Maths Reflection Diaries

Maths Reflection Diaries

I have been uber excited about this idea I thought I'd trial with my class this year.  I wanted to find a way that would honour the learner's need to reflect and process their thinking in a free way that works best for them and at the same time could become a useful assessment tool for me to gain a glimpse of what is going on in their mathematical minds so I can better help them in their journeys.

As we all know, as teachers, we take risks all the time in trying new ideas- lots of flops and sometimes amongst them some great successes.  At the moment, I am feeling this 'maths reflection diary' is going to be a wonderful success.

The idea is pretty simple, I cut in half some exercise books ( I figure if the space the children reflect in is small, then it is less intimidating- there is that sense of satisfaction when we feel we use up a whole space) and on the front taped some thinking symbols we could trial:

We have discussed the John Dewey quote, but as reflecting in maths is new to these learners, we didn't go into so much depth about the meaning behind the quote, but in a month or so after they have gathered more diary entries, we will revisit the quote and see how our thinking about it has changed.

After a maths learning experience a few times a week, we spend 5 minutes reflecting about the learning and thinking we have done.

We choose symbols from the table to help visualise the reflecting we are doing. We are encouraged each time to remember that visualising and creativity in maths are key and some children are picking up on that are drawing their thinking, but most at this stage are still more comfortable in writing sentences with the symbol beside each.  That is where they are at and that is perfectly fine.

From this simple routine, I am gaining so much insight into each child's mind. I am able to see where they feel they are being successful and where they may be struggling.  But more importantly, this routine gives each child the valuable time to actually reflect on their learning and it also gives the a voice that they might not otherwise have in the classroom if we are doing whole class discussions. Those of us who may be less courageous or even introverted have a platform to also share what is going on in their mathematical mind. additionally, it gives them a valuable opportunity to think about who they are as a learner and to process the concepts being explored.

Furthermore, it allows me as the teacher to create a dialogue with each child.

We have only started or diary reflections this week and yet there is so much rich thinking and future potential they could take them.

NB: The fold their paper in half so each time they reflect in their diary is half a page.

This student's creativity and need to visualise in mathematical thinking shines in his entries:

Sometimes it can really valuable to see misconceptions some might be nurturing:

I feel I am building a stronger relationship with each child by giving them the daily written feedback and I can tell that because they know I will respond to their diary entry on the same day, they are progressively putting in more thought and some who are reluctant and taking risks to experiment with visually representing their thinking. 

In  a few weeks, we will discuss ways we could improve our reflection diary- perhaps we could add or improve on the symbols or any other ideas they feel might help deepen their learning through reflection. I'm sure they will come up with even better ways than I have.

Reading and responding has instantly become one of my favourite parts of the day. :)

Here is the link to the cover page if you'd like to have and print:

Cover Page Link

The Value of Pre-Assessing & Using a Central Idea

A solid maths central idea can generate powerful, student-driven enquiries.

To introduce our new unit exploring volume and capacity, each child was given a copy of the central idea.

Occasionally I ask what is the purpose of a central idea to help solidify our understandings. Today some students shared that they saw the purpose is:

- So we know what we are about to enquire into.

- So we can decide what we want to find out.

- It gives us the freedom to investigate what we want to find out, but with a direction.

- They help us to understand what the big concepts are that we will explore

- To give us clues about what we will discover.

- To give us a big picture idea of the maths we will be thinking about.


That's pretty stellar understanding I think. And this from a class that at the beginning of our year only perceived a central idea to be a topic to learn. 

To unpack it, we determined which were the key words in it:

We then thought about what we knew for UNITS. Students recorded on their copy any units they knew for measuring volume / capacity.

We then looked at VOLUME and recorded our understandings- what is volume? How and why do we measure it in real life situations?

We repeated for CAPACITY and then for OBJECTS we recorded things people in real life might want to measure the volume/capacity of and why.

Finally, we then looked at STRATEGIES. What strategies do we know? Eg, Perhaps we know a strategy for measuring the volume of a cube etc.

In between some of these, we were reminded that this is a PRE-assessment simply to find out what we already know.  Some of us might have a lot of experience with volume / capacity in previous years and some not.  It's great if you know a lot and its great with you don't know a lot. 

After this, we then looked at those key words and recorded wonderings we have.  

- What do we want to find out?

- What are we curious about?

- To become an expert on our central idea, what things might we need to find out?

A students began recording their wonderings, some began sharing.

This created some really interesting discussions.

One student wondered whether air has volume.

This got us all thinking. Does it?

- We can't see air so it can't have volume. I think an object can only have volume if you can see and touch it.

- When we turn a fan on, we can feel air push against our face. So, because we can feel it, maybe it does have volume but I'm still not sure. 

- I heard that when astronauts return to earth their muscles are weak because there is no air in space. My theory is that air must have volume because it pushes on our muscles on Earth, but not in space.

- Since volume is the amount of space an object takes up, I can't hold air in my hand and measure its mass. I think an object must have mass to have volume.

We coincidentally had a blown balloon in our classroom from a group enquiring into science.

I held it up.

What does this make us think about whether air has volume or not?

- It doesn't show that air has volume. It shows the capacity of the balloon.

- But then the air must be a unit of measurement. If you wanted to blow the balloon up more just before it bursts you would know its capacity. That must mean air has volume.

- Could you measure the air in the balloon? If we could find a way to measure if it has mass, it would tell us if it has volume.

- Why does volume have to have mass? I don't think it does.

- In space there is black matter everywhere between all the objects. I think black matter is like air on Earth. I think you could measure how much black matter there is so that must mean we can measure the volume of air in our classroom. 

More amazing ideas and justifications of theories being formulated were discussed.  We decided to put this on our our wonder wall to come back to and hopefully find a way to get an answer.



Another interesting wondering was whether we can measure the volume of an object without knowing the capacity and vice versa. The student said she felt their might be a connection like there is between area and perimeter. As she explained, we can't find area without knowing the perimeter, but we can measure the perimeter without knowing the area. She wondered if there was a similar connection between volume and capacity.

We liked that wondering a lot and also put it on our wonder wall to investigate later. 


Walking around during their wondering recordings, I noticed how defining capacity was a bit shaky for quite a few students.  I hoped this wondering would up in sharing and it did.  The kids in my class aren't afraid to ask big or little questions- not that 'what is capacity' is a little question, but because we always value any wonderings that are shared the students do feel all their questions are valued.  This is really a key component of creating an enquiry-based classroom culture. If students are intimidated to ask questions, a lot of valuable learning potential is lost.

When a student shared how they wondered what capacity really meant, I asked for a show of hands who else felt that way. Quite a few raised their hand.

It's not often that I can feel like a magician pulling out a rabbit at a perfect moment, but this is was one of them.  I had a slideshow activity to help determine the difference between volume and capacity I had used created and used last year.  I had kept it opened in a tab for when the right moment in our unit it should be used.  This was the moment.

Slideshow:
Is it volume or capacity?

(feel free to copy and use :)    )

We discussed the first few slides and to make it more active, when we came to the images and had to determine whether it was volume or capacity the children raised their hand in a letter V for volume or C for capacity.

It was apparent (and expected) that volume and capacity is tricky to understand.  The liquid scenarios were the trickiest.  

Towards the end and by discussing each, everyone was able to determine a V or C more confidently.  



Pre-Assessment samples:

Looking through the pre-assessments gives me a pretty good clear picture of where each student is at and perhaps where some misconceptions lie. Identifying those gives me a good idea of types of learning experiences I need to create and include to address so those students come to their own understandings rather than being told. 

When it comes time for ability grouping for certain learning experiences I will have a better idea of how to group (Though for this unit the majority of learning experiences will be mixed ability groupings)
The other benefit to this type of pre-assessment is that it also functions as the students creating their own maths planner in a sense.  They are now curious and invested in our unit because they know they will be given opportunities to explore their own wonderings and share their discoveries.

This makes it enquiry-based learning- students having a voice and their wonderings being valued and explored. 

Some students might finish this unit finding out strategies to measure the volume of a hexagonal prism and others why using cubic centimetres as formal measurement is useful.  

I don't think every student needs to leave this unit all with the same understandings and that's why I don't give them stupid maths tests as a summative (see link ideas below)  

They have entered the unit with diverse levels and should leave as such.  What each student should leave the unit with, however, is a strengthened understanding that they are successful mathematicians and with stronger problem solving and enquiry tools.

For this particular unit, the formative and summative is open-ended (like always). The central idea has been printed on the reverse side of the pre-assessment. As we progress through the unit, they will reflect on what they have learnt and record by unpacking the central idea further and also adding new wonderings to explore.  Instead of a stupid test, that will be their formative and summative assessment task for our maths unit. They will chose two PYP attitudes they feel they developed and reflect how.  My feedback will be a rubric assessing what they showed themselves to be as communicators and inquirers from the Learner Profile.

If we don't pre-assess, we don't know the learning direction and needs. If students aren't given an opportunity like this to draw upon prior knowledge, they stumble unnecessarily in the beginning of a unit. We should value this reflecting time that children need to give them the confidence and the tools to wonder and explore. 

As we left for lunch, I overheard a few students coming up with ideas of how they might be able to measure the mass of air in the balloon.  You know you are doing the right thing when your students are still talking about a discussion when the football pitch is calling their names.



Some related links:

No More Stupid Maths Tests!!!

Positive Maths Assessments

Strategies to Make an Enquiry-Based Maths Classroom

Power of Student Questions for an Enquiry-Based Maths Classroom

Idea for Reflection & Pre / Formative / Summative Assessment

Visible Learning = Pre / Formative / Summative

I had what might be an effective idea today.

I was thinking of different strategies we could use to show our wonderings and enquiries into ratios and proportions. 

As these are new concepts to most of the children, using a KWL chart might not be so effective as their wonderings might be forced rather than natural. I thought of using the central idea for children to record findings and understandings like we have done several times this year, but wondered how to give it more directon.

So, I came up with this idea.  We would create a learning journey about our central idea.  Below is a beginning sample we started today:

The idea is that as we continue through our unit, we will pause and reflect on what we have discovered. This will give each child time to gather their thoughts, perhaps solidify them and also hopefully arouse some wonderings to explore about our central idea.


We began by recording what we already knew about our central idea and then placed a line on the learning journey to indicate where our unit enquiry began.

This was a really useful and effective pre-assessment tool for me to gain insights to who understood what.

We are taking our thinking further by also focusing on our lines of inquiry.

As the children record their understandings OR write wonderings in thought bubbles for their own investigations, they circle each according to each line of inquiry it belongs to.  The children have allocated a different colour for each line of inquiry like this sample:

I was and still am curious how this will pan out.

I asked the children what they thought of this strategy so far and they all thought was an interesting and helpful tool.

A few remarked how they liked the idea of being able to see their growth.

Others commented that it helped them see that they are on a learning journey and were interested to see where this will take them.

I think this might be a winner, but we are at the beginning stages of it.

(And for those of you who follow my blog, you would know by now my thoughts on the negativity of maths tests)

At the completion of our unit, at the bottom of the page, they will have time to reflect on two PYP attitudes they felt they showed the most during their enquiries and give examples of how they showed them.

I'm hoping this will be a really effective continual formative assessment that will help me track where each child is at and this help me to guide them into deeper enquiries.

As PYP teachers, we encourage our students to be risk-takers and so we as their teachers should model this also......

(I'll blog some thoughts and my students' thoughts later when we complete them)

How have I changed as a mathematician?

We have our Three-Way conferences next week where the child, parents and I discuss how we feel we have been progressing as a learner during the year and to set some possible goals for the remainder.


Today, we reflected on who we are as mathematicians.

We thought back to how we saw ourselves before entering Year 6 and how we perceive ourselves today.

Checking on on children's self-perceptions as mathematicians (and readers, writers, enquirers etc) is a paramount practise of an enquiry-based teacher I think. 

By finding out how each child sees themselves, that gives far more valuable data than any test, formal assessment etc can give.  

A standardised test can possibly give us a snapshot idea of where a child might be at in relation to year level standards, but meeting standards doesn't create a passion or wonderment to the world of mathematics.  That innate and ever growing amazement of how that simple idea of creating a number system based on 10 can lead to so many incredible patterns, connections and relationships with numbers and other maths concepts is what we should really be wanting to assess in a child.

That curiosity and amazement when a connection or pattern is found (by the child authentically) and wondering why that is is what our focus should be in nurturing in children as mathematicians.  If they happen to hit a year level standard along the way, that's a bonus, but it certainly should not drive maths learning. 

Passionate mathematicians working at universities; I'm sure, are not holed up in their rooms trying to meet standards.  I like to think they are creating their own hypotheses to explore and get a buzz out of connections or patterns they discover and play with.

That, together with enjoying proving and equally disproving our own theories we create in maths in class is what I am hoping to foster in each child.  I think that is a key difference to traditional maths learning and enquiry-based maths learning.  

How, though, can we assess that in a child?

Today, was a pretty good first attempt at trying to find out  abit more formally rather than from class observations.


For our reflection on how we saw and today perceive ourselves as mathematicians was pretty simple (as are the best reflections I think).

Self-Reflection on My Learning Development

Me as a mathematician	Before Year 6…….	Now…..	Action I feel I should take to further strengthen my skills or understandings

We have already done two check-ins throughout the year about how they see themselves as mathematicians, but this one had a different spin: how have we changed.

If you are reading this and you currently use the Everyday Math textbook or use an overload of those 'valuable' worksheets, you might find this of particular interest as these kids grew up with it before entering Year 6.  

The intention of including some of my students' reflections is not to brag. 

Really. 

I'm Australian so it's not in my culture to be like that - we grow up quickly understanding the tall poppy.....

The intent is to highlight how quickly we can change a child's mindset to come to appreciate mathematical thinking once we start using student-voice and curiosities to drive maths learning in an enquiry-based way.

From textbooks and worksheets, here are how my kids have changed- most a lot and some a bit.

Oh! And to bring some creative thought into the reflecting, they chose images to represent how they perceive themselves as mathematicians. 

Link to reflections:

How our perceptions have changed as mathematicians reflections

Some of the interesting images they chose to represent how they perceived themselves before as mathematicians and today:

Before Year 6, as a mathematician I saw myself as....


I felt like I was a box without any real knowledge. I knew how to do things in maths, but didn't understand the whys.

I thought maths was overrated and boring!!!




I felt a bit like a Hippo dragging through the mud and trying to pull my body through it all.







Now, as a mathematician, I see myself as......



I feel like maths is magical and that if I dig deep in it, I come out with a lot of amazing understandings!

Now I feel literally on top of the world!!

Checking in: How are you feeling these days as a mathematician?

How are you feeling these days as a mathematician?

I gave this question to my students via a google form just before we left for our winter holiday.  

Before our autumn holiday 6 weeks ago, I had asked them:  How have your feelings changed since being in Year 6?

That simple question gave me tonnes of valuable insights into each of my students which I've used to help enrich their maths learning experiences this term.

To help my students feel more successful and to enjoy mathematical thinking, I hoped this question would also provide some good insights.

Looking at their responses, I started playing around with how I could group the children.

Children who are feeling successful / enjoying maths:

° Good because I like how we always have choices of activities. It makes me more interested and it helps me to learn with different partners.

° Good because I have learnt how  to play around with maths and learning interesting strategies to make maths more fun

° Way better because I hated maths and never thought I was smart at it. Now I know I am! I like it when we create problems for others to solve because I can now make more challenging ones.

° I am feeling like an awesome mathematician because in every unit I did last year, I struggled but now I have good a proud feeling. I think I am asking myself deeper questions to enquire into.

° I feel more confident because I do more challenging questions and in Y5 I got the same activities as everybody else which was a bit too easy. Now I can take my learner deeper.

° These days I am feeling really good as a mathematician because I understand what we are talking about and I don't always get confused. I can learn from my partner and I can teach my partner too.

° For me this is my biggest change. Maths was a mystery to me before but now it is starting to all make sense.

° I feel fine about fractions (positive and a bit shaky negative, addition fine, subtraction fine, multiplication fine, division I keep forgetting [I think you swap the first fraction's numerator with denominator and then multiply the 2  fractions]), decimals (same as addition, subtraction, multiplication and division in normal numbers), percentages(positive fine, negative fine except a bit shaky on inventing negative percentage questions), addition and subtraction (positive and OK negative)  multiplication (big numbers and small numbers), Division (positive fine, negative OK)

° I feel that in maths I'm doing better because the way mr.A explains what to do helps me a lot and I understand what to do so I do it well because when I don't understand what to do I do it badly but for the rest I think I'm doing well.

° Being free to choose the activities challenges my thinking, and working with a partner like Pavi or Kayla is fun because we have similar minds so working together is not a difficulty. We try and expand and deepen our understandings when we learn together.

° The maths activities we are doing are fun because as well as doing math it is a hands on activity!!!!! Now I feel I am a successful mathematician and I want to learn more and more.

° I have started to think deeper about it and understand more, I am also starting to enjoy maths more.

Children I need to conference with and monitor more closely:

° Good but I think I can do better next time in maths activities.

° I think I am now a low mathematician compared to my school last year,because we do stuff here that I've never done before.

° In feeling OK... In class we are doing interesting things but I might need a bit of help on the Khan Academy when I do that for home learning. My maths skill have improved a lot and I like it this year.

This simple 'check-in' has given me some good insights into where my kids are in maths. I know who I need to conference with after the holiday to find out exactly why they feel they are struggling so I can help with providing different strategies or learning experiences.

What do you think your students would say?

Positive Maths Assessments- Fractions / Decimals / %

What is the purpose of a summative assessment?

If we don't use the data gathered from them because that unit isn't going to be reviewed, what's the point of them?

I have wondered this a lot over the years.  I believe the only real purpose of any form of assessment is to inform the teaching or learning.  Data gathered in traditional maths test summatives are often not used. 


A traditional maths summative:

° set of closed questions with distinct right or wrong answers

(These right / wrong questions send the wrong message to our young mathematicians. 

 Maths is not a right/wrong subject. It is scientific thinking where we create a hypothesis and      test the strategy/s and then assess their effectiveness.  When we give right/wrong          questions to kids in maths, we are sending the wrong message about what mathematical thinking is about. Failed strategies we test should be celebrated because they teach us why another strategy could be more useful. Enquiry-based maths classrooms are not about getting 'right' answers. They are rich, celebratory learning environments where kids are testing and creating strategies, making connections, questioning and reasoning. Shouldn't our summatives reflect the type of learning during the unit?)

° catered to a grade level expectation average

( Just as we all learn to ride a bike, lose our first tooth, say our first word etc at different ages, comprehending mathematical concepts comes to us all at different ages when our minds are ready. When we give the traditional maths test, we are telling children- If you can't answer this question I have given you, you are clearly dumb at maths because this is the type of question a student in Year 6 should be able to answer.  When we then think about the way we differentiated the learning activities during the unit to cater to each child's levels of understanding, why would we then give them questions we know they aren't likely to answer? Is it not a form of abuse on some level to the child?  Equally, the students we know who are exceeding that grade level expectation are being dumbed down in traditional maths tests except for those typical 2 or 3 questions at the end of the test that cater to those at that end of spectrum)

° gives negative feedback to what they learnt in the unit, not celebrate

( We then traditionally send these maths tests home to share with parents.  What parent celebrates with their child the correct answers they got in the test?  Surely the majority of parents focus on the crosses and discuss these with their child.  So, the child firstly receives negativity towards their learning and who they are as a mathematician from the teacher who gives them a test they know they won't be able to answer and then receives extra negativity from their parents who focus on their wrong answers.  How is this beneficial? And because it is a summative, the student isn't given the opportunity to improve on those 'wrong' answers because we aren't exploring that particular maths concept again for the year.  And so the continual negative cycle keeps spinning for each child.)

Maths pre and formative assessments are crucial to helping student learning. We can use the data to help students with misconceptions or what steps they are mathematically ready for.

To make the summative meaningful, I give the students our maths central idea and during the unit, we pause and reflect on what we have learnt about it as we progress.  In addition to providing valuable reflecting time for the learner to consolidate their own learning, it allows me to see during the unit what each student is gaining in understanding and so it serves as a key formative. 

Here are some samples from our recent fractions,decimals, and percentages unit exploring the central idea: 
        

The reverse side reflection always fascinates me as it gives a big tell of what really stood out the most for each learner:

Another sample:

         It fascinates me to read what stands out in each child's mind as the unit progresses.

Though it is simple, the 'I used to think.......but now I know........' thinking strategy helps children to think deeply about their own learning and can be a powerful reflective tool.  It also fascinates me what stood out the most in the minds.

Some other student samples:

I used to think that fractions were of no use and you can't use them in real life situations.

Now I know that fractions, decimals and percentages are used all the time in real life situations such as shopping, banking etc

I used to think it wasn't possible to add, subtract, multiply or divide fractions.

Now I know that they can be added etc and that we do add them in real life.

I used to think that percentages are not easy to calculate in your mind (mentally)

Now I know there are easy strategies we can use to calculate them mentally.

I used to think that when you see a % discount, you just subtracted that number from the price.

Now I know you need to calculate the % of that number first and then subtract.

I used to think that fractions, decimals and percentages were completely different.

Now I know that they are basically the same thing- just expressed in different ways.

I used to think that decimals were useless and only used at school.

Now I know we use decimals in so many different ways in our real lives- length! map ratios! even money! and so much more!

I used to think that adding or subtracting decimals in our head is too challenging.

Now I know they are really easy to do especially when you use the compensation strategy. 

I used to think that if you add, subtract. multiply or divide decimals, you can just forget about the decimal point

Now I know that you can't forget the decimal point because it changes the size of the number!

Reading through these, it also makes me realise the power of a PYP central idea especially when it includes a reference to how we use that maths in real life.  So many children in my class now seem to be seeing the relevance of enquiring into fractions, decimals & %.  When children don't have the opportunity to see how a maths concepts can relate to their lives, so much engagement is lost. 

An EAL student sample:

This style of summative:

° is open-ended so each child is able to explore their own mathematical concepts they are building

° allows each child to feel like a successful mathematician

° provides valuable reflecting time during the unit to strengthen understandings of the concepts or skills

° allows the learner to see their learning develop as they add more ideas during the unit

° provides a growth mindset as mathematicians

° caters to all levels of mathematical understandings- not just the grade level expectations. It caters to all the differentiating that occurred during the unit.

° allows the learner to celebrate their own learning rather than being judged by a set of grade level expectations

° gives me, as the teacher, valuable insights as a formative during the unit that helps me see where each learner is developing and where they can be helped for their next steps

° the parent sharing is a celebratory experience.  Parents cannot focus on 'wrong' answers with their child. they are forced to focus on what they child has achieved and so additional positive messages are given to the child as a mathematician.  

° the last question asking the child to think about what they feel they need to focus on next time they learn about that particular maths concept provides an opportunity for the child to take ownership and responsibility towards their own learning

This is essentially a positive celebration of learning and helps instill the values we create in our maths learning.