Don't count on calculators
There is a problem with the children of today. They are born of the digital age. The imagery is all around them; they see hi-tech films with flashing digital screens, their watches are digital and their pocket calculators are capable of higher-level functions than the first computer (certainly much faster). In schools they are often supplied with digital timers, thermometers, volt meters, pH meters and ammeters. The problem is that they place implicit trust in the digital read-outs that face them everyday. Their feel for number is very, very limited.
The following conversation took place with a bright Year 8 pupil just the other day: Teacher: "How many grams of sugar are there in the cube?" Pupil: "0.23 grams sir."
Teacher: "Are you sure about that?" Pupil: "Yes, sir, that's what the numbers on the balance were; zero, point, two, three, so that's 0.23 grams."
Teacher: " Was the balance on zero before you put the cube on?" Pupil: " I'm not sure, but it must be right because it's a digital balance."
Teacher: " Well I think the cube must be a little heavier than a quarter of a gram, shall we try again?" Pupil: "But it can't be wrong sir, it's digital!" It took a lot of convincing on my part that the digital balance, although accurate, had not been calibrated to zero before putting the sugar cube on the pan. The cube weighed 2.93 grams.
This set me thinking about the accuracy game in science, and, in particular, the implicit faith that some pupils put into their calculators and other digital devices. My conclusions are that a lot of children, sadly, have no feel for number.
Their reliance on the calculator has deprived them of the opportunity to develop the mental skill of estimating size and approximating measurements. This aspect of mental development is neglected because it is so easy to use a calculator, and it is being used for extremely basic functions that could, with a little practice, be done mentally far faster than pushing the buttons on a calculator.
We are mis-using calculators as a tool, for that is all they are, a tool. They are a convenient way of speeding up lengthy calculations and can help with some subjects such as physics, where the chore can be taken out of complex calculations and the students are allowed the freedom to concentrate more on the concepts behind the calculations. But, in mis-using this tool we are neglecting an important aspect of education that has far wider implications than weighing sugar cubes in a science lesson.
In life, estimation and approximation are essential skills. As drivers we must get a feel for the length and speed of our cars and a big part of driving is to know where you can safely pass and where you cannot and roughly how fast you are going, without constant reference to the speedometer. On another level, the assistant at a delicatessen counter in the local supermarket learns the skill of estimating the approximate size of a lump of cheddar without resorting to rulers and calculators when a customer asks for a half a kilo of Scottish mature. These skills are developed largely through experience but, how much more quickly would they be learned and how much more useful would they be if they were practised from an early age in schools.
I was taught to use log tables for complex calculations, the previous generation used a slide rule. By using these devices we gained a feel for number as simple calculations were invariably done mentally or quickly in the margin of an exercise book. When using a calculator you could soon spot that the wrong button had been pressed if it told us that 15 x 30 = 4,500. Common sense and a feel for number flicked a switch in the brain that said "It can't be that big, no matter what the calculator says!" That switch is not activated enough times in today's pupils.
The problem with digital technology is that it is too precise and the way that the answer is presented on the liquid crystal display almost defies challenge. Why should a pupil challenge the answer that 135 divided by 7 = 19.28571428571? For instance, do you know that the answer is or is not correct? The full answer is out of the range of mental arithmetic, although 135 divided by 7 being roughly 19 or 20 is not.
We are in danger of producing a generation that is dependent on digital technology being accurate because they cannot, or will not, trust their own mental capabilities. I am not against the use of digital equipment and calculators, but, before they are used, we should ensure that our pupils do use their mental facilities to "guesstimate" the answer. In order to do this in class I try to turn it into a "beat the calculator" game that encourages children to try to work out an answer in their heads and compare that with the calculator answer. If there is a big discrepancy then the obvious thing is to do the calculation again. In this way the pupil will get a feel for number over their school years that will stand them in good stead when they leave for the world of work.
When going over science investigations with pupils one notable thing relates to where there may have been error in their observations andor measurements. Rarely, if ever, do pupils say that the instrument may be at fault. Professional scientists make a big fuss about the calibration of any equipment, but, how often do we calibrate our own equipment with known standards, such as the melting and boiling point of water, a known potential difference of current? Rarely, if ever.
Pupils must be taught that error is acceptable in science to a certain degree. I am not encouraging sloppy work, but, an acknowledgement that the beaker of water first boiled in week two of Year 7 will not always boil at 100C, according to conventional thermometers let alone the digital ones that some can afford. What message are we giving to pupils when we say that water boils at 100C and ice melts at 0C regardless of what the thermometer is actually reading? Surely it is better to explore the "error" and make use of the fact that in this instance the reading was correct, given that error can, and will, occur for a number of reasons.
Another common problem is the number of significant places that a pupil will use when putting down an answer derived from a calculator. They often put down all eight decimal places without using two significant figures, more than accurate enough for most investigations. This, however, is much more easily cured than failing to recognise that the calculator answer is an order or two of magnitude out!
We don't need to ban calculators, but we must encourage our pupils to acquire the mental arithmetic capabilities we had no choice but to develop when log tables and slide rules were the order of the day. Having said that, as professionals, we must also learn to recognise those individuals who, far from being handicapped by the use of a calculator for basic arithmetic, are helped to overcome specific learning difficulties in arithmetic operations.
The charge is often brought that children are lazy and will not do mental arithmetic as a matter of course. In reality how much do we as teachers encourage the use of mental arithmetic? As the old saying goes "a bad craftsman always blames their tools". In this instance should we not point the finger at the master craftsman who relishes the power that the tools hold, a power denied to them when they were apprentices using slide rules and log tables.
James Williams is head of the science faculty at The Beacon School, Banstead, Surrey.