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

mathematicsand 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

mathsor physics at A level, and 1 in 3 physics teachers have themselves not studied the subject beyond A level.

This lack of take-up in the

mathsand sciences is particularly acute amongst female pupils. Whilst nearly half of boys who gained an A* grade at physics GCSE in 2011 went on to study the subject at A level, only around a fifth of girls did so.However, before I begin to sound too gloomy, there are significant reasons to be cheerful. One of the achievements of the previous government of which I am most proud of is this: there were 38,000 more entries for science and

mathsA levels in 2015 compared with 2010 – a 17% increase.Due to the government’s focus on STEM subjects, there has been a 17% jump in entries for physics A level since 2010, a 19% jump in entries for chemistry, and a 28% jump in entries for

further maths. Today,mathematicsis by a stretch the most popular A level subject, with 92,000 entries in 2015.We are already well on the way to achieving the aim of the government’s YourLife campaign. Launched in November 2014, this campaign aims to increase the number of students studying

mathsand physics A levels by 50% within 3 years. We hope that themathsand physics chairs programme within Researchers in Schools will play a central role in this campaign.

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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) x

^{2}– 3x +2 =0

(d) x^{2}– 1 = 0

(c) x^{2}– 2x +1 = 0

(d) x^{2}+ sqrt{2}*x – 1 = 0

(e) x^{2 }+ x – sqrt{2} = 0

(f) x^{2}+ 1 = 0

(g) x^{2}+ 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 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 to 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.

]]>A common impairment with lifelong consequences turns out to be highly contagious between parent and child, a new study shows.

The impairment? Math anxiety.

Means of transmission? Homework help.

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“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’?

]]>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.

- A. Borovik, The Quadratic Formula in Malta’s Learning Outcomes Framework, 26 August 2015
- A. Borovik, The Great Mystery of Malta’s Learning Outcomes Framework, 23 August 2015, updated 24 August 2015
- V. Gutev, Outcome Based Education, 13 August 2015 (+ 2 comments)
- J. Lauri, Response to “Malta: new Learning Outcomes Framework”, 12 August 2015 (+ 3 comments)
- A.Borovik, Malta: new “Learning Outcomes Framework”, 7 August 2015 (+ 8 comments)

Malta’s new Learning Outcomes Framework 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.

An attempt to study the official website

http://www.schoolslearningoutcomes.edu.mt/en/pages/about-the-framework

immediately leads to a question:

Who had actually developed the Framework?

According to Wikipedia, population of Malta is about 445,000. When compared with the City of Manchester (about 514,000), it becomes clear that development of the Framework is a job beyond capabilities of a small nation.

So, external consultants were hired, some institutions or companies from English speaking parts of Europe. Taking into consideration traditional cultural connections, this part of Europe is likely to be the UK.

**Added 24 August 2015: ** Indeed I could not locate contractor’s names using advanced Google search on gov.mt,but serendipitously discovered their logos in the document Joint Venture Presentation dated 28 Jan 2015:

Outlook Coop is a company on Malta specialising in project management with expertise in EU funded projects.

East Cost Education Ltd is a small private company based in Northumbira with specialism, judging by their website, concentrated mostly in vocational education and training. In recent years, they worked on Malta on several projects in vocational training.

Institute of Education, London, is

]]>the world’s leading centre for education and applied social science.

]]>The inspection paradox is a common source of confusion, an occasional source of error, and an opportunity for clever experimental design. Most people are unaware of it, but like the cue marks that appear in movies to signal reel changes, once you notice it, you can’t stop seeing it.

A common example is the apparent paradox of class sizes. Suppose you ask college students how big their classes are and average the responses. The result might be 56. But if you ask the school for the average class size, they might say 31. It sounds like someone is lying, but they could both be right.

The problem is that when you survey students, you oversample large classes. If are 10 students in a class, you have 10 chances to sample that class. If there are 100 students, you have 100 chances. In general, if the class size isx, it will be overrepresented in the sample by a factor ofx.That’s not necessarily a mistake. If you want to quantify student experience, the average across students might be a more meaningful statistic than the average across classes. But you have to be clear about what you are measuring and how you report it.

“So now, with a decade of confusion about OBE behind us, I would encourage my South African colleagues to stop referring to OBE in any form. It never existed in 1997, and has only faded farther from the scene since. The real issue facing the country is to mobilize behind educational practice that is sound and makes a significant difference in the lives of ALL South African learners. Empty labels and flowery rhetoric are no longer needed; but principled thinking and constructive action are.”

Educational experts may argue whether it was Outcome Based Education, or some kind of Education Based on Outcomes. These experts may further argue on the terminology, but the fact remains it was supposed to be transformational OBE. A close look at their mathematics curriculum reveals that it is not so different from the proposed new Learning Outcomes Framework (LOF) for school mathematics in Malta, and in some aspects is even better. What is however completely identical in both is the educational utopia of outcomes coming from nowhere.

Essential mathematical skills are not just about a computational answer, for it is not the answer that is of the greatest importance to school children’s mathematical development. Rather it is children’s ability to apprehend mathematics as a conceptual system. Many education systems are emphasising on this, here is an excerpt from the Secondary Mathematics Syllabuses in Singapore:

“Although students should become competent in the various mathematical skills, over-emphasising procedural skills without understanding the underlying mathematical principles should be avoided… Students should develop and explore the mathematics ideas in depth, and see that mathematics is an integrated whole, not merely isolated pieces of knowledge.”

Unfortunately, in Malta’s case the design falls far short of such goals. Here is an example from level 5:

(COGNITIVE LEARNING) 16. I understand that multiplication is repeated addition.

Accordingly, a factor can only be added to itself a counting number of times. In Singapore’s Primary Mathematics Syllabus, multiplication and division are conceptualised gradually, and still on that level are introduced area and various square units. In contrast, square units are not present in Malta’s LOF for school mathematics. In fact, the proposed LOF is teeming with conceptual deficiencies. For instance, there is some kind of misconception between “equation” and “function”. Equations were never related to unknown variables, while functions are assumed to be somehow equations between the variables “*x*” and “*y*“. Use of radian measurement is not present, but learners are supposed to “plot graphs of trigonometric functions”.

Perhaps, Malta can learn from Singapore’s remarkable success since independence and the policies underlying its achievements in mathematical education.

]]>If the thought of a math test makes you break out in a cold sweat, Mom or Dad may be partly to blame, according to new research published in *Psychological Science*, a journal of the Association for Psychological Science.

A team of researchers led by University of Chicago psychological scientists Sian Beilock and Susan Levine found that children of math-anxious parents learned less math over the school year and were more likely to be math-anxious themselves—but only when these parents provided frequent help on the child’s math homework.

Lead study author Erin A. Maloney is a postdoctoral scholar in psychology at UChicago. Gerardo Ramirez and Elizabeth A. Gunderson co-authored the article, along with senior authors Levine and Beilock.

Previous research from this group has established that when teachers are anxious about math, their students learn less math during the school year. The current study is novel in that it establishes a link between parents’ and children’s math anxiety. These findings suggest that adults’ attitudes toward math can play an important role in children’s math achievement.

“We often don’t think about how important parents’ own attitudes are in determining their children’s academic achievement. But our work suggests that if a parent is walking around saying ‘Oh, I don’t like math’ or ‘This stuff makes me nervous,’ kids pick up on this messaging and it affects their success,” explained Beilock, professor in psychology.

“Math-anxious parents may be less effective in explaining math concepts to children, and may not respond well when children make a mistake or solve a problem in a novel way,” added Levine, the Rebecca Anne Boylan Professor of Education and Society in Psychology.

Four hundred and thirty-eight first- and second-grade students and their primary caregivers participated in the study. Children were assessed in math achievement and math anxiety at both the beginning and end of the school year. As a control, the team also assessed reading achievement, which they found was not related to parents’ math anxiety.

Parents completed a questionnaire about their own nervousness and anxiety around math and how often they helped their children with math homework.

The researchers believe the link between parents’ math anxiety and children’s math performance stems more from math attitudes than genetics.

“Although it is possible that there is a genetic component to math anxiety,” the researchers wrote, “the fact that parents’ math anxiety negatively affected children only when they frequently helped them with math homework points to the need for interventions focused on both decreasing parents’ math anxiety and scaffolding their skills in homework help.”

Maloney said the study suggests that parent preparation is essential to effective math homework help. “We can’t just tell parents—especially those who are anxious about math—‘Get involved,’” Maloney explained. “We need to develop better tools to teach parents how to most effectively help their children with math.”

These tools might include math books, computer and traditional board games, or Internet apps that “allow parents to interact with their children around math in positive ways,” the researchers wrote.

###

This work was supported by grants from the U.S. Department of Education Institute of Education Sciences (R305A110682) and the National Science Foundation (NSF; Career Award DRL-0746970) to S. L. Beilock and by awards from the NSF Spatial Intelligence and Learning Center (SBE-0541957, SBE-1041707) to S. C. Levine.

The article press release and abstract are available online.

The APS journal * Psychological Science* is the highest ranked empirical journal in psychology. For a copy of the article “Intergenerational Effects of Parents’ Math Anxiety on Children’s Math Achievement and Anxiety” and access to other

Thank you Alexandre for taking an interest in the curriculum being developed for the Maltese schools. (As a matter of information, this curriculum is being developed by a consortium of foreign “experts” supported by a European Social Fund grant. What is shown on the website is work-in-progress, and one hopes that the final product will be a more coherent curriculum and banalities like the one you pointed out will have been removed.)

So, let me share my answers to the same question you ask, basically why does this draft curriculum contain such a statement: I can use equivalent fractions to discuss issues of equality e.g. gender. I agree with your two responses, namely mis-use of vocabulary and the strictures imposed by an Outcomes Based (OB) curriculum. But allow me to elaborate further.

In my view, the above statement would be banal whether one uses the term “equivalent fractions” or “similar fractions” or any other notion which extrapolates from 1/2=2/4=3/6=etc to anything having to do with gender equality. The problem, in my opinion, is that some people do not realise that, in science, we expropriate a word from everyday vocabulary to use in a context which does have some similarity to the everyday use of the word, but whose meaning becomes something technical which cannot be exported back to the everyday sense of the word. I sometimes taught classes of Arts students who felt they needed to use some mathematical jargon in their essays (a few years ago the fashionable thing to do was to drop the words “chaos” and “fractal”). One of my usual examples of how wrong this is involved the use of the word “work”, as used in science and in everyday life. Translated into the context of curricula, the analogous banal statement could be something like: *I can calculate the work done by a given force moving an object through a given distance and I can use this to discuss the conditions of work in factories and industry. *

What surprises me when statements such as the one on gender equality are made is that while the ambiguity of language is appreciated outside science, in fact it can be a wonderful tool in the hands of a good writer, when transporting scientific vocabulary back into the everyday world, this variegated meaning of the same word in different contexts is sometimes forgotten. I have no explanation why this happens.

But another problem with curricula written in OB style and which could have a bearing on such wording is the necessity that the statements should be written in a way that the learning child would write them, for example, by starting the description of each outcome with “I can…” That sentences such as the one you quote about gender issues crop up is not, in itself the main problem, in my opinion. Such sentences can be edited out when reviewing the curriculum. The problem, as I see it, is that this style excludes the possibility that the curriculum contain concepts to guide the teacher but which the student would not likely be able to express. So take your improved statement of how mathematics can help understand social inequalities:

I believe in the power of mathematics and I am convinced that comparing numbers (for example, salary) reveals a lot about gender inequality (and other, frequently hidden, inequalities in the world — just recall the Oaxaca Decomposition and its role in fight against discrimination of any kind).

It might be reasonable to expect a Level 5 student (aged 7-8) to express such a statement up to “gender inequality”, but hardly the rest of the statement, although the writer of the curriculum might very well want to make a reference to the Oaxaca Decomposition to give the *teacher* an example of a highly non-trivial use of mathematics in this context.

This OB format, I believe, betrays a fallacy about the teaching of mathematics, namely that teaching elementary mathematics to 7-year olds, say, does not involve deep knowledge of mathematics, certainly not deeper than what a 7-year old can express.

I look forward to reading other comments, especially by readers of this blog who are more familiar with OB curricula than I am.

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