“Level 1 maths students would have had to know the formula for quadratic equations to answer one of the questions in yesterday’s exam – a formula that maths teachers say is not taught until Level 2.

The exam left some students in tears and has sparked an open letter by maths teachers to the NZ Qualifications Authority complaining that the exam was too hard for Level 1 of the National Certificate of Educational Achievement (NCEA). The *Herald* has been inundated with complaints. However, NZQA is standing by the test. Deputy chief executive Kristine Kilkelly said the authority was still “*confident in the quality of the Level 1 Mathematics examination*“. Here is the question in question:

1. **Don’t know my times tables** (or have them handy). Yes, I know that I have calculators on my devices, but my brain is a quicker one and I need to use it more often – it gets me through the work quicker, helps my confidence and also exercises my thinking muscles! Perhaps if I really tried to learn my tables and math formulae I might prevent an early case of alzheimer’s??

2. I **didn’t read the question** properly (use highlighter to show each step required).

3. I **forgot to plug my answer back into the question** to see if it actually worked!!

4. Darn it – I **hurried the working** and forgot to write down “x = answer” and I **left out the units required** (e.g. cms or ft). DOH!

5. Heck, I thought I would remember all this math stuff so I **forgot to write any notes** on the topic the teacher was talking about. Then I **never spent any time looking through the textbook** (and copying the worked examples that I needed) and was **too busy on social media** to check out the online resources and similar problems found on (e.g.) khan academy!

6. **Sport is toooo important for me** to miss, so **I skipped checking over the answers** and making corrections. By the time I got home it was too late to do any math.

7. Someone (I forgot who) said that it is impossible to revise for math exams. Of course, they must be right?

8. “*If at first you don’t succeed, give up*.” is a good motto? Well, perhaps **I should ask for help** from the teacher or TA when I need it – it might actually improve my understanding of new math concepts and how to use the formulas correctly after all.

9. Always **sit next to someone who will disturb my working**! NOPE – a DUMB idea!! I say, “sorry but I have to work on this alone now, but will see you after class.” They respect my attempts to do well in THE, repeat THE most important subject for my future income and career opportunities.

10. So, make friends with those kids who do well in Mathematics – you need them! Otherwise, use the oldest math excuse in the universe – “**My dog ate it!**“

David Joyce, Professor of Mathematics at Clark University makes the distinction like this – “Mathematics studies numbers, geometry, and form. Those require definitions and axioms to characterize the subjects of study. Logic is used to prove properties based on those definitions and axioms. Logic can also be used in ordinary discourse of non-mathematical topics. Typically, the logic needed for mathematical investigations is great while that for non-mathematical investigations is small.”

Try this puzzle for example – it is based on pure logic, rather than Mathematics. Which if the following cups will fill with coffee first (answer will appear on the post support column soon)?

]]>*Find the total volume of a rectangular prism with base side lengths of 6 cm and whose height is 8 cm.*

Knowing the formula is V=BxH, students will often solve this by simply multiplying 6×8. Even if they multiply correctly, this shows that they do not understand what volume is measuring, or that they have to find the *area of the base*. Therefore, I prefer to teach “Volume = any end area x length.” But first, use the following steps to work towards a solution (highlight key terms–>find shape (or equation, etc.)–>put in values–>use formula (or equations steps) –> solution (add units if required). This is what it might look like for the earlier volume problem:

Hint: When you have the correct shape (Google it if not sure) and have added the dimensions, shade the ‘end area’ that has two measurements for finding the area, then times by the height (or length). Your diagram with shaded end might look like this:

Try this: Assuming your classroom is also a rectangular prism (aka ‘box’) you can find the volume by using the area of one end and ‘pushing it through’ the length of the room. That is, ‘End Area x Length.’ Or, you could find the area of the ceiling and ‘drop it down’ the height of the room – again this is ‘End Area x Height.’ So, the rule of ‘* End Area x Length*‘ is a really good formula to find volumes.

This image shows a student’s working. What went wrong?

Well, obviously x should = 17, not the square root of 17. Perhaps a better way to explain the error is like this:

When solving equations (expressions with an equal sign) like this, remember the algebraic rule:

**“Whatever you do to one side, do to the other!”**

So, here the student undid the sq. root on the LHS but did not undo it on the RHS. So, there should be no root sign on the RHS. “If you sq. root one side, sq. root the other too! (from mathmistakes.org)

]]>A great comment although, as it turned out, they were going up the wrong road, so I wonder how large the ‘big number’ was?

]]>[As an aside, I have figured things out so often in my sleep that it’s become my preferred way of working on problems. I often wonder if there’s a “math part” of my brain which I don’t have normal access to but which furiously works on questions during the night. That is, if I’ve spent the requisite time during the day trying to figure it out. In any case, when it works, I wake up the next morning just simply knowing the proof and it actually seems obvious. It’s just like magic.]

So here’s my advice to you, high school kid(s). *Ignore your surroundings, ignore the math competitions, and especially ignore the annoying kids who care about doing fast math.* They will slowly recede as you go to college and as high school algebra gives way to college algebra and then Galois Theory. As the math gets awesomer, the speed gets slower.” Great words!

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So read the headline (source here), followed by this brief explanation: “ANZ senior economist Sharon Zollner noted the New Zealand dollar had taken a pounding, post-election, but shot up from US68.49c before the data’s release to a peak of US69.14c on the back of the data, later settling back to about US69c.” Wow, it sounds like a dramatic statistical change, especially when supported by this impressive graph:

But, discerning mathematicians, check out the actual change. The dollar “shot up from 68.49c to 69.14 before finishing at 69c” a change of only about half a cent – hardly a spike? Of course, the graph has a scale that is vertically exaggerated to a considerable degree. In fact, a rise in the Kiwi to US exchange rate of just a few cents would not fit on the graph at all.

This is a great example of “fake statistical news” or, at least, poor use of data and graphs to “sell” a weak story. Of course, it would not get past the eagle eyes of our junior mathematicians, would it?

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