It’s been a few weeks since the HSC 2020 Examinations concluded. Mathematics teachers and the students who sat the exam have had time to ruminate over what went into each paper.

The Sydney Morning Herald printed a story about the Standard 2 Mathematics examination with a title that can only further scare future students from wanting to study the subject: “One of the hardest exams I’ve Come Across…” Here is the link:

One does not need to run a sentiment analysis on the comments made by many teachers in the MANSW Facebook Page to conclude that reception of the examinations on the new syllabuses is mixed:

It’s disappointing that the paper looked nothing like the NESA specimen paper, particularly booklet 2. There seemed to be things that were not covered in any of the 3 textbooks we used to deliver content.

Thought the exam was fair but what pushed it to the harder side was the lack of scaffolding in some questions and the language. Even though it was explained, the use of ‘recurrence relation’ was interesting. I have never used that language anywhere but Ext 2. Guess that is going to have to change now!…lol

The pulley and the reaction force… why not explicitly tell us they can be tested!? Pretty annoyed by those.

Fairly light on Complex Numbers.

A cricket is an insect.

So on that last quote… The infamous 5 mark question in the Advanced and the Standard 2 paper that started with “A cricket is an insect” required students to comprehensively wade through a thick bog of text to extract the information required to solve the question. Un-scaffolded, i.e. no lead in questions to help. Understandably, this question quickly became a meme and many students in the state left it alone.

## Extension 1 Question 14.a)

One of the most controversial questions appeared in the Extension 1 paper, Q14.a)

I’ll only include the first part of the question which requires a proof of a binomial identity:

Use the identity \( (1+x)^{2n} = (1+x)^n (1+x)^n \) to show that:

\[\begin{align*}

\binom{2n}{n} &= \binom{n}{0}^2 + \binom{n}{1}^2 + \ldots + \binom{n}{n}^2

\end{align*}

\]

This question received many criticisms in the community:

**Content Criticisms**

The Extension 1 Syllabus on page 45 states the following for binomial expansions and deriving ** simple** identities.

In what way does Question 14 address a binomial expansion for *small* integers *n* ? The question is an outright expansion for a *general* value for *n*.

If this is an example of “deriving and using *simple* identities”, I am curious to see what a *non-simple* identity proof looks like. The problem with using adjectives to describe the type of work expected from students is that there is no standard for how to interpret those adjectives. In this particular case, the word *simple* is open to interpretation.

What is simple for one student or teacher may not be simple for another.

If we assume that this examination sets a precedent for how the syllabus is to be interpreted, then I’m afraid “simple” means expanding double binomial products for a general *n*, dear readers.

### Marking Allocation Criticisms

Question 14.a) was allocated 8 marks for binomial proof and a combinatorics interpretation to its meaning.

Eight whole marks. Out of 70. That is 11.4% of the paper on one topic. This is not counting all the other marks in the paper also dedicated to the Combinatorics topic.

Students would no doubt have studied other parts of the syllabus, especially the Year 12 components of the course, in more depth than this Year 11 topic (that as previously mentioned is meant to be “simple”).

Do you remember that time when you studied so hard for a test and then none of what you studied was assessed? Instead, the tiny thing you glossed over the night before makes up the largest portion of the test?

Picture that, but for students under high stakes examination conditions, sitting their first penultimate examination of their life – only for 8 marks of the paper dedicated to something “simple” from the Year 11 course. An 8 mark question on something related to Calculus in the Year 12 syllabus would have been much preferred. Or, better yet, no 8 mark questions at all.

## Binomial Approximation by Normal Distribution

Consider the syllabus for the Extension I Mathematics course on Page 61:

Note that the course mentions using a normal approximation to the *sample proportion* – **NOT** *binomial distributions*.

Question 12.b) of the paper has this to assess these dotpoints:

Note the marking allocation – one mark. This implies less working required and should be straight forward.

There are two ways to do part iii) – it is a very simple process if one approximates the Binomial Distribution with a Normal Approximation: By using \(E(X) = 60\) and \(Var(X) = 25\), then \( X \overset{approx}{\sim} N(60, 25)\) and so \(P(55 < X < 65) = 68\%\).

However, there is **no** mention of approximating a binomial distribution with a normal approximation in the syllabus! This method is not the prescribed method that is within the syllabus dotpoints.

By using sample proportion method, it can be done like this:

\[\begin{align*}

E(\frac{X}{100}) & = 3/5\\

Var(\frac{X}{100}) & = \frac{1}{100} \times \frac{3}{5} \times{2}{5} = 0.0024\\

\sigma & = \sqrt{Var(\frac{X}{100})} = 0.04898979486\\

P( 55 < X < 65) & \approx P( 0.55 < \frac{X}{100} < 0.65)\\

& = P(\frac{0.55 – 0.6}{0.04898979486} < Z < \frac{0.65 – 0.6}{0.04898979486})\\

& = P(-1.0206207261 < Z < 1.0206207261)\\

& \approx 68\%

\end{align*}

\]

Compare the amount of working for approximating a sample proportion vs. approximating a binomial distribution. Which method do you think the person who wrote this question intended it for – considering it was for *one* mark and had lead in questions about the binomial distribution?

What does this show about the mathematical (in)competence of NESA with regard to Statistics?

I’ve always thought these dot-points were incomplete with their failure to even mention nominally the Central Limit Theorem, and then fail to mention in a dot-point that a normal distribution can be used to approximate a binomial distribution, only to then jump straight to a sample proportion approximation.

These syllabuses are not complete. Given questions like 14 and 12 in the Extension I paper, the examination is assessing a syllabus that is not yet in writing or distribution – things need to change.

## Extension II Topic Misguidances

### Pulleys

Here is an issue found in the Extension II: Is mechanics of a system involving pulleys as found in Question 16 fair game?

The Syllabus Topic Guidance document yields no results when searching for the word “pulley”. Nor are there any results for “tension”. There is no reference officially to these concepts. How were teachers meant to know that this would be assessed? Where was the guidance?

Are other systems that aren’t mentioned in the syllabus or topic guidance documents now also to be expected? Circular motion? Banked tracks?

### Projectile Motion

The Extension II questions for Projectile Motion were disappointingly a regurgitation of proving the equations of motion.

In the past (old syllabus papers), the equations of motion in **Extension I** were almost always given and then proceeded by a capitalised notice: DO NOT PROVE THESE. A student would need to use these equations to answer the question.

This year’s projectile motion question found in the Extension II paper had 6 marks dedicated to proving these results.

Followed by 2 marks on applying them.

It’s suddenly gone from a rewarding mathematical exercise in the past in Extension I papers to a lacklustre, boring regurgitation of some Lesson 1 notes on the topic in Extension II.

## Hope For The Future

As with any project or debut of any product, there will always be bugs and problems present. As long as one learns from mistakes, and feedback/criticisms from the customers are listened to and acted upon, the product evolves into something better with each iteration.

In the same manner, I hope that NESA will listen to the teachers’ feedback and in the future, carefully consider the questions – in both content and marking allocation – that are created for the final examinations. I hope they continually update the syllabus to be clearer as to what is expected of students, and what is taught by the teachers.

Let’s hope next year’s examination papers are written with more careful consideration.