This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.
Calling all mathematics and bloggers! Get your math posts published!
Mathematics for Teaching will be hosting the 12th edition of the Math and Multimedia blog carnival which will go live on 13th June 2011.
Click here to send your posts/articles. The 11th edition of Math and Multimedia Carnival was hosted by Love of learning blog.
Mathematics is already a difficult subject so let us not make it more difficult to students to make sense of with the confusing rules we tell them. For example, we insist that the square root of 36 is 6 and -6 yet we also insist that ?36 is equal to 6 and not -6. A colleague said that by convention, when you ask ‘what is the square root of 36?’ the answer is 6 and -6 but when you just write the symbol ‘?36 =?’ the answer must be 6. So, I said, do you mean that if you just want the positive root, you do not read the symbol ?36 otherwise it will have 2 values? Here’s an excerpt from an article I read recently about the matter. It’s title is “What root do you want to take?” by Derek Ball.
Teachers and books make the most extraordinary statements about quadratic equations. All quadratic equations have two solutions, they say. How about x2 – 8x + 16 = 0? Does that have two solutions? “Yes”, they say, “it has the repeated solution 4. If you put x = 4 the equation is true.” Fine. “And if you put x=4 …” Yes, I heard you the first time.
Of course, I can mock, but it is quite useful for some purposes to think of this equation having a ‘repeated solution’, just as it is quite useful for other purposes to think of it as having just one solution. Confused? So you should be. To add to the confusion I shall ask you this question: ‘What is the square root of 9?’ Depending on context you might answer ‘3 or -3’. If I ask you: ‘What is the square root of zero’, what are you to answer: ‘Zero’ or ‘Zero and zero’ or ‘Zero and minus zero’?
Anyway, solving quadratics is where things like ?36 really come into their own. If you want to solve x2 – x – 1 = 0, you can use ‘the formula’ and obtain the solution x = (1 + ?5)/2. What I need to remember when interpreting this solution is that ?5 is positive. Or do I? Why do I? And if every collection of symbols is supposed to represent (at most) one number, what about‘+?5’?
And sooner or later you may want to solve equations like z2 – 4z + 5 = 0 and you may perhaps use the quadratic formula and obtain z = (4 +(?4))/2. Now new questions arise. Am I allowed to write ?(-4) and if I am what is its value? Is it 2i or -2i and why? Perhaps you want to say the answer is obviously 2i. In that case how about a quadratic equation whose solution involves ?(3 – 4i). Is this – 2 + i or is it 2 – i? What I am saying is that we use symbols to help us solve problems. If we use + in front of a square root sign this reminds us that in order to solve the quadratic equation completely we need to remember to take two different values for the square root. Knowing that ?36 means 6 and not -6 does not matter at all, unless we are asked silly questions in pub quizzes or GCSE exams like ‘What is ?36?’ and we are supposed (for some unknown reason) to know that we have to answer 6 and not -6.
Still not convinced? Well, am I allowed to write 3?-27 and is its value allowed to be -3? Its value could be other things too, of course. Does that complicate things? Does that make us think that we can only judge what a symbol means from the context? So, far be it from me to defend textbooks, but perhaps they have some justification for using the square root sign inconsistently. As for fractional powers, they seem to raise exactly the same issues as root signs.The moral of this tale is surely that we move away from asking questions towhich we want a correct answer, so thatwe can say ‘Right’ or ‘Wrong’, and instead solve problems that interest us, talk about mathematics and connect ideas together. And I hope we sometimes get confused, because confusion is often a spur to sorting our ideas out.
Amen to that. For more confusing rules that we give to our students read my post on Mistakes vs. Misconceptions. You may also want to know the more about Algebra Errors.
Misconceptions are very different from the mistakes students make. Mistakes are not consciously made. Misconceptions are. Mistakes are usually one-off, while misconceptions, the gods forbid, could be for keeps. Mistakes are made by a few, misconceptions are made by many and, repeatedly.
Students can figure out their mistakes by themselves because mistakes are usually due to carelessness. They cannot do the same for misconceptions. Misconceptions are committed because students think they are correct.
How can misconceptions be addressed? By undressing them, carefully exposing them until the students see it. It cannot be corrected by simply marking them x because misconceptions are usually made with full knowledge.
The following are common misconceptions in arithmetic, algebra and geometry:
1. Did we not learn that multiplication is repeated addition? So, -3 x -4 = -3 + -3 + – 3 +-3 = -12?
2. Didn’t we learn that to multiply fractions we simply multiply numerators and we do the same with the denominators? Didn’t the teacher say multiplication is simply repeated addition so 
3. Did not the teacher say x stands for a number? So in 3x – 5, if x is 5, the value of the expression is 35 – 5 = 30?
4. Did not the teacher/book say to always keep the numbers and decimal points aligned? So if Lucy is 0.9 meters and her friend Martha is 0.2 taller, Martha must be 0.11 meters in height?
5. Did we not learn that the more people there are to share a cake the smaller their portion? So 
6. Did we not learn that by the distributive law 2(a+b) = 2a + 2b? So, 
7. Did not the teacher show us that (x-3)(x+1) = 0 implies that (x-3) = 0 and (x+1) = 0 so x = 3 and x = -1? Hence in (x-3)(x+4) = 8, (x-3) = 8 and x +4 = 8 so x = 11 and x = 4?
8. Did we not learn that the greater the opening of an angle, the bigger it is? So, angle A is less than angle B in the figure below.
9. Did we not learn that you if you cut from something, you make it smaller? Hence in the diagram below, the perimeter of the polygon in Figure 2 is less than the perimeter of original polygon?
10. Isn’t it that the base is the one lying on the ground?
There’s nothing a teacher should worry about mistakes. There’s everything to worry about misconceptions. Good teaching practice exposes misconceptions, not hide them.
You might want to check out this book:
Geometry is a natural area of mathematics for which students should develop reasoning and justification skills and their appreciation of the logico-deductive part of mathematics that build across the grades. Learning tasks therefore should be so designed so that the focus of the learning is on the development of these skills as well and not merely on the learning of facts.
Consider the GeoGebra applets in Figures 1 and 2 below. Which of them will you use for teaching the relationships among the angles made by transversal with parallel lines? Before this lesson of course, the students already learned about linear pairs. Click the figures below to explore the applets before you continue reading.
In the first figure, dragging D or F along the parallel lines, the students will observe that there are angles that will always be equal. Thus from this, they can make the following conjectures:
(1) the alternate interior angles are equal;
(2) the vertical angles are equal;
(3) the corresponding angles (a pair of interior and exterior angles on the same side of the transversal) are equal; and,
(4) the pair of exterior and interior angles on the same side of the transversal sum up to 180 degrees.
In all these cases, the students are reasoning inductively. They will generalize from the measures they observed. Because of this, there seem to be no need for proof since there were bases for the generalizations. The measures of the angles. In this activity students will have learned geometric facts but not the geometric reasoning. Inductive reasoning maybe, but not deductive reasoning.
Contrast the first applet to the second one. Dragging D or F along the parallel lines, the students will observe that the sum of the pair of exterior -interior angles on the same side of the transversal is always 180 degrees. They will also observe that the other angles also changes. The teacher can then challenge the students to make predictions about the measures of these angles and the relationships among them. These will create a need for proof.
And how should the proof look like? My suggestion is not to be very formal about it like using a two-column proof. For example, to prove that measures of vertical angles are always equal they can set up their proof like these:
To prove p = t:
p + s = 180
s + t = 180
p + s = s + t
p = t.
Students can very well set-up an explanation like this. They have seen it when they learned about solving systems of linear equation. What more, it uses the very important property of equality – the transitive property: If a = c, and b = c, then a = b. Great way to link algebra and geometry.
