Students often bring in questions that are taken or adapted from sources outside of the usual textbooks. This is especially true with regards to revision material when leading up to topic test or other major assessment. One question of this type that really caught my attention was from a "Smart Study" guide given to students preparing for a SAC (i.e. major test) on Functions and Graphs. The diagram is shown here:
(note this diagram remains the intellectual property of the copyright holder)
Most secondary school students are taught about the “Scientific Method” during their general science classes. From these early classes, students are encouraged to believe that all scientists follow an objective and almost linear process in their research. The process is generally described as something like this:
an unusual observation is made
an explanation is proposed for this observation
experiments are performed to test the explanation
after repeated experiments a theory is formed
experiments are performed to validate the theory
once validated the theory becomes a new scientific law
the scientific laws are used to predict new observations
During the last few weeks I have been working with some Year 11 students from a local private school that has a fairly good reputation. These students have been working, in particular, on cubic and quartic functions, including the use of polynomial division. The students in question are taught by two different teachers at the school, including their school’s Head of Mathematics, a highly experienced and long serving teacher by all accounts. All of these students are certainly above average students, but like most of their peers, have had little exposure to the long division method in primary school, mostly due to the prevalence of handheld calculators.
Whilst working with these students, each of them has independently demonstrated to me a short-cut method taught to them at school for polynomial division. This short-cut method is well known and quite useful when applied correctly. Unfortunately, when originally demonstrated to the students (judging by each of their theory books), their respective teachers each made the same monumental error and taught the students a technique that would work only in certain circumstances. The following are some of the examples given to these students in class:
Mark Twain is credited with stating that there are three kinds of lies, "Lies, damned lies and statistics". As a statistician, I often hear people quoting this to me to which I often point out that Twain was only partially correct. There are three kinds of lies, "Lies, damned lies and lies told by misusing statistics". It appears that once again in Australia, vital funding choices are about to be made through the use of a faulty statistic, the citation count.
During this past week I’ve been happily working away with my students, teaching them all I can about the wonderful world of mathematics and statistics. In particular, we’ve looked at what inferences you can draw from a set of descriptive statistics. One particular example in their textbooks asks the students to compare the pulse rates of 21 adult males and 22 adult females and make an inference about male and female pulse rates. As expected from the data supplied, these students conclude that females, in general, have higher and more variable pulse rates than the males. Whilst that is a correct conclusion from the supplied data, I always ask the students if there is anything they might be concerned about with regards to this data and how it was collected.