## Saturday, May 10, 2014

### Bringing "childishness" into the mathematics classroom

Why would you show a Sesame Street video to a senior Year 12 mathematics classroom? Is there any value is doing paper folding exercises with such a class?  Here's my thoughts on the value of bringing "childishness" into every mathematics classroom - no matter the age group or the current achievement level, presented at a recent conference.

Games played by children and mathematicians
April 23, 2014
University of Sydney, Mathematics Education Alumni 2014 Conference.

Or just preview the presentation:
 Click on the image for the Google Drive document

## Sunday, March 9, 2014

### Changes

It's been a very long time between posts ...

 CC-BY-NC Gabriela Ruellan
A very big change has happened in my teaching career - I have become the head of a mathematics department. I have been extremely fortunate to move to a school with a terrific faculty, a very supportive executive team and the most amazing students you could ever hope to teach.

However even in the most ideal environment, the time pressures on a new head teacher are unbelievable and there is a very steep learning curve. As such, I'm putting this blog into a dormancy period.

Thank you for your readership in the last few years and hopefully I will be blogging again later in the year.

## Monday, October 7, 2013

### Fruit vectors: checking for understanding in the mechanics classroom

Here's a small idea I had while teaching mechanics that turned out to have very surprising and fruitful results in my classroom.

So our class has been working on mechanics for a few weeks now, we think we know the basics from a physical and a mathematical view. It's time to look at something harder now: circular motion.  But just before we do that, let's check we really understand.

Can you draw vectors, any sort you like, and tell a story about what is happening here?

.. and this one ....

It was surprising just how much discussion resulted from just these three diagrams, by asking students to draw velocity vectors, acceleration vectors and force vectors - and then tell a narrative in mathematical and in physical terms. Many misconceptions dealt with!

And now for the reward, setting things up for circular motion. What are the acceleration vectors (and hence the force vectors) for this picture?

By the time you have finished working out the vectors it's very clear what is required to move in a circle, even at a constant speed. What I love about this activity is that the students have developed an intuitive mathematical sense for what the result should be even before we start doing the detailed analysis to get the acceleration and force equations.

Resource: Here's a version of the diagrams I gave to students to scribble on:

## Sunday, September 15, 2013

### MANSW 2013 Presentation

A quick post for those wanting to see a copy of my presentation this morning at MANSW. Thanks to all who attended and gave such enthusiastic support - all the more so given it was 9AM on Sunday morning after a very late night conference dinner!

Three tools: The ABQuiz, the Tracking Sheet, the Feedback Form
September 15, 2013
MANSW 2013 Conference, Terrigal.

Links to the Feedback Form tools:
Feedback Form template (Word doc)
Feedback Form analysis (Excel spreadsheet)

Read it now in Scribd:

## Friday, July 12, 2013

### Video helpers in the mechanics classroom

This is the second post in a sequence about teaching the NSW (Australia) HSC Mathematics Extension 2 Mechanics topic. The first post looked at some initial challenges teaching mechanics and ways to use Felix Baumgartner's historic freefall jump in 2012.

How wonderful it is to be teaching in the age of the internet - being able to draw on the work of so many talented and inspirational teachers - and better yet, bring their insights and passion directly into your classroom to share with your students! Like hundreds of thousands of other people, I've been following the work of Derek Muller and his incredible Veritasium YouTube channel for some time, however it's only now that I'm planning lessons for a sequence on mechanics that I get to draw on his work for my mathematics classroom. As I designed my lesson sequence, I was stunned just how well the Veritasium videos fitted into my lesson design. Here are a few ways I think it's going to be a winner to have Derek in my classroom this term.

Misconceptions about falling objects
It's all too easy for students to agree with the statement that every object falls with acceleration g, but do they really believe it? The truth is they don't - not even some students who have studied physics at university. This engaging and challenging video will do the trick:

Introducing force concepts with an interesting problem : dropping a slinky
Choice: draw some boring static diagrams - or watch Derek's intriguing video about dropping a slinky?  No brainer! What I love about this sequence is the way it's designed for deeper teaching and learning: it's not just a passive "sit and watch" session - instead we are presented with an intriguing problem and challenged to decide on a response. I'm certain my students are going to respond enthusiastically - and provide me the perfect hook to introduce free body diagrams as a way to better understand the situation.

Then in the next video we get to watch what happens - and it is surprising!

And then extend the idea in several ways: What if we attached something to the slinky? What if we used a SUPER MASSIVE slinky?

As a side benefit, this video communicates positive messages to students about studying science at University. You'll get to do interesting work, and work with people like Rod Cross.

After the slinky videos, we'll take a look at this excellent discussion of reaction forces - which is also going to support understanding of how to work out free body diagrams:

The next video isn't Veritasium, but so powerful I have to share it. This high definition footage from a camera on the Space Shuttle booster rockets, tracking the rise and fall of the boosters is going to make for a exciting exploration of terminal velocity ( 2,900 mph at timestamp 5:15 down to 220 mph at 6:45)

Again - I want to inspire as well as educate. Maths and science is so much more than school work - it's an exciting and rewarding pursuit - with great career options.

So many videos to choose from. The challenge is to choose videos that serve both the needs of the teacher and the student - the video has to do much more than just entertain in order to justify taking time away from "the regular program". It has to serve the learning goals and promote specific outcomes - as well as being engaging and memorable - a gift that keeps on giving in the classroom.

What makes the Veritasium videos so good in the classroom? It's not just the sheer enthusiasm and fun of the presentations, or the fact they are short and sweet and fit nicely into a lesson segment, it's the fact Derek has grounded them in quality pedagogy. Presenting information in videos as a statement of facts turns out to have very little benefit - most definitely for science content, and quite possibly of limited value for mathematical understanding. If you're a fan of using video for teaching, definitely check out Derek's research work - you might be a little surprised at what he found:

Finally, back on topic, if you dare, this contradictory Veritasium video "Three Incorrect Laws of Motion" would be a wonderful basis for class discussion and perfectly demonstrates what makes a richer educational video - provided there is good support in the classroom (you better really understand the correct laws!).

So thanks Derek for your amazing work and generosity - and welcome into my classroom!

## Wednesday, July 3, 2013

### Freefalling into Extension 2 Mechanics

This post refers to the NSW (Australia) Mathematics Extension 2 course - the highest level mathematics taught in our high school system, but should hopefully have relevance for anyone teaching introductory mechanics.

My class is just beginning to explore the harder mechanics content in our senior mathematics course, exploring the equations of resisted motion, so it was with great delight I brought the exciting work of Felix Baumgartner into my classroom:

Here are some reflections on introducing this topic:

Did you lay a strong foundation right at the start of learning calculus? Coming from a physics/engineering background, I taught all the calculus concepts right from the start as a study of rates of change, using this as a motivation for the mathematics. My classes started with motion using a motion detector even before we looked at the concept of the derivative. When we studied the properties of the derivative, the meaning of increasing, decreasing and stationary points, it was all done in the context of motion (lots of roller coasters!). When we studied the second derivative, we asked the question "Why?" - and looked at how much of the physical world works on the second derivative rather than the first. So by the time we moved on to specific topics of applying calculus to the physical world, my students already had wide exposure to the link between the mathematics and the physics.

Did you explain why we have equations of motion that connect acceleration to displacement? One of the hardest parts of the earlier work in our course on motion is understanding and working with these equations:

$$a = v \frac{dv}{dx}, a = \frac{1}{2} \frac{ d(v^2)}{dx}$$

By showing that acceleration is caused by forces, and that in turn forces are often dependent on position rather than time gives the motivation for these more complex equations. It's an excellent opportunity to tap into students' current science knowledge on gravitational and electrical fields - and again reinforces Newton's Laws of Motion (this time the second law). For students with a little more physics knowledge, it's very interesting to link the second form of this equation to the equation for kinetic energy $$KE = \frac{1}{2}mv^2$$ using some integration.

You can't emphasise Newton's First Law often enough.  It's amazing how many studies show that while university physics students can describe and use Newton's First Law of Motion, deep down they remain firmly wedded to the Aristotelian world view. I make it a point to emphasis the First Law of Motion each and every time I start a problem - and I keep an eye out for anyone hesitating or wavering. If necessary, I repeat my stories about ice-skaters and about the Voyager spacecraft continuing on their journey even though they have run out of fuel.

More diagrams, more diagrams! It is sad to see how few diagrams are presented in most mathematics textbooks when presenting the theory of dynamics. While there are diagrams in most worked examples, they aren't explicit in the construction of the diagram, leaving it for students (and teachers) to try to interpret why the diagram was done that way. I spent most of the introductory lessons on this topic just drawing pictures.

"That's why it's called RESISTANCE sir!" My favourite quote from one of my students. We were drawing diagrams and I was trying to find a clear way to show that the resisting force always opposes the current velocity direction. I was saying the word "opposes" a few times when one of my student yelled this out - we won't forget that in a hurry!

"Bait-and-switch" constants.  Your students are probably used to the little bait-and-switch games we play with constants of integration. The same games are played with the constants used for resistance forces:
$$R = kv, R = mkv$$
Sadly many of our standard text books just switch on the fly between the two forms without explanation, adding in the mass whenever it's needed. It's very confusing for students (and teachers) when this is done so arbitrarily. And then it hit me: this is a totally legitimate game - we're just using a different constant to make our life easier. It sure would be nice to write:
$$R = k_1v, R = mk_2v$$
I shouldn't complain since I happily went along with the game when integrating!  This is yet another small but cumulative thing that makes this topic challenging. I think it's important to be explicit about this little game.
Update: As explained in Robin's comment below, this trick is a bit too clever - verging on not being legitimate - because it gives the false impression that the resistive force depends on the mass - it doesn't. The trick only works for a specific case of the mass in this problem.

Students have difficulty seeing that the physics has nothing to do with the coordinate system. And it's not their fault - our teaching and our text books rarely show how arbitrary coordinate systems are - we just happily keep changing them to our convenience, potentially confusing our students. I think it's critical to draw lots of vector diagrams without any coordinate systems, and then make a clear and obvious choice with the class that we can choose any coordinate system that works for us.  We had a deep-learning moment in my class last week when I unwittingly applied a different orientation of the axes than was in our text book - a great opportunity to highlight this issue.

Terminal velocity is a really fun concept. Students are absolutely fascinated with it - the physical understanding is interesting, and the mathematical development is revelatory. We had some great discussion on different terminal velocities for different situations and these led directly into a more rigorous discussion of Felix's jump.

Don't think that girls aren't interested in watching extreme sport events. My class of fifteen girls was were absolutely riveted watching Felix make his jump. They insisted on watching the full length 10 minute video, totally transfixed for the duration. I recommend reading the Wikipedia page on Felix's "Red Bull Stratos" jump with your students prior to watching the video - it provides an excellent opportunity to discuss the language of motion, examine the different stages of the jump and provides meaningful context for this thrilling event.

Hold your breath and enjoy the whole jump:

The next post in this series looks at the wonderful Veritasium You Tube resources available for teaching mechanics.

## Monday, June 10, 2013

### Still alive ...

It's been along time between posts - just a short note to regular readers to say I'm still breathing. It's been a very hectic year so far - teaching Year 12 Extension 2 mathematics for the first time (18 hours a week of very intense mathematics just with one class), and this term taking on the role of relieving head teacher. So not much time for reading blogs let alone writing them. Hopefully I will get back to blogging next term - lots of posts I would like to write based on recent experiences.

Some forthcoming topics I think readers may find interesting:
• Helping students (and parents) when faced with a poor result in a Year 9 half-yearly exam,
• The joy of the binomial gnomes - and the mystery of Yang Hoi's triangle,
• Teaching ideas for geometric series and inverse functions.