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Why is IB HL Maths so hard?

If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

This is a question that nearly all students who take the subject will ask themselves at some point during the course – and they’d be right to do so because it’s a question that most teachers ask as well.  The table below shows the students entered for the May 2015 IB exams:

Only 14% of IB students take IB HL,  an incredibly low take up.  Even more remarkably half of this group will only get a level 2-4.   Looking at the top end, to get a level 6 or 7 at HL you probably need to have a maths ability in the top 3.5% of all global IB students.  Out of a year group of 60 you would expect on average only 2 students to be good enough to get a 5 and 2 more to get a 6 or 7.  Given than universities asking for HL maths will invariably be asking for level 5+, this means that the IB have designed a course which is only really useful for 4 students in every 60.

It’s not as if the IB aren’t aware that they’ve created a course that hardly any students get benefit from.  The most recent release acknowledges that “students struggle to reach their full potential” in the subject – with a plan to reduce the marks on the paper to give students more time.  But this is failing to address the overall cause of the problem, i.e that examiners persist in producing bad exams which don’t take into account the needs of the students taking them.

You can see this failure easily enough by looking at past paper grade boundaries.  Given that only half of HL students will come out with more than a level 4, the grade boundary for a level 4 is normally pitifully low – around 40% on paper 1.  There’s absolutely no justification for this when 70% is low enough to produce a level 7.  What is the final 30% of the paper – too difficult even for level 7 students – hoping to achieve?   This is evidence of a bad exam and nothing more.  And yet examiners seem to be incapable of doing anything to change this.  An exam designed expecting level 4 students to be getting 50% should be an absolute minimum requirement.  Hard questions and low grade boundaries simply result in demotivated and disillusioned students who feel like their 2 years slogging through HL has been wasted.  Is this the legacy that IB want to achieve?  That students who start the course with enthusiasm and love for the subject get gradually crushed and demoralised?  It seems a pretty poor outcome.

The option paper for Calculus also shows the same lamentable failings.  The November 2015 Calculus paper required only 38% to get a level 4 and 56% for a level 6.  Given that so few students achieve level 6 or 7 this means the paper was so badly designed that probably close to 75% of the students taking it got less than 50%.  How can this be anything other than a failure of the examiners to actually produce a fair test?  This bunching up of all grades meant that a slip on one question could cost a student 2 grades.  17 marks out of 60 was a level 2 – but 23 marks a level 4.  Equally a student getting 28 marks or 34 marks would have got level 4 or 6 respectively.  This is a terrible test!  Small mistakes are massively penalised and all students leave the exam room feeling like they have failed.

Examiner Mindset

The examiner mindset especially in evidence in the Calculus option unit is to purposely avoid all topics that they think the students will be able to do, and instead to find parts of the syllabus that they expect to catch students out on.  On P1 and P2 where examiners have 240 marks to play with, these exams are a good reflection of the syllabus content.  For P3 this isn’t the case – large chunks of the syllabus are ignored completely in exams, which makes it all the more unfair when examiners decide to deliberately look for problem areas, rather than concentrating on making a test which reflects the overall content of the course.  There also seems to be the desire to make the Calculus option a university undergraduate level maths paper – as though there is a pride to be taken in making it as difficult and rigorous as possible.  But HL maths is not an undergraduate  course – and the students taking it are not university mathematicians.  Designing a paper which is aimed only at the needs of level 6 and 7 students is incredibly unfair to the rest of the cohort – and yet this appears to be what happens.

Essential Resources for IB Teachers

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

This is a fantastic passage – which is part of the Mathematician’s Lament by Paul Lockhart.  He goes into a lot more detail in the pdf (available to read here ).  He really highlights some of the absurdities in how mathematics is both viewed and taught in society.

A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.”

Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer. Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.”

It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.

As for the primary and secondary schools, their mission is to train students to use this language— to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”

In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”

In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.”

Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one.

“To tell you the truth, most students just aren’t very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable— every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She’s going to make one hell of a musician someday.”

Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. “Of course!” he reassures himself, “No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!”

The whole pdf is well worth a read.

If you enjoyed this you might also like:

Maths and Marking – Why are the current accepted school methods lagging behind evidence of what best raises attainment?

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

The Battle over Homework: Marking in Mathematics

Within five minutes of any teaching inspection from OFSTED, the inspector will be leafing through students’ exercise books in search of evidence of regular and meaningful marking. If it’s not there then they will probably already be penciling in the “requires improvement” column. With no-notice inspections now in the UK, and with such high stakes for Senior Management, there is ever greater pressure and expectation on UK teachers for “OFSTED ready” marking – with all the significant increase in workload that this entails. Yet, how much discussion ever takes place as to how effective such marking is for mathematics?

Dr Harris Cooper of Duke University, one of the few academics to really research homework policies in depth, provides compelling evidence of the need to rethink marking strategies through a meta-analysis of research on the subject. He cites 47 studies which demonstrate that the act of setting homework itself, regardless of whether grading or comments were provided has a positive benefit for students.

There are much fewer studies specifically looking at grading strategies – but here again they run contrary to accepted norms. Three studies which looked at different homework grading strategies – marking every problem, marking a random sample, marking for accuracy or marking for completeness were found to have no difference in student attainment. Five more studies meanwhile suggest there is little difference in attainment when students are provided with comments as a pose to grades.

One study cited was that by J Austin looking at whether comments on homework affected student performance. Researchers found that in 7 of the 9 classes in the study, there was no significance in raised mathematical performance between students who received comments instead of simple grades.

This probably makes sense to anyone who has ever marked maths homework before – either students will get everything correct, or most students will get the same question wrong. Rather than spending 40 minutes writing out 20 explanations of how to use Pythagoras’ rule correctly, surely it makes more sense to spend 2 minutes at the start of the next lesson explaining (or getting a student to explain) it to the whole class?

Mathematics marking is also an area which computer programs can provide huge benefits. Websites like MyMaths allow students to be set targeted homework, which they can have marked in real time. These homework tasks are accompanied by lesson content so that students who do get questions wrong can then go back to the lesson, review the material and then resubmit their homework again. The evidence clearly shows that specific, immediate feedback raises attainment – and such programs not only do this, but also encourage students to become self-learners in the process.

A couple of recent studies looking at the effectiveness of web-based marking versus traditionally graded marking in university students found no significant difference in overall student performance between different approaches – and found that web-based homework both increased the students’ time spent on the task and provided greater opportunities for them to recognise and correct errors.

Given such evidence it is depressing that so many school leaders and OFSTED inspectorate remain wedded to the importance of using comments in exercise books as a core method of assessing teaching.

Central to this discussion should instead be the concept of greatest teaching efficiency – i.e. what is the most effective use of teaching time to maximise student attainment. Current policies effectively ignore this question completely, and work on the assumption that there is no upper limit to working hours in a day and that all new strategies will have no detrimental effect on other ones. The reality is the reverse, new strategies squeeze out old ones, overwork leads to increased stress and an overall drop in both student-teacher relations and teaching standards.

A more honest approach would be to start with a fixed idea for the number of hours per week staff members should be expected to work – whether that be 45 hours, 50 hours or more. Once that figure is fixed then it simply becomes a case of working backwards – filling in those hours with various teaching responsibilities. Doing this then forces a genuine debate as to what is the most effective use of teaching time – 5 hours a week marking books or an extra 5 hours a week planning lessons, organising gifted and talented events, or holding after school revision sessions. Which strategy actually has the greatest impact on attainment?

Immediate, targeted verbal feedback within the classroom setting has been shown to to have one of the greatest impacts on attainment. Comments for maths homework do little to raise attainment over simple grading, and web based programs offer both immediate feedback and the opportunity for students to become self-learners. None of these appear as evidence in a two minute flick through of a student book – but what evidence does this actually provide? Maybe it’s time to challenge the existing marking paradigm.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

### New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

### Free HL Paper 3 Questions

P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.