Modelling ideas

Ideas that focus on using technology to support looking at different mathematical structures, to support or reveal a deeper understanding..

The resources described by Pete Sides at the White Rose Maths Hub conference on bar modelling can be found here.

Power point presentation.

PDF of presentation at White Rose Bar Model conference, Salford (Feb 2017).

Spreadsheets

10 Bar random sums <10 illustrated on a bar model (any part of the number sentence can be hidden)

10Bar1

NB comparative bars & subtraction are also available on the same file

 

Fraction as a number a mixture of random and self-input fractions showing equivalence in different denominations and mixed number & vulgar fractions  

frac1frac2

Fractions "of" & ratio 

frac3frac4

frac5

NB- Values of parts or ratios can be selected to be shown or not

 

Geogebra

10 Bar interactive model of x+y=10 as a bar and as a line graph

geo1

 

x-squared interactive model of y=xshown as bars and as a point on graph

Geo2

 

Circle thoerems: a series of simple geogebra files to allow interactive investigation of theorems

cyclic quadrilaterals

same chord

angle at centre double

geo3geo4geo6

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Arithmetical Structures

Representing the structure behind mathematics can help develop an understanding of arithmetic and/or algebra. This can also be linked with multiple representations of a concept as each of the illustrations below can be shown in alternative ways including number lines, sets & arrays of counters amongst others.

These are intended to provoke discussion amongst colleagues rather than to be used without thought or reflection with learners but SYMH would be happy to hear of colleagues' thoughts on these and if they have used any or similar models with learners to good effect.  

You can also find here a series of resources that can be used with colleagues to discuss conceptual development and structures underlying some of the common techniques taught in number and algebra.

Feel free to use these resources as a catalyst for professional dialogue - should you wish a professional development session to be facilitated in your school, feel free to contact This email address is being protected from spambots. You need JavaScript enabled to view it.

Click on each below link to download PDF

Additive structures

Multiplicative Structures

Linear equations as bar models

Simultaneous (linear) equations as bar models

mr1

 

mr02

 

MR03

 

mr04

 

mr06

 

mr07

 

mr08

 

mr09

 

mr10

 

mr11

 

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Modelling negatives

 

Just a few images that some may find helpful when describing calculations with negatives

aneg1

aneg2

aneg3

 

 

 aneg4

 

Multiplying involves the concept of an "enlargement" operation.

amneg1

 

Then the distinction between the multiplying effect of 1 (stays the same) and -1 (flips it over)

amneg2

 

Which also works on negatives

amneg3

...and so we can multiply by any sized positive or negative number.

amneg4

amneg5

Now I know I haven't included "taking away" negatives but my explanation takes more than one image to explain - but consider 5 - (-2). Since we know that 5 = 7-2, then 5 - (-2) can be written as (7-2) - (-2). Using this argument alongside appropriate diagrams with learners has proven to be succesful.

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IRIS Connect

What is IRIS Connect?

Originally set up as a system for teachers to record their own teaching through video capture on a secure and convenient platform IRIS Connect now has the additional feature of being able to host training materials in a secure and useful format. It is now easier to sign up for the FREE content licence via this webpage. http://www.irisconnect.co.uk/order-your-group/

IRISnew

How will this benefit my school?

This will allow all schools to access free of charge training materials developed by the Hub and a platform for schools to benefit from previous workgroups as well as engage with new ones.

 barlogik icon

The KS2/3 "Developing Problem Solving using Bar Models" research work group materials are now available.

Additional work groups being developed that intend to share exemplar material and professional development videos include; 

  • Creating teaching for mastery resources
  • Mental agility & subitising research
  • Lesson Study
  • Arithmetic development

In addition to Primary & Secondary based teaching materials.

Problems getting your licence?

Schools can email Katie Adams at This email address is being protected from spambots. You need JavaScript enabled to view it.

 

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Planning & resource development

All we want is .....

More than any other aspect the most common request from teachers is one of resources and teaching materials. This can be frustrating if you hold the opinion that resources alone are not the answer to effect the change we hope to achieve in the current evolution of maths education. However often the provision of a teaching idea in the form of an activity, problem or series of questions can allow a teacher to explore a new strategy for learning which they may not have done.

So the focus for sharing of resources and teaching materials could be to act as a catalyst to encourage colleagues to try new and occassionally innovative methods of teaching in a bid to generate greater reflection and collaboration amongst the region's professionals. We very much hope that teachers are inspired to tell us how new strategies fare in the classroom (good and bad) and share resources of their own. SYMH will be most effective if it can act as a conduit to collaboration across the region as well as providing access to new ideas from across the country.

Teaching for mastery

With respect to teaching for mastery we hope the focus of sharing ideas and materials will, over time increasingly shift to incorporating elements of teaching for mastery principles and pedagogy. One of the most impactful strategies is collaboratively planning and in all work groups there are oppotunities available for colleagues to participate with other teachesr from other schools.

Collaborative planning, where do we start?

Different methods being suggested for departments or groups of colleagues to try include;

  • Card sort of increasingly complex questions
  • "S-plan" of small step development
    • Taking a "big idea" and exploring connections

 

Increasingly complex questions/concepts

This strategy for professsional development collaborative discussion involves choosing a topic area within the curriculum (for example area of simple shapes) and presenting staff with a mixture of cards each with different questions that may be typically asked of learners within that topic. Staff are asked to arrange the cards in such a way so that they represent a development of learning from the fundamental skills to the more complex. Staff are then asked to consider the learning journey and in particular the conceptual development as well as the procedural development the learners undertake from moving from the simpest question to the most complex.

The ensuing discussions revolve around how we as teachers can guide rather than instruct learners along this "journey", with particular focus on small but clear steps of progression. Once one topic has been addressed teachers can then be asked to create their own sets of cards for a new topic, which adds a further element to the process.

Exemplar sort cards

Addition : these are an example of cards that illustrate concepts

Areas of triangles : these are an example of cards that illustrate questions

Once a group of colleagues has done this a few times, it can be enough to start introducing just challenging questions and ask them to consider, What do learners need to get to this point where they are able to attempt this type of question? This can form the basis of the S-plan style of collaborative discussion.

 

S-planning

This method addresses a theme - often a national curriculum statement and aims to strip it down into smaller steps by which teachers can refelect on the strategies to move learners along the "learning journey" of these small steps. Once the steps have been identified teachers can focus on how best learners can master the conceptual & procedural skills and knowledge of each one in turn.

If colleagues which to save time on identifying these steps they may wish to use the resource created by GLOW Maths Hub (which also includes some White Rose Maths Hub material) namely MathsNAV (a Sat Nav for Maths!) By selecting an appropriate topic there are suggested smaller steps that could be placed on the S-plan NB these are suggested steps not pre-scribed

The journey may start with a "hook" and end with a "deeper element" or "problem solving" aspect. A collaborative planning group may also reflect on the prior learning required for learners and focus on these end points as mathematical rich activities for encouraging dialogue with students.

splan

Some schools are preparing for this sort of collaboration by focusing on finding/creating these quality final challenging questions, which can then be shared and discussed.

Sharing questions might help because

  • Questions are a resource that every teacher needs, but teachers can use them in a way that fits with the way they teach. Sharing whole lessons and teaching approaches can give the impression that experienced and successful teachers, who know their classes best and have their own style, are being told what to do.

  • They help teachers see all the possible ways that students need to be able to think about the topic, and make them reflect on what the key ideas are and best to use explanations and examples to build a deep understanding of the topic.

  • They free up teacher time from creating or finding the questions (when this work has already been done by others), and gives them time to concentrate on how to structure a unit of work into small steps that give students the understanding and confidence to tackle harder problems. ​

 

 

Developing a big idea

This involves a much wider discussion about the curriculum as "big" ideas often permeate across a number of traditional topics. Proportion and the concept of multiplication is one example that can be considered. The professional dialogue may begin with considering which aspects or topics in maths this big idea contributes to. Questions that may be considered are; Does the big idea or concept reveal itself explicitly? How is the concept developed over time as learners meet the different topic areas?  What is the focus of learning?

Often the discussions reveal a possible shift in emphasis of the focus of learning in topics linked by a "big idea" from the individual procedures of any one topic towards the conceptual development of the wider understanding which in turn lead to learners making links across the curriculum.

For an example of how a concept may provoke discussion in this way : see Psychobabble blog 1

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Modelling ideas

15-02-2017 Hits:580

Ideas that focus on using technology to support looking at different mathematical structures, to support or reveal a deeper understanding.. The resources described by Pete Sides at the White Rose Maths Hub conference...

Read more

Arithmetical Structures

08-11-2016 Hits:615

Representing the structure behind mathematics can help develop an understanding of arithmetic and/or algebra. This can also be linked with multiple representations of a concept as each of the illustrations...

Read more

Modelling negatives

03-11-2016 Hits:612

  Just a few images that some may find helpful when describing calculations with negatives         Multiplying involves the concept of an "enlargement" operation.   Then the distinction between the multiplying effect of 1 (stays...

Read more

IRIS Connect

22-04-2016 Hits:1069

What is IRIS Connect? Originally set up as a system for teachers to record their own teaching through video capture on a secure and convenient platform IRIS Connect now has the...

Read more

Planning & resource development

01-10-2014 Hits:5066

All we want is ..... More than any other aspect the most common request from teachers is one of resources and teaching materials. This can be frustrating if you hold the opinion...

Read more

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