Posted in Math blogs

Math blog carnival

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This site will be hosting the 25th edition of Math and Multimedia Carnival which will go live at the end of this month, October 31.

A math blog carnival is a collection of articles from various math blogs and sites. So if you are a blogger, this is an opportunity for you to promote your favourite or latest posts and yes, your blogs for free. Below is a collection of blog carnivals I previously hosted.

If you have articles about math problems, puzzles and games, tips for teaching math and specific topics in math, videos, tutorials, lessons, curriculum materials and book reviews, math trivia especially about the the number 25, etc, you may email the permalinks to me or use the math and multimedia blog carnival submission form.

Please share, like, and tweet so more bloggers will know. Thank you.

 

Posted in Algebra, Number Sense

The many faces of multiplication

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The following table is not meant to be a complete list of ideas about the concept of multiplication. It is not meant to be definitive but it does include the basic concepts about multiplication for middle school learners. The inclusion of the last two columns about the definition of a prime number and whether or not 1 is considered a prime show that there are definitions adapted to teach school mathematics that teachers in the higher year levels need to revise. Note that branching and grouping which make 1 not a prime number can only model multiplication of whole numbers unlike the rest of the models. Multiplication as repeated addition has launched a math war. Formal mathematics, of course, has a definitive answer on whether 1 is prime or not. According to the Fundamental Theorem of Arithmetic, 1 must not be prime so that each number greater than 1 has a unique prime factorisation.

If multiplication is … … then a product is: … a factor is: … a prime is: Is 1 prime?
REPEATED ADDITION a sum (e.g., 2×3=2+2+2 = 3+3) either an addend or the count of addends a product that is either a sum of 1’s or itself. NO: 1 cannot be produced by repeatedly adding any whole number to itself.
GROUPING a set of sets (e.g., 2×3 means either 2 sets of three items or 3 sets of 2) either the number of items in a set, or the number of sets a product that can only be made when one of the factor is 1 YES: 1 is one set of one.
BRANCHING the number of end tips on a ‘tree’ produced by a sequence of branchings (think of fractals) a branching (i.e., to multiply by n, each tip is branched n times) a tree that can only be produced directly (i.e., not as a combination of branchings) NO: 1 is a starting place/point … a pre-product as it were.
FOLDING number of discrete regions produced by a series of folds (e.g., 2×3 means do a 2-fold, then a 3-fold, giving 6 regions) a fold (i.e., to multiply by n, the object is folded in n equal-sized regions using n-1 creases) a number of regions that can only be folded directly NO: no folds are involved in generating 1 region
ARRAY-MAKING cells in an m by n array a dimension a product that can only be constructed with a unit dimension. YES: an array with one cell must have a unit dimension

The table is from the study of Brent Davis and Moshe Renert in their article Mathematics-for-Teaching as Shared Dynamic Participation published in For the Learning of Mathematics. Vol. 29, No. 3. The table was constructed by a group of teachers who were doing a concept analysis about multiplication. Concept analysis involves tracing the origins and applications of a concept, looking at the different ways in which it appears both within and outside mathematics, and examining the various representations and definitions used to describe it and their consequences, (Usiskin et. al, 2003, p.1)

The Multiplication Models (Natural Math: Multiplication) also provides good visual for explaining multiplication.

You may also want to read How should students understand the subtraction operation?

Posted in Number Sense

Test your understanding of irrational numbers

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The following is a set of tasks which I think are great questions for assessing understanding of irrational numbers. These tasks were from the study of Natasa Sirotic and Rina Zazkis. The responses were analysed in terms of algorithmic, formal, and intuitive knowledge described at the end of the post.

Set A

This set of tasks assesses the formal and intuitive knowledge about about the relative sizes of two infinite sets – rationals and irrationals.

  1. Which set do you think is “richer”, rationals or irrationals (i.e. which do we have more of)?
  2. Suppose you pick a number at random from [0,1] interval (on the real number line). What is the probability of getting a rational number?
Set B

This set assesses knowledge about how the rational and irrational numbers fit together in relation to the density of both sets.

  1. It is always possible to find a rational number between any two irrational numbers. Determine True or False and explain your thinking.
  2. It is always possible to find an irrational number between any two irrational numbers. Determine True or False and explain your thinking.
  3. It is always possible to find an irrational number between any two rational numbers. Determine True or False and explain your thinking. 
  4. It is always possible to find a rational number between any two rational numbers. Determine True or False and explain your thinking.
Set C

This set investigate knowledge of  the effects of operations between irrational numbers

  1. If you add two positive irrational numbers the result is always irrational. True or false? Explain your thinking.
  2. If you multiply two different irrational numbers the result is always irrational. True or false? Explain your thinking.

You may want to analyse the responses using Tirosh et al.’s (1998) dimensions of knowledge:

  • The algorithmic dimension is procedural in nature – it consists of the knowledge of rules and prescriptions with regard to a certain mathematical domain and it involves a learner’s capability to explain the successive steps involved in various standard operations.
  • The formal dimension is represented by definitions of concepts and structures relevant to a specific content domain, as well as by theorems and their proofs; it involves a learner’s capability to recall and implement definitions and theorems in a problem solving situation.
  • The intuitive dimension of knowledge (also referred to as intuitive knowledge) is composed of a learner’s intuitions, ideas and beliefs about mathematical entities, and it includes mental models used to represent number concepts and operations.

At the conclusion of the study, Sirotic and Zaskis reported this short exchange:

What do you think of the teacher’s answer?

You may want to share your responses to the questions in the comment section below.

Posted in Algebra

Equations, Equations, Equations

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Students deal with a different ‘types’ of equations: equations in one unknown, equations in two unknowns, and the equation representations of function. There are others like the parametric equations but let’s talk about the first three I just enumerated. Is there a connection among all these three apart from being equations?

Let’s take for example the equation 4x – 1 = 3x + 2. To solve the equation, students are taught to use the properties of equality. When the topic gets to equation in two unknowns, this equation is learned independently of the equation in one unknown especially in  finding the solutions. When the topic gets to solving systems of equation say 3x+y = 4 and x–y = 5, the methods for solving the system of linear equation – substitution,  elimination, graphing – are also learned without making the connection to the methods of solving equation they already know. Then, function comes in the scene; the y‘s disappeared and out of nowhere comes f(x). Most times we assume the students will make the connection themselves.

How can we help students make connection among these three? To solve equation in one unknown, I think we should not rush to teaching them how to solve it using the properties of equality. There are other ways of solving these equations one of which is generating values which I’m sure you use in introducing equations in two variables. Using the example earlier, students can generate the values of 4x – 1 and then 3x+2. This way, the question “What is x so that 4x-1 = 3x+2 is true?” will make sense to students. They will have to find the value of x that belongs to the group of numbers generated by 4x-1 as well as to the group of numbers generated by 3x+2.

equation in one unknown

Now, why go through all these? Two reasons: 1) to reinforce the notion that algebraic expressions is a generalized expression representing a group of numbers/values and 2) to plant the seed of  the notion of function and equations in two unknowns which students will meet later. Of course this does not mean we should not teach how to solve equation using the properties of equality. I just mean we should teach them other solutions that will help students make the connection when they meet the other types of equations.

function machineAnother popular tool is the input-output machine which is the same really as the table of values. For some reason they are used mostly to introduce equations in two unknowns or to introduce function. Why not introduce it early with equations in one unknown? Of course you need a second machine for the other expression. The challenge for the students is to find what they need to input in both machine so they will have the same output. The outputs can be represented by the expressions on each side of the equal sign but later you get to the study of function you may introduce y provided that y = 4x-1. Students need to see that this equation does not just mean equality but that it also means the value of y depends on x according to the rule 4x-1. Since every x value generates a unique y value, y is said to be a function of x, in symbol, y=f(x). Since y = f(x), we can also write f(x) = 4x – 1.

In most curricula, the formal study of function comes after systems of linear equation so there’s no hurry with the f(x) thing. The use of the form y = 4x-1 would be enough. If students understand equations this way I think they can figure out the substitution method for solving systems of linear equations by themselves. Graphing would therefore also be a natural solution students can think of. Equation Solver is a simple GeoGebra applet I made to help students make the connection.

Wouldn’t it be nice if students see 4x-1=3x+2 not just a simple equation in one unknown where they need to find x but also as two functions who might share the same (x,y) pair? This will really come in handy later. Solutions #2 and #3 of solving problems by equations and graphs are examples of problems where this knowledge will be needed.

I recommend that you also read my post What Makes Algebra Difficult is the Equal Sign.

To understand is to make connections. This has become a mantra in this blog. Students will not make the connection unless you make it explicit in the design and implementation of the lesson.