## Basic maths

This post tries make to one simple point. It refers to a statement made by David Cameron on 7 March as part of his major speech on the economy. The statement was also made by Schools Minister David Laws on Channel 4 News earlier in the week. The error on which it is based has been repeated by every Education Secretary and Prime Minister back at least as far as David Blunkett and Tony Blair. It is a fallacy that is so commonplace that it is accepted as unquestionably reasonable by the entire national media, print and TV.

It is that 'only 50 percent of pupils reaching the expected level in maths and English is self-evidently not good enough'.

There must be many experts in maths and statistics that read these posts. You don't even have to be an expert. Even the most basic exercise in logical thinking should sound the alarm.

Please, expert readers, have I got this wrong or not?

1. The ability and attainment of a large population of school children at any given point in time is continuously variable.

2. Therefore the marks on any valid test of either must also be continously variable.

3. Therefore if 50 percent of pupils attain a particular benchmark score in a test then 50 percent must have scored less than this benchmark.

4. If the next cohort of children to take the same test are more able and/or better taught then a higher proportion of pupils will attain the previous benchmark score.

What would be the significance of this? Yes, standards would be rising, a good thing. But what would be the new status of the former benchmark score? Are we to believe that that particular score on that particular test could describe an 'expected level' in any valid or meaningful sense?

PISA and TIMSS do international tests. They also publish examples of test items.

For PISA: Download PISA 2009 key findings Volume 1.

For TIMSS: Select Grade 8 Released Items (bottom of page)

(I am indebted to Terry Wrigley for pointing me to these sources)

I defy any teacher or educationalist to study these or any other reputable tests of knowledge and understanding, and find a test item that represents 'the expected level' that pupils should achieve. It would have to be an item that 'expected level children' would get correct, but 'below expected level children' would not. For 'expected level' to have any meaning in KS2 and GCSE tests the same argument applies. Show me a test item that has this 'expected level determining' quality.

Of course it is nonsense. Children, bless them, get easy questions wrong and harder questions right. So if you can't find an 'expected level' test item then is there an 'expected level' test mark? This is where the notion of 'average' so frequently intrudes and misleads. There can certainly be a 'mean mark', +/- 1 SD marks, a 25th percentile mark etc etc. But all are completely arbitrary. What is certain however is that when standards rise then the 'average' also rises so attempts to increase the proportion of pupils that are 'above average' is plain silly.

Yet has this common fallacy really been the basis of every government's education policy since the 1988 Education Reform Act?

If the Prime Minister can spout fallacies like this, despite his Eton education, HM Opposition can fail to notice and the media can blithely repeat such nonsense as if was obviously correct then we are all truly Alices in Wonderland.

Who will challenge the Mad Hatter and wreck the tea party?

It is that 'only 50 percent of pupils reaching the expected level in maths and English is self-evidently not good enough'.

There must be many experts in maths and statistics that read these posts. You don't even have to be an expert. Even the most basic exercise in logical thinking should sound the alarm.

Please, expert readers, have I got this wrong or not?

1. The ability and attainment of a large population of school children at any given point in time is continuously variable.

2. Therefore the marks on any valid test of either must also be continously variable.

3. Therefore if 50 percent of pupils attain a particular benchmark score in a test then 50 percent must have scored less than this benchmark.

4. If the next cohort of children to take the same test are more able and/or better taught then a higher proportion of pupils will attain the previous benchmark score.

What would be the significance of this? Yes, standards would be rising, a good thing. But what would be the new status of the former benchmark score? Are we to believe that that particular score on that particular test could describe an 'expected level' in any valid or meaningful sense?

PISA and TIMSS do international tests. They also publish examples of test items.

For PISA: Download PISA 2009 key findings Volume 1.

*What students know and can do*. Maths and Science samples start on pages 122 and 137.For TIMSS: Select Grade 8 Released Items (bottom of page)

(I am indebted to Terry Wrigley for pointing me to these sources)

I defy any teacher or educationalist to study these or any other reputable tests of knowledge and understanding, and find a test item that represents 'the expected level' that pupils should achieve. It would have to be an item that 'expected level children' would get correct, but 'below expected level children' would not. For 'expected level' to have any meaning in KS2 and GCSE tests the same argument applies. Show me a test item that has this 'expected level determining' quality.

Of course it is nonsense. Children, bless them, get easy questions wrong and harder questions right. So if you can't find an 'expected level' test item then is there an 'expected level' test mark? This is where the notion of 'average' so frequently intrudes and misleads. There can certainly be a 'mean mark', +/- 1 SD marks, a 25th percentile mark etc etc. But all are completely arbitrary. What is certain however is that when standards rise then the 'average' also rises so attempts to increase the proportion of pupils that are 'above average' is plain silly.

Yet has this common fallacy really been the basis of every government's education policy since the 1988 Education Reform Act?

If the Prime Minister can spout fallacies like this, despite his Eton education, HM Opposition can fail to notice and the media can blithely repeat such nonsense as if was obviously correct then we are all truly Alices in Wonderland.

Who will challenge the Mad Hatter and wreck the tea party?

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