# Maths Cafe

## Conservation of Number Checkup

What are you trying to do?

- Spot check your
*Low Block*Foundation/Year 1 (Early Stage 1 or Stage 1) students for their ability to conserve number, In readiness for a focus on simple adding and subtracting. These students should already be able to 1-1 match oral number words to objects accurately.

- To conserve number means that a young student understands a quantity does not change if it is rearranged, covered up or hidden behind a box. A student who can conserve number is ready for the concept of addition where you can count on to find the total of 2 groups. A student who can conserve number can also count back to find the difference between 2 quantities.

- A young student who cannot conserve number doesn’t understand that a quantity remains the same when it is rearranged. This student is able to 1-1 match number words to objects but continues to recount when objects are rearranged. These students solve addition problems by counting all objects one by one. This is a Count All strategy, which is NOT an addition strategy. You can’t expect these students to participate effectively in any addition or subtraction classroom experiences. You need to identify these students before you focus on any addition or subtraction experiences.

Suggested Checkup Process

You need:

- 12 colourful counters e.g. frogs, cars, fish …
- A few minutes for each student interview
- F/Year 1 (ES1/Stage 1) Conservation Checklist (available to subscribers in the Counting activity files)

What do you do?

- Ask the student to count out 8 objects for you and to place them on the table.
- Next tell the student that you are going to put these “frogs” together in pairs e.g. rearrange the 8 objects into 2 lines of 4.
- After the objects are rearranged, ask the student; “How many frogs are there now?” Notice the student’s response. Some students will automatically say “8” so you can confidently mark their name off as a conserver on your checklist.
- Some students will look at you as if to say “…but I already told you that…” but they will still recount. You need to rearrange the objects a third time, e.g. in a curve. Again ask: “How many objects are there now?” If they are a conserver they will definitely tell you “8”. You can then mark them off on your checklist.
- But if a student recounts all the objects this third time, it is a strong indication that they cannot conserve number. These students are not indicating their readiness to experience adding 2 sets together. They need to experience counting objects, covering them up and then saying how many there are altogether. They need to “trust” objects. It may take some time focusing on similar activities to this before they realize that the number of objects does not change if they are rearranged or hidden. These students are definitely not ready to investigate adding and subtracting objects from a given group.

## Blockages

When we talk about student ability levels, we often use terms such as low, medium and high ability. In itself a term like “low” conjures up images of students who do not have the capacity to learn as much as others. It implies that we as teachers have not much control over such a student’s learning, as if they are missing a “maths gene”. But what happens if a low ability student now understands that concept that was previously beyond their grasp. How do they go from being a low ability student in front of your very eyes to being a medium or high ability student, for that concept?

I find a more useful maths language term is “blockage”. Again we can categorise our students as low, medium or high block, but the terms “high” and “low” are now opposite in meaning to our previous focus on “ability”. A new image reveals itself. If a student has a high blockage, it implies far more action on our part as teachers. There are multiple things that can be blocking this student, including emotional and physical as well as conceptual issues. How can we unblock this student? What strategies have we tried already? What other strategies may be effective? What language structures have I used? What terms does this student misunderstand? And if they are able to tackle a misunderstanding effectively and now “get” what it is we are talking about, they are unblocked. Their ability hasn’t magically jumped from low to medium or even high. The emphasis becomes on what we can do and provide as teachers to remove each student’s blockage.

A medium block student has several misunderstandings and after some one-to-one questioning and discussion with you or fellow students they may discover their incorrect thinking. A low block student may have just one or two misunderstandings and can more quickly see how to change their thinking.

It’s a little bit crazy, but I think of blockages as transparent clingwrap. Imagine your students are sitting in front of you with different layers of clingwrap covering their eyes. They can all see something, but those with only one or two sheets of the clingwrap, your low block students, can see most clearly and can see perfectly once those few sheets are removed. Everyone has the possibility of understanding more. It does not mean that your high block students WILL understand, but the language of blockages gives us more confidence that they are capable of learning.

There is no maths gene that some students have and others don’t. What can we all do to help each student learn more effectively?

## SMART Tests Victoria

This is a new project developed by leading mathematics educators. You can access quick targeted maths assessments to reveal your students thinking at www.smartvic.com. The suggested age group is Years 7, 8 and 9 but you’ll find the test levels can also fit with the needs of some Year 5 and 6 students too. Read the accompanying article “Getting Smart about Assessment for Learning.” It is still a work in progress but it will keep you up to date with the latest thinking about assessment.

## Australian Primary Mathematics Classroom Journal

Does your school already subscribe to this fantastic journal from AAMT (Australian Association of Mathematics Teachers)? There are 4 issues each year of the Australian Primary Maths Classroom with plenty of practical articles to help you reflect on your primary classroom practice. It is only $60 per year. Or if you are already an AAMT member you get 50% off your yearly subscription. The editors are Linda Marshall and Paul Swan.

## Worldometer Statistics

If you haven’t discovered this website yet, the Worldometers are a fantastic resource for studying large numbers. They continuously update number facts related to populations, births and deaths, the number of cars and bicycles and even the number of mobile phones in the world.

You can investigate at the same time each day and ask students to create their own statistical facts, such as “Since yesterday, the world’s population has increased by …” or “There are almost 2 billion google searches made each day”.

www.worldometers.info

## Australian Maths Associations

Are you a member of your local Maths Association?

Australian Association of Mathematics Teachers (AAMT)

**Northern Territory** Mathematics Teachers Association of the Northern Territory (MTANT)

**NSW** Mathematical Association of NSW (MANSW)

**NSW** Primary Association of Mathematics (PAM)

**Queensland** Queensland Association of Mathematics Teachers (QAMT)

**SA** Mathematical Association of South Australia (MASA)

**Tasmania** Mathematical Association of Tasmania (MAT)

**Victoria** Mathematical Association of Victoria (MAV)

## Girl with macaws

Here’s another great photo to stimulate your students’ thinking about real-life maths. Enya has 2 macaws on her shoulders. Why is this part of our maths resources photographs? How can this be used to promote mathematical thinking? Most of our photographs at first glance don’t shout “Mathematics” to you. Try to leapfrog from the photo to a wide range of mathematical possibilities. This photo could be the start of a class brainstorm. How many macaw species are there in our world today? What is the heaviest macaw species? The lightest? Which macaw species has the longest wing span? In what countries can you find macaws in the wild? What percentage of the earth is this?

What is the heaviest macaw species? How far do they fly on average? What territory do they cover? What do they eat? How much do they eat in a week? How many macaws, on average, would you need to balance your own body mass? How many years do they live? What might a model macaw kite look like?

Each of these suggestions could be the basis of mathematical research in your classroom.

## Why did we include this graphic?

Although each graphic in itself may not appear too mathematical, let your imagination fly. Challenge your students to create a wide range of possible questions related to each one.

Look at this hen. How many different mathematical questions can you record in 1 minute?

How many hens can possibly fit in one square metre of a chook pen? Even so, what is the official number of hens allowed per square metre? How many eggs does a hen lay on average in one week. How many weeks would it take to lay 2 dozen eggs? How heavy is a hen? Is a rooster just as heavy or heavier? How long does a hen live? How many different breeds of hen are there in our world? What type of hen is the largest? The smallest?

How much food does a hen eat in one week? If a farmer has 100 hens, how much food is this in one month? If chook food costs $2.50 per kilo, how much does this cost this farmer for 1 month of chook food?

And I am sure there are plenty more questions you can think of!

## Why did we include this photograph?

When you look at each photograph it may not at first be obvious why it can be used for maths. We will leave it up to your imagination in most cases.

If you look at this sign, how many different mathematical possibilities do you see?

How many kilometres will you travel if you leave Orange and drive to Cowra? Is it further to drive from Bathurst to Orange or from Orange to Cowra? If you start at Bathurst in which direction is Orange? When was each of the 3 towns established? What is the current population of each town? If you keep driving past Cowra, what town will you pass through next? How far from Cowra is this town? Where does Highway 33 go to and from? What about Highway 31? About where would you be if you are 49 km from Bathurst?

What other challenges can you create using this photograph as the stimulus?

## Where to now with subtraction?

Bev wrote this article many years ago when she discovered so many students continued to find subtraction difficult to tackle even at the end of Year 6. It seems to boil down to 6 key issues that all schools need to consider.

read more...## The Gilded Image

In another life, Bev has been painting in her studio, The Gilded Image, since 1996, specialising in medieval-style images with gilded backgrounds, inspired by the beauty of gold-leaf and the luminous colours in medieval paintings. She is passionate about creating and recreating images from medieval life in general, archangels and angels in particular.

You can also browse through Bev’s non-maths photograph collections with favourite images from across the world.

Click here to see examples of her medieval-style paintings at www.thegildedimage.com.