Posted in Math blogs

Math Teachers at Play blog carnival #50 – submission

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Hello  bloggers and teachers. This blog is hosting the 50th edition of Math Teachers at Play (MTAP) blog carnival this 18th of May 2012. Promote your favorite posts/articles by submitting the links using the MATP submission form before the 18th.

  1. Do you have a game, activity, or anecdote about teaching math to young students? Please share!
  2. What is your favorite math club games, numerical investigations, or contest-preparation tips?
  3. Have you found a clever explanation for math concepts and procedures? E.g. how to teach bisecting an angle, or what is wrong with distributing the square in the expression (a + b)^2.
  4. How do you make an upper-level (high school) math topics come alive?
  5. What is your favorite problem? (I hope not the students:-))
  6. What kind of math do you do, just for the fun of it?

Click here to see past editions of MTAP Carnivals.

Don’t be shy — share your insights! If you do not have a blog, just send your ideas and short articles at mathforteaching@gmail.com. I’ll find a way to publish it in the carnival.

The last math blog carnival I hosted was Math and Multimedia Carnival #17.

Posted in Algebra

Math knowledge for teaching tangent to a curve

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I am creating a new category of posts about mathematical tasks aimed at developing teachers’ math knowledge for teaching. Most of the tasks I will present here have been used in studies about teaching and teacher learning. Mathematical knowledge for teaching was coined by J. Boaler based on what Shulman (1986) call pedagogical content knowledge (PCK) or subject-matter knowledge for teaching. I know this is a blog and not a discussion forum but with the comment section at the bottom of the post, there’s nothing that should prevent the readers from answering the questions and giving their thoughts about the task. Your thoughts and sharing will help enrich knowledge for teaching the math concepts involve in the task.

The following task was originally given to teachers to explore teachers beliefs to sufficiency of a visual argument.

The task:

Year 12 students, specializing in mathematics, were given the following question:
Examine whether the line y = 2 is tangent to the graph of the function f, where f(x) = x^3 + 2.

Two students responded as follows:

Student A: I will find the common point between the line and the graph and solving the system

math

The common point is A(0,2). The line is tangent of the graph at point A because they have only one common point (which is A).’

Student B: The line is not tangent to the graph because, even though they have one common

tangentpoint, the line cuts across the graph, as we can see in the figure.

Questions:

a. In your view what is the aim of the above exercise? (Why would a teacher give the problem to students?)

b. How do you interpret the choices made by each of the students in their responses above?

c. What feedback would you give to each of the students above with regard to their response to the exercise?

Source: Teacher Beliefs and the Didactic Contract on Visualisation by Irene Biza, Elena Nardi, Theodossios Zachariades.

Posted in Number Sense

What can the representations of numbers tell us?

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Numbers can be represented in different ways. The kind of representation we choose can highlight or de-emphasise the properties of the numbers.

Studies about understanding mathematics discuss about two kinds of representations of a mathematical idea: (1) transparent representations and (2) opaque representations. A transparent representation has no more and no less meaning than the represented idea(s) or structure(s). An opaque representation emphasizes some aspects of the ideas or structures and de-emphasizes others.

Examples:

  1. Representing  the number 784 as 28^2 emphasizes – makes transparent – that it is a perfect square, but de-emphasizes – leaves opaque – that it is divisible by 98.
  2. Representing the 784 as 13×60+4 makes it transparent that the remainder of 784 on dividing by 13 is 4, but leaves opaque its property of being a perfect square
  3. For a whole number k, 17k is a transparent representation for a multiple of 17, as this property is embedded or ‘can be seen’ in this form of the representation. However, it is impossible to determine whether 17k is a multiple of 3 by considering the representation alone. In this case we say that the representation is opaque with respect to divisibility by 3.
  4. An infinite non-repeating decimal representation (such as 0.010011000111. . .) is a transparent representation of an irrational number (that is, irrationality can be derived from this representation if the definition adopted is its being non-repeating, non-terminating decimal; It becomes an opaque representation for the definition of irrationals as numbers that cannot be expressed as quotient of two integers.)
  5. 2k+1 and 2k are transparent representations of odd and even numbers, respectively.

But what about prime numbers and irrational numbers in general? What are their representations? P for prime is not a representation.  In the article Representing numbers: prime and irrational, Rina Zaskis argued these two numbers have something in common: they both cannot be represented. Don’t we say irrational numbers are those that cannot be represented as a quotient and prime numbers are those that cannot be represented as a product? The examples I listed above were from the same paper. The author used them to argue the importance of representations and how the absence of it can become a cognitive obstacle to understand the concept.

Posted in GeoGebra worksheets, Geometry

The house of quadrilaterals

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In Investigating an Ordering of Quadrilaterals published in ZDM, Gunter Graumann shared a good activity for developing students mathematical thinking. The activity is about ordering quadrilaterals based on its characteristics. He gave the following list of different aspects of quadrilaterals as possible basis for investigation.

  1. Sides with equal length (two neighbouring or two opposite or three or four sides)
  2. Sum of the length of two sides are equal (two neighbouring or two opposite sides)
  3. Parallel sides (one pair of opposite or two pairs of opposite sides)
  4. Angles with equal measure (one pair or two pairs of neighbouring or opposite angles, three angles or four angles)
  5. Special angle measures (90° – perhaps 60° and 120° with one, two, three or four angles)
  6. Special sum of angle measures (two neighbouring or opposite angles lead to 180°)
  7. Diagonals with equal length
  8. Orthogonal diagonals (diagonals at right angles)
  9. One diagonal bisects the other one or each diagonal bisects the other one,
  10. Symmetry (one, two or four axis’ of symmetry where an axis connects two vertices or two side-midpoints, one or three rotation symmetry, one or two axis’ of sloping symmetry). With a sloping-symmetry there exists a reflection – notabsolutely necessary orthogonal to the axis – which maps the quadrilateral onto itself. For such a sloping reflection the connection of one point and its picture is bisected by the axis and all connections lines point-picture are parallel to each other.

The house of quadrilaterals based on analysis of the different characteristics of its diagonals is shown below. Knowledge of these comes in handy in problem solving.

House of quadrilaterals based on diagonals

Read my post Problem Solving with Quadrilaterals. You will like it.:-)