Hung-Hsi Wu articles

I was never all that sure about joining Twitter, but it’s paid off.  My most recent post was about a very interesting article I saw mentioned on Twitter.  A request to the relevant twit pointed me in the direction of heaps more by the same author.  Thanks, @dcox21!  I shan’t want for something to read for quite a while.

(Hung-Hsi Wu is a member of the Mathematics Department at the University of California, Berkeley.  The linked articles are about high school mathematics education.)

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The mathematics K-12 teachers need to know

From a tweet by @dcox21:

…then they will realize how misguided it really is to teach the addition of fractions using the LCM of the denominators. p62

The link is to a 70-page article by H. Wu (2008) called The Mathematics K-12 Teachers Need to Know.  I’m indebted to David Cox for pointing it out, and my skimming of it reveals much to be read and thought about as time goes on.  I was particularly interested in the alleged misguidedness of using the LCD (lowest common denominator) to add fractions.  Seems like a great method to me!  Anyway, on p. 62, Wu gives the example

\frac{x^2}{3x^4+x+2} + \frac{6}{x^2+5}

and notices that the addition is carried out using the formula

\frac{k}{l} + \frac{m}{n} = \frac{kn+ml}{ln}

Well, my senior students don’t have any great problem with fractions like that, and they use LCDs routinely to simplify algebraic fractions.  What’s more, they’ve never seen a formula like the klmn one above.  Adding fractions is a first-principle process of finding a common denominator (lowest or otherwise), modifying each fraction to have that denominator, then combining.  For hairy algebraic ones like

\frac{5}{x^2-4} - \frac{3}{x^2-4x+4}

it’s a relief to find a common denominator because it makes the overall working and answer simpler.

I’m not seeing why I shouldn’t teach Year 7 students to add numeric fractions by finding a common denominator, nor teach Year 8-11 students to add algebraic fractions in the same way.  It seems to me it’s a great opportunity to reinforce understanding with every example given.

One thought that occurs is that textbooks in NSW emphasise the addition of complex algebraic fractions with worthwhile LCDs, but perhaps textbooks elsewhere do not.  Or perhaps Mr Wu doesn’t think such emphasis is warranted.  (I could be pressed into agreeing with that…)  But as it stands, for building algebraic skills and reinforcing fractional understanding, I fail to see what I’m doing wrong.

Reminder: I see much of interest in the 70 pages Wu offers, including other aspects of the teaching of fractions.  I just happen to disagree with this point.  At the moment.

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Christmas pudding and ratios

I love Christmas pudding, and boiled up a nice little 100g number for dessert tonight.  The instructions contain cooking times for all three sizes available.  The sizes are 100g, 400g and 900g.  The cooking times for oven are 20, 35 and 45 minutes respectively.  The cooking times for boiling are 25, 45 and 80 minutes respectively.  This is clearly not a case of linear variation, as 100:900 does not equal 20:45.  Kids are likely to be at least a little surprised by that.  I can imagine all sorts of activities springing from this data, depending on the year group and purported topic.

Year 7 number plane. Plot the data on a number plane (determine the scale yourself!) and join the points with a curve.  Should the curve go through the origin?  Predict the required cooking time for other weights.

Year 10 ratio of volumes. This is the most obvious application.  Given that 100g must be boiled for 25 minutes (and given none of the other data), predict the required boiling time for the other weights and see if it matches the instructions.  Required insight/knowledge: volume is proportional to mass; the different puddings are very likely to be similarly shaped; cooking time is likely proportional to the width; the width varies with the cube root of the volume (and therefore the mass).  The specific language in that sentence is advances for Year 10, but the idea is simple enough.

Year 11 functions. As per Year 10 above, but plot the points, join it up and determine the equation of the curve (it “should” be y=k\sqrt[3]{x}, for some value of k, at least to my mind).  Or work out what the equation “should” be with nought but algebra, then plot it on a computer and see if it goes through all the points.

Year 8 ratios. Give the 900g cooking time and ask for a prediction for 100g.  When it’s (very) wrong, give the correct answer and talk about why it might be like that. Then solicit a cooking time for 400g. Observe that cooking time is not proportional to mass, but tell them it is proportional to something to do with mass, and see if they can guess what.  (Hint: it’s something on your calculator…) If possible, give an introduction to ratios of volumes of similar solids using simple diagrams, algebra and equations.

The Year 8 idea is the most handwavy, but there’s nothing wrong with that: it’s stretching the Year 8 syllabus beyond breaking point, so it depends entirely on the ability and interest of the class.

I look forward to using this example in class this year.

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Goodbye 2010; Hello 2011

2010 was a good year, my fifth year of teaching.  The year my first calculus class matriculated; my first experience of a very bright (indeed, accelerated) class; and I developed a very good rapport with several middle-year students who I’ll miss next year when they’re not in my class.  The first teaching day of 2011 is in 14 days.  I’ve had about six weeks’ holiday, enjoyed it, and not done a whole lot of schoolwork.  While I’m not aching to get back to it (the beginning of a new teaching year is always daunting), my mind is definitely starting to concentrate on what the new year may have in store.

I have been putting some serious effort into reorganising my desk: building a shelf, moving the computer, and, most importantly, getting rid of reams of paper.  I create and use a lot of paper resources for students, and when something is good, I don’t want to throw it away, but it’s hard to keep these things organised at the time due to the volume and slippery structure.  (Is a Year 9 algebra sheet more to do with Year 9 or algebra?  How to integrate third-party resources with my own?)  Digital resources have always been quite well organised, but they’re only part of the picture.  I’ve also committed many ideas and scraps of questions to email, Evernote, scraps of paper, text files, and so on.  With the experience of five years, it’s time to consolidate.

So I’ve gathered piles of paper that have been sitting in document wallets or binders and am gradually filtering and cataloging them into a Word document that spells out, year by year, topic by topic: what resources are particularly good for the topic; what approach I like to take to teaching that topic; particular lessons or activities that have worked, or not; ideas for future activities; ideas for future resources.  This feels like a very valuable thing to do, and something that I couldn’t have done any earlier.  I just hope I get it finished before the school year starts again.

Here is a small example:

Plan for teaching percentages (excerpt)

Plan for teaching percentages (excerpt)

That’s all I have for Percentages at the moment, so it’s very much a work in progress.  But the thoughts captured there represent an important idea that’s always occurred to me at the wrong time, which equates to “any time I’m not teaching Percentages”.  Having it in writing means I can review it at the appropriate time, and plan my lessons accordingly.

(Note: that Algebra heading is a placeholder.)

Taking time to review my notes before starting every topic is probably unrealistic, but ensuring my notes are good quality and tailored especially for me is the best way to encourage me to do it.

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Hello, World!

This is cos squared theta (CST), a journal of ideas and experiences in Mathematics teaching. Intended audience: myself in six months when I can’t quite remember that great idea I had or saw. Plus anyone else who is interested. We’ll see how we go.

I’m in Sydney, Australia and am one week into the Christmas holidays. So no teaching for a while, but I am hoping to get some serious work done these holidays, especially creating tests for my standards-based grading (SBG) initiative I’m planning to roll out in 2011. There’ll be more on that in the near future, for sure.

I also blog on matters of general interest at Midnight Rambler ( “nosedog” is an online handle I have used for years. It came from an Australian joke/story published on the Internet in its very early days. I don’t remember the joke now, but it was pretty funny. Anyway, it’s as well to keep Maths teaching and other matters separate.

I use Tumblr for general blogging because it’s shiny and new and has ideas I like, but WordPress has support for mathematical content.

To end this introductory post, here are two of my favourite mathematical xkcd cartoons:

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