RAMR Deadly Maths: Adding Fractions

The RAMR cycle originates from Chris Matthews, an indigenous man who has a PhD in maths.

RAMR is a new model for teaching maths, or at least, new to me. I've appended a link to a video at the bottom where Tom Cooper from the YuMi Deadly Centre at the Queensland University of Technology (QUT) explains the RAMR cycle. I adapted his talk to a lesson I developed about adding fractions. YuMi is a Torres Strait Islander word meaning "you and me" and Deadly is an aboriginal word meaning smart.

The RAMR acronym stands for Reality, Abstraction, Mathematics, Reflection as illustrated by this graphic:

REALITY

Teach maths the way it is created or invented. Start with a problem. The problem I posed to the class was how do you add one half and two thirds.

The elements recommended in this Reality phase of the cycle are start with a real life problem, draw on local knowledge and construct kinesthenic activities.

PIZZA DIAGRAMS half and two-thirds

Present this problem as a real life exercise. John eats half a pizza. Jermain eats two thirds of a pizza. How much pizza have they eaten altogether?

Bring real pizza into the room and cut it up. It is a recommended part of the RAMR cycle that the teacher constructs such an activity, preferably kinaesthetic.

Prerequisites: The teacher needs to be aware of prerequisite knowledge required for the problem. In this case the denominator (bottom number) tells us how many equal parts the pizza is divided into. The numerator (top number) tells us how many part of each pizza are being eaten. This had been covered in a previous lesson.

ABSTRACTION

Abstraction means moving from the real world (a pizza cut into various pieces) to a representation of that reality in words, pictures and / or symbols.

We have already begun this above by using the words, symbols and pictures for half and two-thirds. In practice the various phases of the RAMR cycle overlap as well as having some distinctiveness.

One aspect I need to improve on is that of adding in creativity by inviting students to create their own representations of fractions. I didn't do this in the class but later when running the session for trainee teachers it did energise the session with some imagination.

Can you develop your own representation of fractions?

MATHEMATICS

In this phase of the cycle we stress the formal language and symbols of mathematics, practice the concepts a lot (most students need lots of practice) and connect to other maths ideas that have been taught earlier.

When asked how to add ½ plus 2/3 many students will add the numerators and denominators to get the answer 3/5ths. Explain why this is wrong. You don't add denominators because they don't represent something that ought to be added to solve this problem. Rather they represent how many pieces each pizza has been cut into.

The trick to solving this problem is to cut both pizzas in such a way that the parts are equal. This can be solved either
(a) visually or
(b) arithmetically by multiplying the denominators or
(c) by finding the lowest common number in the two times (2, 4, 6 ...) and three times tables (3, 6 …)

So, we divide both pizzas into 6 equal parts

Now we can add the fractions 3/6 + 4/6 = 7/6 = 1 and 1/6

The denominators are not added since they represent how many pieces the pizza was cut into. The numerators are added since they represent the parts of the pizza which are eaten.

Transforming ½ into 3/6

How do we get 6 from 2? Multiply by 3.

Now if we multiply the denominator by 3 we then have to multiply the numerator by 3. ½ x 3/3 = 3/6

This doesn't change the value of the fraction since 3/3 = 1 and multiplying by 1 doesn't change the value of the number. The technical name for this is compensation, the numerator 3 compensates for the denominator 3, etc. This technique is really valuable and can be used over and over again in the future, so it needs to be reinforced. Multiplying by 1 doesn't change the value of a number.

Repeat this process to transform 2/3 into 4/6 by multiplying by 2/2

More Prerequisites: (which were covered in earlier lessons) Equivalent fractions: ½ = 2/4 = 3/6 etc.

Improper fractions (7/6) and mixed numbers 1 and 1/6

REFLECTION

The goals here are to:

• set problems that apply the new idea back to reality
• enable students to validate and justify their own knowledge

The concepts being covered can be extended at any point throughout the cycle. It doesn't have to be confined to the end.
Extension: Represent 7/6 or 1 and 1/6 on a clock? Answer: 70 minutes or 1 hour and 10 minutes.

Inverse: How else could the pizza have been divided b/w John and Jermain to get the same answer? ie. What other two numbers (fractions) would add up to 7/6 or 1 and 1/6?

Generalise: How could you add any two fractions? Looking for answer here about achieving a common denominator.

For the class I was teaching I found that they struggled to understand what was required for the inverse section but with prompting they got it. Overall, I was happy about the response to the challenge posed by the Reflection section.

CONCLUSION:

Maths textbooks are notoriously dull. Direct Instruction as developed by Rhonda Farkota (link to her PhD thesis) is very useful. But how do we further develop maths curriculum in a rich way once the basics are established?

I found that the RAMR cycle challenged me as a teacher to develop my delivery further. There were some elements in the cycle which made me think hard before I could deliver them. I did find that students responded well to those elements of the cycle in which I harboured a hidden belief that they may not cope with. The cycle integrates real world, creative elements and traditional elements of maths in a manner which I found very satisfactory. I think it is a very good model.

Is this maths which incorporates indigenous culture or simply good maths teaching? Good question! I think both but mainly I lean to the latter view. But really good teaching adjusts itself to take the individual needs of all the current students in the class into account. This requires more analysis and thinking. LINKS
Tom Cooper explains the RAMR cycle here: Professor Tom Cooper - YuMi Deadly Maths

teaching fractions through instructional design

I'm helping some home schooling parents teach their kids fractions - using the Idit Harel Instructional Software Design Project (ISDP) method. It's the best way I know to teach fractions, which is acknowledged to be one of the worst taught things in primary school ("a weeping sore" article by Ellerton and Clements)

For an overview read this article on my website --> Educational Software: Designed by Kids for Kids. This time around we're using Scratch rather than LogoWriter.

It is available as a book - Children Designers: Interdisciplinary Constructions for Learning and Knowing (some extracts available from this link)

Today I took the parents through the pretest and the general approach. The pretest contains questions like this:




The way I explained this is that we are learning the deep structure of fractions, in contrast to a worksheet algorithm approach made up of questions like what is 3/4 of 16. The problem with the worksheet alogorithms is that the children might be able to get the correct answer but still not have a clear understanding of what a fraction is.

By the deep structure of fractions I mean the ability to transform between picture, word and symbolic representations:


We discussed the questions after the adults had done the test. Naturally, this focused mainly on the questions that someone had got wrong. This is modelling the process the parents will go through with their children.

The next step is to pose these questions to your kids:

• What do you or your siblings find difficult about fractions?
• Make up your own multiple choice questions to test your sibling with, something that will improve their understanding.

There is a lot of meta learning involved in this process - asking and assisting the kids to finding the zone of proximal development for themselves and their siblings.

I then showed the parents how to represent such a question on the screen using Scratch

That's where we got up to. After I went home I looked up Idit Harel's thesis and realised I had missed out her "What is a fraction?" interview, which is another good way to start. This isn't completely represented in her thesis but I'll quote part of what she says here, so the parents can work this into their approach as well.

PURPOSE OF INTERVIEW IS TO INVESTIGATE THE CHILD'S CONCEPT OF WHAT FRACTION IS, their "favourite" representations of fractions, their way of "talking about" fractions and their ability to link or translate representations. Add to these questions if you want, whatever comes to mind.

General questions:
What is a fraction?
Can you describe anything in this room as a fraction?

Activities:
Use this set of pegs (or play dough, paper, blocks etc.) to show an example of the fraction 2/3

Probing questions:
One child told me that this red block and this yellow block together equal one-half. What do you think?
If this yellow block is the unit, what are these two yellow blocks and three red blocks together (the red blocks being half the size of the yellow blocks)?
Then probe further depending on what they come up with

Related: Questioning Research