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 ....

How about 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:

or get this from Google drive: Thinking about motion (free download)

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.

Google docs - free download:

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.

Sunday, March 17, 2013

How to get your math class SCREAMING

I know students are not supposed to be using mobile phones in class for private communications*, but I couldn't help but smile when one of my students showed me a text message she had just received from a friend in the class next door : "What are you guys doing in there? We can hear you screaming!"  

If you don't know who these five boys are,
 then you're definitely not teaching at a girls school.

So a little context first (context is everything!): This class doesn't really like maths that much - they tolerate it - I try my best to make it relevant and pleasant, trying to raise their confidence and skill levels. We've been studying a fairly dry topic for the last few weeks - they've done reasonably well in the topic test but need more practice. Looking for something engaging to make the second half of a long double period interesting, I turned to Stu Hasic's Quiz Boxes.

Download Stu's Quiz Boxes at

Quiz Boxes offers a Jeopardy! style game with questions of increasing complexity organised into categories, with a high stakes question at the end. Students love this game - and with careful planning and implementation (you will need to design the questions) it makes for a terrific fun period with high levels of engagement and gets students doing a lot more maths revision than they might have otherwise intended :-). There are many ways you can use Quiz Boxes so I would like to share an approach I have found that works well for classes of all levels of maths achievement. 

Quiz Design
  • Choose categories that students are interested in.  Current hot topics are "One Direction", "Justin Bieber", "Beyonce", "You Tube Hits" and "In the Movies".  Find whatever your class is interested in. Once they play the game, they will suggest topics to you. Since I don't know that much about One Direction, I go to Wikipedia and collect the factual information I need. Find some obscure information for the harder questions. Your students will be amazed you know something so detailed about One Direction - and infuriated most of them don't know it.  I like to use student interests for half the categories, and use more explicit math topic categories for the rest.
  • Work maths into the "non-maths" categories. For example, my third question on One Direction was "What percentage of One Direction are boys?". OK - it is a simple question - but it reinforces the idea that 100% means "all". One question I found generated interesting responses was "How many records has Beyonce sold?" - which gave a good opportunity to explore estimation. Another One Direction question: What is the name of the band member who is last in alphabetic order?"  Again - it's easy, but it gets some mathematical thinking happening.
  • Make the maths category questions easy at the start You want students to engage with the maths categories. I always start with easy questions - if you make them too hard, students will turn off - it's not a game any more. I save the harder questions for the 800 and 1000 point questions. I make the end-game question a more challenging - but doable - math question on the current topic.

The Quiz editor in Stu's Quiz Boxes.
 I find I can reuse the quizzes across many grade levels,
 making this an efficient use of lesson preparation time.

Playing the game
This game is so much fun, and the students get so excited, it's essential to have a management strategy.
  • Every group gets a chance to answer the question. This is perhaps the biggest change I make to playing the game: I don't have a "first-answer-wins" approach. In a classroom of 30 students, it's impossible to work out who gave the first answer and the noise levels are impossible if you go this way. Instead every group has a mini whiteboard to write their answer (you could use just a sheet of paper).  Once I see a group has a quality answer (doesn't have to be correct - just interesting), I yell "2 minutes" and give all the other groups time to complete. When I call "time up", we look at all the answers and every group that has a correct answer gets the points.
  • Encourage group checking of answers. As the questions get harder and are worth more points, I ask each group to ensure everyone agrees on the answer before presenting it. This gives the group a chance to teach the content to each other. It's wonderful to see students try to convince each other their answer to a maths question is correct.
  • Noise level management. This is hard because it's so exciting. Never have you seen a class so interested in knowing what 8% of $200 is! As the noise level rises you'll have to calm the class down.
  • Prizes. I confess to motivating with a very small chocolate prize. I give one to every student at the end and don't buy into "but we won...." discussions - as far as I'm concerned everyone is a winner if they participated :-)  Waving the packet at the start of the game gets their attention - but it's amazing how quickly the students forget about the chocolate and become obsessed with winning game points.

Special thanks to Stu Hasic who so kindly donated Quiz Boxes to the education community.  I highly recommend you try Quiz Boxes with your classes. And over time you will develop a bank of quizzes which you can share with other teachers in your faculty - or maybe even at Stu's website.

Here's what you need:
  • A data projector (or an Interactive White Board)
  • A copy of Quiz Boxes - free download from Stu's web site
  • A pre-prepared quiz. It can take a good hour to design a quiz, but you will find you can reuse quizzes across many year levels and they stay current for several years.  You might like to challenge your class to design quiz questions for a category - although this will take some time and planning.
  • Students arranged in groups - maximum six groups for Quiz Boxes.
  • Mini-whiteboards OR a pad of paper per group.
  • Solid walls between you and the classroom next door. Close your windows :-)

Sunday, February 3, 2013

Getting the most out of graphing software

"GeoGebra is your friend!" - my students must have heard me say it a hundred times.  If a student asks me about a homework question, they know my immediate response : "Did you check what it looked like in GeoGebra?". If they haven't, then I will usually ask them to sit with me while we explore it together using the software.

Some teachers worry using mathematics software will weaken student's skills, but here's a mantra I recite in class which I believe not only develops mathematical skills but also stimulates deeper learning:

I believe the essential ingredient in using graphing software to answer questions is to stop and think before using the software and then predict what you expect the software to display. If you are fortunate, you'll find the software doesn't match your prediction. I say fortunate because you have discovered a misconception, an error - or in some cases, managed to confuse the software. Prediction and the subsequent reveal of an incorrect prediction is a powerful learning tool.  With a positive attitude to the error monster this revelation will stimulate questions and further exploration.

Another key learning idea I advocate is to take a few extra minutes once you have your answer to extend the problem with some "what if?" questions: "What if I changed that positive x to a negative x? What if that was to the power 3, not power 2? What if that parameter was 4 not 5? Can I reflect that curve?" Here the power of the software comes to the fore: we can ask many questions and rapidly get answers - something not possible in reasonable time without the software. Of course students won't have the time to do this for every question, but even just doing this once in a study session is rewarding.

One more powerful pedagogical factor is at work when students use a graphing tool to help with their homework: they are forced to translate their problem into a representation suitable for the tool. For example, an algebraic equation has to be split into two (or more) graphs and intersections found. This serves to build and reinforce understanding of the links between the different forms of mathematical representation. Often a student needs break down the problem into steps, introducing parameters and intermediate results or constructions, providing 'hooks' they can use to explore how the problem changes as parameters are changed. 

A topic I recently taught was based totally on drawing graphs by hand - and students have to be able to do this in an exam situation, without software.  For a course like this, I think the graphing software is an even more valuable learning tool. Why check your answers in the back of the book when you can do this:

This approach means students are still learning to work by hand - and maximising the benefits of having software during the learning of the topic - without becoming dependent on it - a bad thing at exam time!

So to my way of thinking, there's no question dynamic geometry software is a powerful learning tool: when coupled with a mindset that thinks and predicts prior to using the software, and then extends a problem through questioning and exploration with the software - it's like having a personal tutor. GeoGebra is indeed your friend!

Practicalities: There's lots of good quality dynamic geometry and algebra software available to students: I'm a big GeoGebra fan, and I also like the Desmos tool. I'm beginning to really appreciate AutoGraph - but sadly the cost factor rules it out for most of my students.  For intensive algebraic work, I point my students at WolframAlpha - especially the WolframAlpha iPad app which is great value.