The Secret Question (Are We Actually Good at Math?)

Posted on the AMS Blogs on September 1, 2015 by Ben Braun

“How many of you feel, deep down in your most private thoughts, that you aren’t actually any good at math? That by some miracle, you’ve managed to fake your way to this point, but you’re always at least a little worried that your secret will be revealed? That you’ll be found out?”

Over half of my students’ hands went into the air in response to my question, some shooting up decisively from eagerness, others hesitantly, gingerly, eyes glancing around to check the responses of their peers before fully extending their reach.  Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren’t raised looked around in surprised confusion at the general response.

Preterm Birth Linked With Lower Math Abilities and Less Wealth

September 1, 2015
For Immediate Release
Contact: Anna Mikulak
Association for Psychological Science
amikulak >>at<<

People who are born premature tend to accumulate less wealth as adults, and a new study suggests that this may be due to lower mathematics abilities. The findings, published in Psychological Science, a journal of the Association for Psychological Science, show that preterm birth is associated with lower academic abilities in childhood, and lower educational attainment and less wealth in adulthood
“Our findings suggest that the economic costs of preterm birth are not limited to healthcare and educational support in childhood, but extend well into adulthood,” says psychological scientist Dieter Wolke of the University of Warwick in the UK. “Together, these results suggest that the effects of prematurity via academic performance on wealth are long term, lasting into the fifth decade of life.”

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Latest speech by Nick GIbb

Nick Gibb speaks at the Researchers in Schools celebration event, 25 August 2015.

What follows are paragraphs from the text containing the words maths or mathematics.

The Researchers in Schools programme prioritises recruiting teachers in STEM subjects, in particular mathematics and physics. Nobody needs reminding that British employers face ongoing skills shortages in these areas.

One in 10 state schools have no pupils progressing to either further maths or physics at A level, and 1 in 3 physics teachers have themselves not studied the subject beyond A level.

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The Quadratic Formula in Malta’s Learning Outcomes Framework

What I see as a deficiency of the Learning Outcomes Framework is that it does not specify learning outcomes in a usable way.

There are several references to quadratic equations in Levels 8–10, for example

Level 8
Number – Numerical calculations
18. I can solve quadratic equations by factorisation and by using the formula.

If a student from Malta comes to my university (and I have had students from Malta in the past, I believe), I want to know what is his/her level of understanding of the Quadratic Formula.

There are at least 7 levels of students’ competencies here, expressed by some sample quadratic equations:

(a) x2 – 3x +2 =0
(d) x2 – 1 = 0
(c) x2 – 2x +1 = 0
(d) x2 + sqrt{2}*x – 1 = 0
(e) x2 + x –  sqrt{2} = 0
(f) x2 + 1 = 0
(g) x2 + sqrt{2}*x + 1 = 0

These quadratic equations are chosen and listed according to their increasing degree of conceptual difficulty: (a) is straightforward, (b) has a missing coefficient (a serious obstacle for many students), (c) has multiple roots, (d) involves a surd, but no nested surds in the solution, (e) has nested surds in the answer, (f) has complex roots, although very innocuous ones, and (g) has trickier complex roots. Of course, another list  can be made, with approximately the same gradation of conceptual difficulty.

I would expect my potential students to be at least at level (d); but LOF tells me nothing about what I should expect from a student from Malta.

And one more comment: a comparison of the statements in the LOF Level 10:

I can solve quadratic equations by completing a square

and in the LOF Level 8:

I can solve quadratic equations by factorisation and by using the formula.

apperars to suggest that at Level 8 the Quadratic Formula is introduced to students without proof or proper propaedeutics which appear only at Level 10. In my opinion, this should raise concerns: at Level 8, this approach has a potential to degenerate into one of those “rote teaching”  practices that make children to hate mathematics for the rest of their lives.

The use of the term ‘Expected Frequency’

The June 2015 GCSE Subject Level Conditions and Requirements for Mathematics includes (P3)

“relate relative expected frequencies to theoretical probability, using appropriate language and the 0 – 1 probability scale”

and this leads to questions like

“If you rolled a die 600 times, how many sixes would you expect to get’.

which is taken from the CIMT MEP Pupil’s textbook on probability, and is given the answer

‘You would expect to get a 6 in 1/6 of the cases, so 100 sixes’.

This seems a confusing and misleading term. What exactly is an ‘expected frequency?’ The obvious meaning is the frequency that you expect. But we are trying to support the concept of a random variable, with ideas that a random variable is unpredictable in terms of value, that values do not form patterns or sequences, and can only be forecast and predicted in some general ways.

If you roll a die 600 times, I do not expect any value for the number of sixes. That is the most significant aspect of a random variable.

The implied sub-text is that

Expected frequency = probability X number of trials

So that, for example, if we toss a fair coin 100 times, what is the expected frequency of heads? Well, 50. So does that mean we expect to get 50 heads? This is a Bernoulli trial, and the probability of getting precisely 50 heads in 100 tosses is about 0.08. So we would need to say to a pupil

‘The expected frequency is 50; but it is unlikely that you would get 50 heads’

which hardly makes sense.

The probability of 51 is about .078, and 52 is .074. So, of course, 50 is the most likely frequency.

The phrase ‘most likely frequency’ is straight-forward, makes sense, and says what it means, unlike ‘expected frequency’.

Please can we stop using the phrase ‘expected frequency’?

Malta’s Learning Outcomes Framework: a Discussion

Malta’s new Learning Outcomes Framework for school mathematics is an important case study of the European Union’s approaches to implementation of its education policies in member countries. For that reason the Framework deserves a close attention.

The original post of 13 August generated more responses than it was anticipated, and it is useful to collect them all at a single page.