Tag: problem solving

Posted in What is mathematics

Math is not easy to learn – that is a fact.

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I think it’s a waste of time trying to make math easy and fun to learn if your idea of fun does not involve challenge.

Mathematics is not an easy subject and it is not easy to learn it. That is a fact. The sooner the teacher accepts this, the better for her students. The challenge to us teachers is not in how we can make math easy to learn but in how we can make it makes sense and how we can make our students love the challenge that mathematics presents. Can math be challenging if students feel that what they are expected to do in the class is to follow the teacher’s method, the teacher’s way of thinking, and the teacher’s way of doing things? Where is the fun in that?

Mathematics is not fun to learn if the idea of fun is like playing bingo! However, if ‘fun’ is a function of the challenge a sport or a game presents, then indeed learning mathematics is fun. We love a sport because of the challenge it presents, the opportunities it gives us to make prediction, analyze, strategize, make our stand and defend it, etc and not because it is easy to play!

Everything in mathematics makes sense. Everything in mathematics is connected to everything else. I think this is where we teachers should be devoting our time to. And this is what this blog is about!

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.
Posted in Assessment, Curriculum Reform

Features of good problem solving tasks for learning mathematics

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To develop higher-order thinking skills (HOTS) the mind needs to engage in higher-order learning task (HOLT). A good task for developing higher-order thinking skills is a problem solving task. But not all problems are created equal. Some problems are best suited for evaluating learning while others are best suited for assessing learning that would inform teaching. This post is about the second set of problems.The difference between these two sets of problems is not the content and skills needed to solve them but the way they are constructed.

What are the features of a good problem solving task for learning mathematics?
  1. It uses contexts familiar to the students
  2. What is problematic is the mathematics rather than the aspect of the situation
  3. It encourages students to use intuitive solutions as well as knowledge and skills they already possess
  4. The task can have several solutions
  5. It challenges students to use the strategy that would highlight the depth of their understanding of the concept involved
  6. It allows students to show the connections they have made between the concepts they have learned

It is this kind of problem solving task that is used in the strategy Teaching through Problem Solving (TtPS) which I described in the previous post. Here is a sample task:

Students solutions to the task can be used to teach area of polygons, kinds of polygons, preserving area, and meaning of algebraic expression. You can use the task to facilitate students construction of knowledge about adding, subtracting, multiplying and dividing algebraic expressions. Yes, you read it right. This is a good problem solving task for introducing operations with algebraic expression through problem solving! The problem above is also an example of a mathematical tasks that links algebra and geometry. Good mathematics teaching always links concepts.

Posted in Curriculum Reform, Mathematics education

Teaching through Problem Solving

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Problem solving is not only the reason for teaching and learning mathematics. It is also the means for learning it. In the words of Hiebert et al:

Students should be allowed to make the subject problematic. … Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students. (Hiebert, et al, 1996, p. 12)

For years now, UP NISMED in-service training programs for teachers have organized mathematics lessons for teachers using the strategy we call Teaching through Problem Solving (TtPS). This teaching strategy had also been tried by teachers in their classes and the results far outweighed the disadvantages anticipated by the teachers.

Teaching through problem solving provides context for reviewing previously learned concepts and linking it to the new concepts to be learned. It provides context for students to experience working with the new concepts before they are formally defined and manipulated procedurally, thus making definitions and procedures meaningful to them.

What are the characteristics of a TtPS?

  1. main learning activity is problem solving
  2. concepts are learned in the context of solving a problem
  3. students think about math ideas without having the ideas pre-explained
  4. students solve problems without the teacher showing a solution to a similar problem first

What is the typical lesson sequence organized around TtPS?

  1. An which can be solved in many ways is posed to the class.
  2. Students initially work on the problem on their own then join a group to share their solutions and find other ways of solving the problem. (Role of teacher is to encourage pupils to try many possible solutions with minimum hints)
  3. Students studies/evaluates solutions. (Teacher ask learners questions like “Which solutions do you like most? Why?”)
  4. Teacher asks questions to help students make connections among concepts
  5. Teacher/students extend the problem.

What are the theoretical underpinnings of TtPS strategy?

  1. Constructivism
  2. Vygotsky’s Zone of Proximal Development (ZPD)

Click here for sample lesson using Teaching through Problem Solving to teach the tangent ratio/function.

The best resource for improving one’s problem solving skills is still these books by George Polya.

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volume I