Who's that giving effective feedback?

 I wrote the following piece of pedagogical documentation back in 2015 as my final project for Project Zero's course, "Making Learning Visible: The Power of Group Learning and Documentation in Classrooms and Communities." This course is run by the Harvard School of Graduate Education. It seems appropriate to share it now, not only because it's Halloween but because all of the children have moved on to high school and cannot be identified.

This piece of writing marks a pivotal point in my understanding of how even very young children can be taught to provide each other with effective feedback, and how that feedback can have a profound effect on the quality of their writing.

Context

This term during library lessons, Year One have been collaborating to write a new book every week. The aim is to have each child contribute to a page. Not all of the children are able to write independently yet, so children were provided with opportunities to write with a partner. To further support students, the collaborative texts were modelled on the language structure and images of a mentor text. 

The lesson documented here is the third in this series of lessons. 

In this lesson, the mentor text was I'm Green and I'm Grumpy! by Alison Lester.
(You can find a video of Alison reading her text aloud here.)

We began by reading the book and talking about how the clues allowed us to guess which character was hiding behind the door.

On page 1 we learn who is hiding in the cupboard; 

On page 2 (the speech bubble on the door), we are given clues about their costume to help us guess

Opening the door reveals the answer on page 3

Next, the children took turns to describe to each other the costumes they wore for Halloween while others used the clues to guess who they were.

The writing task

The students were asked to divide themselves into groups of two or three. I gave them the separate pages of the scaffold modelled on the original text.

They had fifteen minutes to:

1. Copy the language pattern in the book by filling in a name and the appropriate pronoun (page 1).
   
   
2. Write some clues on a speech bubble on the door that could be used by the reader to correctly guess who was hiding behind the door (page 2).
   
   
3. Decide who was hiding in the cupboard, draw them and write their character name in another speech bubble (page 3).
   
   
4. Borrow two library books each!

After our fifteen minute introduction, they began work. 

Last week's book was based on a much simpler task, yet few students finished it in the time available and so the book was quite short. The children's eagerness to complete the task and have their work included for this week's book drove them to concentrate. There was universal engagement with their work.

Where to next?

At the end of the lesson, about half the class had finished the writing component of their pages. Of the other half, most were almost finished but not enough for their pages to make sense within the context of the book. Students' commitment to the task was obvious so I decided that rather than creating another new book next lesson, we would finish this one. But before they could do that successfully, they needed some feedback about their work in progress.

I made the completed pages into a book and looked at the remaining unfinished pages to identify what feedback students would need to receive in order to be successful. I wrote a series of post-it notes to guide students' efforts. However, after discussing this approach with a colleague, I decided to take the post-it notes off and instead give students the opportunity to give and receive feedback from each other.

Hattie (2012) identifies feedback as crucial to student achievement. His analysis of multiple studies has shown that on average, it has twice the impact of all other schooling effects; however not all forms of feedback that students receive are equally effective. He cites Nuthall's conclusions that while as much as 80 percent of verbal feedback received by students comes from their peers, much of it is incorrect (Nuthall, 2007, cited by Hattie op.cit.).

I wondered:

• How can I support students to give each other effective feedback? 
• What impact will the children's feedback have on the work of their peers? 
• How effective would their feedback to each other be, when compare to teacher feedback?

In the next lesson, instead of reading another commercially-published picture book and starting a new book-making activity, I read the completed pages of our picture book: Who's that hiding in the cupboard?

The research of Mark Gan (2011, cited by Hattie op.cit.) showed the effectiveness of providing students with prompts to guide them as they gave feedback to their peers. To prepare students for giving feedback, I modelled the thinking routine of I see... I think.... I wonder...

Image courtesy of Silvia Rosenthal Tosilano @langwitches

As we read our book we described what we saw, thought about the connections we could make with our model text and asked and answered wonder questions that would help students figure out what they had to do next to meet our learning objective: to complete a page for our class book using the same language structures and features as our modelled text.

As I read, we tried to see if we could figure out who was hiding inside the cupboard. If we felt we could guess correctly, we thought about which part of the clue had made it easy.

If it was hard to guess who was hiding behind the door, we wondered, "What other clues would have helped?"

Giving each other feedback

After modelling the thinking routine, students whose work was already in the book were teamed up to provide feedback to students who were yet to finish. Once again, the children worked independently while I scanned their library books. In the heat of the lesson there were few opportunities for me to take photographs or document conversations. However from the circulation desk I observed groups of children drawing and writing together, huddled around the same pieces of paper. I wondered what impact the feedback would have on their work and whether their collaborations would be fruitful. By the end of the lesson fifteen minutes later, all the pages of the book were completed - and everyone had borrowed two library books.

Determining the impact of student feedback

The following images show student work before and after our feedback lesson. The 'Before' images include the post-it notes that I wrote to identify the areas where I thought they needed to improve. 

I removed all post-it notes before the lesson and did not share them with the students. 

• There are sometimes fewer pages shown in the 'Before' image because I only photographed those pages where I was intending to give feedback: I assumed the other pages were already finished. 
• The 'After' images show the students' work after they received peer feedback and their pages were bound into our book.

As well as the written clues, many other visual embellishments were added. For example, the pages are all highly coloured with multiple layers that the girls completed together. On the first page it is possible to distinguish three separate illustrators, each person drawing another character.

This group of six were proud of their collective effort: all added their names to the back of the completed pages to claim joint ownership of their work (not shown) and wrote, "we're the best girls," inside a read circle in the top left corner of the first page.

During our feedback class the boys added a detailed image that matched their original clue. However they also added the second written clue. I can see that it's written in the same handwriting as the first clue, which makes me think that both were written by the same person. Even if this clue was added by the writer in response to our class discussion, I wonder what role an audience of his peers played in his decision to add an additional clue?

The left page and the clues on the door have been completed in the same handwriting as the first word and using the same blue pencil. This points to continuity in authorship across time, suggesting that it was the original author who added their ideas during our feedback lesson. The words on the final page are written in red and the handwriting is different to the the other two pages - note the difference in the formation of the 'a' and the 'm.' While written by two different authors, the clues and the answer match. This shows that the collaboration between Will and George that started in the first lesson continued in the second, and suggest that changes were made based upon feedback. The door has been coloured in and additional details have been added to the drawing, showing that the students considered the aesthetic appeal of the page to be important - even thought this was not explicitly stated by me.

After the feedback lesson, the boys were able to complete all aspects of the task, despite my post-it notes having been removed. In addition, the zombie has been redrawn and coloured and now occupies a much larger proportion of the space, showing that they have thought carefully about how this page will be viewed by the reader. More significantly, the specific clues leave the reader in no doubt about who is hiding behind the door.

Analysis

At the end of this process, I still found myself wondering, "Whose feedback was most useful to the children: the whole class feedback given by me, or the student-to-student feedback the children gave each other?"

Hattie (2012) identifies three different levels of feedback that help advance student achievement: task/product feedback, process feedback and self regulation. 

Feedback at the task level helps students determine if they have completed a task or if a response is right or wrong. It helps students build 'surface knowledge.' 

Process feedback concerns itself with the thinking processes a student must use in order to complete a task including providing students with strategies to identify errors and explicitly teaching them how to learn from their mistakes. 

Feedback that focuses on self-regulation provides students with the skills to self-monitor and to achieve a deeper level of learning.

When students can monitor and self-regulate their learning, they can use feedback more effectively to reduce discrepancies between where they are in their learning and the desired outcomes or successes of their learning (Hattie, 2021 p 134).

 This form of feedback is most often in the form of reflective or probing questions.

Then came the realisation that it was the lesson in its entirety that had enabled the students to achieve a high level of collective success: the collaboration between teacher and students used all three levels of feedback as identified by Hattie.

• "What I can see..." related to the students' work as they attempted to complete the task and involved identifying which elements were present/missing. My intended post-it notes to give students directions that would enable them to complete their work are an example of feedback at this most basic level. By removing them and replacing them with our whole class discussion, students were able to revisit the task instructions then collaborate to ensure that everyone completed the task without written scaffolding from me.
  
  
• "What it makes me think..." related to the thinking processes that a piece of student work inspired. For example, as students read each others' clues, they were engaged in evaluating the effectiveness of the clues. Our class discussion gave students the skills to be able to do this effectively for themselves and each other.
  
  
• As they asked, "I wonder..." questions, students engaged each other in self-regulatory feedback because they challenged each other to reflect upon other strategies they might use to improve their work.

Implications

This documentation shows the positive impact that teaching students to give each other feedback can have on their work. Revisiting the completed pages of the book at the beginning of the lesson not only accorded value to student work already completed, it also gave students another opportunity to rethink their own work in relation to the learning goals. By using completed work samples to model the feedback routine of See- Think- Wonder, students had a tool that they could use to respectfully critique each others' work that traversed all three levels of effective feedback.

Without the evidence of recorded conversations it's hard to judge whether students' feedback to each other followed the thinking routine. This makes it hard to separate out which specific parts of students' successful editing was due to the additional input from me at the beginning of the lesson, and how much to the feedback from other students. Either way, it is clear that the lesson on giving feedback lead students to create work of a high standard. 

The feedback I had planned to give focused on only one or two elements of the task. However, when the responsibility for offering feedback on specific work was given back to the students, the quality of editing went beyond my expectations. Students gave greater consideration to their readers as they completed all parts of the task and added or edited clues that would help them infer correctly, and changed text structures so that they more closely echoed those of our mentor text.

It also appears that the feedback students gave went beyond the elements of the task that were explicitly taught, to include aesthetic and social considerations. Students redrew, added and coloured images. They valued the opportunity to think critically together: to improve the standard of their own work and to offer feedback that would assist their peers. Their sense of shared ownership of the edited work showed that they often saw the act of feedback as also an act of collaboration.

As I reflect on this from the vantage point of 2021, I find myself asking new questions:

• What did the students giving feedback learn as a result of their experience? (Self-regulation)
• To what extent might students who received feedback be able to identify how it helped them? (Process)
• What did students learn about how to provide effective feedback to each other? (Task)

If only I had been able to listen in to their conversations!

Reference list

Hattie, J. (2012). Visible learning for teachers: Maximizing impact on learning. Routledge/Taylor & Francis Group

Rainy Days and Student Surveys

I wrote the following blog a while ago but waited to publish it until now to allow time to put distance between the evaluations and the students who gave them. As student evaluation time is coming around again, now seemed the right time to share it.

It turns out that bad weather is good for encouraging deep thinking. Which is just as well, because student evaluation surveys have the potential to make me feel a little under the weather.

Last semester I taught a number of different courses in science education across multiple campuses to preservice teachers including First Year Bachelor of Education (Primary) students and Fourth Year Bachelor of Education (Primary).

At the end of the semester, all students are encouraged to complete surveys. I know the responses are collated and emailed to all the lecturers and tutors, and I suspect they go much further afield too because we use them as part of our quality assurance processes.

The email with survey summaries for last semester's courses arrived today, complete with handy graphs. Here are two of mine:

Table 1. Evaluations by 4th Year Bachelor of Education (Primary) students

Table 2. Evaluations by 1st Year Bachelor of Education (Primary) students

 The first thing I noticed was that the responses of the first year students are overwhelmingly positive (Table 2), while the fourth year students are much more ambivalent about my teaching (Table 1).

• Is this because the first year students have lower expectations and the fourth year students are more discerning? 
• Was the content of the tutorials clearer in my mind for the first year course? 
• What criteria do students use when making these judgements? Are the first year students comparing me to their high school teachers while the fourth year students compare me to other tertiary teachers; or do they have some sort of criteria that they apply to all teachers? 
• Not all students responded. How would the results be different if there was a higher response rate: better or worse?

As well as responding to questions on a Likert scale, students are invited to make comments. The comments I received are such a mixed bag: some say my teaching was the best thing about the course; others say it was the worst thing and that I didn't know what I was talking about most of the time! It's hard to hear some of their comments. At times it's because they obviously come from a place of pain. On other occasions it's because they are painfully true.

I feel the danger of wondering how closely students' evaluations of our teaching reflect the marks they achieved. While the responses are anonymous, I sometimes imagine I hear the voices of the students behind them because I have already heard these concerns; but perhaps instead it is someone else making the same comment. The university's support website reminds me:

In addition to student feedback, teaching staff should draw on a range of evidence, including peer and self-reflection, and current literature on on effective teaching and learning.

I try to keep all of this in mind as I plan how I will teach this coming semester.

Some students find it hard to use the professional tone that teachers aim for in report comments: that even-handed voice that tries to give an objective, measurable account of a student's understanding rather than offer personal criticisms or praise. Instead I can hear their anger, their frustration and their disappointment through their comments about my teaching. (There are positive comments too - thankfully - but it's the negative ones that I feel the greatest need to understand.) I try to stand above the emotion to see the deeper message and to learn from it. Sometimes the comments are so cruel that it's hard to take the criticism at face value. It reminds me of the need to give feedback to students that is kind, honest and professional so that they will be more receptive and be better able to use it to improve.

My personal research tells me that preservice teachers will teach in the manner in which they were taught. In my classes, I try to model for my students that as a teacher it's OK to say, "I don't know the answer to that yet," because this is what will happen to them in the classroom too. Teaching and learning go hand-in-hand and I want them to expect that they will need to learn alongside their students, just as I am learning alongside them. I think I made this point often. Still, some students criticise me for not responding immediately to their questions about science. However, I also acknowledge that I did not always follow through: I did not always go back the following week with answers I had promised. I'm going to have to put in place systems to help me track my promises and ensure that I improve.

Others complain that I didn't give them immediate feedback on their presentations like all the other tutors, even though I explained that I needed time to check ideas they shared that were new to me so that my grades were fair and accurate. I wonder how I can help them see that teachers need thinking time too?

Students do not have to make comments but many do and there seems to be no set length. The long, impassioned efforts of students to critique the course cut me to the core because I can see them: I can see our students and their desire to do well, I can see the effort they put into their work, I can see their yearning to help us understand where their difficulties and frustrations stem from, so that future students may benefit from their insights. I can see the teachers they are striving to become.

I can also see the children that they will one day teach, I can see the colleagues they will work with, and the parents and administrators they will have to be accountable to. As tutor to these future teachers, I feel my responsibilities towards all these groups most keenly. When I mark their assignments, I feel the responsibility to ensure that they know enough to start with, because I know that they will continue to learn and know more the longer they teach.

In this moment, I see myself reflected in their feedback and know that I, too, will continue to improve; and I know that in this moment, I am good enough to be their teacher.

• How do you solicit information from your students about your teaching? 
• What filters do you apply as you seek to interpret their answers?
• What comments have help you the most to improve your practice?

From Learned Facts to Playful Maths: my Mathematical Autobiography

Having read the maths autobiography of my friend Amie Albrecht and inspired by those written by many others, I have finally taken the risk to publish my own.

Early maths experiences
Other than starting school learning about sets, I don't remember much about my early maths experiences; although I do remember my dad having a small box of cuisenaire rods in his study that I liked to play with, and I vaguely remember a pack of maths flash cards that were a partner pair to the sight word flash cards. I remember having some problems learning how to read, but none with maths so this can't have been an issue. (I do remember that my younger sister used to always miss 13 when she was counting until Mum reminded her that her birthday was the 13th May, so she better remember or she might miss out on presents!)

Our family is good at maths
There was a family history of being "good at maths." My mum used to work in the local TAB betting office in the days before computers or even calculators. She always claimed that she could have a conversation while adding up two rows of figures and still get the right answer at the end of the page. Someone even picked an argument with her once while she was tallying the balance sheet and not only did she get the right answer, it didn't even slow her down. (Or so the legend goes!) Mum liked to play the odds and would sometimes have a punt solely on the grounds that there was no-one else betting on a horse and if it did win, she could score a lot of money for very little outlay.

My paternal grandfather was a supervisor in a glass factory on week days, but on weekends he was a bookie's penciller at the local racetrack. It was his job to keep track of the bets and manage the risk for the bookie by continually calculating and readjusting the odds. By all accounts, he was able to do these complex calculations in his head. Dad has a disdain for gambling: his father taught him that bookmakers ensure that the odds are always in their favour.

School maths: get it right, do it fast.
My first memory of maths in Primary school is from Year 3 when Sr Pauline would put a whole stream of sums up on the board and show us how to do them. Once I'd seen where she was going, the only challenge was waiting for her to move out of the way so that I could see the numbers she was blocking with her body. I think she found it quite frustrating that she had only just finished explaining the first problem and I'd already finished the set, so she stopped setting out the board ahead of time and I had to wait for her to write each one. At least that way we finished together and she didn't need to occupy me for as long before everyone else finished.

In Year 4 we would have a maths warm-up each day where we would line up in two parallel lines down the classroom and approach the teacher in pairs. Like an elimination round in a gameshow, each pair would be asked a maths fact. The first partner to give the right answer went to the back of the line and stayed in the game, while the losing partner returned to their desk. I don't remember ever winning, but I usually made it through the first round at least. My report of that year states, "[Sally] has an accurate knowledge of tables though her delivery is a little slow." Perhaps that's why my parents got me one of the first ever electronic teaching tools for Christmas that year: The Little Professor.

It was marketed as a "backwards calculator" in that it would randomly generate a mathematical expression and the user (me!) had to provide the answer. If the answer was incorrect, it would display what I thought of as a set of "three backwards 3s" but was in fact 'E E E' for "Error Error Error." I might have misunderstood the exact reference, but I knew it meant I'd got the wrong answer. It was one of the earliest efforts at a digital teaching tool.  There were three chances to get the correct answer, but if you still got it wrong, it would tell you the correct answer. At the end of each set of 10 questions there was a score: the number answered correctly on the first attempt. There was no time limit however - or not that I remember anyway - just the jolt of seeing the backwards 3 3 3 when an answer was incorrect. My nemesis was always 7 x 8. Was the answer 54? or 56? (Even today I have to think this through: it's as if the correct answer just won't 'stick' in my mind.)

My Year 5 teacher was newly minted and also used the pair strategy to regularly test us on our tables. I clearly remember a fellow student who found school difficult and was often the first person eliminated. In the way that kids do, people used to push to line up next to him because they were assured of at least making it to round two. I remember him getting very upset and frustrated one day, and our teacher telling him that anyone could be good at tables with enough practice. His parents must have heard the message, because he came back after the holidays and we knew something had changed. Darren not only avoided elimination in the first round, he won! As a class we were stunned the first time it happened, and I even remember spontaneous applause as we recognised a fellow student who had overcome a monumental personal challenge to achieve a seemingly impossible goal. From that point on, he won every time we played. The dynamic quickly changed as everyone tried to ensure that they weren't matched up with Darren, because it was a guaranteed exit in the first round.

High school maths: more of the same
I attended an all girls high school from Year 7 to Year 10 and there were about 90 students in my year. Mrs Webber taught the top maths class. She was a diminutive, soft-spoken woman who's response to classroom noise was to whisper - and it worked. In my first ever high school maths half yearly, Mrs Webber explained that she had planned the marking so that all the scores would add up to 100 to make it easy to convert our scores to a percentage. But during the marking process she had realised they added to 99. So, she had decided that our final score would be whatever mark we had plus one mark to take it up to 100. Her plan was to give us all an extra mark 'for free' to make her life easier. It's the only time it ever happened, but I can lay claim to scoring 101% on a maths test. I was happy, but it didn't win me any friends.

I consistently topped the grade in every maths test during those years, but that was the only exam where I got full marks. When we would get our tests back, my friends couldn't understand how I could be disappointed with a final score that they yearned for; and I couldn't understand how they could be happy with just a pass. I also felt that I needed to hide my excitement when I did well. Instead of asking, "How did you go in the exam?" I eventually learned to ask, "Are you happy with how you went in the maths exam?" which allowed us all to express our joys and our disappointments on a personal level. And I stopped sharing my test scores.

My teachers figured out a way to use all the time I had left over in every maths class: I became a second teacher and would roam the class helping my classmates. We did a unit on circular geometrical proofs and I remember figuring each one out in my head, then helping others to see what I could see. I did this for the whole chapter and the teacher never checked my work once so I got away with never actually recording any written proofs.

I changed schools in Year 11 and found myself no longer on top of the heap. My Year 11 maths teacher was a former Kindergarten teacher. While I loved her, I found Mrs Tully very frustrating. I'd call her over to ask a question about one small aspect of a problem, and she'd start at the beginning and work her way slowly through the entire problem. When I complained(!) she told me that she needed to do this for herself.

I wasn't coming first but I did well enough to be allowed to take on the hardest maths course in Year 12. This meant that four out of my eleven units of study were maths. We started classes at the end of Year 11, just before the summer holidays and Mrs Rapp gave us Imaginary Numbers to learn over the holidays because it was "easy." It wasn't easy for me, and for the first time, I really struggled with maths.

A new way to learn: thinking like a mathematician
Mrs Rapp was unlike any other maths teacher I'd had before. Our maths classes were less about a demonstration followed by practise, and more an open class discussion. We derived our own methods - and then did the practise for homework. She challenged us to find clever, faster, more insightful ways of solving problems. Grinding through things was to be a last resort. Think first, then solve. In our discussions she would allow us to suggest approaches and help us work through them to find a solution. But she didn't always follow through student suggestions, especially if she knew it wouldn't go anywhere. As soon as she'd gone far enough that most people could see it was a dead end, she would stop and move to something else.

The people in my class that year were extraordinary: the girl who topped our year came 3rd in the state and 2nd in 4 unit maths. My classmates are now specialist doctors, psychiatrists, journalists and mathematicians. I did my best to keep up, I tried hard to be insightful and find a faster, better approach. Sometimes I was the person who found an interesting approach. Often my ideas would lead to dead ends that everyone else saw before I did. Much to my dismay, Mrs Rapp would move on without me seeing where I'd made a mistake. My saviour was my best friend, Nicole, who would patiently sit and teach me. She would always talk through at recess what Mrs Rapp didn't have time for in class.

Tertiary mathematics
I studied Pharmacy for a year at university. First semester was a repeat of much of the content of 4 Unit maths covered in lectures at twice the speed and devoid of discussion. It was revision for me but all new for my friends so it was my turn to tutor them.

Second semester was statistics, the only branch of maths I ever hated. There were so many terms whose meaning seemed so similar, and all these 'squared tests' that I never saw the point of. I just scraped through and finally understood what it meant to be grateful for a pass. Then I made the decision to swap pharmacy for an arts degree and fell in love with philosophy and history.

I didn't study maths again until I started my Masters of Teaching degree more than 20 years later. Our tutor was a former primary teacher who talked a lot about teaching maths. His stories were interesting and in hindsight, I learnt a lot from them - but we were very frustrated that we didn't get to do very much maths ourselves. Even the major assessment was an essay. When were we going to be asked to do some maths?  Our final exam was a group practical assessment, with manipulatives all laid out around the table. We were able to discuss ideas in whispers within our group and were marked as a collective. It's hard to bow to peer pressure when the other two members of your group are convinced that their answer is right and you're equally convinced that yours is right!

Learning how to teach maths
I don't think I learnt how to teach maths until I was in the classroom. I was lucky to go to a school where the maths pedagogy continued to evolve while I was there. I learnt how to design open-ended tasks although at first they were written into a 'contract' that students worked through at their own pace. Every kid in the grade did identical pre- and post- tests to measure if students had learnt anything, rather than to modify our instruction. I knew it was hard for the kids who struggled, but that's what we had to do, right? While we did a whole class warm-up, there wasn't an expectation that we would engage students in a whole class reflection. In the Year 5 & 6 classes, kids were separated according to ability across the four classes - always a disaster for the kids in the bottom class.

Early in my teaching career our school was involved in a research project using the DS Brain Training program. In the beginning, the kids were given a page of simple equations using all four operations. They did them as quickly as possible and recorded the time taken to finish the page. Then we were given a class set of Nintendo DS machines. Every morning, the kids would spend five minutes on the DS playing Brain Training. At the end of the year the kids redid the paper test. The results?  They were definitely faster at finishing - but still made the same mistakes at the end of the year that they had at the beginning. Their speed had improved, but they didn't know any more than they had in the beginning. So much for Brain Training!

The next year, I planned my first family maths task: a plastic jar full of small chocolate easter eggs with the lid super-glued shut, another identical jar with only ten eggs, and a journal for students to record their guesses. Kids would take the kit home overnight and figure out a way to use the loose eggs and the empty jar to estimate how many eggs were in the full jar. Each kid (and their family) tried to find a different way to solve the problem OR checked someone else's thinking. Our goal was to find an increasingly more accurate estimate, or to find multiple ways of getting approximately the same answer. Kids would share their approach every morning. I still remember the story of the family that went to the local supermarket in their pyjamas to use the scales in the fresh produce department because they didn't have scales at home! At the end of the term we cut to top off the jar, counted the eggs and the kids figured out how to share them equitably.

In the years that followed my professional practice developed in partnership with my colleagues. We started looking for authentic contexts and created PBL-style maths tasks. For example, we did a unit at the end of the year which involved students needing to develop their understanding of multiplication, division and mass. They had to figure out if it was possible for one teacher to carry plastic shopping bags containing all the materials for making tiny gingerbread houses for every person in our Year 3/Year 4 classes (104 people, including teachers.)

Another time we planned a class party. The kids had to do a whole range of tasks related to volume and capacity to ensure that everyone got a cup of jelly, half a cup of soft drink and a treat box of lollies. In true PBL style, they developed a list of "Need to Knows" that included such questions as:

• What's a reasonable volume of jelly for one person? 
• How many packets of jelly would we need to make enough for everyone? 
• What might a treat box of 120 cubic cm look like? 
• How many servings of 125mL of soft drink are in a 2L bottle? 
• What does 125mL look like in a cup? Is this enough?

There were some tasks that everyone did and others where different kids worked on different parts of the problem and then had to justify their answers to one another. At the end of the unit, we held a party!

It became obvious that finding the right context could have a huge impact on kids' willingness to engage with the maths. Because we did these tasks during maths time, the kids had no problem thinking of these tasks as "doing maths" however as a teaching team we had to make sure we took the time to guide them reflect on what they had learnt each day, or the maths got lost in the excitement of the task. Parents were surprised and delighted that it was possible for their kids to have fun learning maths.

In my final three years at this school, I was given the role of an EMU teacher (Extending Mathematical Understanding). This meant that every day for 30 minutes at a time, I worked with the same group of three kids who were considered 'vulnerable' in different areas of maths. I had another 15 minutes to reflect on the learning and plan for the following day. It was such a privilege to help them make sense of their thinking and to develop their confidence and understanding. It also made me realise the importance of exploring the same concepts in multiple contexts and using a range of different materials. For example, just because a kid knows that a bundle of ten popsticks and 3 more is 13, doesn't mean that they can rerepresent this using MAB, or that the quantity should be written as '13' and not '31' or even remember that the quantity is called, 'thirteen' and not 'thirty.'

#MTBoS: Math Twitter Blog-o-Sphere
In recent years, the people who have had the most influence on my experience of doing and teaching mathematics are people I know on Twitter. David Butler's mathematical challenges reminded me of how much I liked to solve maths problems before I encountered statistics and was scared away; and playing with David, Paula Beardell Kreig and others to explore geometrical concepts is so much fun! Tracy Zager's blog posts and her most excellent book inspired me to be a better teacher of mathematics. Christopher Danielson's ideas, books and maths toys have been shared and enjoyed by family and friends as well as students and colleagues. Amie Albrecht is the applied mathematician I secretly wish I'd been brave enough to be, and her insights into maths, teaching, people and books inform and challenge my thinking. Working with Simon Gregg on our #PatternBlockProject gave my students an authentic audience for their thinking and helped them find their voices as mathematicians. Charles T. Gray's journey towards world domination of R-stats and ongoing encouragement has made me wonder whether I could possibly go back and master statistics after all.

There are so many people who make up the MTBoS community who have shared their ideas, insights, resources, fears, beliefs, pedagogical approaches, successes and amazing mathematical creations. You have helped me see myself as not only a maths educator, but an amateur mathematician. Thank you.

Leveraging feedback

I wish I worked in a system where I could get paid to give feedback but not grades.  Dylan Wiliam cites research that students who receive just feedback grow more substantially than those who receive just grades. Give students both at the same time and they tend to focus on the grade at the expense of the feedback. If we want students to pay attention to feedback and use it to grow, we are better off not giving them grades.

I wish I worked in a system where I could give feedback but not grades. But I don't. When I raised this problem with Wiliam at a conference earlier this year, he suggested a possible solution: give the feedback first, then give the grades a few days later.

Sounds great; but it's not something I have the power to implement. At my university at least, grades and feedback are delivered as a package.

This semester I was given the opportunity to co-write a new course including the assessments with my good friend and colleague, Stella. We wondered,

"How can we create the right conditions within our context so that students give greater attention to feedback?" 

As a minimum, we wanted them to at least read the feedback we gave them. But what we really wanted was for them to think about the feedback, seek to understand it and apply what they had learned.

We split our major assessment into two separate but related parts, each with its own marking rubric.

• In Part 1, students would complete a task and we would provide feedback and a grade. 
• In Part 2, students had to reflect on Part 1 and submit a written piece of work as well as deliver a verbal presentation to their peers about their learning.

We told them that in Part 2, they would be marked on their ability to critically reflect on Part 1 in the light of their feedback. They didn't have to agree with our comments to get a good grade: we made space for them to be critical of our feedback just as we asked them to be self-critical. We knew that it was possible that some students would only tell us what they thought we wanted to hear; but even so, we would know that at least they had read the feedback.

All went according to plan: the students submitted Part 1; we graded them, provided written feedback and a grade, and returned their work. We acknowledged their efforts, asked them to be open to our feedback and told them we were looking forward to Part 2. We gave them one week to write and submit their written reflection. To show them how much value we placed on self-reflection in the light of feedback, we set aside 10 minutes per student (1/4 of all of our workshop time for the semester) for students to present their reflection to their peers in subsequent weeks.

At first, those students who didn't do as well as they had hoped in Part 1 were worried about sharing their work with their peers; but we made it clear that they were not required to share their grades, and that any comments markers had made were theirs to share - or not. Stella and I feel strongly that even when students don't do well, they deserve to preserve their dignity. This wasn't about shaming; it was about creating a community of reflective practitioners. We reminded all of our students that no-one had received full marks, and no-one had received zero. Everyone had something to offer, and everyone had something more they could learn.

Over the past few weeks we have had the privilege to hear and read those reflections. 

Every person went about the task in a different way, but all of them responded to the feedback. Some students echoed back the feedback they were given - almost word-for-word - and then chose one or two points to elaborate on. Others have redone whole sections of their assignment and presented "before-and-after-feedback" versions of their work. As markers, we can clearly see the extent to which our students read and thought about our feedback, and used it to further their understanding.

There were some unexpected consequences: in the past, students have wanted to argue about the marks they received for their work. But by making everyone's work visible, those students who didn't get the mark they were hoping for were able to see why. There have been students who have asked for clarification of some comments but no-one has asked for their work to be remarked.

As an added bonus, the level of honesty and vulnerability our students have chosen to reveal to each other about their learning, coupled with their commitment to make a safe space in which to share those revelations, gives us great faith in the future development of these amazing people as reflective teaching practitioners.

Letter to my students

Dear students,

We know that you worked really, REALLY hard on this assignment. We know you agonised over your fair test, striving to write a testable question, to identify independent and dependent and control variables. We know you were thinking about how to make the best use of what you learnt in your last science unit to do it *just right.* (We even know when you fudged your results - because you left it too late or the plants didn't grow the way you thought they should or you only bought enough materials to do one repetition.) 

We realise that for many of you, Aboriginal and Torres Strait Islander histories and cultures are not something you are familiar with; but it's important so you took the time to learn more and to find connections with your topic wherever you could. Ditto sustainability. 

We know you spent a lot of time poring over the syllabus, trying to figure out which concepts to address and which skills were relevant for which Stage. We appreciate the effort you put into identifying alternative science conceptions. We recognise that you put considerable effort into creating and drawing and finding images that would engage students while also helping them understand science concepts. We get that it took time to prune your text down to fit the word limit so that every single word was important. We can see where you searched long and hard for relevant resources and worked to decipher APA6 formatting when compiling your reference lists. 

We accept that putting all of this together into a picture book required a HUGE effort.

You may not realise it yet, but we know that your effort was worth it. We know you will have learnt a lot about a great many things. 

         "But," we hear you say, "if I have come so far, why do I feel that there is still so much to learn?"

(Spoiler alert: teaching will always be this way.)

We want you to do well! 

This is why we have taken such time and effort to give you feedback: to help you recognise the things you did well and to identify the places where you fell short - or fell over. We want you to know yourself and to know what to fix, where to go next and even when to start again. 

Just as you learned a lot from making the book, we hope that you will learn from our comments. 

Be open to our feedback. It's far more important than your mark because it shows you how to grow.

We believe in you. 

We trust in your ability to work hard and to learn from your mistakes. We know you are capable of great things because already we can see how far you have come. 

No matter what happened in the first part of this assignment, know that we are eager to see you share your thinking with others in Part 2. They too know how hard this was, they know what you've been through and can't wait to see your work. Know also that each of you has something to offer your fellow students that they need to hear. 

We can't wait to see how you will continue to build your understanding of science and how to teach it. 

You've got this. 

We know you do. 

Thinking about thinking

This is a blog about thinking. More importantly, it will be concerned with thinking about thinking, and how that can advance learning.

It's not a Maths blog, but it will contain reflections about mathematical thinking because I'm a teacher who has trained as a maths specialist and I also work as a maths tutor to preservice teachers at two Australian universities.

It's not an English or literature blog either, but there will be discussions about literature because I'm a school librarian and I love using picture books as a springboard for thinking.

It's not a Science blog, even though I did a year of a pharmacy degree and have always loved teaching science: to my primary students, my fellow colleagues and even now in my role as a science tutor at ACU and at Macquarie University. But there will be posts about what it means to learn to think scientifically.

I've spent many years working in museums and historic houses across Sydney, so there will be thinking about what it means to undertake historical and geographical inquiry; but it's not a social studies blog.

I like to use technology in my teaching: to capture ideas, promote collaborative thinking and encourage reflection. There will be posts about using technology to support thinking; but it's not a technology blog.

I've experimented with Project-Based Learning, Philosophy with Children, Reggio-style documentation, Inquiry-based Learning and Project Zero's Visible Thinking Routines. All have made me realise that the children we work with are much deeper thinkers than Piaget gives them credit for.

For this reason, my blog will be devoted to uncovering, understanding and making visible the thinking of others. I'm also looking forward to seeing these ideas through your eyes. In fact, this is both the most exciting and the scariest part of sharing this blog.

It will also be about making my thinking visible to me. Writing has always allowed me to find order in my ideas, to give substance to the myriad of thoughts that float around and only come into proper focus when I tie them down.

It's a blog for thinking aloud, thinking again, asking questions, presenting arguments, exploring solutions. It will range across the thinking of students, colleagues, and self-reflections but always with the same purpose: to explore thinking and its relationship to learning.


This blog is inspired by my encounters with the ideas of other people who have taken a risk and shared their thinking with others, including the wonderful people from #MTBoS. With special thanks to:

• Tracy Zager, who's blog Becoming the Math Teacher You'd Wish You'd Had has inspired me since its very first post and continues to challenge my thinking. 
• Amie Albrecht who is a kindred spirit and writes an insightful blog that crosses the boundaries between tertiary, secondary and elementary maths. Thank you for encouraging me to make my thinking visible.  
• David Butler whose provocations on Twitter and subsequent blog posts made me remember how much I love doing maths for its own sake.
• Daphne who showed me the joy of starting a conversation with people you don't know about things that are important to you. I'm looking forward to your post about Sydney!
• My husband who has had his own blog for many years now and keeps reminding me that it doesn't need to be perfect. Thanks for all the moral and technical support! (NB I reserve the right to change my blog template until I'm happy with it.)
• My colleagues, the children and parents across the various schools where I've worked and continue to work. I look forward to sharing our thinking with others.

A special mention to the 4th year students in my science and maths ttuorials who posted photos of experiments and maths activities in my classes with the hashtag #ACU_edu (under the leadership of David Lee). Thank you for tagging me too. Without you, I would not have known how to engage with Twitter and would have missed out on so much learning and fun!