Maths Reasoning and Pupil Voice Part 2

Yesterday I was able to sit with a few of my learners to discuss their approaches to some of the reasoning questions.  Whilst some of my initial hypotheses were correct about why some low scores were attained – not reading the question properly being the top one – it was a useful exercise in gaining insight about the children’s approaches.

I interviewed four pupils who all scored below ten on the KS2 Mathematics Reasoning Paper 3 2016.  Their scores ranged from 1/35 to 9/35.  Their procedural fluency scores were higher with a range of 12/40 to 27/40.  These scores would render these children working below Year 6 standard, if they were the actual results in May.

“I read it through and didn’t get it”

shay

This response to the above question was quite straightforward, but it made me think of the emotional response in addition to the cognitive response.  Compared to the previous question, which the pupil knew he could do because he “knows time”, I wonder how a lack of confidence with this non-routine problem affected his potential to attack the question. With guidance from me prompting him to look for certain information and a scaffold for how to re-represent the problem, there were no arithmetic barriers and he was successful.  This suggests that the children need a richer diet of problems to build experiences which will support them in the SATs.  In addition, perhaps another pedagogic implication is to plan for individual thinking time, before collaborative working, when dealing with problems in class so that I can more easily identify which pupils rely on others more.

“Reading the protractor made it difficult”

Jaccob angles.png

This response showed that the pupil was confused between the different directions of the protractor.  He did not score on this question at all, and yet independently defined acute and obtuse for me and was able to identify, without a protractor, which angles were acute and obtuse.  This implies that he, and perhaps others, need to be build confidence in their capabilities and be more reliant on checking strategies.

“I thought the answer was too big”

jaccob marbles.png

The pupil achieved the correct answer in his working out but did not write it in the answer box. He explained that he thought it was too big an answer.  This was an interesting response and I suggest that more experience of problems involving “larger” answers would support this pupil.  Kudos to him for considering how likely the answer was – showing he is thinking about what he is doing.

“I thought smallest means the least digits”

lewis

This pupil applied the wrong strategy despite practising a very similar question days before our practice paper.  He explained that the language was confusing.  Given the pupil’s history, language is an issue – however, it shows that providing experience with these type of problems cannot be taken for granted that they will prepare the children for similar problems in the future.  It shows for these children who score poorly, I need to have a stronger focus on them explaining or proving that they understand why they have answered the question in this way – especially if they have worked with a partner.  This ‘explanation’ will be better feedback for me than the response by itself

Summary

This was a really useful task, and one which I think I should do more often.  It has given me a real insight into the students’ approaches and has allowed me to get to know the pupils better.  The implications for preparing these pupils for success are:

  • Insist on explanations for answers in lessons
  • Ensure that collaborative learning tasks are structured in ways which allow for individual assessment
  • Provide a broad range of problems to build pupils’ confidence (knew this anyway, but glad to see it is important)
  • Provide instruction on how to re-represent problems to support understanding – bar modelling in particular.
  • Provide more opportunities for children to hear other children’s working out and then attempt strategies themselves.  I am quite interesting in using the Rally Coach Kagan structure for this.

 

Knowledge Organisers in Primary Science

For some time now, I have been really interested by the concept of a knowledge organiser. I first came across this idea on Joe Kirby’s blog a few years ago and have tried over the years to understand how it might be applied to primary (in amongst getting on with the day job and everything else).  More recently I have seen examples of knowledge organisers being used at primary level, and it led me to experiment with creating a few.  This blog is about my thoughts on how I think they might be used and how they might support teaching and learning.

My understanding of a knowledge organiser

A knowledge organiser is a document which details all the important facts and relevant information needed for learners to understand the topic they are learning.  These are listed on a simple document and children are given copies at the start of the unit, encouraged to spend 10 minutes a night reviewing this information, and can be regularly tested on the content in a similar fashion to times tables tests.  They ensure that all members of the school community understand the expectations of what is to be learned in a unit of learning.

Do knowledge organisers represent didactic teaching?

I am certainly not against didactic teaching; I think it is a tool that suits certain situations better than others and can be used very effectively.  My concern would be if only didactic teaching were used in primary science because I think children, like adults, need to encounter concepts in different ways and have multiple experiences with those concepts to truly develop understanding.

I did initially think, perhaps due to my perception of some of the practices at the school where Joe Kirby works, that knowledge organisers represent rote learning.  Having been trained at university by a science department that held at its core a social-constructivist view of learning (as I feel I myself do), I wasn’t sure how I felt about this.  However, I am moving towards the idea that knowledge organisers in their simplicity would provide a grounding in the topic children are learning.  A grounding in vocabulary and facts that children could then use to make sense of more practical and stimulating classroom lessons.

For example, if children came to a science lesson having an awareness of terms such as force, friction, air resistance and gravity, any experiences with parachutes or feathers or hammers or whatever is dropped from a height would then be able to fit into a conceptual framework and develop understanding.  Even if children had no understanding of the definitions they had learned beyond recital, further experiences would then develop this understanding by providing examples and deepening their knowledge.  I would argue that both examples align with a social-constructivist view of learning, where the core knowledge is the starting point and surface knowledge can more readily be established so that deeper understanding can more readily be obtained.  Moreover, if children regularly review knowledge organisers, their understanding of each term would be reshaped and refined as they bring their experiences to their mind whilst revising the core knowledge.

Knowledge organisers as a shared language of learning

What really excites me about knowledge organisers is that the knowledge components of the curriculum can be made visible to everyone: teachers, parents and most importantly pupils.  Moreover, I hope that these components might support subject knowledge for teachers who have not had specialist science training.

Our curriculum has deconstructed the National Curriculum objectives and aligned them to the ASE’s Big Ideas of Science so we have a conceptual curriculum which is progressive in each of the main ideas year-on-year.  The reasons for this were numerous but one key reason was to give teachers autonomy over how they structure science learning in line with the rest of the curriculum – to prevent them from being restricted about what is covered and when, which they might find if we used a scheme of work.  This intended freedom has then created a work-load issue for staff in terms of building content knowledge for each objective.  By specifying the knowledge content of each objective, this will create consistency for all children but hopefully reduce teachers researching time so they can spend more time planning and preparing exciting lessons.

How will knowledge organisers change the format of science lessons?

I envisage that if children review, revise and quiz themselves (using their chrome books) on knowledge components of the curriculum, lessons can be dedicated to developing these ideas through activities which require higher order thinking.  For example, lessons in Year 6 on the circulatory system could be spent comparing the circulatory system and the respiratory system, rather than spending time labelling parts.  Role play of change of state would be more meaningful if children had the language to discuss what their actions actually represented.  If Year 3 learned about the different theories of gravity, not only could they develop the understanding that the scientific view of the world can change, but they can also then spend lessons recreating famous experiments and distilling key aspects of why scientists such as Galileo made the theories he did.  In conclusion, lessons can be more ambitious, challenging and motivating because all children will have the chance to engage in higher order thinking. It goes without saying that this would raise attainment for all pupils, particularly those at risk of underachievement.

Closing thoughts…

I have created 20% of the knowledge organisers for our curriculum and it has taken me about 10 hours in total so far (an example of one for geology is here).  I do not want to be the one to exclusively make them because I value the contribution of others and truly believe the knowledge organisers would be better constructed in collaboration with others.  However, everyone is extremely busy and a start is better than nothing at all.  I want to trial and investigate the impact the knowledge organisers have on science teaching and learning and gather pupil and teacher voice to check how they fit with the many assumptions I am making about the benefits I hope they will bring.  I think all of the important aspects of science teaching – particularly in gathering what children’s pre-existing ideas about concepts are to identify misconceptions which need to be addressed – can sit alongside knowledge organisers, which will provide a consistency and commonality that should increase engagement and attainment.

I would absolutely love and welcome and feedback, thoughts or comments anyone has.

Thanks,

Marc

Maths Reasoning and Pupil Voice – Part 1

Today our Y6 children completed Paper 3 of last year’s maths SATs.  Reasoning has been a focus in every single lesson, every single day since September.  Some children did very well, with around 10% showing achievement within greater depth.  However, the test held deep challenges for many children.

This evening I’m thinking about why some children were completely flummoxed by the questions.  We teach in a very talk-led, collaborative style with children having ample and regular opportunities throughout each and every lesson to discuss their learning and learn with and from each other.  Children demonstrate successful application of their learning with the White Rose Maths Hub activities – which are superb – and teachers assess that the learning has been very successful.  Then, the children sit a SATs paper and for some reason, every ounce of ‘sense-making’ seems to escape.

One of the tasks for my master’s degree was to interview children about a piece of work.  I think rather than trying to use literature to interpret why some children struggle so much in the tests, I will make time next week to interview some of the children who really found this paper difficult.  I’ll have three main questions:

  1. How did the paper make you feel?
  2. What strategies did you think you’d need to use?
  3. What strategies did you use for (a range of questions)?

I’ll post the results next week but I would be very much interested in other Y6 teachers’ experiences and how their children perform in tests in comparison to daily lessons.

Have a great weekend.

Marc

Maths Reasoning, Growth Mindset and Language

After the SATs, I was really bothered about how my teaching impacts children’s ability to reason mathematically.  It felt like we had nailed arithmetic, with historic low-achievers (Level 2As at EOY4) achieving 60% or more on tough practice papers.  However, for reasoning, it was quite a different story.  The challenges of teaching children how to understand and represent mathematical relationships, and teaching the language, preparedness and resilience to reason with them, has been quite significant.  It is an area of my practice that I want to drastically improve and develop next year.

I’ve started off this half term by planning a unit of learning around Algebra, using the planning guidance from the NCETM.  I must say I found the planning process quite lengthy, however it was extremely valuable.  The planning process explores necessary prior knowledge, opportunities for exploration and play, consideration of possible misconceptions and difficult areas to teach, and how children can demonstrate their understanding along their learning journey.

What is the focus of the teaching sequence?

Children are able to represent linear sequences using algebraic expressions; reason with algebraic expressions in a range of contexts; and understand how their understanding of number can help them solve equations.

KEY CONCEPTUAL IDEAS AND MAKING CONNECTIONS

  • What are the steps in understanding needed along the journey?
    • Seeing patterns – numbers, shapes, sounds,
      • Square, triangular, rectangular numbers – Pascal’s Triangle
    • Generalising rules from observed patterns with justification
    • Understanding rules as being recursive or ordinal
    • Representing sequences using algebra – Finding the nth term
    • Representing variables using letters
    • Four operations using variables
    • Proof – use of Sudoku – deductive, counter-example, exhaustion, contradiction
    • Finding the value of variables
  • What is the best way to order these steps?
    As above
  • How are the steps going to be connected?
    Through building on previous steps
  • How is this journey going to be connected to prior learning?
    • Seeing patterns in nature and maths
    • Area/Volume + perimeter
    • Missing number problems

DIFFICULT POINTS

  • What are the common misconceptions within this area?
    • Seeing the = sign as equivalence on both sides
    • Understanding letters to mean numbers
      • Confusing letters with units of measurements
      • Confusing letter with initial of object
      • Not understanding that letters are variables
    • Visualising mathematical structure in word problems
  • Which parts are difficult to teach and difficult to learn?
    • Generalising patterns – conceptual step
    • Understanding variables – children need to see symbols as the pronouns of maths and then substitute symbol for value; then finding values is the reverse and deepens understanding
    • Solving equations with two unknown variables – use Numicon/Cuisenaire to support
    • Visualising mathematical structure in word problems – will need lots of practice and varied representations

REPRESENTATIONS

  • Which models and images will best support understanding of the different parts of the journey?
    • Function machine – representing patterns
    • Matchsticks – representing patterns
    • Square, triangular, rectangular numbers – representing patterns
    • Number squares – representing patterns
    • Numicon – representing and naming patterns finding nth term PPT
    • Tangrams – introducing letters for variables
    • Follow me cards e.g. 3 + x = 7 – introducing letters for variables
    • Bar model/cubes – mathematical structure
  • Which models and images will expose the difficult points and misconceptions and support understanding in these areas?
    • Numicon
  • Which contexts will support the children to make sense of the maths and give the maths meaning and purpose?
    • Patterns in times tables
    • Pascal’s triangle
    • Fibonacci sequence – in nature
    • Playing and representing ‘Nim’
    • Number square
    • Paving slabs around a garden
  • What language will the children be expected to make sense of and use?
    • Next, before, after, first, fifth, repeat, again, continue, guess, check, predict, if, then, so, because, but
    • Variable
    • Equation
  • How will children be expected to represent their thinking and understanding and different points on the journey?
    • Describing patterns
    • Generalising patterns using words and justifications – function machines
    • Recognising different ways of expressing sequences
    • Finding the nth term
    • Balancing equations using the equal sign
    • Calculations using formula – match up the symbols with the activity
    • Developing reasoning through Sudoku – how do you know?
    • Finding the value of variables in different contexts

VARIATION

  • How can variations be used to support the understanding of the structure of the mathematics?
    • Use contexts to support understanding of patterns and variables Days of Christmas (Children compare how they would write and how it is written conventionally; and then set targets/feed forward).
    • Using children’s own ideas about sequences
    • Replacing missing number boxes with different shapes first, then letters
    • Using Numicon and envelopes to work with unknown variables
  • What needs to be varied to expose the difficult points and misconceptions?
    • The side of the equation with the calculation – to support understanding of the equals sign
    • The types of sequencing
    • Representing word problems visually using bar model and then symbolically using algebra
  • How can variation be used to ensure depth of understanding?
    • Using slightly different examples to develop understanding
    • Use examples and non-examples to develop understanding

GOING DEEPER
What opportunities are there for demonstrating creativity and imagination?
What contexts would provide opportunities to explore and generalise about the mathematics at a deeper level
Patterns in times tables

I’ve really enjoyed teaching this unit, and I dare say that the children have enjoyed discovering patterns and finding rules.  What has been challenging but doable, is persuading the children that the wrong answer is equally as important as the right answer, in terms of learning.  We have talked extensively about being in the pit, and how the learning dispositions can help us get out with new learning; extending this metaphor to us being at a higher point in our learning even if all we have found are ‘wrong’ solutions.

This has been achieved through the use of success criteria which provides feedback on reasoning processes such as ‘working systematically’ and ‘making and checking predictions’, in addition to perfecting my poker face so that answers right or wrong are both welcomed in the same way.  Giving children open-ended investigations, along with scaffolding the language of generalising and justifying, has also brought about rewards in making the learning more successful.

There have also been key learning points for me:

  • Language has a vital role in maths reasoning and next year I must develop children’s oracy in mathematics much more deliberately than this year.
  • Patterns exist everywhere in maths, and identifying and continuing patterns is relevant and valuable throughout the year – not just with algebra.
  • Opportunities to develop generalisation and justification capacity must be created across the year, so that children are always thinking mathematically.  For this reason, mathematical reasoning must be the focus of most lessons.  This will enable children to develop their relational understanding.
  • Assessment of how children think about maths is extremely useful for planning.  For example, children who still resort to additive thinking need teaching and training with multiplicative thinking if they are to begin to understand relationships such as ratio, proportion, fractions etc
  • Self-efficacy in maths can be created through use of success criteria of the reasoning thinking processes.  This is a slow process and needs implicit messages to be controlled.  Regular discussion of what it means to be in the pit, using the learning dispositions and celebrating all learning, rather than correct answers, are important tools which can be used to achieve this.

I’m looking forward to increasing my bank of straightforward reasoning tasks which can be used across the maths curriculum, particularly for arithmetic practice and effective starters (in the Shirley Clarke sense of the phrase).  I’m going to try to store them on a Pinterest board as there are so many good ideas out there.

Teaching Morphology – Reading and Spelling

Having recently completed a literature review and evaluation of a morphology intervention on reading and spelling, I’ve decided to blog to share some of my learning and welcome comments and feedback.

What is morphology?

Morphology is the study of word parts.  Teaching morphology can improve reading and spelling as it develops children’s ability to link meaning with orthography.

Morphology and Reading

Morphology can benefit reading in the following ways:

  • Word reading – children will be able to read syllables more automatically e.g. they will recognise affixes such as ‘inter-‘, ‘ment’ or ‘ious’ without needing to decode each sound and blend.
  • Word understanding (grammatical knowledge) – children will be able to use knowledge of affixes to determine the function of the word e.g. words ending in ‘ment’ or ‘ness’ are nouns
  • Children can use their knowledge of root words and affixes to hypothesise about unfamiliar words, using the context of the sentence to check their hypothesis e.g. ‘The crowd erupted with applause.’ erupt = ‘ e(x) [out] + rupt [break] + ed [past tense]’ meaning ‘broke out’ + ‘applause ‘means clapping = The crowd broke out with clapping = The crowd made a lot of noise by clapping.
    Obviously there are some inferences to be made here, but the strategy gives the children a starting point to infer the meaning of the word.
  • Vocabulary size – understanding that affixes and root words can be combined to generate other words means that children can know the meaning of many words without being directly instructed in each one.

Morphology and Spelling

It is argued that English spelling more closely reflects meaning than sounds.  This is in direct conflict with a phonological approach to spelling being taught through phonics in KS1.  Whilst this approach is sufficient as a starting point, it is increasingly insufficient for children learning more advanced vocabulary.  For example ‘sign’ and ‘significant’ both use the root word ‘sign’ yet the root word is pronounced completely differently.  Explicit teaching of how root words retain their spelling, or follow a pattern, when inflected with affixes can support children by providing a wider knowledge of spelling strategies.  This would be particularly useful for children who read less as they are less likely to generalise from reading how inflections affect spelling.

My intervention found that my Y6 pupils had a primarily phonological approach to spelling – to the point that even when all complete the Lexia programme on a daily basis, which teaches children about derivational and inflectional morphology, they have not developed, or are metacognitively aware of, a morphological approach to spelling.

Creating awareness of and developing this key knowledge could support these children to become better readers and spellers.  It is interesting that this approach may be particularly beneficial for dyslexic pupils.

How will this be implemented in my practice?

It’s advised in the literature to use morphological approaches alongside other approaches such as phonological and visual strategies.  From next year, I will teach root words and inflections to children to read and spell new words.  This will require teaching for connections in spelling (relational thinking) and explicit word hypothesising skills in reading.  Additionally, I will teach it as a word reading strategy to improve children’s automaticity and grammatical knowledge.

Teaching Evolution

BethBudden@school

Last year I attended an excellent lecture at the IOE by Professor Martie Sanders on teaching evolution to young children. I’ve been meaning to share this.

As evolution now has to be taught in Key Stage 2,  I think it’s really important for teachers to think carefully about how to do this, and do it well. To begin with I’m going to use a quote Martie used:

“Nothing in biology makes sense except in the light of evolution”            

  (Dobzhansky, 1973:125)

You can also watch a quick Ytube clip below explaining this in a snappy set of clips, but in essence life processes and living things begin and end with evolution. If children understand the basic concept of evolution it will mean they have a fundamental foundation for understanding all the other biological concepts. Even if pupils don’t become scientists, which most won’t do, understanding…

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What does ‘secondary ready’ look like in history?

Clio et cetera

It is a good time to be thinking about transition. In a few weeks, a new cohort of pupils will arrive at secondary schools, ready to continue their history education. The word ‘continue’ is crucial here. In England, history is compulsory for only two or three years of secondary education (in contrast to nearly every other European country) and a significant component of what a pupil learns about history has to happen in primary school. Yet, as secondary school history teachers, I would wager that we rarely think much about what it is that they learned at Key Stages 1 and 2.

This matters because what we teach in secondary school needs to build on what has been learned in primary school. One cannot make sense of the Norman Conquest and its impact on England without understanding the nature of Anglo-Saxon society before it. We miss a trick if…

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