The phrase ‘problem solving’ has different meanings in mathematics education. Even its role in mathematics teaching and learning is not clear cut. Some view problem solving as an end in itself. Others see it as starting point for learning. Nevertheless, here are some of the types of problem solving tasks we would see in textbooks and teaching. They are arranged according to cognitive demand. Remember that learners will only consider something a problem if they experience some sort of barrier in a situation they are curious about. There’s a difference between an exercise and a problem. Continue reading “Types of Problem Solving Tasks”
When is it algebra and when is it arithmetic?
In the post Algebra vs Arithmetic, I distinguished between arithmetic and algebra by arguing that it has nothing to do with the use of letters. That algebra is about letters and arithmetic is about numbers is an oversimplified view of algebra and can create misconceptions. Here are more ways of characterizing algebra. Continue reading “When is it algebra and when is it arithmetic?”
Free Fractions Pamphlet
I just want to promote in this post James Tanton’s latest pamphlet on fractions. It’s FREE for download. Just click Pamphlet on Fractions. Tanton writes:
If fractions are pieces of pie, then what does the multiplication of fractions mean? (You can’t multiply pie!)
If fractions are proportions, then what are their units? Amount of pie per student (and not just pie)?
If fractions are points on the number line, then what does half a pie mean?Fractions are slippery and tricky and, in the end, abstract. It is actually unfair to expect students to have a good grasp of fractions during their middle-school and high-school years. This pamphlet explains why, and offers the means to have an honest conversation with students as to why this is the case. Their confusion and haziness about them is well founded!
What do you say about the statement I highlighted above? What a relief to know that? 🙂
I highly recommend that you also checkout his collection of past MATH ESSAYS.
You may also want to read a couple of my posts on fractions:
FOIL Method is Distributive Law
The FOIL method is not that bad really for teaching multiplication of two binomials as long as it is derived from applying the distributive law or more officially known as the Distributive Property (over addition or subtraction).
The FOIL method is a mnemonic for First term, Outer term, Inner Term, Last term. It means multiply the first terms of the factors, then the outer terms, then the inner terms and the last terms. I would suggest the following sequence of examples before the teacher introduces product of two binomials: Click here for the complete description of how to teach this with meaning.
Simplify the following expressions
- 3(2x-1)
- 3x(2x-1)
- -3x(2x-1)
- (x+3)(2x-1)
The FOIL method is also related to Line Multiplication. However, while the latter is applicable to any number of terms in the factors like the Distributive Law, the FOIL method is not. It only works for getting the factors of binomials. This is why it is not a powerful tool. The most powerful knowledge is still the distributive property of equality. Before the teacher should introduce any fancy way of calculating, he or she should make sure this knowledge is in place. Sample lesson on how to do this is presented in Sequencing Examples.
Want a more challenging problem for distributive law? Click Application of Distributive Law.