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:
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.
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 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?
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
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
The goals here are to:
- set problems that apply the new idea back to reality
- enable students to validate and justify their own knowledge
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.
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