If y = 3, then 6y = 63 ... or is it?

Calvin & Hobbes is copyright
Bill Watterson and UPS

Here is a little worksheet I made for students to explore common algebra misconceptions. I'm planning to use it as part of a small-group activity, as a way to elicit discussion about algebra concepts.

https://drive.google.com/file/d/0ByVkChxwrC4DSUl3OGV3Q210S0U/view?usp=sharing

Can you see why someone might think (Calvin in the last panel) that

n + n = 28

even though they have been given no value for n ?

I sourced the algebra misconceptions from

MacGregor, M., & Stacey, K. (1993). What is x? The Australian Mathematics Teacher, 49(4), 28-30.

and Calvin just forced his way onto the page (apologies to Bill Watterson!).

The Funny Side of the Crying Game

I've had an amazing response from colleagues, mentors and friends to the last post about my shameful crying episode. Most surprising was to discover just how common these feelings are for new teachers - and just how many of my peers are out there having their own cry :-) A good friend told me they've even been crying in the car on the way home, and a mentor who I hold in the highest esteem told me they nearly resigned in their first term because they thought they were a terrible teacher.  It seems it comes with the territory - and you can sort of see the funny side if you imagine all those secret crying corners. So, if you're reading this because you've had a cry in the classroom (hopefully not in front of the students!) - take heart - it's part of the process! And don't be afraid to share the story with trusted mentors and peers. Chances are they've been having a secret cry too.

So what advice did I receive? Here's a summary and some further thoughts:

A certain level of control is necessary before you can do the exciting stuff

This response from friend and fellow new teacher Beth Bevan echoed the experience of many of my peers:

One thing I have learnt though is that you need to build the strong relationship with a challenging class before you can break out the really exciting stuff. If you don't have that certain level of "control" you won't be able to reign them back in when it's time to discuss the learning, or time to work quietly on a task. It doesn't mean you need to be the scary disciplinarian, you just have your find your way of telling the kids they're off track and bringing them back to focus. And it certainly doesn't mean your wonderful teaching strategies go out the window, it might mean you introduce things slowly, remembering that they need to get used to you and your style being a different way of learning too and this will take them out of their comfort zone. By the end of the year, your most challenging class could well be your favourite.

See students as individuals - not 'that class'

Some very helpful and practical advice I was given was to stop thinking about my challenging class as "that class" and instead view it as composed of unique individuals - each of them with their own story, their own personality, needs and responses. When you do that, it becomes so much easier to forgive their individual transgressions - you're not facing a whole class, with a mob personality that's attacking you (even if it feels that way sometimes!).

Who do you think you are?

In the nicest possible way: Don't kid yourself that you've got something special that dedicated teachers who came before you haven't tried. It makes good movies, but it ain't true. Accept your limitations and the reality of the current context.  I'm reminded of a Buddhist teaching called "To Care and Not to Care" - or to use the lingo, getting the right balance between Compassion and Equanimity. We need to have compassion with our students, but also not be so caught up in emotion we can no longer help. Some things can't be changed just by us wanting them to change - we need equanimity to accept things as they are, while still keeping our heart open so that when and if students are ready for us, we are still there for them.

"I hate you" - but it's not personal

In her wonderful book "Relating to Adolescents: Educators in a Teenage World", Susan Porter repeatedly warns teachers not to take students' admiration and interest seriously: "they don't care about you". Teenagers are interested in themselves - and their interest in you is only based in how it relates to them. I'm coming to realise this caution also applies when students say they dislike you, or are rude to you - it's the same: "they don't really care about you" - the emotion isn't that personal - you the teacher just don't figure that much in their personal lives. So next period, they could be all different again.

And how is this class going now? I have changed my teaching strategies to use more 'working quietly' activities (book work, worksheets) and once the class is settled, I walk around the class to students and teach individually or in pairs.  We're not at the stage yet (me included) that we can have sustained whole class discussion about mathematics. I also offer more differentiation options, allowing students to choose the level of difficulty they want to start at. The wonderful thing is now that the shouting (theirs) and the crying (mine) has stopped, I'm really taking pleasure in the variety and humour of the class.

The Crying Game

Sometimes the adventure gets a little rough (with apologies to MGM).

I wrote this piece a month ago as a private reflection - at the time I was hesitant to share it more widely because it is a little emotional and I didn't want to take the chance my students would see it while I was feeling this way. However the response from the people I showed it to at the time suggests it might be helpful to other new teachers - even if just to realise how normal this sort of reaction is. And I think it's important for balance to show some of the low points as well as the high points of first term of teaching.

One week in March 2011

A shocker of a week - an absolute shocker of a week - the low point: finding myself crying uncontrollably - fortunately alone in my classroom at recess with the door shut. It was the end of a dreadful period with my "challenging class" - their rejection of my attempts to engage with them and the aggressive rudeness of some students finally got to me. I was angry and upset - upset with them, and then upset with myself that I was so upset - what sort of crappy teacher was I? Is this what I gave up a career in IT for? What I studied so hard for two years without pay for? I'm kind of laughing now - at the irrationality and intensity of it ... but it really did feel that way. I just bawled and bawled like a baby for a five minutes. I couldn't go into the faculty staff room - I didn't want anyone to see how low I had gone - or see my red eyes.

The day before, I had another dent in my confidence when the school executive decided not to support an initiative I thought I had their support for. Looking back I can see I didn't take the smart approach of getting genuinely solid support for the idea before it went to the meeting - I had forgotten my skills from the corporate world how to play that game. The combined effect of what felt like a rejection from both the school and the students with weeks' of sleep deprivation meant that by Wednesday night I really was feeling it was "Game Over" for me.

Fortunately I've got some great support in my faculty. After I confessed to my pathetic crying session to two colleagues, they shared their own stories of when they had their "crying time" and gave me support and many practical ideas how to deal with the challenges. As one put it to me : "All teachers go through this stage - and then you can go one of several ways: some teachers face it and work out how to deal with it and go on to be good teachers, others leave the profession, and the worst option, some never deal with it, but stay in the profession". I felt very sorry for myself - but decided I wanted to be in the first category of teachers: work it out.

Two days later I faced the test again: and a tougher test - a double period with the challenging class. Different teaching strategies, I stayed very calm, and ... amazingly .... the students (mostly) worked and (some) learned new material. Two students actually apologised to me for the previous period (they must have seen the start of my breakdown!), and the student who had been the most aggressive was polite and civil to me (one of my colleagues said that was their way of apologising). It went to so well, I actually let the students continue doing maths exercises rather than bring out some games I had planned - save that one for another day! Maybe we can start from this base, and slowly introduce some of those more engaging and deeper learning activities I perhaps foolishly tried on Day One. So the week ended on a high.

So why the uncontrollable flood of tears? Apart from just fear of failure, and the ego hit of being rejected, I really have been resisting the pressure to conform and follow more traditional approaches to the "teacher-student-role" - I didn't want to be the adult who dominates the classroom, who uses well-tried techniques to achieve command and control - techniques that I believe often create barriers and inhibit learning. In a way, those tears were for the loss of that of idealism, tears that I might have been terribly mistaken - and that to survive I would have to become what I don't want to be. What I've learnt though is it's not quite that black-and-white: students and the system itself are not always as open and welcoming to your idealism (naiveté?) as you may hope. So you do have to toughen up - this isn't a game for the weak. Now the challenge is to learn how to be strong and resilient, and carefully use those command-and-control techniques, and "play the system"- without losing the idealism, hope and commitment that are the reasons I want to be a teacher.

It's Saturday morning now - a great night's sleep - and the comfort of knowing I survived my first really challenging week. I'm so grateful for the support and friendship of colleagues. And also a little embarrassed to think of my reaction given I only have one class out of five that is challenging - I can only imagine what it's like for teachers who have maybe four out of five challenging classes - period after period, every day of the week.

I don't think I'll be crying again.

One month later: Looking back on this, part of me is chuckling at how sensitive I was at the time - but another part of me sees the importance of going through the experience.  I will be sharing some of the teacher responses to this story in the next post.

Take a piece of paper ....

What does it take to learn the Index Laws? Surprisingly little if you (i) can fold a piece of paper (ii) can multiply by two. I first observed my head teacher do a version of this lesson last year with his Year 7 class, and had the joy of doing a more advanced version with my Year 9 this year.  I have a feeling the idea goes back to the dawn of maths teaching, but I haven't seen a resource explaining it, so thought it would be fun to share my take on it.

Index Laws Folding Paper v1

Approved for "Free Cultural Works"

Urged on by Chris Betcher's post "If you want to share, say so" exzuberant is now CC-BY-SA.

Creative Commons Attribution-ShareAlike 3.0 Unported License.

Previously I had this blog set at CC-BY-SA-NC, but  this way anyone can use my work now, so long as they accept to open their own work unencumbered by copyright. Welcome to the Commons :-)

Explaining Standards Based Grading to Students and Parents

A little something to give back to the wonderful SBG community on the web:  my version of a letter explaining "the how" and more importantly, "the why" of my SBG implementation.

  

I've adapted sentences from similar letters shared by other teachers using SBG (apologies - I mixed up the material to the point I can't identify individual authors any more!), and then extended for my own context. I also felt I should add some FAQ style paragraphs at the end of the document to help address any concerns. When I wrote this letter I very carefully avoided using the SBG terminology or any sense this was something to make a fuss about. I think it's important to reassure people this is a normal way to work with a class - the only thing that is different is that the final topic test result is no longer the main form of grading and that students can do repeat attempts to improve their grades.  Amazing how such a small change in the way of thinking about assessment can have such a massive impact on teaching and learning!

SBG Explained

Standards Based Grading: 11 weeks later

It's been eleven weeks now since I started using Standards Based Grading with my Year 8 class. It was tough going at times, not because of SBG, but rather because I'm a new teacher, still working out how to manage five classes, teaching topics for the first time while at the same experimenting with SBG and trying not to confuse my Year 8 students. In hindsight I probably bit off more than I could chew!

Did SBG work as hoped for? I'm very happy with the results so far. SBG certainly 'did no harm' to student test results in the most recently completed topic. The mean topic test score was 80%, with a standard deviation of 13%. Based on student feedback and my observations, I believe SBG helped students gain these results - although I can't actually prove it.  Looking at the actual SBG grading I can say that 12 out of 24 students achieved full mastery on all eight outcomes for the topic, and 22 of 24 obtained at least a B+ for every outcome, with almost every student who did not get all As on a quiz coming back to do repeat attempts. Students regularly told me how much they liked doing assessment this way and often told me how they "finally got it" after doing the quiz a second or even a third time.

Test anxiety appeared to be greatly reduced. I was able to constantly reassure the more anxious students they had control over 80% of their mark by doing repeats on the quizes and that if they did that, the test which was only worth 20% would take care of itself. When I did give test results back, we were able to discuss specific strengths and weakeness in their test results in terms of the outcome grid.

SBG helped me with my teaching: I received continuous feedback throughout the topic as to what students could and could not do, and which of my lessons were actually working. In one case I realised the last two lessons I had taught had failed to help students meet an outcome - so I knew I was teaching it the wrong way for them. I tried a different approach and suddenly the quiz results for that outcome soared. Without SBG I could have had a nasty surprise at topic test time.

The key SBG components I used were: an outcomes grid; a sequence of diagnostic quizzes made in multiple versions (slight variations allowing quick production of multiple tests for repeat attempts); student pair-marking and self-tracking to emphasise student control and focus on outcomes; linking the traditional topic test to the outcomes grid. However did I fall behind one week in the program because of my inexperience and I've learnt I need to explicitly plan the SBG components ahead of time and fit them into the program - hopefully in a way that allows the SBG activities to double up as learning activities. I think it will be smoother next topic.

Was there a discrepancy between end of topic test results and the quiz marks? In most cases, student quiz marks matched their performance in corresponding outcomes in the topic test. However for some students there were some differences. As I marked the tests, I compared each student's quiz outcomes to how they did in the test on that outcome - this was time consuming - but I think worth it.  An idea that came about while marking the test, which may not be in the spirit of SBG, was to modify the quiz marks where I saw major difference between the corresponding test outcome. If the student did better in the test for an outcome, I raised their quiz result to match what they did in the test - since they had clearly mastered the outcome and I think it's fair to view the test as another quiz. If the student did worse in the test for an outcome and clearly demonstrated they no longer met the outcome, I reduced their quiz mark down to a B or a C. This didn't happen for many students, and in the rare cases I made these adjustments, I later explained to them what I had done and offered them the chance to repeat the corresponding quiz - thus giving them the chance to get back a B+ or an A for the outcome - and revise the skill. 

Will I do SBG next term with Year 8? Absolutely!  Will I try it with another year group? Not just yet as I'm working through other first-year-teacher challenges.


More details for those really interested in the nitty-gritty:

How did I implement SBG?

• Use of an Outcomes Grid:  I gave each student a sheet like this to glue in their exercise books:

• Use of a letter based grading scheme.
• A - Fully mastered - perfect answers
• B+ - Almost fully mastered - minor improvements or refinement needed
• B - Developing - at least some skill/understanding demonstrated
• C - Not demonstrated - doesn't look like this was understood.
  Note well: I never told a student they couldn't do, or didn't know the outcome - I just said they hadn't demonstrated it. A subtle difference, but a huge difference for self-esteem.
• Use of a weekly diagnostic quiz in two versions. Most weeks I prepared a short quiz that tested two to four of the outcomes. I prepared two versions of the essentially the same quiz (A) and (B).  On the first attempt for each quiz I handed out papers (A) and (B) to alternate students. This helped ensure I was assessing the student's skills and not their neighbours' and made sure I always had a second quiz available.  Students were enouraged to just write "IDK I don't know" if they really didn't know how to do a question and move on to the next question. The goal was to get through the quiz without too much stress or time wasted. Here is how the quiz  looked for the first four outcomes:
Percent Diag 01
• Student pair-marking of papers: Unless I was really strapped for time, I asked the students to swap papers and mark the quiz. The specific instruction was "look at the colour pen your neighbour used - now please use a different colour pen to mark them". We then worked through the quiz, with me asking random students to help solve the (A) version of the problem on the board.  If I found a student who could not do the (A) version of problem, we worked through it, then I asked them to work out the (B) version answer for the class. The students graded their neighbour's paper, and wrote letter grades for each outcome at the top of the paper.
• Quiz results guided my lesson planning: I collected the papers, looked for student misconceptions, checked the student allocated marks and recorded the results. Based on the overall class results I then made a call if we could move on to the next outcomes, or if more whole class work was required. Depending on that call, I might make a (C) and (D) versions of the quiz for a later lesson, or I made these papers available for any students wanting to repeat the quiz.
• Repeating Quizzes: I didn't impose any strict procedure for doing repeat attempts - I played it by ear. I had a stack of (A) and (B) papers available any time - I gave the student the other paper than they had done for their second try. If need be I made (C) and (D).
• Student continuous self-monitoring of outcomes: I handed back the quizes after checking and gave the students (mostly) clear instruction how to update their outcomes sheet with their marks. The idea was to encourage them to track their own learning.
• Follow up material made available: I put copies of the (A) and (B) versions of the quizes with worked solutions on the class edmodo, and sometimes prepared specific revision material for those students needing more support.
• Rinse and Repeat: We repeated the process until we completed the topic.

Sheesh - I'm exhausted just writing this out ... but it's harder to explain than to do .. mostly.

Lessons learned

• It's hard work to implement SBG the first time you are teaching the topic - there is a lot of material to prepare for SBG. Probably a bit crazy to do this the first year teaching,  but hopefully most of the resources built can be reused the next year.  
• Next term I will set a regular period for doing the quizzes, and a regular weekly recess period for doing repeat attempts.
• For this last topic I used a premade test which didn't sufficiently test one item on my outcome grid, so next time I will take more care to ensure full coverage.

Year 7: Children or Young Adults?

The final in a series of reflections about my first time experiences teaching a Year 7 mathematics class.

While the end-of-term survey showed that most of my Year 7 are enjoying or at least tolerating maths (let's be honest, many students don't like maths), some unexpected feedback appeared in the comments from several students: "Treat us like Year 7 instead of 5 year olds". When I first read this feedback I wondered if students were referring to the "fun lessons" we've been doing - which do sometimes look like primary school activities, even though they explore Year 7 content, but then I realised the comments probably reflected something deeper.

Is it possible I've been treating my Year 7 like small children without being aware of it?

How old is Spongebob Squarepants?  Spongebob definitely isn't a child - he sat his
 driving test 58 times.  Age estimates range from 13 to "50 sponge years".
And yes - my Year 7 class is slightly chaotic!

On the last day of term, I presented the summary of the feedback to the class and asked them, if they didn't mind, to help me understand the comments. They seemed surprised to see critical comments about my teaching so openly presented, but were keen to explain.  At first some students mentioned they found the maths too easy and that made it childish, but then some students told me that sometimes I talk down to them - and that opened the conversation up - clearly most students felt I was treating them like little kids.  I think it must have taken some courage to tell me this to my face and I'm very grateful to them. And they probably are right. Sometimes I take too much pleasure in the childish aspects of Year 7 - and play to it.  (Frustrated primary school teacher?) I asked the students: "So what should I do to treat you more like Year 7?"  I treasure the response: "Do what you're doing now - talk to us like this". I think it was the first time these students had seen a teacher share student feedback with them - and they were quite surprised how open I was to discussing how I could improve. And in return, they were being very mature in helping me understand their feedback.

So are Year 7 children or young adults? I think the answer is both - they have the amazing ability to be both childish and mature at the same time. Adding to the confusion is the wide range of physical size, behaviour and mathematical skill across the class. Then throw in the fact you've maybe just come from teaching a Year 11 class, where students are so much older - it's easy to get confused.  I'm guessing for the Year 7 students, who are going through the messy transition process from being the 'big kids' at primary school to the 'little kids' at high school, having a teacher talk down to you is not helpful. They want to feel they are in the adult world now.

I'm resolved next term to act as if my Year 7 class are all young adults - not children. And for the times they act more like children - well, I'll remember that last day of Term 1 when we talked together as adults about what constitutes good teaching.

I can't overstate how valuable getting student feedback on your teaching can be. See my post on Student Voice for details how I do it in ways to encourage constructive feedback and protect student anonymity, along with some resources you may wish to use for your own student surveys.

Year 7 Maths : Extreme Differentiation Needed?

This post is the second in a series of reflections on my first experience of teaching a Year 7 mathematics class.

I'm almost embarrassed to say it, but it's taken me almost ten weeks to realise the extreme variation of mathematical ability in my Year 7 class. I knew there were differences across students, but underestimated just how wide that range is. What made the scales drop from my eyes? A second round of summative assessment (topic tests) and feedback from the end of term anonymous class survey. And I don't think there's anything different in my Year 7 class to any other Year 7 class in a comprehensive school.

Staggering mathematical bio-diversity in Year 7!
© David Hall seaphotos.com Used with permission.

Here's an extract from the survey that demonstrates the challenge:

I've never seen such variation in any class I've surveyed in the past - and the student self-reported feedback matches the most recent round of test results - which ranged from 3/30 to 30/30. How devastating to self-confidence must it be to receive 3/30? I don't buy any argument this will encourage them to "try harder" - especially since "try harder" just won't help when the content and skills are so far ahead of where the student is now.

In a follow up session I explored this with the class and confirmed that the more mathematically advanced students are getting frustrated and feel like they are being treated like babies (the math is too easy), and that other students are struggling - which may explain some of the work avoidance patterns beginning emerge. Looking at primary school records (something I recommend all high school teachers do for their Year 7 students - and wish I had done earlier) revealed a vast range of difference in mathematics learning outcomes - some students have already mastered Year 7 outcomes (through tutoring?) and others are still mastering early Primary School mathematics.

Looking ahead, I'm thinking it's time to implement an SBG approach to assessment (as with my Year 8 class), but even that isn't enough: extreme differentiation is called for!  I've asked many experienced teaches for advice and here's what I'm going to try out next term:

Extreme Differentiation: Ideas for Term 2

• Design a pre-topic diagnostic that indicates student readiness for this topic. I've come to realise I need to design my own diagnostic test - the standard "Are you ready?" diagnostics in the textbooks aren't always up to scratch!
• Communicate the result of the pre-topic diagnostic for this topic to the student. I'm keeping in mind that a student who is not ready for, say, algebra, may well be ready for geometry.
• Keep a copy of the pre-topic diagnostic on file. I need to be able to justify why I offered this student the option to work on different, easier material.
• Differentiate the topic into three levels: Essentials, Development and Challenge. The Essentials level will also include material from earlier 'stages' of mathematics (ie: some primary school material).  I'm considering using an easier text book for the Essentials material.
• Course material will contain a level indicator: one star for Essentials, two stars for Development, three stars for Challenge. Each lesson and each assessment tool will offer students material at each level.
• Offer students the option to select the level they want to work at in the topic. I believe most students will make the appropriate choice. More advanced students will be able to skip the Essentials and go straight to Development and Challenge. Of course I'll be watching for what happens, and encourage students making the wrong choice to consider the alternatives.
• Class summative assessments will report marks for each level the student attempted.

In the final post of this series, I'll consider the question  "Year 7: Children or Young Adults?"

Having too much fun with Year 7?

Year 7 in my state (NSW, Australia) is the first year at high school (USA: middle school). Students are aged 12-13 yrs. This is the first in a series of three posts where I reflect on the experience of teaching my first full time Year 7 class.

During my first (ever) term with my Year 7 class, I was keen to create a positive learning environment for students starting on the long journey that is high school mathematics. Like most keen-and-eager new teachers, I didn't take the "start out tough with them" advice of more experienced teachers and I eagerly tried many ideas  to make maths more engaging than perhaps traditional approaches take. To give full credit to my teachers at university, they did caution us to use balance and wisdom - and not to overdo the 'creative' lessons.

Adapted from http://www.alex99.co.nr/
Alex99's free Sponge Bob screensaver

At the end of the first term, I think we do have a happy and positive class - but after analysing test results, looking over student exercise books and thinking carefully about the student feedback from the end of term anonymous survey, I'm asking myself: Have I actually been helping my Year 7 students develop good habits? Have I been reinforcing the need to apply effort to their mathematics? Remember Andrew Martin's critical factors for building academic success: Effort, Attitude and Strategy. I think I've been helping with the attitude and the strategy - but probably neglecting the effort. I'm coming to the conclusion that not following enough traditional techniques undermines the effort aspect - and in so doing, I risk undermining academic outcomes. We have been doing "real" maths, but at least some of the class isn't doing enough of it.

Encouraging Effort Strategies for Term 2:

• Be explicit about the role of effort in improving academic outcomes.
• Ensure most lessons to have a 15 minute "working silently" component - with an option on the board for those who don't have books (a perennial problem that causes distractions and chaos).
• Provide a clear and explicit message about homework: keep it reasonable (15 mins per lesson), keep it consistent (every lesson).
• Seating-plan based observation sheet to check and reinforce text books, exercise books, Homework.

I'm not intending to abandon the fun lessons - but temper them with calmer lesson segments where the focus is on application of effort and practice.


A Seating-Plan Based Observation Sheet

Here's an idea I developed in the last few weeks to help me be more organised in my Year 7 class - something I can find very difficult when there are so many students asking for attention.

Idea for a seating-plan based observation sheet

The idea is to make a pad of these and put them in a clipboard which I carry as I walk around the class. A very quick set of ticks as I move allows me to record: who is present (no need to do separate roll call), who has brought books to class (and reinforce the message it's important), who has made an effort at the homework (and reinforce the message it's important), as well as allow a spot survey across the class who is on-task and who is off-task. I'm thinking if I can do all this on one sheet, then I can quickly collate some notes from it at the end of the day. (Darn it I want an iPad :-)  I'll try this out for a few weeks and if it works I'll publish the template.

Beyond the Effort issue, an even more pressing factor has become prominent with my Year 7 class - a vast difference in mathematical skill levels - and how I need to do a radical overhaul of my teaching with this class to accommodate it. But that's for the next post.

Waving, Not Drowning

Mother Sea Otter with pup (cropped)
Mike Baird, bairdphotos.com

It's been a very long time between posts - the first eleven weeks of my first year teaching has been exhilarating but also totally draining - making it very hard to find the energy to write more frequently. But I'm pleased to report I survived the first term and still smiling, despite a major hurdle around week eight (aka "The Crying Game").

It's term break now, so time to catch up - and lots to share.

Posts coming up during the next two weeks:

• An update on my SBG work with Year 8
• What you didn't learn at University about Year 7
• The Crying Game
• Teaching index laws using a blank piece of paper
• Prime Numbers with Willy Wonka and the Liquorice Factory
• The joy of Co-operative Teaching
• "Mr Zuber, why do you have art work in your maths room?"