Tuesday, 12 May 2015

Feedback / Assessment Revolution

Viva le Mathematical Mindset Revolution!!!


Inheriting a new class at the beginning of the year and asking them their thoughts and feelings about maths tends to have a similar pattern by the time they reach the upper primary years:

° about half of the class have a dislike for maths

° a third have been lead to believe they are 'bad' at mathematical thinking

° a few enjoy it

What happens in a child's primary school history for them to so quickly form this misguided and often negative self reflection of themselves as mathematicians? What are we doing wrong?

Ask a class at the beginning of the year their thoughts and feelings about science, and we find a very different pattern:

° you might have an odd year when 1 child in your class says they don't really like science

° the rest of your class loves it

Such huge differences and yet, mathematical thinking is very similar to scientific thinking. When we are thinking mathematically we are often formulating a hypothesis in our mind, testing it out and reflecting on its success or failure. Whether we are investigating what strategy is best to apply to a given maths problem or the number of edges on a cube, the thinking process we undertake is the same as a scientist: hypothesis - test - reflect.

As we know, maths IS a science; it is a branch of science like biology or physics. It is also the language of science. They are closely intertwined & cannot exist without each other.

I think if we can present mathematical thinking in our classrooms akin to how we present a science activity, I think we would be raising our students' interest and self-value as mathematicians.   

As it is, we continue to follow the traditional approach to maths where it is essentially a 'right / wrong' subject.  Those red ticks or crosses we habitually flick beside a student's maths' answers, I feel, undervalues the scientific thinking that happened and instead places too much value on achieving the 'correct' answer.   During scientific enquiries in the classroom, we value the thinking processes.  We don't just give ticks and crosses; we give comment feedbacks. So why aren't we doing the same in maths?

Guiding children in maths like they are following a scientific process is a really interesting way to approach it. Throughout the year, we often formulate a hypothesis of a particular strategy we think we should best use, we test the strategy out and then draw a conclusion as to its effectiveness.  As we learn in science, even if an experiment disproves a hypothesis, it is still a success because it shows something negative which still gives evidence and information. Approaching maths in a similar way, I think helps children to relax more with mathematical thinking and alleviates those frustrating moments when we might not choice a successful strategy when trying to solve a problem. 


No more ticks or crosses for the rest of the year.........

Maths isn't a 'right / wrong' subject? 

I discussed this with my class a few months ago to see what they thought.  It was quite a revolutionary rethink for most of them.  

When I then told them that for the rest of our year, I am not going to give them any ticks or crosses, they were curious and somewhat surprised. Our discussion then went something similar to this:

- But how will we know if we got our answers right or not? 

When you do a science experiment, how do you know if your result 
       supported your hypothesis or not?

- We reflect back and can clearly see if it makes sense or not.

- We double check if the results match the hypothesis.

So, since maths is a science, should we not be doing the same?

- Oh, yeah.........

Since then, I haven't given any ticks or crosses in maths activities and the transformation in our classroom has been a slow, but powerful revolution in our perception towards mathematical thinking.  

° There isn't a fear of failure as much as there used to be in their minds.  

° They take more risks by trialling different strategies to see if they are more effective than others.  

° Sometimes, students will deliberately choose a strategy they think is not effective just to find out what makes it a bad choice (amazing!) and then they will compare it to a more effective strategy (without me prompting!). 

° The value has shifted from getting the answer (it's just a minor bonus) to the thinking process.  The trial-and-error that happened to reach an answer is what is being valued. 

° Students seek feedback from their peers to test whether an answer they got could be correct or not. This then requires another student to see a different approach to solving a maths question through that peer's mind.

° When looking at answers to simple number equations, they need to test if their answer is correct by finding a different strategy to help solve it OR they rationalise and evaluate why that answer is plausible or not.

° They are slowly understanding that 'mistakes' in maths are valuable. They help us to become more knowledgeable and help us to deepen our understandings of the concept being explored. 

Viva la Mindset Revolution!!!

Changing the mindsets of children in how to perceive mathematical thinking after years of 'ticks and crosses' being ingrained in them is actually far easier than one might expect.  In just 5 or 6 months, most of my students have had their own personal revolution towards maths.  Most actually enjoy maths now. Most now believe they are successful mathematicians. And perhaps even slightly more important as it creates the framework for the previous two changes, most now think maths is actually really interesting (gasp!).

We can transform children's thinking and attitudes towards maths in a very short space of time no matter how narrowly-valued their history has been.


The recipe for the revolution:

These are some of the key aspects I've found that help revolutionise the attitude & understanding of maths in my classroom:

° Give equal airtime to discussing the concept as you do to the skills. Maths is a concept. Ask your class:  If humans didn't exist, would numbers exist?  It quickly becomes a fascinating discussion and helps them to understand maths is a concept. One could argue it doesn't actually exist. This becomes a huge light bulb moment in children's minds. Maybe it would still exist if humans never existed or maybe it wouldn't. The point is though, in our classes we should be exploring the mathematical concepts with an equal amount of time that we might give to skills or strategies. It's the concepts that make maths interesting and it is this can arouse children's curiosity.

° Always ensure students understand the relevance of what they are doing and regularly discuss how they or someone might use this mathematical thinking in real life. My personal philosophy is if they can't see themselves using a particular math skill or concept in their lives, then it doesn't belong in a primary curriculum. As a Year 6 (Grade 5) teacher, maths curriculums can tend to have quite a few expected elements we should teach that children struggle to connect their lives to. The solution? Just don't teach it. Leave that to the secondary teachers; if the secondary maths teachers deem those elements to be important, the children will pick up those concepts just as quickly and probably far more easily by being older than we can try to instill in them in primary. I feel my goal as a Year 6 teacher is to send them to middle school with confidence, appreciation and curiosity towards maths. When children can connect with maths, they will appreciate it more. Just like adults, if we can't see the purpose in knowing something, our learning of it isn't effective at all. In any maths activity, if you feel you couldn't give a response to 'how would we use this in real life?' that the child asking could connect with, then that maths activity or concept shouldn't enter our classroom.

° No more ticks or crosses given- as tempting as it is and as habitual as it has become stop that red pen flicking; instead give feedback (see below)

° Continually connect mathematical thinking to scientific thinking in discussions throughout the year.  Regularly discuss the 'hypothesis - test theory - reflect on results' thinking process we do as mathematicians.

° Don't focus on asking what answers they got; ask what strategy did they use to get an answer and if they thought it was the most effective strategy they could use for that particular question.

° Have students create their own activity sheets for number practise.  When children need to practise number patterns or number strategies etc (the main reason we feel compelled to give a worksheet), have them create their own questions to solve as a home learning activity.  This gives them ownership of their learning. It will allow students to set questions for themselves that will challenge their own thinking because they know far better than us what level of understanding they are at. In addition, and perhaps most importantly, by creating questions based upon a strategy you want them to master, they have to really think how that strategy actually works in order to create a question that they will need to use that strategy for. Their thinking has just deepened a lot and the understanding of using that number strategy would surely stay with them far longer than any number of activity practise sheets we can photocopy and fling at them. 

° Any sort of number practise or reviewing, have them do for home learning (homework). If they don't need peers in their maths group or a teacher to discuss the activity, free up your classroom time by assigning that type of learning to be done at home. We are then also respecting those students who need a bit of additional time to compute numbers by allowing them to do so at their own pace at home without a time pressure they feel in the classroom. If they can comfortably do maths number skills at their own pace, surely that is effective learning rather than feeling compelled to complete another activity sheet in class time whilst observing others who finish it faster than them. 

° Stop using the word 'work'.  We aren't doing work when we are learning, we are learning so use the word learning or thinking instead. It gives a more positive portrayal of what we are doing and takes away the notion we are doing a 'task' like doing the laundry or washing the dishes. We want children to enjoy learning, but we negate that by using terms like 'do your work' or 'worksheet' or 'who has finished their work?'  They aren't working, they are thinking and learning and those are positive things! :)

° Continually praise 'mistakes' as a friend to our learning. 


Would we give students the same sort of feedback from a science activity as we do for maths?

I have thought about this a lot in the past few years and have figured, if I use the PYP Learner Profile Attributes & Attitudes to give feedback and assess my students in science or units of inquiry, why shouldn't I do the same with maths?

So, that's what I do now and it completely transforms both my students and my own approach to maths. They aren't given feedback that tells them they are a strong or weak mathematician like maths tests or ticks and crosses do. 

Instead, they now see themselves as being responsible for the learning and continually developing a better understanding of what we need to do to become more confident mathematicians.  

So, here is what I have been experimenting with:

Using PYP Learner Profile & Attitudes

This year, with the 'no ticks or crosses' revolution underway I have continued experimenting with using the PYP Learner Profile and Attitudes to assess and give feedback to my students in maths.

The more I've been doing it, the easier and faster it becomes to create simple rubrics to give as an assessment or as feedback from a maths enquiry or investigation they have done.

When your children are doing a mathematical investigation, which Learner Profile attributes would you hope they are developing?

Image from: issh.wikispaces.com

Better still, before beginning a maths enquiry, asking our students which they feel they will need to develop will empower them to further understand why they are about to do that activity.

Which attitudes would we hope they are displaying in a maths enquiry?

Image from: www.sps186.org

Again, asking our students before commencing will instill in them what is expected of them and why. 

Either when I select a few Learner Profile Attributes and Attitudes or when my class and I select some (it is a time issue), we tend to gravitate towards using:

° Communicator - ability in communicating the thinking processes

° Risk-taker - willingness to explore different strategies even if they might not
                     be effective or helpful

° Reflective - takes time to reflect on the strategies used to see if they were the
                    most effective / uses different strategies to see if an answer is 

For the Attitudes, I often use:

° Committed - challenges self by choosing maths investigation that suits own
                      level / tries best / challenges thinking by exploring concept 

° Creativity - experiments with different approaches in investigations and in 
                    ways to show thinking processes

Then it is just a matter of putting them in a rubric.  I use four columns:

    emerging     developing     consolidating     expanding

After assessing how well they demonstrated selected Attributes or Attitudes, I then add a brief comment below that focuses on their thinking processes.

Some feedback examples:

Measuring Elapsed Time Investigation                                  Mathematician: XXX

Able to identify multiple strategies for measuring elapsed time in years, decades, centuries and millenniums

Ability to visually show mathematical thinking in a clear, neat and well-spaced out manner


XXX, you showed some good thought in spacing out your timeline, though you should have rethought how close the year 1656 is to year 1 compared to how close you had placed 600 BC.  However, your decision to spend extra time redoing your timeline so it was better spaced out shows your wonderful attitude in wanting to achieve your best in your learning. It’s a great attitude you have! You show you have a strength in determining how to use different mental strategies and analysing which you personally find more effective.  A very good investigation and an excellent effort in presenting your thinking neatly. 

Maths Enquiry Assessment                                           Mathematician:  XXX

                           Function: How we convert metric units of mass

Enquiry: To investigate how we convert mg to g, kg or tonnes according to self-assessed level of understanding.  To deepen understanding, try to explain the place value meaning of using decimals and what happens to the place value of digits when we convert.

Learner Profile Attribute: Inquirer
-          Ability to conduct own mathematical investigation successfully
Learner Profile Attribute: Communicator
-          Ability to effectively communicate findings and understandings

You were able to explain the place value reasons well XXX. Next time when you are investigating, try to challenge your thinking a little bit more.  You are a strong mathematician so I think you could have investigated converting mg to kg or perhaps g to tonnes. However, I know you were wanting to really gain a deep understanding of why rather than how we convert which is impressive. Whilst in Year 6, it has been remarkable how much progress you have been making in maths- both your conceptual understandings of the topics and your skill building XXX.  You just need to focus a bit more during our discussions to help you take yourself to those higher levels of understanding you are more than capable of reaching. 

Measuring Elapsed Time Investigation                                          Mathematician: XXX

Ability to visually show mathematical thinking in a clear manner


Ability to clearly explain thinking process (and neatly so it is easily communicated to the reader)


Willingness to select an investigation that will challenge according to own level of understanding


XXX, this is a very well presented and clearly communicated investigation. You conducted your investigation in a logical and effective way. You showed how you experimented with more than one strategy and could determine which was more effective which was great. It was also interesting how you thought about simplifying your number line to make it easier for the reader to comprehend what you did. Spacing out your writing and number lines could have made your paper a bit easier to read.  Try to remember this in future activities. It was great to see you choosing a more complicated question to solve. This shows your great attitude in wanting to challenge your thinking and achieving your best! 

1 comment:

  1. Dear Graeme,

    I love how you are changing the thinking of your students by comparing science and math skills. Maths is so much about attitude, I find particularly in the upper grades where sadly many students feel they are already bad at maths. The way you provide feedback for your students is fantastic. I also try to use open ended assessments that provide students with personalized feedback. Wiggins & Mc Tighe (2005) speak of the importance of assessment being personalized and relevant. From experience, I find that when students are given a number grade that is all they focus on, however with personalized feedback they take time to think and set goals for the future. I love how you have combined the learner profile and attitudes into your assessment for maths, you have inspired me to attempt this for our upcoming school year. I am a 5th grade teacher in an international school in Indonesia, also working under the PYP. May I ask how do you report information to administration and parents if you do not grade students work? This is a challenge that I face in my current school. I feel grading has a negative impact on my students, however for reporting am required to have evidence of learning.

    Thank you for sharing such an outstanding blog. It is truly wonderful to read about so many of your fantastic approaches to mathematical thinking. You are an inspiration.

    Wiggins, G. & McTighe, J. (2005). Understanding by design (2nd ed.). Alexandria, VA: Association for Supervision and Curriculum Design.

    Warmest regards,


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