Aim: The idea of this unit is to bring out the maths in the culture of the students, rather than bringing the culture into the maths for the students.

Basically the unit is an overview of maths as a discipline, as a day-to-day tool and as a natural system that exists in the real world of the students and links to worlds beyond their own. The aim is to position math concepts positively in the worldviews of the students, aligning with familiar patterns of logic and customary habits of thinking, then extending and transferring these to multiple contexts/purposes.

Objectives: Your objectives are diagnostic as well as didactic. You are identifying culturally familiar or appropriate logic patterns and applications in your students. You are also connecting these with the necessary concepts and logics needed for success in maths at school. The planning here consists of broad suggestions rather than prescribed content and activities, as the most important part of this work is that you remain responsive to your students. Remember that you are mapping basic math concepts onto their pre-existing frameworks of customary logic, whatever these may be.

Year level: This unit has not been written for any particular year level – it is up to you design it flexibly in response to the knowledge and needs of your students. It may not even be a formal unit of work, but rather a social activity you do for half an hour each day with your class. The key is to make and find patterns through a social activity organised around visual stimuli and tactile materials, which arguably could be a useful exercise for any year level. Teachers can design different images and patterns at different levels of complexity to match any stage of learning.

Equipment and setting: Ideally this activity would take place outside in a sandy area where patterns could easily be sketched on the ground with a stick. Teacher and students work with a number of generative patterns comprising dots and lines (see images below). Instead of dots, it is desirable to work with objects like seeds, gumnuts, shells, beads or rocks. But depending on your circumstances and resources, it could be adapted to working with chalk on cement, or crayons on butcher’s paper. In the images given as examples here, we have simply used permanent markers on printer paper.

Learning Sequence: Each lesson follows basically the same process. Before the lesson begins, the students must be prepared to focus completely and silently upon what the teacher is doing for the first few minutes of the lesson. No words are to be spoken in that time. The students must understand this to be an important cultural protocol, and even a cultural skill.

1. The students closely observe the teacher (in complete silence) creating an image on the ground, in clear stages marked by significant pauses. (Duration and complexity increase each lesson.)

2. The teacher shares a story with the students, based on a pattern, shape, sequence or number from the design.

3. The students recreate the design individually.

4. The students identify the shapes, symbols and patterns they see.

5. The teacher identifies the shapes, symbols and patterns he/she sees.

6. Whole class discusses where and how these patterns/shapes etc. (or similar) occur IN NATURE locally or in other places they have been or seen/heard/read about.

7. Whole class discusses where and how these patterns/shapes etc. (or similar) occur IN COMMUNITY LIFE locally or in other places they have been or seen/heard/read about.

8. Teacher identifies and explains key mathematical concepts and processes in relation to the information and contexts students have shared. (Where possible, map processes onto some aspect of the image as a memory device, like “BOMDAS” but using images instead of letters.)

9. Students take one key concept and use it as a stimulus to create their own image/design on the ground. Students then present and explain their design to the class.

It is advisable for teachers to keep records of what the students produce in these sessions – photographed images, work samples and teacher notes that can be used as data for analysis of student learning needs, but just as importantly as resources for lessons. Wherever possible, content in subsequent maths lessons should refer explicitly to examples from the contexts, habits and images contributed by the students in these sessions. In fact, because these sessions do not use digits or letters in any way (focusing instead on material, oral and visual concepts), it is important to follow up in the classroom with abstract print/numeric knowledge linked to the concrete learnings from the session.

Generative Images: See images below, in order (top to bottom) from one to ten. You will notice that written numbers and words do not appear at any time, that students and teachers are required to work with concrete images and materials. Numbers can be spoken, just not written. And they must be linked to/supported by stories and experiences drawn from life beyond school. Below each image you will see some brief examples of explicit math content that may be drawn from interactions with the images. For most of the images you will see a progression showing at least the first and last step for making the pattern.

One, zero, circle, centre, circumference, radius, whole.

Three, six, nine, counting in threes, three squared, multiplication symbol, semicircle, square shape, thirds.

Four, four squared, hexagons, arcs, quarters, 12 hour clock. On most of these designs now, if the dots on one side of the diamond are numbered, they can be added with the number at the other end of the arc (like an abacus) with the answer being one more than the total in the row. E.g. for this one, the arc sums are 1+4=5, 2+3=5.

Five, ten, counting in fives, fifths, five squared, odd and even numbers, symmetry.

Six, six squared, sixths, right angled triangles, multiplying and dividing by 2, 3 and 6, arc sums: 1+6, 2+5, 3+4.

Seven, seven squared, sevenths. In these diagrams you can also count the arcs and multiply by four and eight. E.g. in this one you have 3x4=12, 3x8=24. Could explore hours in the day or 24 hr clock.

Eight, eighths, eight squared, rectangles. Arc sums: 1+8, 2+7, 3+6, 4+5 (note these sums preview the next image…) Arc products: 4x4, 4x8.

Nine, ninths, nine squared, prime numbers, review half, quarter, third. Divide by 2, 4 and three. Arc sums: 1+9, 2+8, 3+7, 4+6.

Ten, tenths, hundred, ten squared, percentages (quarter=25, half=50, etc.)

Just drawing these patterns is a pretty intense mathematical procedure. First you're building a triangle from a single point (then two points, then three etc.) until it has equal points on each side of the number you're exploring in the image. Then you're expanding that beneath the triangle to turn it into a diamond with equal sides. Then you're finding the central point of the outer rows to place the first small arc. If it is an even number, it will incorporate the middle two dots. If it is an odd number, it will span the middle three dots. Finding this middle group requires calculation.

Becoming proficient at drawing these patterns draws from and develops basic maths skills that can then be extended to other purposes and abstract expressions.

The idea of this unit is to bring out the

maths in the cultureof the students, rather than bringing theculture into the mathsfor the students.Basically the unit is an overview of maths as a discipline, as a day-to-day tool and as a natural system that exists in the real world of the students and links to worlds beyond their own. The aim is to position math concepts positively in the worldviews of the students, aligning with familiar patterns of logic and customary habits of thinking, then extending and transferring these to multiple contexts/purposes.

Objectives:

Your objectives are diagnostic as well as didactic. You are identifying culturally familiar or appropriate logic patterns and applications in your students. You are also connecting these with the necessary concepts and logics needed for success in maths at school. The planning here consists of broad suggestions rather than prescribed content and activities, as the most important part of this work is that you remain responsive to your students. Remember that you are mapping basic math concepts onto their pre-existing frameworks of customary logic, whatever these may be.

Year level:

This unit has not been written for any particular year level – it is up to you design it flexibly in response to the knowledge and needs of your students. It may not even be a formal unit of work, but rather a social activity you do for half an hour each day with your class. The key is to make and find patterns through a social activity organised around visual stimuli and tactile materials, which arguably could be a useful exercise for any year level. Teachers can design different images and patterns at different levels of complexity to match any stage of learning.

Equipment and setting:

Ideally this activity would take place outside in a sandy area where patterns could easily be sketched on the ground with a stick. Teacher and students work with a number of generative patterns comprising dots and lines (see images below). Instead of dots, it is desirable to work with objects like seeds, gumnuts, shells, beads or rocks. But depending on your circumstances and resources, it could be adapted to working with chalk on cement, or crayons on butcher’s paper. In the images given as examples here, we have simply used permanent markers on printer paper.

Learning Sequence:

Each lesson follows basically the same process. Before the lesson begins, the students must be prepared to focus completely and silently upon what the teacher is doing for the first few minutes of the lesson. No words are to be spoken in that time. The students must understand this to be an important cultural protocol, and even a cultural skill.

1. The students closely observe the teacher (in complete silence) creating an image on the ground, in clear stages marked by significant pauses. (Duration and complexity increase each lesson.)

2. The teacher shares a story with the students, based on a pattern, shape, sequence or number from the design.

3. The students recreate the design individually.

4. The students identify the shapes, symbols and patterns they see.

5. The teacher identifies the shapes, symbols and patterns he/she sees.

6. Whole class discusses where and how these patterns/shapes etc. (or similar) occur IN NATURE locally or in other places they have been or seen/heard/read about.

7. Whole class discusses where and how these patterns/shapes etc. (or similar) occur IN COMMUNITY LIFE locally or in other places they have been or seen/heard/read about.

8. Teacher identifies and explains key mathematical concepts and processes in relation to the information and contexts students have shared. (Where possible, map processes onto some aspect of the image as a memory device, like “BOMDAS” but using images instead of letters.)

9. Students take one key concept and use it as a stimulus to create their own image/design on the ground. Students then present and explain their design to the class.

It is advisable for teachers to keep records of what the students produce in these sessions – photographed images, work samples and teacher notes that can be used as data for analysis of student learning needs, but just as importantly as resources for lessons. Wherever possible, content in subsequent maths lessons should refer explicitly to examples from the contexts, habits and images contributed by the students in these sessions. In fact, because these sessions do not use digits or letters in any way (focusing instead on material, oral and visual concepts), it is important to follow up in the classroom with abstract print/numeric knowledge linked to the concrete learnings from the session.

Generative Images:

See images below, in order (top to bottom) from one to ten. You will notice that written numbers and words do not appear at any time, that students and teachers are required to work with concrete images and materials. Numbers can be spoken, just not written. And they must be linked to/supported by stories and experiences drawn from life beyond school. Below each image you will see some brief examples of explicit math content that may be drawn from interactions with the images. For most of the images you will see a progression showing at least the first and last step for making the pattern.

One, zero, circle, centre, circumference, radius, whole.

Two, three, four, addition, subtraction, triangle, diamond, half, twice.

Three, six, nine, counting in threes, three squared, multiplication symbol, semicircle, square shape, thirds.

Four, four squared, hexagons, arcs, quarters, 12 hour clock.

On most of these designs now, if the dots on one side of the diamond are numbered, they can be added with the number at the other end of the arc (like an abacus) with the answer being one more than the total in the row. E.g. for this one, the arc sums are 1+4=5, 2+3=5.

Five, ten, counting in fives, fifths, five squared, odd and even numbers, symmetry.

Six, six squared, sixths, right angled triangles, multiplying and dividing by 2, 3 and 6, arc sums: 1+6, 2+5, 3+4.

Seven, seven squared, sevenths. In these diagrams you can also count the arcs and multiply by four and eight. E.g. in this one you have 3x4=12, 3x8=24. Could explore hours in the day or 24 hr clock.

Eight, eighths, eight squared, rectangles. Arc sums: 1+8, 2+7, 3+6, 4+5 (note these sums preview the next image…) Arc products: 4x4, 4x8.

Nine, ninths, nine squared, prime numbers, review half, quarter, third. Divide by 2, 4 and three. Arc sums: 1+9, 2+8, 3+7, 4+6.

Ten, tenths, hundred, ten squared, percentages (quarter=25, half=50, etc.)

Just drawing these patterns is a pretty intense mathematical procedure. First you're building a triangle from a single point (then two points, then three etc.) until it has equal points on each side of the number you're exploring in the image. Then you're expanding that beneath the triangle to turn it into a diamond with equal sides. Then you're finding the central point of the outer rows to place the first small arc. If it is an even number, it will incorporate the middle two dots. If it is an odd number, it will span the middle three dots. Finding this middle group requires calculation.

Becoming proficient at drawing these patterns draws from and develops basic maths skills that can then be extended to other purposes and abstract expressions.