Australian Geometry Test Too Difficult?

[Gerald H]

Students complained:

https://www.theguardian.com/world/2017/n...s-teachers

[JimP]

Ah, but this isn't an Australian test. It's for Kiwis.

[AlexFekken]

I like 1c in particular: so much (obvious) misdirection... Very naughty for an exam question though.

The solutions can be found with just a couple of basic techniques, nothing too fancy (e.g. properties of a triangle inscribed in a circle). If even the teachers were struggling with this then presumably they never mastered or taught those techniques themselves. That would explain things...

[pier4r]

I never liked this argument:

"One student who studied for weeks in preparation for the exam said she was thrown by the difficulty of some of the questions, which tested skills she hadn’t been taught."

So either someone spoonfeeds the technique to you, or you are brainless. Although I can understand that finding innovative solutions may need more time, I really dislike when people act like robots.

Anyway thanks for sharing, the questions are interesting.

[Paul Dale]

I suspect this test required a little creative thought and that stymied the examinees rather more than the problems' difficulties. The knowledge required to solve these problems was contained within the high school curriculum (in oz a couple of years ago when my daughters were doing it):

• angles in a regular polygon
• angle inscribed in a semicircle is 90°
• triangle angles sum to 180°
• angle on a line is 180°
• tangent to a circle and radius meet at 90°
• similar triangle properties
• Pythagoras's theorem
• the Pythagorean triad 1:√3:2 has angles 90°, 60° and 30° (this one isn't necessarily taught)
• perhaps some of the basic plane geometry theorems (not sure if they are required)
  

Pauli

[Thomas Okken]

(11-22-2017 08:50 AM)pier4r Wrote:  I never liked this argument:

"One student who studied for weeks in preparation for the exam said she was thrown by the difficulty of some of the questions, which tested skills she hadn’t been taught."

So either someone spoonfeeds the technique to you, or you are brainless. Although I can understand that finding innovative solutions may need more time, I really dislike when people act like robots.

So you think it's fine to ask students to translate some German in a French test? Calculus in an algebra exam? Just figure it out, be creative?

I took a look at that Guardian article, and I was quite surprised, too. "Geometrical reasoning?" I remember that being touched very lightly in my high school (Netherlands, late '70s to early '80s), before moving on to Cartesian coordinates and solving geometrical problems in terms of such coordinates, using not geometrical reasoning, but trigs and algebra. (Which is an approach that has served me well ever since). If after that kind of lesson plan, I had gotten an exam with a three-part question asking for clear geometric reasoning, I would have felt I was getting screwed, too.

[Gerald H]

Well spotted, JimP, I apologize for my sloppiness.

[pier4r]

(11-22-2017 02:43 PM)Thomas Okken Wrote:  So you think it's fine to ask students to translate some German in a French test? Calculus in an algebra exam? Just figure it out, be creative?

Of course not. But if you studied French all the time building short sentences, filling gaps in text where some words are omitted, studying grammar and learning words all the time. I won't be surprised if the test would ask "please write a letter of 50 to 100 words to your friend" (given that you have a dictionary with you and enough time assigned for the test). You never wrote such long letters, but you have the building blocks to do it.

Nonetheless be sure there will someone saying "We never wrote letters of 50 words, what is this for a test!?"

My point is: if one has the blocks to solve it, but one needs to find the proper arrangement of those to solve it (and one never saw the arrangement before) then it will be a good test - in my opinion - to give out the mark "with honors".

I think it is ok to reserve the top score, and with "top" I mean really over 90% or so of the total marks, only to those that solve something that requires creativity. Disclaimer: not that I would have been able to achieve those top scores, but I would have accepted them.

[Thomas Okken]

(11-22-2017 09:06 PM)pier4r Wrote:  

(11-22-2017 02:43 PM)Thomas Okken Wrote:  So you think it's fine to ask students to translate some German in a French test? Calculus in an algebra exam? Just figure it out, be creative?

Of course not. But if you studied French all the time building short sentences, filling gaps in text where some words are omitted, studying grammar and learning words all the time. I won't be surprised if the test would ask "please write a letter of 50 to 100 words to your friend" (given that you have a dictionary with you and enough time assigned for the test). You never wrote such long letters, but you have the building blocks to do it.

Nonetheless be sure there will someone saying "We never wrote letters of 50 words, what is this for a test!?"

My point is: if one has the blocks to solve it, but one needs to find the proper arrangement of those to solve it (and one never saw the arrangement before) then it will be a good test - in my opinion - to give out the mark "with honors".

I think it is ok to reserve the top score, and with "top" I mean really over 90% or so of the total marks, only to those that solve something that requires creativity. Disclaimer: not that I would have been able to achieve those top scores, but I would have accepted them.

You're still saying it's OK for the test to be harder than what the students were taught. I don't know what kind of school you attended, where you never wrote an essay before suddenly being required to do so on a test. I'm sure that kind of approach works great for identifying those students who are ahead of the curriculum, but that's not what schools should be doing.

[AlexFekken]

After reading the "evidence" somewhat carefully:

There is a difference between students being tested on skills they weren't taught and students claiming so. Any half-decent journalist would not report such a claim by itself but at least try to confirm with the teachers and report their response or lack of response. In other words, reporting such claims says more about the journalist than the curriculum.

Also, the teacher who was mentioned did not confirm the claim that students were tested on skills that had not been taught (just that "she and other maths teachers struggeld") and presumably she would have confirmed the claim if it was true.

Paul's list confirms that nothing fancy is needed to solve these problems. Perhaps this student (and certain teachers) are having the same problem as one of my former class mates. After weeks, if not months, of solving variuos instances of the quadratic equation a*x^2 + b*x + c = 0, he complained that he could not solve an equation of the type a*t^2 + b*t + c = 0 because he had never seen such an equation before. He didn't start crying or escalate it to the minister of education however...

[toml_12953]

(11-22-2017 08:50 AM)pier4r Wrote:  I never liked this argument:

"One student who studied for weeks in preparation for the exam said she was thrown by the difficulty of some of the questions, which tested skills she hadn’t been taught."

So either someone spoonfeeds the technique to you, or you are brainless. Although I can understand that finding innovative solutions may need more time, I really dislike when people act like robots.

Anyway thanks for sharing, the questions are interesting.

Some of the questions asked for proof. In order to prove something you use axioms. If the necessary axioms had never been taught, how is a student supposed to know how to prove something?

[Paul Dale]

I would say that given theorems are acceptable in a proof as well as axioms.

I've gone through the process of beginning with the Peano's set theory axioms and constructing the complex numbers (and beyond). However, I'd never start proving anything from these axioms.

Likewise, I'd never start from Euclid's axioms of geometry when attempting a proof. The list of knowledge required I gave above includes none of these axioms, however they all derive from them so a purely axiomatic approach is possible.


As a positive contribution, I'll reference a recent theorem in geometry that was proven in the late 1800s: Morley's theorem. This one stunned me when I first saw it.


Pauli

[pier4r]

(11-23-2017 04:02 AM)toml_12953 Wrote:  Some of the questions asked for proof. In order to prove something you use axioms.

No? Otherwise every new proof will be super long. The existing proofs accepted as valid can be used as building blocks for the next steps.

@Thomas Okken: yes, in my view a test should verify that one knows the curriculum and at the same time it may rank students in terms of "this student has mastered this better than the other".

Therefore in my view would be ok that until 90% of the maximum mark the test is based on "are you able to solve by yourself what was solved in class?". While the rest 10% of the marks are "are you able to solve by yourself this problem that we did not solve in class, but that you have the building blocks to solve it?"

Plus also the observation of AlexFekken is important, I have similar anectdotal evidence.


@Paul: thanks for the link.