas before students can solve the problems like the ones above using the fundamental counting principle. Permutations and combination formulas are only powerful to the extent to which students understand the multiplication principle.
  3. Another pedagogical tip is to ask students to identify what makes a problem similar and different in structure from the problems they previously solved. If they can specify which problem, the better.
  4. As a consequence of tip#3, problems are best given per family with family defined in terms context (e.g., the family of problems given in the example above) and not in terms of similarity in solution structure.

Please add yours. Thanks.

You may want to read my two other posts on combinatorics:

Some useful references:
Posted in Combinatorics

Pascal’s triangle and Counting Permutations

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This is the second in my series of posts in combinatorics. The first post links the Fundamental  Counting Principle, Powers of 2, and the Pascal Triangle. This second post connects the Pascal’s Triangle and the formula for counting the number of permutations with identical objects. The context for connections is a puzzle about counting the total routes of a rook to squares in a chessboard.

The Puzzle

Trace the shortest route the rook can traverse from its corner position to the opposite corner. How many such routes are there?

Solution 1

A useful problem solving strategy here is to simplify the problem: Count the number of distinct paths the rook will land to the nearest square.

To determine the shortest path, the rook should either go north or east or left or right only. That is, the rook has only two choices each time it moves through a square as shown in the figure below. You will not reach the third or fourth line before you figure out that the pattern generates the Pascal’s Triangle.

Thus there are 3432 distinct shortest paths the rook can traverse from one corner to the opposite corner.

Solution 2

Another way to solve puzzle 2 is shown below. The arrows trace some possible shortest paths for the rook.

The paths of the arrows in the figure can be represented this way: (N = North, E = East)

  1. N N N E N E E N E N E N E E
  2. E E E E N N N E E N N N N E
  3. E E E E E E E N N N N N N N
Each shortest path from one corner square to the opposite corner square is made up of 7 N’s and 7 E’s. Thus, counting the total number of shortest paths is the same as counting the total number of distinct arrangements of 7 N’s and 7 E’s in a row. The formula for counting the permutation of n objects a, b, c, … of which are identical will be useful here. This is given by the formula: P = \frac{n!}{a!b!c!...}. So,
P = \frac{14!}{7!7!} = 3432.
Click here to see solutions to this problem using the multiplication principle and combination formula.
If you want to practice your skill in solving permutation and combination problems, Examrace Permutation & Combination Made Easy (Examrace Aptitude Series) will be useful.