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The Progression of Addition and Subtraction

The more I create… the more I learn.  Here is the 3rd installment of this whole Making Sense Series which has truly forced me to be a better teacher.  A more educated teacher. I can’t stress enough how much I’ve learned from diving into the progressions found here. There are many intricate pieces of learning […]

http://gfletchy.com/2016/03/04/the-progression-of-addition-and-subtraction/

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The teaching of fractions

There are certain prerequisites for children to develop a solid understanding of fractions.  First, they must understand, through work on additive reasoning, that a whole can be split into parts and that the sum of those parts is the whole.  There’s a short step into multiplicative reasoning from here – that a whole can be split into multiple, equal parts and that the whole is the product of the size of each part and the number of parts.  Once this is understood, children can begin to think about the whole being worth one and the parts being fractions of one.  The ideas that follow are broadly sequential in terms of conceptual development.

Prior concepts

Children will need to manipulate various representations of fractions, for example making them with fraction tiles (as both bars and circles); taking strips of paper and ripping them in to equal parts; and drawing bars and circles, dividing them into equal parts.  It is worthwhile to get children to do lots of judging by eye and marking equal parts of a whole as well as using squared paper to do so accurately.

Of course, there is a lot of language to work on whilst manipulating these models of fractions.  Children need to be shown clearly the link between the total number of parts and the language (but not yet necessarily the written form) of the denominator: two parts – halves; three parts – thirds; four parts – quarters etc.

With a secure start in the basics of splitting a whole into equal parts, children can work on the idea that fractions always refer to something.  A third, for example, doesn’t stand alone.  It might be a third of an apple or a third of twelve sweets or a third of one whole.  Modelling these full sentences and getting children to speak in this way should solidify their understanding of proportion.  Through the sharing out of objects, even very young children can work on the concept of fractions of numbers – sharing six sweets between three children means that each child has the same number of sweets and that two sweets is one third of six sweets.

sweets

Once children are comfortable with the idea that an object or a set of objects or a number can be split into equal parts, and that each of those equal parts can be described as a fraction of something, that object or that set of objects or that number, they can go on to work at greater depth.  By comparing strips of paper or bar models that are the same length yet are split into different fractions, children can look at the relationship between the size of each part and the number of parts.  That is, the greater the number of equal parts, the smaller the size of each part.  Children should be expected to think about how ¼ is smaller than ½ because ¼ of one whole is one of four equal parts whereas ½ of one whole is only one of two equal parts.  Then, questions like this should be relatively straightforward:

Covered strips

The understanding that unit fractions with larger denominators are smaller than unit fractions with smaller denominators will contribute significantly to work in comparing fractions later on.

Children could begin to look at improper fractions and mixed numbers next.  Using ¼ fraction tiles, they could make one whole and then see what happens if you add another ¼.

Mixed number

This lends itself to counting in unit fractions but we should exercise caution.  Children may be able to chant ‘Three quarters, four quarters, five quarters…’ but early conversion to mixed numbers as well should help to secure their understanding of the relationship between them.  Manipulatives like fraction tiles and multi-link cubes are great for representing improper fractions because they can trigger accurate mathematical talk to describe the improper fraction (the total number of cubes as the numerator and how many cubes in each whole as the denominator).  The same can be done to describe the mixed number (the number of wholes, then what is left over as a fraction of a whole).

Returning to additive reasoning, children could generate complements to 1 whole and record them as addition and subtraction statements.

Complements

A slight change to the representation used can support children to work with complements where denominators are different:

Complements 2

Placing two bar models of equal length one on top of the other is great scaffold for comparing fractions.  When the denominators of the fractions are the same, the bars should not even be necessary but when they are different, the image can help to structure thinking.

Comparing

When dealing with fractions with different denominators, the practice that children had earlier of judging by eye to split a whole into equal parts and marking the divisions themselves becomes crucial, otherwise, things like this could happen:

Inaccurate bars

A standard fraction wall is all that is needed to begin work on equivalence and the first step is of course shading one fraction and looking up or down the fraction wall to find fractions of equal size.  When children are comfortable with that, they can begin to look at patterns in the abstract representations, particularly the link between times tables, numerators and denominators.

Fraction wall

Using the language of simplifying or cancelling fractions without first talking more generally about the concept is a mistake.  If children are well versed in using a fraction wall to find equivalents to a given fraction, it is only a slight tweak to talk about finding the equivalent fraction that has the fewest total parts.  It would be tempting to talk about finding the equivalent fraction that is ‘closest to the top’ of the fraction wall but this would be a mistake too.  The language of simplifying or cancelling can be used to attach to the concept of finding the equivalent fraction with the fewest total parts to get children thinking conceptually soundly.

One further aspect of thinking of fractions is to consider them as numbers.  To do this, plotting fractions on a number line directly beneath the bar model is a good way of linking the two representations.

Number line

Representing fractions as a proportion of one, as a part of a quantity and as a position on a number line significantly supports children’s development of proportional reasoning and ensures that future tricky concepts such as calculating with fractions can be built on a secure foundation.

10 Research Based Principles of Instruction for Teachers

Belmont Teach

I recently read an American Educator article from 2012 by Barak Rosenshine that set out 10 principles of instruction informed by research, with subsequent suggestions for implementing them in the classroom. It was also one of the articles cited in the “What makes great teaching? Review of the underpinning research” by Rob Coe et al and provided further elaboration on one of their six components of great teaching thought to have strong evidence of impact on student outcomes, i.e. quality of instruction.

Here’s my summary of the key messages from each of the 10 principles.

1: Begin with a short review of prior learning

Time-for-Review

Students in experimental classes where daily review was used had higher achievement scores. A 5-8 minute review of prior learning was said to strengthen connections between material learned and improve recall so that it became effortless and automatic, thus freeing up working memory.

Daily review could…

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Great teacher talk

Evidence into Practice

The ‘war’ on Teacher Talk Time was one of the more depressing developments in my teaching career. Obviously secondary students will rapidly get bored if they are subjected to lectures – no one is advocating that teachers should drone on and on for hours – but the obsession with cutting down teacher talk became dysfunctional under the Ofsted framework.

The profession became awash with strategies to prevent teachers from talking. Here’s an example from TeacherToolkit (a typically excellent resource bank) that I found dismaying at the time:

http://teachertoolkit.me/teachertalk/

“Teacher Talk can often be the root-cause of poor behaviour and debilitating progress during a lesson. … “Reducing Teacher-talk” will be my sole focus for 2012/13 in my own classroom, as well as observations of other colleagues and CPD training throughout the school year.”

Some of the Top Tips show the extent to which ‘teacher talk’ became taboo.

  • “Give the students a…

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#MathsCPDChat on times tables strategies

Pre-reading: Strategies for learning, remembering and understanding the times tables. Some additional thoughts for starting out teaching the times tables with year 2s onwards, prompted by a #MathsCPDChat These are the things I think are important for mastery of the tables (most of which, I suspect our primary colleagues are doing): 1. Begin with manipulatives…

http://mrreddy.com/blog/2015/01/mathscpdchat-on-times-tables-strategies/

Contemporary educational ideas all my staff should know about

teacherhead

 

Key ideas from different sources. Key ideas from different sources.

As I look ahead to starting my new job at Highbury Grove,  I’m thinking about all the conversations we are going to have about learning.  To a large degree I want my teachers to be as up-to-date as possible within their own subject domains. They should know the latest OfSTED position ( eg with Moving English Forward or Mathematics: made to measure ) and be up to speed with exam specifications and assessment requirements.  Subject knowledge and subject-specific pedagogical knowledge are going to be key drivers of everything we do.

However, in order to fuel the collaborative effort of reaching the ambitious goals we have for the school, we’ll need to establish a shared conceptual language for talking about teaching across the school as well as within departments. Inevitably, different teachers will have engaged to different degrees with certain ideas depending on the books…

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On meeting your new class…

Grow your teacher reputation

Teacher-student relationships exert a strong influential role in personal development. Establishing relationships that will allow the children in your class to learn effectively take time and effort.

  • Assure them that you are great at helping children to learn and that you believe that they can all improve. Your assurances will be backed up by teaching great lessons.
  • Quickly learn and use their names.
  • Tell them stories of past students who have had success with you.
  • Speak with a warm, friendly tone but patrol boundaries consistently and fairly. Have a controlled, firm tone ready for appropriate times.
  • Tell them exactly what behaviours you expect from them – that everything you ask them to do will be chosen carefully and you’ll never waste their time or let them waste theirs.
  • Explain what they can expect from you.

Talk and model the vision

At Penn Wood, better never stops. Having sky high expectations from the first moment will pay dividends later in the year.

  • Explain what ‘Better never stops’ looks like in your classroom.
  • Mastery – the urge to get better and better at something that matters.
  • Telling a story is a great way for them to get to know you and to see that you embody ‘Better never stops’ too.
  • Tell them about your class tree, making links to mindset and vision

Expect a growth mindset

Working harder makes you smarter. It is through struggle that children learn more deeply and they should expect to find work desirably difficult.

  • You’re going to practise lots because that’s how you get better – sustained effort is the route to success.
  • Share stories, photos of work or books that show excellence from your current class.
  • Describe the behaviours for learning that you expect to see.
  • Explain the expectation of standard of work.

Set your working expectations

First impressions last a lifetime. The quality and amount of work that you expect in this session should set the tone for the year ahead.

  • Books are at the heart of our curriculum. Choose quality texts to convey your messages.
  • Expect children to think hard about content.
  • Explain how you expect them to work in your classroom so that everyone can work hard and concentrate.