Standardised tests for school calculators?

My first posting so, a bit of personal background: I'm a semi-retired consulting systems engineer with a granddaughter rapidly approaching school age. Educated at St. Dunstan's College in the late 1960s, when it had a good claim for having the best school maths department in the country. Later professional education (BCS Exams) at South Bank University (then Borough Polytechnic).

I'm dispirited by what I think are the poor standards of secondary-level maths education in this country and want to give something back to good maths teachers. A current hobby-horse is the dire quality of scientific calculators at school level. Some useful papers by Prof. Harold Thimbleby, (now Univ. Swansea) are attached. Some might find them surprising (even alarming).

I would like to suggest something that might help improve things, namely a set of standardised functional tests for calculators sold for use in secondary schools. The idea is a web-based collaboration of interested individuals to devise such a set of tests. The tests should be made freely available so that any interested teachers can try them on calculators that they propose to use or recommend to pupils. Also the results of tests on individual devices could be published to spur calculator developers to do better.

Added 27/10/2016: To start the ball rolling I'm posting in this thread the results of various tests on calculators that I own. Feel free to contribute your own experiences.

Olwen Morgan

 

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Subject(s)Computing, Computer Networks, Creating Media, Design & Development, Mathematics
Age11-14, 14-16, 16-19, FE/HE
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Olwen Morgan

Texet FX1500 gets sums wrong!

The Texet FX1500 is an older style of calculator that performs chained arithmetic operations. Thus if you enter 2+3= you get 5. Subsequent presses of the = key repeat the last operation.

For addition you get:

2+3=5

=8

=11, etc.

For subtraction you get:

2-3=-1

=-4

=-7, etc.

For division you get:

2/3=0.666666667

=0.222222222, etc ...

... but for multiplication you get:

2*3=6

=12

=24, etc.

So here repeated pressing of the = key is repeating the last operation but apparently with the wrong operand! Also a device that I think is an unbranded version of the Texet FX500 has the same behaviour.

Should we consider such calculators fit for use in schools?

Olwen Morgan

Texet FX1500 rounding

Here's another idiosyncrasy of the Texet 1500:

Power it on, then key in:

[ON/C]         to power on and clear the calculator,

[2ndF] .         to set the (apparently truncating) rounding mode,

1/7=               displays answer as 0.142857142

Now enter:

[2ndF] 9        to set rounding to nine decimal places (apparently)

1/7=               displays answer as 0.142857143

The trouble here is that there is no visual indicator on the LCD screen to show what the rounding mode is. It is obvious if the mode is set to give fewer than nine decimal places but not obvious otherwise.

Surely a good calculator should always show details of modes that affect the accuracy of displayed results? An experienced professional scientific calculator user would most probably be aware of potential problems in this area but I'm guessing that few school-age children would (my evidence being that, in my experience, most graduate engineers are unaware of such things).

Try also the following:

[ON/C]       to power on and clear

[2ndF] .        to set truncating mode

1/11=            displays 0.09090909

and now:

[ON/C]       to clear

[2ndF] 9      to set 9 decimal places precision

1/11=            displays 0.090909091

which makes one ask why only nine digits are displayed in the truncating mode. I suspect that this is due to the phenomenon known as "wobbling precision" (WP) which can arise if the calculator is doing arithmetic four bits at a time. With WP, precision can be lost four bits at a time in some cases, depending on the values of the operands. Again there is no displayed indicator that would alert the user to potential WP problems.

These behaviours of the Texet FX1500 are also shown by what is apparently an unbadged Texet FX500. I regard such behaviours as further demerits against these devices as regards their suitability for use in schools, but what do teachers think?

Olwen Morgan

Texet FX1500 half-baked algebraic-cum-RP formula entry

Try the following with this device:

[ON/C]       to power on and clear

press the [DEG] key until the angle mode indicator shows "DEG"

now enter:

45[sin][x2] + 45[cos][x2]=

and the answer is one as expected

Note that for chains of unary operators or functions the formula entry is the same as Reverse Polish but then has to be infix for binary operators. The bracket keys allow bracketted formulae to be entered, but this falls short of full algebraic formula entry since postfix key-ins are still required for unary operators and functions. As before, it also seems that the FX500 behaves in the same way.

IMO the device should do either RP or normal algebraic formula entry (or better still be switchable between the two like the HP50g). As it stands the Texet FX1500 stands half way between RP and algebraic. My gut feel is that this would be confusing to most schoolkids. What experiences do teachers have of students' use of devices that behave in this way?

Olwen Morgan

Casio fx991ES-PLUS curiosities

The Casio fx991ES-PLUS seems to be widely used in schools but it has various foibles that students should know about:

The Rec function converts polar to rectangular coordinates. The Pol function converts from rectangular to polar. Both require their first and second arguments to be in the X and Y memory locations.

Store 1 in both X and Y, then enter Pol(X, Y) and you get the square root of two in the X memory and 45 in the Y memory (assuming you've set trig functions to work in degrees).

Now enter Rec(X, Y) and you are taken back to where you started with 1 left in both the X and Y memories.

Now enter Rec(Pol(X, Y,)) and you get "Math Error" (whatever that's supposed to mean) on the display ... but you have made no mathematical error at all. The same happens with Pol(Rec(X, Y)). What you entered was perfectly correct mathematical notation and should function as an identity function on the ordered pair of memories (X, Y).

It is mathematically incorrect to call this an error. It is a limitation of the calculator. IMO Teachers should point this out to students.

Olwen Morgan

Casio fx991ES-PLUS complex number non-calculation

Try this one:

Enter "COMPLEX" mode and calculate √(-1). The result is i.

Now try to calculate  i by the same means. The calculator displays, "Math ERROR" but the result is the perfectly well defined 1/√2 + i/√2.

It is mathematically incorrect to call this an error. It is simply a limitation of the calculator. IMO Teachers should point this out to students

Olwen Morgan

Casio fx991ES-PLUS trigonometric functions

Power on the calculator and set it up to calculate trigonometric functions with arguments given in radians. Now try this sequence of keyins:

[tan] 4 5 [o ' "] =

You will find that the display now shows:

tan(45o) in the top line and 1.619775191 in the answer line ... 

... but tan(45o) is actually 1.

What is happening here is that although the keyins give the argument as 45o, this does not override the radian setting for trigonometric functions. The displayed result is the value of tan(45) with the argument in radians but the display leads you to think that the result is actually tan(45o). The [R] mode indicator is displayed but this is small in relation to the result display and very easy to miss.

I imagine that it would be easy for students not to spot this and end up making mistakes. (In fact when I was checking the result while writing this post, I initially failed to notice that the calculator was set to Grad rather than Rad). Again, I think teachers should draw students' attention to this aspect of using the calculator.

Olwen Morgan

Casio fx991ES-PLUS equation solving

Try the following to solve e-x = sin(x)

1. Power on, clear all and set to COMP mode.

2. Enter:

[SHIFT] [ex] [(-)] [ALPHA] X ) [ALPHA] = [sin] [ALPHA] X ) [SHIFT] [SOLVE] =

(Note that the '=' following the [ALPHA] key is on the [CALC] button, not the normal '=' button.)

The screen goes blank and eventually displays:

e^(-X)=sin(X)

X= 0.588532744

L-R = 0

Substitution confirms that e-0.588532744 - sin(0.588532744) = -2.5156 x 10-11 ...

... but the original equation has not merely one but a countably infinite number of solutions, as is easily seen by sketching graphs of e-x and sin(x).

Here what the calculator says is correct to appropriate accuracy but the device gives no indication of the limitations of its root-finding algorithm. This is a general problem that is seen even in professional-level calculators. Professional users are wary of such things but I have known students to be quite oblivious to them. IMO it would be good teaching practice to draw to the attention of students that this limitation exists.

To solve an equation really means to find all of its solutions, not just the solutions that a particular algorithm arrives at first.

Olwen Morgan

Casio fx991ES-PLUS equation non-solving

Power the calculator on, press [AC] to clear and set to COMP mode.

Try to solve x2 + 1 = 0 by entering:

[ALPHA] X [x2 ] + 1 [ALPHA] = 0 [SHIFT] [SOLVE] =

(Note that the '=' following the [ALPHA] key is on the [CALC] button, not the normal '=' button.)

After (presumably) having tried to solve the equation for around 10 seconds, the calculator displays: "Can't Solve".

Now press [AC], set to CMPLX mode and repeat the above key-ins.

The calculator responds with, "Syntax ERROR".

It appears that:

1. In COMP mode the calculator takes 10 seconds to decide that it cannot solve an equation that most students (hopefully) can solve immediately in their heads.

2. If in COMP mode the keyed-in equation is syntactically correct but beyond the capability of the calculator to solve, there is no mathematically logical reason for it to be pronounced syntactically incorrect in CMPLX mode.

3. While one might expect COMP mode not to give the two imaginary solutions, one could reasonably expect CMPLX mode to do the job. Yet it appears not even to try.

4. You have to go into equation mode and select the quadratic solver option to get the solution. Yet the device seems to have no difficulty with getting a root of e-x=sin(x) (see previous post) using the general-purpose solver.

Anyone else baffled?

Stephen Lyon

The New to teaching A level summer school was originally funded by the Clothworkers foundation. Casio then funded the summer school  for the next three years. In this time 150 teachers, new to teaching A level mathematics, were able to access high quality CPD thanks to the generousity of Casio. The funding for the summer school has been taken on by Project Enthuse, enabling many more teachers access to this support. Details of next year's summer school can be found at www.stem.org.uk/my500 

Casio now fund, amongst other things, the MEI/Casio Teacher Networks and the excellent MEI conference which I thoroughly enjoyed last year. I feel Casio should be congratulated for the support they are offering to teachers of mathematics.

In the interests of balance, Texas Instuments support the IMA scholars programme. The excellent Peter Ransom delivered a training session at the National STEM Learning Centre in the use of graphic calculators. The training is supported by Texas Instuments and will be repeated in the coming year.

Footnote: my Casio calculator performs the above calculation correctly and even rationalises the denominators.

Olwen Morgan

For further balance ...

I've posted things above about Casio and Texet calculators. For balance, my next posts will look at Sharp calculators. If you have any further examples of confusing calculator behaviour, please feel free to share them. The more pitfalls people are aware of, the better placed they are to tell students how to avoid them.

Olwen Morgan

Sharp EL-W506 bafflers

Off, then, with the baffling oddities of the Sharp EL-W506 calculator.

Power on the calculator and clear it by pressing [ON/C]. Now enter:

1 + 1 3 [2ndF] %

and the calculator displays 1.13, having increased 1 by 13%, apparently having proceeded as if it had a compound binary operator %= defined by:

x %= y = x + x(y/100).

There's nothing mathematically wrong with a compound operator that straddles its operands. Such operators are used in programmng languages. The C conditional operator ? : is an example.

But now press = and the calculator displays 14. AFAI can see the press of the = button undoes the effect of the % operation and then proceeds as if the user had entered:

1 + 1 3 =

Three points:

1. It is very easy to make errors using this feature. An inadvertent press of the = button is easily missed when keying in a long expression.

2. Implementing the add-percentage operator as a straddling operator is unnecessary and IMO confusing to students. It would be better implemented it as a compound infix operator %=.

3. There is in this case neither need for nor useful purpose in having one operator undo the effect of a preceding one. If an UNDO function is provided, then it should be on a button labelled UNDO and be applicable to any operation.

IMO this is at best exceptionally poor HMI design.

Olwen Morgan

Sharp EL-W506 - the tan 45o problem again:

Power on the calculator, clear it with [on/C] and set the trig. function mode to RAD. Now enter:

[tan] ( 4 5 [DMS] ) =

The calculator now displays tan(45o) in the top line and 1.619775191 in the bottom line. It behaves in this respect exactly like the Casio fx991ES-PLUS. Just as with the Casio device, students need to be made aware of this less-than-helpful behaviour.

Olwen Morgan

Sharp EL-W506 operator precedence

Power on the calculator, clear it with [on/C] and set the trig. function mode to DEG. Enter:

[sin] ( 4 5 ) [x2] =

The calculator displays sin(45)2= in the top line and -0.707106781 in the bottom line. Instead of computing sin2(45o) it has computed sin((452)o). This shows that the [x2] operation has higher precedence than function application, which is contrary to what most programming languages do.

There is no mathematical reason why the sequence of keyins:

[sin] [x2] ( 4 5 [DMS] )

should not produce the result sin2(45o). In this case the calculator does not accept a mathematical expression how one would normally write it.

The usual caveat: Make sure your students know about such behaviours.

Olwen Morgan

Sharp El-W506 - hexadecimal operations

Power on the calculator, clear it with [on/C] and set it into hexadecimal mode with [2ndF] [HEX]. Now enter:

[NOT] A 5 =

The calculator display now shows: NOTA5= in the top line and FFFFFFFF5A in the bottom line.

Two comments:

1. The output makes clear that the calculator interprets A5 as a hexadecimal non-reduced numeral, i.e. it has leading zeros. There is no mathematical reason to assume a non-reduced numeral.

2. It is easy to make the keying-in mistake:

[NOT] [ALPHA] A 5 =

which results in the calculator displaying: Error 01 Syntax.

While it is logical not to require [ALPHA] to precede A, B, C, D, E or F in hex mode. It is illogical not to ignore [ALPHA] if it is used in this context.

Once again we see an example of the poor HMI design of school-level calculators.

Olwen Morgan

Sharp EL-W546 degrees minutes and seconds

Power on the calculator, clear it with [on/C] and set the trig. function mode to DEG. Now enter:

4 5 [DMS] 9 0 [DMS] 9 0 [DMS] =

The top line of the display shows 45o90o90o= and the bottom line displays 46o31'30." Clearly the calculator has correctly understood that 45o + 90' + 90" = 46o31'30", in which case two questions arise:

1. After the device sees the keyins 4 5 [DMS], it has all the information it needs to show the degree, minute and second symbols correctly on the top line. So why don't successive presses of [DMS] make the top line display in degrees minutes and seconds?

2. When 4 5 [DMS] 9 0 [DMS] 9 0 [DMS] has been entered, the device has enough information to know that the seconds amount is integral. So, why does the bottom line show what looks like a decimal point after the seconds figure?

Ideas anyone?

Olwen Morgan

Sharp EL-W546 deg/rad/grad conversions

Power on the calculator, clear it with [on/C] and set the trig. function mode to DEG. Now enter:

4 5 [2ndF] [DRG]

The top line now displays 45RAD and the bottom line 0.785398163, which is the value of 45o in radians. This shows that the [DRG] button functions in this context like a compound operator effecting both a unit conversion and an implicit press of the  = button - but it also has a pernicious side-effect. The angle mode indicator at the top edge of the display now shows RAD. Modedness is bad enough where it exists unnecessarily but to have a computing operation change it as a side effect is far from good HMI design.

If you think that's idiosyncratic, try this:

Clear using [on/C] and set the angle mode to DEG. Now enter:

[tan] ( 4 5 [2ndF] [DRG►]

The top line now displays: tan (45►RAD while the bottom line displays 0.017453292.

Not only does the [DRG] key force immediate evaluation, together with a mode change to RAD, but it does so without having a closing bracket after the argument of tan.

Sharp calls its data entry and display system "WriteView". Do you write formulae with unbalanced brackets?

Olwen Morgan

Sharp EL-W506 non-decimal arithmetic

Power on the calculator, clear it with [on/C] and set it into hexadecimal mode with [2ndF] [HEX]. Now enter:

F F / 5 =

The device gives the answer 33.

Now clear with [on/C] and enter:

F F / 7 =

The device gives the answer 24, having performed truncating integer division. Behaviour is the same for base 2, 5 and 8, all of which are offered by the EL-W506. The Casio fx991ES-PLUS has the same limitation.

There is no mathematical reason why division in bases other than 10 need be truncating integer division. The trouble with this kind of limitation is that it can create the impression in students' minds that only integer division is defined for bases other than 10. I remember in the 3rd form doing base 7 long division manually to fractional places, so nearly 40 years on a modern calculator has little excuse for not attempting it.