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There are too few words to describe the complexities of our number system.
I am sure there is nothing new about what I am trying to explain here, much of it is to get my own head around some fundamental issues.However the concepts are often poorly understood through a lack of clear and unambiguous language.
The framework I am trying to describe is this..
Additive operation
The operation suggests a before and after, the number sentence (or equation) suggests you start with 5 and there is an operation of 3 added to this to make a new value of 8. This is AUGMENTATION.
The reverse operation is taking away the 3 from the new starting number 8 so the new value is 5. The diagram could be drawn left to right but here I am emphasising the REVERSE operation which is the INVERSE of adding on, that being taking away.
Interesting to note that you can do the same operation (adding on) but with an INVERSE number (in this case negative 3) which also results in getting back to where you started with.This isn’t a point I would necessarily raise with learners early on but is important for comparing the nature of additive and multiplicative systems.
Of course it is also perfectly possible to start with 3 and add on 5 which is another “story” in whatever context the equation is representing but illustrates the COMMUTATIVE nature of addition.
Additive: same amount expressed different ways
Rightly or wrongly I am using a horizontal bar to describe this but this is just for my own benefit and there is nothing mandatory about this.
Consider the amount 8. This can be PARTITIONED into 5 + 3 among many other ways of course. And recombined by AGGREGATING to find the original sum.
There is no before and after, just a value of 8 (or items if we are considering a context) which can be “split” into two or more separate values and then recombined to make the original value.
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Additive Comparison
Again there is no before and after, but two values occurring at the same time. The comparison is not specifically linked to adding on or taking away but depends which value is your initial focus.
If we consider 5 first, then 8 in comparison is three more than 5.
If we focus on 8 first then 5 in comparison is 3 less than 8.
NOTE that we can add 3 on to 5 to result in 8 or take away 3 from 8 to result in 5.
We can also find the difference by taking away 5 from 8 to get 3 but we do not (normally) take 8 from 5 to get a difference of -3, the difference is always positive.
So the difference between 8 & 5 is the same as the difference between 5 & 8, this has implications for interpreting the number sentence.
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Multiplicative operation
Here we have a before and after situation where 5 has been multiplied 3 times. In effect we have an ENLARGEMENT of SCALE FACTOR 3 to produce a new number or PRODUCT which in this case is 15
We can do the REVERSE operation with the same number (this being division – whatever that means) but it looks like we are dividing the starting number 15 into 3 equal sections and choosing just one of them. Another way to think of this could be to say 15 (the new starting number) represents 3 parts and I want to know what one part is.
Of course we can (like additive) do the same operation with an INVERSE number (in this case multiply by 1/3) but language helps if we say what is a “third of”.
Multiplicative: same amount expressed different ways
Most easily shown by an array. There is no starting number or new end result, just in this case 15 items but the can be arranged in such a way that shows 5 groups of 3 or three groups of 5 are equivalent to 15. Similar to partitioning and aggregation this form of expressing a value as a product of its factors is called FACTORISING.
Note there is a language issue which can confuse.
Three lots of five in context is not the same as five lots of three, but the product is the same. Five lots of three is the same as 3 multiplied 5 times the latter of these suggests an operation whereas the former leans more to factorising?
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Multiplicative comparison
For this example I am going to choose the numbers 10 & 15. Like the additive comparison it depends on which is your initial focus.
In terms of 10, the number 15 is 1 ½ times this.
We can see this by comparing equal parts and in this case can conveniently simplify the comparison of 3 lots of 5 with 2 lots of 5 to 3/2 or 1 ½ .
Note that this also finds us the scale factor to turn 10 into 15 using the multiplicative operation.
But if we focus on 15 first, (I have swapped the bars around) then in terms of 15 the value of 10 is only 2/3 this.
Like the additive comparison there are two ways of looking at it (more than, less than), the multiplicative comparison can also be done both ways, although the comparative RATIO should be written in a specific way depending on the context or focus number, unlike the difference which is always given as a positive number.
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Dividing into a ratio
Now we are combining both additive and multiplicative structures such that the two parts of an additive model are in a ratio.
This opens up a whole new set of issues including fractions of amounts.
The whole is divided into equal portions where the word part could be used to describe each of these smaller, equal parts or indeed each of the combined section of portions that constitute the part, part whole of the additive model.
If we take a particular example
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This example can be described many ways, e.g. three-fifths of the whole 30 is 18
18 as a comparative multiplicative portion of 30 is 3/5 and 3 is to 5 is in the same proportion as 18 is to 30.
Whereas the ratio of the additive parts 2:3 gives us the constituent parts of the whole in an additive sense, being 18 and 12 which are also in the same proportion as the ratio 2:3
So to go back to my original issue for writing this piece….
There aren’t enough words to describe the complexity of our number system, or perhaps there are but they are not all used commonly. Those that are used do not always clearly express the distinctions that need to be made to gain a full and deep understanding.
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Generic terms, particularly the four favourites of Addition, Subtraction , Multiplication & Division and perhaps added to these the words Part & Proportion on their own are not sufficient to distinguish the subtleties that at least need to be understood by teachers if not learners and whilst we continue to use them in a general way ambiguous meanings and therefore misunderstandings will prevail.
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The use of diagrammatical models certainly allow us to clarify what we mean and can act as a frame of reference to all when clarifying our meaning but perhaps there is also a need for a clarifying language of terms so we can explain our deeper understanding.
Although I doubt we will ever see KS2 questions quite like this…
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“Describe the how the multiplicand and the product in a multiplicative relationship can be used in a comparative way to find the multiplier and the significance of this quotient in describing the proportion between the aforesaid multiplicand and product.”
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Although if you have read and understood this blog, you may well be able to give it a go.
Psychosides 2017
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Ideas that focus on using technology to support looking at different mathematical structures, to support or reveal a deeper understanding..
Spreadsheets
10 Bar random sums <10 illustrated on a bar model (any part of the number sentence can be hidden)
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
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
x-squared interactive model of y=x^{2 }shown as bars and as a point on graph
Circle thoerems: a series of simple geogebra files to allow interactive investigation of theorems
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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 admin@symathshub.org.uk
Click on each below link to download PDF
Linear equations as bar models
Simultaneous (linear) equations as bar models
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 the same) and -1 (flips it over)
Which also works on negatives
...and so we can multiply by any sized positive or negative number.
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.
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/
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.
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;
In addition to Primary & Secondary based teaching materials.
Problems getting your licence?
Schools can email Katie Adams at katie@smartmoveit.co.uk
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;
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.
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