Posted in Mathematics education

Levels of Problem Solving Skills

Here is one way of describing students levels of problem solving skills in mathematics. I call them levels of problem solving skills rather than process of reflective abstraction as described in the original paper. As math teachers it is important that we are aware of our students learning trajectory in problem solving so we can properly help them move into the next level.problem solving

Level 1 – Recognition

Students at this level have the ability to recognize characteristics of a previously solved problem in a new situation and believe that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. A student operating at this level would not be able to mentally run-through a solution method in order to confirm or reject its usefulness.

Level 2 – Re-presentation

Students at this level are able to run through a problem mentally and are able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. Such solvers could imagine using the methods and could even imagine some of the problems they might encounter but could not take the results as a given. At this level, the subject would be unable to think about potential methods of solution and the anticipated results of such activity.

Level 3 – Structural abstraction

Students at this level evaluates solution prospects based on mental run-throughs of potential methods as well as methods that have been used before. They are able to discern the characteristics that are necessary to solve the problem and are able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought.

Level 4 – Structural awareness

A solver operating at this level is able to anticipate the results of potential activity without having to complete a mental run-through of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution.

The levels of problem solving skills described above indicate that as solvers attain the higher levels they become increasingly flexible in their thinking. This framework is from the dissertation of Cifarelli but I read it from the paper The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra by Tracy Goodson-Espy. Educational Studies in Mathematics 36: 219–245, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

You may also be interested on Levels of understanding of function in equation form based on my own research on understanding function.

Image Credit: vidoons.com/how-it-works

Posted in Humor

Two Plus Two Apples or Why Indians Flunk


I found a piece of paper with this little poem inserted in my old notebook. It was written by Beverly Slapin. I realized I was born an Indian and will always be.
two apples

All right, class, let’s see who know what two plus two is. Yes, Doris?

I have a question. Two plus two what?

Two plus two anything.

I don’t understand.

OK, Doris, I’ll explain it to you. You have two apples and you get two more. How many do you have?

Where would I get two more?

From a tree.

Why would I pick two apples if I already have two?

Never mind, you have two apples and someone gives you two more.

Why would someone give me two more, if she could give them to someone who’s hungry?

Doris, it’s just an example.

An example of what?

Let’s try again—you have two apples and you find two more. Now, how many do you have?

Who lost them?

YOU HAVE TWO PLUS TWO APPLES!!!! HOW MANY DO YOU HAVE ALL TOGETHER????

Well, if I ate one, and gave away the other three, I’d have none left, but I could always get some more if I got hungry from that tree you were talking about before.

Doris, this is your last chance—you have two, uh, buffalo, and you get two more. Now how many do you have?

It depends. How many are cows and how many are bulls, and is any of the cows pregnant?

It’s hopeless! You Indians have absolutely no grasp of abstraction!

Huh?

-by Beverly Slapin

Posted in Mathematics education

Theories and ideas behind the math lessons in this blog

I have put together in this post some of the ideas behind the kind of mathematics teaching I promote. As I stated in the subheadings of this blog, the articles and lessons I write here are not about making mathematics easy because it isn’t but about making mathematics makes sense because it does. Before reading any of the articles below, I suggest read the About page first and what I think mathematics is. I hope I make sense in the following articles. Click here for the list of math lessons.

To understand mathematics is to make connection

Mathematics is an art

Teaching through Problem Solving

Mathematical habits of mind

What is mathematical investigation?

Exercises, Problems, and Math Investigations

Why it is bad habit to introduce math concepts through their definitions

What is reasoning? How can we teach it?

What is the role of visualization in mathematics?

Making generalizations in mathematics

What is abstraction in mathematics?

I also have a new blog about research studies in mathematics teaching and learning.

Posted in Teaching mathematics

Making generalizations in mathematics

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.