Posted in Conferences and seminars

Announcement: Seminar-Workshop on GeoGebra

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The University of the Philippines National Institute for Science and Mathematics Education Development (UPNISMED) will conduct a three-Saturday seminar-workshop on using GeoGebra in the teaching and learning of high school mathematics on August 13, 1320, & 27, 2011 at UP NISMED. This is a first level seminar and will cover the basic tools of GeoGebra. Sample lessons, activities,  applets will be presented. The participants are expected to develop at least one activity/ GeoGebra applet for high school mathematics lessons as output.

For more information, fees, and registration procedures visit this link. To Filipino math teachers, please share this post in your Facebook.

Posted in Geometry

Twelve definitions of a square

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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 Elementary School Math

Teaching the meaning of equal sign

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Here’s how I sequence my lesson to develop pupil’s understanding of the meaning of the equal sign. Actually the lesson uses the context of the meaning of equal sign to introduce the students to the meaning of variable intuitively. The students enjoyed this lesson and they said they loved the way I made them think. Scaffolding was done through questions that engages pupils in reasoning and making decisions. Note that the emphasis of the lesson is not on computations but on thinking and problem solving. This is also an example of teaching algebraic thinking in the grades.

I first wrote the equal sign on the board then said What does the equal sign mean? You may use an example to explain your answer. One boy said it means you add or do the operation and provided this example 2 + 10 = 12. I asked the class who agrees with him and 25 out of 35 showed hand.

What about in 15 + ____ = 21 + ____? One girl said “It means balance” and explained that 15 plus a number balances with 21 plus another number. When I asked the class who agrees with her 30 out of 35 raised their hand. Everyone’s eyes was on me, waiting for me to say which meaning of equal sign is correct. I just gave them a wink to heighten their curiosity.

Now that I got them all thinking, I asked: Do you think you can put just any two numbers in the blanks? With this question I successfully divided the class into two camps: those who say yes and those who say no and everyone is challenged to prove themselves right or prove the other wrong.

Click here for the slide version of this post.