To be explicit or to inquire that is the question!
Here is my take on the latest maths wars. What do I mean? With the revised Australian Curriculum in the news, the war between the proponents of explicit teaching and inquiry learning has been hotting up.
Personally, I do not see this as a dichotomy. Today there was an article in The Age written by Adam Carey. It outlines the two schools of thought and closes with a teacher describing how they ignited students’ interest through inquiry and plugged gaps in knowledge using explicit teaching. This approach seems sensible to me. Below is my response to the article on The Age website.
In a classroom, many different types of pedagogy are used every day. A good teacher will tailor the learning to the students in front of them. They are teaching the same material in different ways. Some explicit teaching is always necessary, and some inquiry/investigation is also needed. For example, being flexible andthinking about numbers and number manipulation in different ways is essential to use numbers in everyday situations and at higher levels. But, in order to approach a new problem, you need to have the inquiry/investigation skills to transfer knowledge and think creatively. There are techniques that I feel are not used often enough in the classroom. These include: using physical objects (manipulatives), encouraging students to visualise and draw diagrams, and to estimate before they calculate.
A curriculum should not be dictating a pedagogy. It should outline content to be learnt but not how it should be taught.
We need to trust and respect our teachers to decide the best way to teach their students.
In Part 2 of this series, I will demonstrate how Algebra Tiles can be used to build expressions, model addition and subtraction of expressions, and model substitution into algebraic expressions.
In the previous blog post, I looked at how to make Algebra Tiles and use them to model integer operations.
Building an algebraic expression
If you have used Algebra Tiles to model integer operations you can now introduce the x-tile and the x2-tile.
In the video below I show how to build an expression. It is important that the students recognise the different shapes and colours and what they mean. I would take some time on this stage as you want them to be very familiar and comfortable using the tiles.
Using Algebra Tiles to build an expression
Collecting Like-Terms
Once students can build a simple expression you can move on to more complex expressions and looking how they can be simplified by collecting together like-terms. Students often find it difficult to recognise what is meant by a like-term. By using the tiles it is easy to see which terms are similar by their size and shape. I would start with positive values only and then when students are confident with those introduce negative values, which are the same size and shape but a different colour.
Collecting like-terms
Multi-variable expressions
You can use Algebra Tiles to model expressions that have more than one variable. Below are videos of two ways this can be done. The first is using different coloured tile-sets for each variable. This can get a little messy and I would suggest the students write a key for each variable showing which colour is positive and which negative. The beauty of this method is that the students can really identify what a ‘like-term’ is. They can collect the tiles of the same shape and related colours.
The second method uses only one set of tiles as follows:
How to use Algebra Tiles to illustrate multiple variables
Using the square tiles to represent different variables can help the students to appreciate that each variable represents a different value.
Using Algebra Tiles to represent a multi-variable expression
Addition and Subtraction of Expressions
Once students can collect like-terms together you can move on to addition and subtraction of expressions. Below I use the same expressions with each operation and show how the result is different.
Adding expressions
Addition of expressions
Subtracting expressions
Subtracting multivariable expressions
Using the second method of modelling a multivariable expressions I showed above, I will demonstrate how you can use Algebra Tiles to subtract two expressions.
(Spot the mistake!)
Subtraction of multivariable expressions
Substitution
I have found that students often struggle with substitution. They do not understand what it means. Using Algebra Tiles the students can physically exchange the variable tiles for the unit tiles. The x2-tile is replaced by a square with side length of the given value. This allows the students to build a clear visual representation of what they are doing when they substitute the variable with a given value.
Substitution
Other ideas and suggestions
You may find that students will move away from using the tiles quite quickly and start drawing diagrams or using algebraic notation. I would suggest that you encourage them to use the tiles for the first few examples of each new type of question/activity. Students should feel comfortable to use the tiles whenever they feel the need.
I would also allow students access to the tiles during tests as well. Tests and quizzes can cause students increased stress level. Just knowing that they have the manipulative available can reduce their anxiety.
In Part 3 of this series, I will explore solving simple linear equations.
There is a Padlet to accompany this blog series with links to research, virtual manipulatives and other resources.
Today I received a really exciting package. I bought my own set of Cuisenaire Rods! For those that have not heard of these they are sometimes called Proportional Rods as they are used to illustrate proportional reasoning, amongst other concepts.
I remember these from my first day at school. We were given these rods and told to play with them. It is a really clear memory and I have loved these rods ever since.
Unfortunately, they have fallen out of fashion since then (it was a very long time ago) and now there are countless boxes at the back of school cupboards gathering dust.
Have a look in your school and see if you can find some. If you can, get them out, dust them off and let’s explore some of the ways they can be used.
The basic concept is that the lengths of the rods are proportional to each other. Note the colours are important too. Although the originals are wooden you can now get them in plastic and virtually too. No matter what material yours are they should all have the following properties:
White – 1cm in length (a cube)
Red – 2cm in length
Light Green – 3cm
Purple – 4cm
Yellow – 5cm
Dark Green – 6cm
Black – 7cm
Brown – 8cm
Blue – 9cm
Orange – 10cm
The connections between the colours is as follows:
White (1) and Black (7) are on their own
Red (2), Purple (4), Brown (8) – Red family – multiples of 2
Light Green (3), Dark Green (6), Blue (9) – Blue family – multiples of 3
Yellow (5), Orange (10) – Yellow family – multiples of 5
When students are first introduced to these rods, they are told the values as 1, 2, 3 . . . But it is important to note that this is NOT the only way they can be seen. By being flexible with the values many more concepts can be explored.
I am going to demonstrate a (very) few of the ways in which Cuisenaire rods can be used below and post links to websites where you can continue exploring. These are NOT tools for the very young grades only. I believe that they should be used in older grades too. Especially when exploring fractions and decimals, as I will demonstrate below.
Ideas for the younger grades:
By laying down a rod and then finding combinations of other rods that make the same length students can explore addition facts, for any number.
Here are some arrangements for Facts to 5
Students can be asked questions such as: Are these two arrangements the same?
They both make 5 and use the same smaller rods but are they the same? This can help students to understand the communicative and associative properties of addition.
Use the rods to explore odd and even numbers and what happens when you add odds and evens.
They can be used to model subtraction problems: Here is 7 – 4
There are several ways to discover the difference:
Multiplication as repeated addition is easily modelled too.
Ask the question – Can we use only one other rod to reach the same length?
What does this tell us?
Other ideas for younger grades:
Partitioning
Make a picture worth 100 or 50 . . .
Barrier games – students describe what pattern/picture they have made and another has to form it from the verbal instructions
Finding areas – great for introducing the area model for multiplication
Factorising
A picture worth 100
Worth 100
For the older grades
When working with older grades we can play around with the values of the rods.
Call the Orangerod ONE – now what are the values of the other rods?
Call the Yellow rod ONE – now what are the values of the other rods?
This is a great way for students to explore fractions and decimals.
If the Purple rod has the value ½ or 0.5, what are the values of the Red rod and the White rod? What is the value of the Blue rod?
There are so many other ways to use these rods, I am sure that students themselves, once they are familiar with them and comfortable with using them, will find uses you had not thought of.
The most important thing is to give the rods to students, allow them to ‘play’ with them for a while. Then use them regularly for specific activities. If they are left out and are easily available students will use them when they feel the need. They are not just for ‘weaker’ students but there as a tool to support all students!
