Posted in Algebra

Fun with ‘definitions’ in algebra

WARNING:  use the following definitions with great caution.

  • number phrase is a mathematical phrase which does not express a complete thought.
  • An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7?’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it’s undefined (in which case you can work that out).
  • An algebraic expression is any grammatically sensible expression made up of any or all of the following:

– specific numbers (called constants);
– letters (or other symbols) standing for numbers (called variables); and
– arithmetic operations.

  • By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a constant, we mean an algebraic expression that contains no variables at all.
  • polynomial is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers.
  • monomial is an algebraic expression made up only from any or all of these:

– Constants;
– Variables;
– Multiplication;
– Taking opposites (optional);
– Division by nonzero constants (optional);
– Raising to constant whole exponents (optional).

  • An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operations, and so forth.
I got these definitions from where else, www. Of course we just want to simplify things for students but … . Anyway, just make sure that you don’t start your algebra lessons with definition of terms, be they legitimate or not legitimate.
Posted in Geometry

Twelve definitions of a square

How does mathematics define a math concept?

Definitions of concepts in mathematics are different from definitions of concepts in other discipline or subject area. A definition of a concept in mathematics give a list of properties of that concept. A mathematics object will only be an example of that concept if it fits ALL those requirements, not just most of them. Further, a definition is also stated in a way that the concept being defined belongs to an already ‘well-defined’ concept. On top of this, economy of words and symbols and properties are highly observed.

Does a math concept only have one definition? Of course, not. A concept can be defined in different ways, depending on your knowledge about other math objects. In a study by Zaskin and Leikin, they suggested that the definitions students give about a concept mirrors their knowledge of mathematics. Below are examples of definitions of squares from that research. Do you think they are all legitimate definitions?

What is a square?

A square is

  1. a regular polygon with four sides
  2. a quadrilateral with all the angles and all the sides are equal
  3. a quadrilateral with all the sides equal and an angle of 90 degrees
  4. a rectangle with equal sides
  5. a rectangle with perpendicular diagonals
  6. a rhombus with equal angles
  7. a rhombus with equal diagonals
  8. a parallelogram with equal adjacent angles and equal adjacent sides
  9. a parallelogram with equal and perpendicular diagonals
  10. a quadrilateral having 4 symmetry axes
  11. a quadrilateral symmetric under rotation by 90 degrees
  12. the locus of all the points in a plane for which the sum of the distances from two given perpendicular lines is constant. Click this link to visualize #12.

 

Making (not stating) definitions is a worthwhile assessment task.

Here’s three great references for definitions of mathematical concepts. The first is from no other than Dr. Math (The Math Forum Drexel University). The middle one’s for mom and kids – G is for Google and the third’s a book of definitions for scientists and engineers.

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Posted in What is mathematics

The heart of mathematics

Axioms, theorems, proofs, definitions, methods, are just some of the sacred words in mathematics. These words command respect and create awe  especially to mathematicians but deliver shock to many students. P.R. Halmos argued that not even one of these sacred words is the heart of mathematics. Then, what is? Problem solving. Solving problems is at the heart of mathematics.


Indeed, can you imagine mathematics without problem solving? It might as well be dead! But why is it that problem solving tasks are relegated as end of lesson activity? When it’s almost end of the term and the teacher’s in a hurry to finish their budget of work, the first to go are the problem solving activities. And when time allows the teacher to engage students in problems solving, the typical teaching sequence goes like this based on my observation in many math classes and from the teaching plans made by teachers.

  1. Teacher reviews the computational procedures needed to solve the problem.
  2. Teacher solves a sample problem first usually very neatly and algebraically (especially in high school)
  3. Teacher asks the class to solve a similar problem using the teacher’s solution
  4. Students practice solving problems using the teacher’s method.

Even textbooks are organized this way!In this strategy, students are given problem solving tasks only after having learned all the concepts and skills needed to solve the problem. Most often than not, they are also shown a sample method for solving the problem before they are given a set of similar problems to work on. I will not even call this a problem solving activity/lesson. How can a problem be a problem if you already know how to solve it? Of course, this particular strategy also gives the students the opportunity to deepen, consolidate and synthesize the new math concepts they just learned. But it also deprives them the opportunity to engage in real problem solving where they themselves figure out methods for solving the problem and using knowledge they already possess.

Another approach to increase students engagement with problem solving is to teach mathematics through problem solving.