We were alerted to this one by a friend currently suffering through a teaching degree. It is a word problem courtesy of AITLS, the guys who set the standards of how teachers should be professional.
AITSL’s proficiency standard 2.5 appears as follows:
In support of this proficiency standard, AITSL has a page, Working with Mathematical problems, the main content of which is a video (below). The page and the video (and the “resource pack“, which in turn links to the old AC content ACMNA057 and ACMNA082) could be a meaty WitCH.* But, let’s focus.
The video is primarily concerned with a word problem presented to a Year 3-4 class (containing a substantial number of indigenous and ESL students). Here is the problem:
I picked 36 apples off my apple trees, there was an even amount on each tree. What were the possible combinations of apples on the trees?
*) I am very much not attacking the teacher here. AITSL, on the other hand, will be first against the wall when the revolution comes.
**) Well, they’ll be somewhere in the single digits.**
I am happy to be guided by professional people. Are we told how many apples were on the trees before the picking therefrom ?
So, did we pick all apples off all the trees, or just the ones that were ripe for picking…?
Beyond the obvious, the choice of words here is quite awful. “Amount” instead of simply “number” is one example.
That’s not the worst of the word choices.
It is but one example. Ironic, given this is a PoSWW…
I could have easily made it a WitCH.
Indeed. I meant ironic that the wording is the source of many issues, given this is in the category “without words”.
Just another case of accidental humor. Which in some ways makes it more funny.
Ah, yes. Very funny. It really is PoS(With)W.
“What were the possible combinations of apples on the trees?”
A few Royal Galas, handful of Pink Ladys, and the rest Granny Smiths.
The statement “there was an even amount on each tree (presumably before picking some)” does not imply that only an even number of apples was picked from each tree unless none of the trees had any apples left on them afterwards.
If they want to teach critical thinking to students, I would argue that this is more likely to achieve the opposite. Any student who may have been rightfully confused about this context is basically told that this question makes perfect sense and that they should just be able to go on with the job and solve something that is inherently unsolvable because there is no reason to assume that there were only 36 apples on the trees before picking.
Also, the statement “What were the possible combinations of apples on the trees?” is likely to cause misconceptions about the meaning of the word “combinations”. “What were the possible number of apples on the trees?” would have been better provided they had clearly stated that no apples were left on the trees after picking.
My two cents
Thanks, MB. A very good “two cents”.
What does the expression “even amount of apples” mean?
Thank you, Marty.
As far as I am aware, “amount of” suggestes a continuous variable which could be quantified in, for example, lots of 100 grams of apples. However, in this context I would be inclined to accept the use of “amount” as a synonym of “number” given that it is used in conjunction with “apples” which suggests an integer quantity.
English is not my first language though, so maybe that conclusion is flawed. What is your take on this matter?
Thanks, MB. Yes, as RF pointed out, “amount of” seems to be a laxy version of “number”.
What about “even”?
Splitting an amount “evenly” means everyone gets the same amount (without any remainder). So, looking at it from that perspective, all trees would have had the same number of apples, and it wouldn’t say anything about whether that was an odd or “even” number.
That’s a nice one and I had totally missed that as a possible interpretation. Thank you for the nudge.
Thanks, MB. How did you first interpret it?
I had jumped to the conclusion that the wording implied that each tree had a number of apples on it (before picking) that was a multiple of two.
That seems like a reasonable jump to me.
P
If the author did, indeed, mean ‘amount’ and not number’, then there could be an infinite number of possible combinations. I could assume that ‘even’ meant ‘equal’, but that, too, would lead to an infinite number of combinations.
If the author meant ‘number’, there would still be numerous, possible combinations. Are we to assume that all the trees had the same number of apples on them, and now have zero apples on them, or each tree could have just had an even number of apples. We have to make so many assumptions for this question, I can only conclude that it is a total stuff up.
The way I interpreted it when I first read it was, how many ways are there to add positive even numbers to get to 36? Isn’t this a partitioning problem or something?
Edit: though looking at the question now, I don’t think my interpretation makes any sense in the context of the question, though maybe neither does the question.
That’s exactly what I thought, and it is exactly what my degree-suffering friend thought.
So, the video suggests using Newman’s Error Analysis to identify why students struggle with mathematical word problems. Unfortunately, the first step is not ‘make sure your stupid question actually makes sense’
It is wild that this question, as written, was given to *any* student, let alone students who most likely struggle with general literacy in the first place. I still have no idea what the question is actually asking of me!
It is even wilder that AITSL considers this thing an exemplar.
I really don’t want to beat up on the teacher. It appears that she is teaching in a tough environment and she is undoubtedly doing her best in very trying circumstances. But she is being led by idiots, as are all teachers.
Oh, absolutelyThe blurb on the video says that an Assistant Principal suggested using Newman’s, which is unsurprising and even expected
The teacher has a hell of a job with primarily Indigienous/ESL students. I imagine that the other Year 3/4 classes at the school aren’t in a similar boat – too many schools love to just group kids together like that
Thanks, Fox. I didn’t mean to suggest you were whacking the teacher. I just feel guilty when attacking such crap, when attacking cretinous organisations like AITSL and ACARA and VCAA, collaterally places some innocent and well-meaning teacher under the spotlight.
An absolute classic! You can just imagine AITSL’s thinking. Right – we’ll tick the indigenous correct line box and do some lovely virtue signalling, (without bothering to check whether this really was appropriate for kids for whom English probably is very much a second language). But then we won’t worry about the content because we’ve done the virtue signalling! But we will slip in a Newman’s – whatever it is, just to show how professionally these things are done…
I hammer the message constantly home that maths is essentially the opposite of English – it is UNambiguous. That’s part of teaching kids to think mathematically. One of the reasons word problems are so hard it that we are making the ambiguous interact with the unambiguous! In fact I just answered a Year 12 student on Friday about this! So to create an incredibly linguistically complex problem for such an audience and THEN to add in ambiguity in the word even is just silly.
Not the teacher’s fault – my guess is she’s doing an amazing job – definitely the fault of the people who should have known better. I think this is definitely worth a Dame Slap award!
This post probably needs a literacy tag…
Use of “even” instead of “equal”, use of “amount” instead of “number”, unnecessary use of “combinations”, poorly (not at all) defined parameters for the question…
As a “numeracy” question, this is poor at best. As a “literacy” question, exploring how people attempt to communicate information, this question may have some merit.
The incorrect use of “even” is inexcusable. It is bad enough when students are not taught appropriate vocabulary in mathematics . What are we to do if they’re being taught wrong vocabulary?
Thank you, Wilba, and tag added.
There are 385 possible ways to partition 36 with only even numbers… Assuming we pick all apples from all trees. It doesn’t specify that each tree had the same number of apples on it. Hardly a grade 3/4 question. This is embarrassing for AITSL to use as an exemplar. The teacher needs some proper guidance from someone Maths trained.
Thanks, Bryce. That’s how I read the question but, watching the video, it is clearly not what was intended. I have little doubt that that the teacher could use some maths-trained guidance, but it’s not the main issue here.