What’s my function? Derivative game to deepen learning

At the recent Educational Changemakers 2016 conference (EC16)  I came across the work of Shane Loader and the Empowering local learners project. After an engaging chat at the post-conference drinks, Shane was generous enough to share with me some of the top blogs that he follows as well as a link to the Empowering Local Learners (ELL) project that he’s currently working on. Whilst browsing the site I found a great lesson plan on surface area and volume that gave me an idea for my Methods Class. The activity: ‘What’s my function?’.

The previous few lessons had been on introducing the derivative function. Students were able to calculate the derivative of a polynomial both from first principals and ‘the rule’. The goal of this lesson was to scaffold students to make the link between the zeroes of a derivative function and the turning points of its parent/primitive function.

I split my (luxuriously small) class into two groups and gave each group one of the following instruction sheets.

more students using desmos to check their functions and derivativesStudents were excited! They were straight onto Desmos to explore their functions


Students using desmos to check their functions and derivatives

 

 

 

 

 

 

 

 

 

 

 

 

Then they jumped into asking their questions. Here’s how the discussion went…

first functions and derivatives questions and answersThe students were taking the activity really seriously, and being very competitive. Due their deep consideration of the questions they were going to ask, they were really taking their time, and we were fast coming up to the end of the period. With about 15-20 minutes to go I took advantage of my teacher privilege with the following change to the rules (I recorded the following in class).

Here’s he chat that followed… (The team on the left knew their original fn was a cubic, and the other knew they were looking for a quartic).

function and derivative questions and answers It can be seen that the team on the left got the hint and jumped into asking for the zeroes of the derivative to find the turning point. The team on the right tried a similar thing, but only on their second turn, and they were playing copy-cat so didn’t quite know why they were asking the question. Results? Here’s what was produced…

What's my function, results By this time we were up against the end of the period so I took a photo of the fns to prompt discussion at the start of the following lesson.

In the following period, I asked one basic question of the team who drew the fn on the bottom right: “Why did the other team ask for the zeroes of your derivative function?”. A long discussion ensued but we managed to boil it down to the following spaced repetition card…

Screen Shot 2016-09-15 at 7.44.41 pmI guess I could have just told them that at the start, but that would have taken out the joy and excitement of making the connection themselves*. We’ll see if it helped it stick… term break starts tomorrow!

 

*Many readers will already be familiar with Paul Lockhart’s ‘A Mathematician’s Lament‘ on this point.

Top 9 Math Question Websites and Bloggers for deeper thinking (Shane Loader)

A big thanks to Shane Loader for the following recommendations.

Shane uses resources from the following sites to promote “problem solving and reasoning skills and the flexible use of mathematical knowledge in students”.

Enjoy! I’ve found them incredible!

The resources:

Which one doesn’t belong                           http://wodb.ca/index.html

Open Middle                                                     http://www.openmiddle.com/

Visual Patterns                                                  http://www.visualpatterns.org/

Robert Kaplinsky’s Tasks                               http://robertkaplinsky.com/lessons/

Andrew Stadel’s Three act tasks               https://docs.google.com/spreadsheet/ccc?key=0AkLk45wwjYBudG9LeXRad0lHM0E0VFRyOEtRckVvM1E#gid=0

Dan Meyer’s three act tasks                       https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit?toomany=true&pli=1#gid=0

Yummy Math                                                     http://www.yummymath.com/

Estimation 180                                                   http://www.estimation180.com/

Would you rather                                            http://www.wouldyourathermath.com/

Maths Card Games to Promote Multiplicative Thinking

One of the most powerful lessons that’s emerging out of my current Masters project on multiplicative thinking is on the power of maths card games in the classroom.

The power of games jumped out to me for two particular reasons; one relating to spacing repetition, and the other relating to formative assessment.

It’s perhaps not surprising that playing games has proved a fun and engaging way to incorporate spaced repetition of traditionally laborious number facts (like doubles of the numbers to 10, complements of 10, and doubling strategies) into sessions with students. But what definitely came as a surprise to me was the notion of using student boredom as a formative assessment tool.

Students would come in at the start of our sessions and ask me, session after session ‘Sir, can we play double trouble bingo* again?’. I was shocked by how many times the students were authentically excited to play the same game. It took me a while to realise that what was happening was that they were still challenged by the maths on which the game was based, and as a result, were still keen to play. With all of the games there came a day when the students would suddenly say ‘this is boring’. Why such an abrupt change?

Original Image Source: Pixbay

Original Image Source: Pixbay

The answer came when I serendipitously had a campfire-side chat with an early childhood educator and, whilst discussing learning through play, she spoke of the analogous situation in her learning space. “Yeah”, she said, “They’ll play with a toy and they’ll struggle with it, then they’ll solve it. They might solve it a few times, but then they’ll leave and go to another toy. That’s when we know they’re ready for us to make it a bit harder”… BINGO! What a revelation!!! (Well, for me anyway). Turns out we can trust students to know when they’re ready for the next challenge. #studentvoice!

As my masters project wraps up, and I think of taking some of the techniques and strategies out of the small group setting and into the classroom, I’m trying to brainstorm ways to bring the power of these games into the mix.

What I’ve shared below is the three games that I’ll be trialling with my VCAL numeracy class tomorrow. I’ve created a quick worksheet as a warm up to each of the games (and as an aid to help the students select the right game for them) and I’ve bought a bunch of packs of cards to get us started. The idea of the videos is not only to share with other teachers some of these games but also as a (non literacy-dependent) reference for students to check if they’re a bit hazy on the rules. I’ve got a whole heap of other games that also powerfully support multiplicative thinking and I hope to eventually compile them all into a nice concise resource. But for now, here’s a taste whilst I experiment with what actually works at the chalkface.

Big thanks to my buddy Holly for helping me with the vids!!! 

A note on modelling and mathematical language: Holly and I have tried to model the kind of mathematical thinking (and talking) that we hope to promote in students. In addition to explaining our thinking out loud, doing our own calculations and keeping our own scores is a really important part of the game play to ensure that everyone learns as much as possible through playing.

Click here for the auto-generating worksheets that complement these games

*I’ll upload my ‘double trouble bingo’ slideshows soon.

Complements of 10 memory

Quickly recognising the complements of 10 (2 and 8, 7 and 3, 6 and 4, etc) is an imperative skill for students addition and subtraction abilities and is a key  element of part-part-whole understanding (students recognising that numbers are made up of other numbers!). I developed this game by making a simple change to the age old ‘memory’ game that I used to play with cards as a kid. (ps: Another game to easily be modified to work with complements of 10 is the game of ‘snap’. Depends on how much you value your cards though as this game will definitely reduce their longevity).

Stop or Dare

This game builds on part-part-whole understanding and place value understanding (e.g., to add a 10 we just increase the number in the 10s column by 1) to help build students’ addition strategies. I can’t claim this game, I came across it through the people behind MrMooreMaths on youtube. This was one of the students’ faves and, as I mention in the video, it can be made more complex by subtracting or adding, or even adding or subtracting doubles to or from 200 (could do triples or quadruples too, the possibilities are endless).

Picking Percentages

Holly and I designed this one together a couple of days ago. I feel it’s a really powerful way to help students to link fractions to percentages and develop some proportional thinking (the fact that the percentage changes as the number of the cards in the round changes has the potential to be a real mind bender!). It’s only been played by 2 people in the world so far so I feel it’s got some tweaking to go. In particular I want to change the scoring system to more of a progressive adding one, this will help build the tension in the game. I’ll try it out in the classroom and we’ll see how we go. Again, this game could be made more complex by introducing more rounds such as 6, 8, 9 and 20 card rounds.

If you have any suggestions on how to improve these games, or any suggestions on other games I’d love for you to hit me up on twitter. @ollie_lovell : )

What others are playing. Some nuggets from twitter…

 

How to pronounce Mandarin Chinese in under 15 minutes.

In late 2013 I set myself a challenge of getting conversational in Mandarin Chinese in one year. One of the biggest barriers to me reaching this goal was the pronunciation of the language, as it has many foreign sounds to a native English speaker.

Throughout my journey I came across a whole host of incredible resources, from the website Hacking Chinese to the Glossika approach, that really helped me to overcome this challenge. Over summer I sat down with my brother (who is now also learning Mandarin) to try to give him a bit of a crash course on what I had garnered from these resources. Please find a video of that lesson below with the associated document and links embedded below that. I hope that Mandarin learners of all stages find this a helpful article.

An additional tip that I didn’t mention in the video is the benefit of recording yourself and playing it back to check how you’re tracking with your pronunciation. I can’t recommend this approach highly enough. Good luck!

WHY CAN’T THEY REMEMBER THIS FROM LAST YEAR??? Help students remember key information: Spaced Repetition Software (SRS).

This post is one of a series detailing my current mathematics lesson rhythm and routine. This one outlines how I use spaced repetition software (SRS) at the start of my lessons to help students to remember key information. There is a video of me teaching with SRS at the bottom of this post. 

Thinking back to my own time at school, I distinctly remember one challenge in particular. I remember feeling that studying  mathematics in discrete topics (or units), made it really hard for me to remember the relevant concepts when it was time to revisit that branch of mathematics again, sometimes over a year later.

Through my post-schooling forays into language learning in particular, I have come across some research backing up those schoolboy intuitions.

What I was feeling was the effects of a cognitive phenomena called the ‘forgetting curve’ (Ebbinghaus, 1913). The forgetting curve (pictured below) is a graph that approximates the rate at which an individual will forget a given unit of information.

the forgetting curve

(image source: https://www.flickr.com/photos/suzymushu/3411344554)

In the late 1800s, a German chap by the name of Hermann Ebbinghaus constructed the first forgetting curve by trying to memorise nonsense syllables (such as “WID” and “ZOF”) and then testing himself at regular intervals, rating his level of accuracy, then plotting these points out on a graph.

Hermann Ebbinghaus

(Old mate Ebbinghaus: https://commons.wikimedia.org/wiki/File:Ebbinghaus2.jpg)

As can be seen in the picture of the forgetting curve, if we want to remember something, we need to be reminded about it at regular intervals*. The good news is that the more times we’re reminded about it, the longer the interval until we need to be reminded about it again!

*(The necessity of reviewing a unit of information at regular intervals is obviously dependent on what the unit of info is, and how it relates to your prior knowledge/how emotionally charged that memory is. For example, It’s highly unlikely you’ll ever forget your first kiss! Ebbinghaus’ original forgetting curve is, however, a great approximation for units of info like; words in a foreign language, or even terms such as ‘perimeter’ or ‘circumference’.)

Such a curve has important implications for teaching and learning. If we want a student to remember the basics of trigonometry when we come around to the topic again a year later (e.g., basic terminology, sum of the angles in a triangle, etc), we had better ensure that several times between now (time of teaching) and next year, they get reminders at key intervals.

The basic idea underlying this reminding-at-intervals is the spacing of repetition. We all know that it isn’t a good idea to cram your study, but a recent meta-analysis of studies, Carpenter, Cepeda, Rohrer, Kang and Paschler (2012) brought together research on the actual effectiveness of spacing repetition. The following excerpt details the results from just one of the studies that they cited in their meta-analysis.

the benefit of spacing repetition

(Carpenter et al., 2012, p. 371)

This is all well and good as a concept, but how can we do it in practice? There are literally hundreds of new words and concepts that a student is expected to grasp in a year, is it realistic for a teacher to keep track of each of these terms and ideas, and remind students of all of them at periodic intervals?

I’m hoping that the answer is yes.

In 2014 I set myself the challenge to learn Mandarin Chinese in a year. As I delved deeper and deeper into effective learning methods,  I came across spaced repetition software (SRS). SRS is a program of digital flash cards (you can make them yourself, or download pre-made decks) that, based on self-ratings, uses an algorithm to calculate the optimum time to review each given unit of information. It is essentially plotting your forgetting curve and reminding you of that piece of information just before you forget!

This software has been notably used to great success by such polyglots as Scott Young (who learnt 4 languages to a very high standard in one year) and Benny Lewis (very famous polyglot). It definitely helped me, and with the help of the SRS program that I use, Anki, I was able to  reach my goal and achieve a conversational level of Mandarin within a year. These days I use it to remember a whole host of things; from people’s names, to new english words, to the countries of the world. I currently have a little over 3000 digital flash cards in my review ‘circulation’ and to keep on top of all this info it only takes between 10 to 15 minutes of my time per day. Here’s a snapshot of my study statistics from the last month.

Anki statistics

(my personal spaced repetition data from the past month)

I was really keen to bring this incredibly powerful tool into the classroom to try to help my students to overcome the memory challenges that I, myself, faced as a student. So I did!

Since I started teaching at the start of this year, I’ve been using an SRS program (Anki)  in all of my classes. We use it at the start of every lesson and I call students’ names with the use of coloured pop-sticks, a method that I’ve written about previously.

The result?

It’s hard to comment on the long term effects as it’s still early days, but student feedback has been good, for example: On the end of Term 1 feedback form that I handed out to students, many of them made comments such as the following:

But hey, I thought that the most helpful thing would be to give readers some eyes into my classroom to see exactly how it plays out. With my students’ permission, I’m sharing below a clip from my  VCAL (Victorian Certificate of Applied Learning) numeracy class. Just for a bit of context, VCAL is a program designed for students who are planning to explore post-secondary pathways into vocational training. I have students who want to be nurses, flight attendants, and many of them aspire to the a position in the military. What you see below is a classic beginning of lesson episode. One of the students (Sharnee) is in charge of the pop-sticks, pulling out student names, and the other students are sitting (with varying degrees of focus), considering what their answer would be, then answering if their name is called up. I’ve found that the students enjoy the routine and it adds a game show like feel to the start of the class. Hopefully this little clip gives you a bit of a glimpse into how Anki works, and how I feel it can help my students to overcome one of the challenges that I myself faced in school.

References:

Carpenter, S. K., Cepeda, N. J., Rohrer, D., Kang, S. H. K., & Pashler, H. (2012). Using Spacing to Enhance Diverse Forms of Learning: Review of Recent Research and Implications for Instruction. Educational Psychology Review, 24(3), 369–378. http://doi.org/10.1007/s10648-012-9205-z

Ebbinghaus, H. (1913). Memory: A contribution to experimental psychology, (3).

 

 

A Problem Solving Spell!

I’m super excited about sharing this ‘Problem Solving Spell’. I modelled it on the problem solving template outlined in Charles & Lester (1984, p. 20), a pic of which I’ve also included at the bottom of this post.

Please feel free to use as you see fit, I’ve included the text below the pic incase you’d like to format in some other way.

Problem Solving Spell

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Text:

A Problem Solving Spell
In a test or in a lesson
Come across a tricky question
No need to worry, fret, or doubt
This solving spell will help you out

Read the question carefully
Which words or phrases might be key?
Do you understand ok?
Now write it in another way

List what you already know
This gives us hints of where to go
Draw a pic, or act it out
Visualise what it’s about

List your info in a table
Spot a pattern, if you’re able
We now are only half way through
The many tricks that you can use

Struggling to make progress?
Working backwards could be best
Make the question simpler
Or solve a problem similar

But sometimes there will come a time
You’ve thought, and thunk, and tried and tried
Eventually say, ‘what the heck!’
And have a crack at guess and check!

You’ve found an answer, good for you!
And showed your working right way through
Before you’re feeling too spellbound
CHECK THE ANSWER that you’ve found!!!

Basis for the ‘spell’ (Charles & Lester,1984, p. 20)

Screen Shot 2016-04-24 at 8.33.25 am

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Reference:

Charles, R. I., & Lester Jr, F. K. (1984). An evaluation of a process-oriented instructional program in mathematical problem solving in grades 5 and 7.Journal for Research in Mathematics Education, 15-34.

Extending linear arithmetic blocks (LAB) to a circle’s diameter:circumference relationship

In the quest to make a circle’s diameter:circumference ratio a little more tangible to my VCAL students I thought of a way to extend linear arithmetic blocks (LAB) to this new context.

Cut out 6 or 7 pieces of string/rope, each as long as your LAB ‘one’ length. Cut out two bits of rope of length equivalent 0.14 w.r.t your LAB resources.

Go outside with chalk. Use one of the one’s to draw a circle of diameter one as pictured below.

Make clear to students that the pieces of string are the same length as one.

Ask students: What’s the diameter of this circle?

Ask students: How many of these (holding up a string of unit one) do you think will fit around the outside of this circle? (You can tally answers if you like)

Get volunteer students to place the string neatly around the outside of the circle, as pictured below.Classify the answer as ‘3 and a bit’. Bring out your ‘bit’ (the length of 0.14). If you’ve done the circle neatly it should neatly fit!

Get some students to use LAB to measure the length of the ‘bit’. It’s equal to  0.14! (reinforce the language, it’s on tenth and four hundredths).Ask: So, when we have a circle of diameter 1, what’s the circumference?

Draw a circle of diameter 2, repeat the process!

Go back inside and draw the learning together on the board, give students 3 or so example questions to check their understanding.

 

 

 

 

Basic Auslan for Maths Teachers

Today whilst observing a fellow maths teacher at my school, I saw her use some Auslan (one form of sign language) with one of the deaf students. I thought, ‘Wow, I want to be able to do that’. So, at recess whilst in the staff room, I asked for a little impromptu lesson…

It’s not a pretty video, but I definitely think it was a worth while 4 minute time investment for me. Pardon the eating, and happy Auslan learning : )

‘The Points System’: One approach to differentiation

When I was in school I remember that I found nothing more boring than the instruction to ‘do questions number 1 to 5, parts a) to h)’. I would get my head around the concept by question b or c of each section, then have to spend what seemed like an eternity working through a whole bunch of exactly the same question with different numbers. With this knowledge of my own experience in mind, I’ve always known that I wanted to try to avoid this same boredom for my own students.

This approach to differentiation and boring repetition reduction was brought to my attention by one of my Masters of Teaching lecturers Nicky Dulfer, who said that one of her friends had implemented an ‘earn points’ system in their own mathematics class. I thought it sounded like a nice idea. Here’s how I approached it…

The slide I showed to students: 

For those unfamiliar with the proficiency strands of fluency, understanding, reasoning and problem solving (‘open-ended’ in this case), here’s an example of a few example Q’s for context (from the Pearson Mathematics 9 Textbook).

 

Did students like it?

This was implemented with a year 9 mathematics class.

Mid placement feedback: In response to the question ‘What should Ollie keep on doing?’ five students nominated the points system. In response to the question ‘What should Ollie stop doing or modify?’ no students nominated the point system. (Highest number of votes for any one category was 8, which was to suggest that I continue making videos of content).

End of placement feedback: In response to the question ‘What did Ollie do that most helped you learn?’, four  students nominated the points system. In response to the question ‘What are some things Ollie shouldn’t do in future, or things to modify’, one student said ‘Point system (15 in 30 minutes was too rushed and stressful)’. (Highest number of votes for any one category was 12, which was to suggest that micro-revisions (post on micro-revisions to come…) were what most helped students to learn).

So, on the whole, the points system was well received by students.

Did it help the students to learn?

source: http://store.discovery.com/img/product/catl/00275198-962183.jpg

Unfortunately I didn’t conduct this points system long enough to be able to tell if it increased student learning. But I can say that it appeared to increase engagement, and as ‘a major precursor to learning is engagement’ (Hattie, 2012, Chapter 8, Section 2, para. 1), it’s plausible that it increased learning.

I do however think that it’s fair to say that this task helped the higher achieving students to learn more. After they had gained their 15 points (some of them would finish this in under 10 minutes), students were able to move onto ‘challenge questions’, like the pythagoras challenge questions that I’ve written about previously. The traditional approach for these students was to get them to do a set of questions from each of the proficiency strands (fluency, understanding… etc), which would take up the whole lesson, and was pointless in many cases as these high achieving students could easily complete the task and weren’t being challenged at all.

The idea of students self-differentiating was also intended to help promote metacognition. Fostering metacognition is a key step to helping students to ‘become their own teachers, which is the core attribute of lifelong learning or self-regulation, and of the love of learning’ (Hattie, 2012, loc 168). But actually scaffolding this metacognition is something that I need to do better in this task and in the classroom more generally. I feel that it’s unreasonable to assume, as I implicitly did, that a year 9 student will be able to select a question at the appropriate level without any help.

Conclusion:

I feel that this task was a step in the right direction and, coupled with the power of spacing repetition of content for students (Carpenter, Cepeda, Rohrer et al., 2012), has great potential to be expanded and improved upon in future. One immediate improvement to make would be to have ‘challenge questions’ (for students who work through the 15 points) of different levels of difficulty, rather than just the one ‘challenge question’ that wasn’t so easily accessible to some students.

Note: Sunshine College has taken a similar approach where students select for themselves a ‘just right’ task and form small groups to solve it. This is an approach that I’ll be exploring more in the near future. You can read a brief paper about it here.

References:

Hattie, J. (2012). Visible learning for teachers. [Kindle version]. Retrieved from Amazon.com.au

Carpenter, S. K., Cepeda , N. J., Rohrer, D., Kang, S. H., & Pashler, H. (2012). Using Spacing to Enhance Diverse Forms of Learning: Review of Recent Research and Implications for Instruction. Education Psychology Review , (24), 369-378.

Teacher Blogger chat at MAV Conference 2015

At MAV Con the other day I sent out a tweet to catch up with other teacher bloggers:

Screen Shot 2015-12-08 at 8.23.14 am

 

 

 

 

 

It was a small meeting (only 2 of us), but I was happy to have the opportunity to meet with fellow teacher blogger Michaela Epstein. Aside from a great chat that touched on our backgrounds, motivations for teaching, and even spaced repetition software,  Michaela and I turned to a few questions that I’d drafted up to get into the why, how, and what of our blogging approaches. Enjoy : )

Screen Shot 2015-12-08 at 8.26.43 amMichaela Epstein

Practicalities

    • twitter handle: @mic_epstein
    • blog address?: https://michaelaepstein.wordpress.com/ 
    • Where are you based (as much detail as you’d like to give)?: Melbourne

Juicy Q’s

  • Why do you blog?: Initially was to give non-teachers an insight into what actually happens in schools and give an understanding of the social side and social justice side of the school context. Since starting teaching it’s taken on a maths/social justice focus.
    • What part/s of your blog/blogging are going well?: Get good feedback from people who read it. A useful way to express some complex ideas that I think about then communicate that with others. Good way to start interesting conversations.
    • Which blogging/social media apps or programs do you find core to the way you do what you do in the blogosphere?: Twitter is being used more and more, easy, quick, can share stuff, low barriers to entry and great way to connect with others.
    • What is a challenge that you’re contending with at the moment w.r.t your blogging/use of social media?: Not doing it /not doing it regularly.
    • What’s your favourite Ed blog?: www.mathwithbaddrawing.com   it’s a U.S dude now in the UK (also social commentary on maths education). His blogs are usually just cartoons with annotations on those cartoons.
    • Why?: This website gets to the crux of the philosophical issues that we deal with as maths teachers.

Ollie Lovell

Practicalities

    • Twitter handle: @ollie_lovell
    • Blog address?: www.ollielovell.com
    • Where are you based (as much detail as you’d like to give)?: Melbourne, gonna be doing research with a Northern Metropolitan Region school in 2016.

Juicy Q’s

    • Why do you blog?: To sort through my own ideas, share what I do in my classroom, and as a way to force me to reflect on books, conference, etc.
    • How would you describe the focus of your blog?: It’s really about my learning journey. I’ve taken the title ollielovell.com so that I can be really flexible just to be me and explore and blog about what’s relevant for me. Main foci are wot-I-got from various conferences/books, what I’m trying out in my own class, and my learning journey more broadly
    • What part/s of your blog/blogging are going well?: I’ve enjoyed particularly sharing some of my classroom approaches recently.
    • Which blogging/social media apps or programs do you find core to the way you do what you do in the blogosphere?: The Google Docs Add-on Docs to WordPress. Allows me to write posts in google docs then export to WordPress. Saves a bunch of time in uploading photos and makes it easier to collaborate on posts.
    • What is a challenge that you’re contending with at the moment w.r.t your blogging/use of social media?:
      • Keeping up with twitter and sorting the wheat from the chaff!
      • Ordering/sorting the content on my website to make it easier for punters to use
    • What’s your favourite Ed blog?: Dan Meyer’s blog and the less well known BetterExplained.
      • Why?: Dan’s creativity and the way that he does things in his class blows my mind. Kalid’s work on BetterExplained has helped me to understand maths better.