Category: Math investigations

Posted in Math investigations, Teaching mathematics

What is cognitive conflict approach to teaching?

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According to Piaget, knowledge is constructed when a learner encounters input from the environment and incorporates the new experiences to his/her existing schemes and mental structures (assimilation). When this new assimilated information conflicts with previously formed mental structures, the result is called disequilibrium – a cognitive conflict. This state of disequilibrium motivates the learner to seek equilibrium. The cognitive conflict approach to teaching is based on this assumption – that learners will seek this equilibrium.

Cognitive conflict approach to teaching exposes students in  situations where some of their existing understandings about an idea or a topic no longer hold. A famous example on this is the Chords and Regions activity:

Find a way of predicting the maximum number of regions created by chords connecting n points.

This activity is usually used to challenge students thinking that patterns observed will always hold true and that patterns can be used as proof. The pattern observed will not hold true for n > 5. You can read the result of this activity in this paper Chords and Regions.

The assumption that learners will seek equilibrium when they are put in a situation of disequilibrium, when they experience cogntive conflict isn’t often the case. In fact, a common challenge faced by the cognitive conflict approach is that students often possess ‘contradictory understandings’ (from a mathematical point of view) but they don’t feel the need to address the inconsistencies in their understandings. This is the reason why it is very hard to correct a misconception. Also, students often do not see the importance (or necessity) to engage in a process of modifying their understandings to resolve the contradictions and they tend to treat the contradictions as exceptions (Stylianides & Stylianides, ICME-11). In the above activity for example, instead of being challenged, students can just accept the fact that the pattern stops after n=5 and not try to think of a more general rule to cover all cases. It is also possible that students can just say Next time I’ll try up to 10 cases before generalizing. This is now the challenge to the teacher. As a teaching approach, the use of cognitive conflict has a lot of potential but it needs more than simply using the appropriate task to create the conflict. Our students can be very resilient.

Mary Pardoe via LinkedIn discussion suggests that a strategy that encourages students to confront, rather than avoid, a cognitive conflict is to challenge small groups of students to reach a group conclusion (about the situation) with which everyone in the group agrees. Students who individually might respond differently to the ‘conflict’ will usually then try to persuade each other that their own points of view are correct, and so they are motivated to explain and discuss their thinking. Sample teaching using this approach is described in Using cognitive conflict to teach solving inequalities.

Common misconceptions are also rich sources of tasks for creating cognitive conflict. Click the link Mistakes and Misconceptions and Top 10 Errors in Algebra for sample of tasks.

You may want to check the book below to get more ideas on teaching mathematics.

Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education)

 

Posted in Algebra, High school mathematics, Math investigations

Teaching combining algebraic expressions with conceptual understanding

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In Math investigation about polygons and algebraic expressions I presented possible problems that students can explore. In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding. Like the rest of the lessons in this blog, this lesson is not so just about learning the math but also making sense of the math and engaging students in problem solving.

The lesson consists of four problem solving tasks to scaffold  learning of adding, subtracting, multiplying and dividing algebraic expression with conceptual understanding.

Problem 1 – What are the different ways can you find the area of each polygons? Write an algebraic expression that would represent each of your method.

The diagram below are just some of the ways students can find the area of the polygons.

1. by counting each square
2. by dissecting the polygons into parts of a rectangle
3. by completing the polygon into a square or rectangle and take away parts included in the counting
4. by use of formula

The solutions can be represented by the algebraic expressions written below each polygon. Draw the students’ attention to the fact that each of these polygons have the same area of 5x^2 and that all the seven expressions are equal to5x^2 also.

Multiple representations of the same algebraic expressions

Problem 2 – (Ask students to draw polygons with a given area using algebraic expressions with two terms like in the above figure. For example a polygon with area 6x^2-x^2.

Problem 3 – (Ask students to do operations. For example 4.5x^2-x^2.)

Note: Whatever happens, do not give the rule.

Problem 4 – Extension: Draw polygons with area 6xy on an x by y unit grid.

These problem solving tasks not only links geometry and algebra but also concepts and procedures. The lesson also engages students in problem solving and in visualizing solutions and shapes. Visualization is basic to abstraction.

There’s nothing that should prevent you from extending the problem to 3-D. You may want to ask students to show the algebraic expression for calculating the surface area of  solids made of five cubes each with volume x^3. I used Google SketchUp to draw the 3-D models.

some possible shapes made of 5 cubes

Point for reflection

In what way does the lesson show that mathematics is a language?

Posted in Elementary School Math, Geometry, Math investigations

Math investigation lesson on polygons and algebraic expressions

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Understanding is about making connection. The extent to which a concept is understood is a function of the strength of its connection with other concepts. An isolated piece of knowledge is not powerful.

To understand mathematics is to make connections among concepts, procedures, contexts. A lesson that has a very good potential for learning a well-connected mathematical knowledge is one which is organized around a mathematical investigation. This is because of the divergence nature of this task which revolves around a single tool or context.

Here is a simple investigation activity about polygons. Change the x by x unit to 1 by 1 unit if you will give this to Grade 5-6 students.

Investigate polygons with area 5x^2 units on an x by x unit grid.

Some initial shapes students could come up with may look like the following:

different shapes, the same area
Figure 1. Polygons with the same area

Note: This is a mathematical investigation so the students are expected to pose the problems they want to pursue and on how they will solve it. It will cease to be a math investigation if the teachers will be the one to pose the problems for them. The following are sample problems that students can pose for themselves.

  • What is the same and what is different among these polygons? How can I classify these polygons?

Possible classifications would be

a. convex vs non-convex polygons

b. according to the number of sides

  • What shapes and how many are there if I only consider polygons made up of squares?

Students will discover that while they can have as many polygons with an area of 5, there are only 12 polygons made of  squares.  This is shown in Figure 2. These shapes are called pentominoes because it is made up of 5 squares. I have arranged it here for easy recall of shapes. It contains the last seven letters of the english alphabet (TUVWKXZ) and the word FILIPINO without the last 2 I’s and O in the spelling.

Figure 2. Pentominoes
  • Is there a way of constructing different triangles or any of the polygons with same area?

Figure 3 shows this process for triangle.

Figure 3.Triangle with same area

Click this or the  figure below to see this process in dynamic mode using Geogebra.

Fig 4 – Preserving area of triangle in Geogebra

Possible extension of this investigation is to consider polygons with areas other than 5x^2.

Click this link to see some ideas on how you can use this activity to teach combining algebraic expressions.

Posted in Algebra, Math investigations

Solving systems of equations by elimination – why it works

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Mathematical knowledge is only powerful to the extent to which it is understood conceptually, not just procedurally. For example, students are taught the three ways of solving a system of linear equation: by graphing, by substitution and by elimination. Of these three methods, graphing is the one that would easily make sense to many students. Substitution, which involves expressing the equations in terms of one of the variables and then equating them is based on the principle of transitive property: if a = c and b = c then a = b. But, what about the elimination method, what is the idea behind it? Why does it work?

While the elimination method seems to be the most efficient of the three methods especially for linear equations of the form ax + by = c, the principle behind it is not easily accessible to most students.

Example: Solve the system (1) 3x + y = 12 , (2) x – 2y = -2.

To solve the system by the method of elimination by eliminating y we multiply equation (1) by 2. This gives the equation (3), 6x + 2y = 24. Thus we have the resulting system,

6x + 2y = 24
x – 2y = -2.

The procedure for elimination tells us that we should add the two equations. This gives us a fourth equation (4), 7x = 22. We can then solve for x and then for y. But we have actually introduced 2 more equations, (3) and (4) in this process. Why is it ok to ‘mix’ these equations with the original equations in the system?

Equation (3) is easy to explain. Just graph 3x + y = 12 and 6x + 2y = 24. The graph of these two equations coincide which means they are equal. But what about equation (4), why is it correct to add to any of the equations? The figure below shows that equation (4) will intersect(1) and (2) at the same point.

Is this always the case? Think of any two linear equations A and B and then graph them. Take the sum or difference of A and B and graph the resulting equation C. What do you notice? This is the principle behind the procedure for the elimination method. But before students can do this investigation, they need to have some fluency on creating equation passing through a given point. The following problem can thus be given before introducing them to elimination method.

Is there a systematic way of generating other equations passing through (3,1)? This will lead to the discovery that when two linear equations A and B intersect at (p,q), A+B will also pass through (p,q). With little help, students can even discover the elimination method for solving systems of linear equations themselves from this. This problem is again another example of a task that can be used for teaching mathematics through problem solving . The task also links algebra and geometry. Click this link for a proposed introductory activity for teaching systems of equation by elimination method.