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The World of Maths
Transcript of The World of Maths
Maths and Numeracy in the Real World
What is Mathematics and Numeracy?
Mathematics and Numeracy within the School Curriculum
In the school curriculum, numeracy is a skill developed mainly in maths lessons, but is also applied and extended in other subjects such as science (when reading scientific charts and graphs), writing (understanding poetry and the number of words included in each line), art (through geometry) and music (musical rhythm often follows mathematical series). The present curriculum in Australia is based on the National Curriculum which aims to ensure the teaching of a common curriculum in all schools and acts as a foundation for high quality teaching in order to meet the needs of all Australian children. The Australian Curriculum is guided by the Melbourne Declaration on Educational Goals for Young Australians, approved by the Ministerial Council in December 2008. It commits “to supporting all young Australians to become successful learners, confident and creative individuals, and active and informed citizens” (The Department of Education and Early Childhood Development, 2011) and to promoting equity and excellence in education .
Mathematics is the study of numbers, patterns, space and change; whereas, numeracy is a life skill. Mathematics encompasses the ability to “put mathematical knowledge and skills to functional use as well as the ability to pose and solve mathematical problems in a variety of situations…” (Mathematical literacy, Australian Council of Educational Research, 2013). Being numerate goes beyond simply ‘doing sums’, it means having the confidence and competence to use numbers and think mathematically in everyday life.
Numeracy in practice: teaching, learning and using mathematics (The Department of Education and Early Childhood Development, 2011) writes that Steen (2001) believes that “the term ‘quantitative literacy’ and ‘mathematical literacy’ are used interchangeably”. However numeracy helps people to cope with the quantitative demands of modern society.
Everyone uses both numeracy and mathematics in their everyday lives such as calculating time, space or distance, using spreadsheets, using timetables, and confidently handling money. Through numeracy and mathematics we learn essential skills such as solving problems, undertaking and explaining solutions, making decisions based on logical thinking and reasoning, and interpreting data, charts and diagrams. “Numeracy, like literacy, provides key skills for individuals to participate successfully in schooling”. (The Department of Education and Early Childhood Development, 2011).
In both the school context and outside of school life, numeracy refers to the ability to use mathematics to “interpret information or solve practical problems, apply their knowledge appropriately in contexts where they will have to use mathematical reasoning processes, choose mathematics that makes sense in the circumstances, make assumptions, resolve ambiguity and judge what is reasonable in the context”.(COAG 2008, p.10)
We can see that numeracy and mathematics are interrelated. All numeracy is underpinned by some mathematics, therefore showing the important role in which school mathematics plays in young peoples lives.
Eight hundred years ago a mathematician named Fibonacci recognised patterns in nature with a particular sequence of numbers recurring. 0 1 1 2 3 5 8 13 21… and so on.
The Fibonacci sequence is created by adding the latest two numbers in the series to get the next like this: 0+1=1 1+1=2 1+2=3 2+3=5 3+5=8 5+8=13 8+13=21
This sequences turns up in some odd places. Pineapples and pine cones have spikes arranged in two sets of spirals, one going clockwise and one anti-clockwise. Depending on how big the pineapple is, there will be either 5 and 8 spirals, 8 and 13 spirals, or 13 and 21 spirals. This Fibonacci sequence can be seen in the picture below. It can also be found in the way that tree branches grow, and the arrangement of seeds in sunflower heads.
“The Fibonacci number patterns encountered here occurs so frequently in nature that we often hear the phenomenon referred to as a ‘law of nature’.” (Britton. J, 2011)
The Law Of Nature
The Honeycomb Conjecture
Teaching this in the Classroom....
In order to introduce the concept of hexagons into the classroom, I would begin by giving each child a nut (nut and bolt). We will then discuss what we notice about the shape and its properties e.g. how many sides it has, if each side is equal and the symmetry of the shape. I will then take out a soccer ball and a picture of a stop sign, discussing their similarities and differences to the bolt. Asking the question, which shape matched the shape of the bolt? We will then compare the shapes and count the sides of each shape.
I will then give the children a worksheet with different shapes on it, getting the children to colour the ones that are shaped like hexagons. Another work sheet I will give the children is a worksheet where they can colour the hexagons in to make a pattern. This activity related to the Australian Curriculum Mathematics Learning Area for the foundation year and year 1 in learning about shapes (ACMMG009: Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment and ACMMG022: Recognise and classify familiar two-dimensional shapes and three-dimensional objects using obvious features).
The next lesson will focus on drawing a hexagon. I will begin the class by taking the children outside and getting them to draw a freehand hexagon on the concrete in coloured chalk. We will then measure the hexagons sides to see who's hexagon has the most even sides (this person will get a small prize). We will then go inside and draw hexagons on graph paper, ensuring all sides of the hexagon are equal to one small square. After this, the children will learn how to draw a hexagon using a protractor. In order to do this, they need to learn how to use the protractor so that each of the six angles equals 120 degrees. This lesson relates to the Australian Curriculum Mathematics Learning Area for year 2 in relation to learning about shapes (ACMMG042: Describe and draw two-dimensional shapes, with and without digital technologies).
“You and me, we all agree —
Math is fun as you will see!
It makes us think, it makes us strong,
It helps us learn even when we’re wrong.
You and me, we all agree —
Math is fun for you and me!”
Engaging Students in Mathematics and Numeracy
In order for students to be competent in what they have been taught, they must first be given opportunities to practice and apply the mathematics they have learned. They are then able to apply this learning to other situations in life. The Australian Curriculum Assessment and Reporting Authority states that “It is important that the Mathematics curriculum provides the opportunity to apply mathematical understanding and skills in context, both in other learning areas and in real world contexts”. Through applying these skills in context, we are bridging the gap between mathematics learned at school and the variety of situations where it needs to be used in everyday life.
From the early years students need to be provided with higher level problems and questions, which help them make connections between key concepts and procedures, rather than instruction that is focused on routine tasks. Students need to see the mathematics they are doing as important and relevant, and themselves as capable of thinking and working mathematically. Working collaboratively in mixed ability groupings is generally to be encouraged with opportunities for students to support one another and to share explanations.
Similarities and Differences of Mathematics and Numeracy
Both Mathematics and Numeracy are Taught in School.
In a honeycomb each cell is hexagonal.
In 36 B.C., Roman scholar, Marcus Terentius Varro proposed that "a structure built from hexagons is probably a bit more compact than a structure built from squares or triangles"( Krulwich. R, 2013).
Marcus Terentius Varro believed that a hexagonal honeycomb would have
the smallest total perimeter. Therefore, the more compact the structure, the less wax will be needed to construct the honeycomb. In 1999 a mathematician named Thomas Hales proved that hexagons are indeed more compact.
This maximizes the space in each one while keeping the gap between them to a minimum.
If circles were used, more wax would be needed to fill the gaps between the cells.
"The honeycomb is a masterpiece of engineering. It is "absolutely perfect in economizing labor and wax." - Charles Darwin
Using real world examples, relating the questions to the students interests and showing students how math relates to their lives will make the subject more interesting and enjoyable. “Learning is enhanced when concepts are presented in the context of their use…thus (the student gains) confidence in his or her ability to solve similar problems outside the classroom” (CORD, 1999). Determining how the mathematical concept can be used to answer real-life questions will motivate students to learn as they are put into stimulating environments that inspire learning by drawing on past experiences and that recognise when they will use this information in the future. These questions make the topic more ‘realistic’ and ‘accessible’ to the students.
Contextual learning is a proven concept that incorporates much of the most recent research in cognitive science. The contextual learning theory focuses on the multiple aspects of any learning environment, whether a classroom, the park, a computer lab, a work site. Clements, D. & Sarama, J. (2013) stated that “high quality early math includes thinking, active experimentation, and talk about mathematical ideas”. Teachers are encouraged to choose and/or design learning environments that incorporate as many different forms of experience as possible in working toward the desired learning outcomes e.g. if learning about addition, get the children to collect rocks from outside then bring them inside to use for their math’s.
Christiansen and Walther (1986) believed that tasks with many possible solutions engaged students in productive exploration; also known as an open ended approach to teaching. “The aim of open-approach teaching is to foster both the creative activities of the students and their mathematical thinking in problem solving simultaneously” a study of "open-approach" (Nohda.N, n.d.). Open ended questions and activities offer students the opportunity to be both creative and challenge their thinking. Clements, D. & Sarama, J. (2013) explored the concept of teaching students different ways to get to the correct answer. Math teachers should accommodate for the different ways in which their students think by giving them different ways to arrive at an answer.
Howard Gardiner (1991) identified 7 different learning styles. He believed “…students possess different kinds of minds and therefore learn, remember, perform, and understand in different ways” (Lane. C, 2013). In order to make mathematics and numeracy more engaging in the classroom, I have ensured I have included all of Howard Gardiner's learning styles in the activities above. This approach ensures I meet the needs of all students as well as helping improve students understanding of concepts through different ways of presenting material. This approach also challenged high performance students. The table to the right shows how each student’s individual needs are met.
As teachers, it is our duty to “improve our awareness of what motivates students to become engaged in mathematics and to maintain this engagement, even when faced with difficult mathematical problems” (NSW Department of Education and Training, 2011). We want students to reach a level of numeracy that enables them to realise their full potential in their personal and social life too. Other ways to engage students in mathematics include:
Students today are growing up with computers, mobile phones, video games and ipads. By incorporating technology and multimedia, such as productive maths games, videos, songs and online activities; into the classroom we will be more likely to capture the student’s attention. Catherine Attard and Maria Northcote explore , the use of mobile technologies as part of our digital repertoire for teaching mathematics. In their article, Teaching with technology: mathematics on the move: using mobile technologies to support students learning (2011), Catherine and Maria write that “the use of mathematics apps can be engaging and of benefit to students in terms of building fluency and increasing motivation through their sometimes competitive nature”.
Mathematics is Abstract and Numeracy is Practical
Mathematics is interrelated with numeracy.
All numeracy is underpinned by mathematics.
Mathematics is the study of abstract concepts. It is not necessarily applied as it is possible to study maths and not be concerned with how it relates to the real world. Through mathematics, you are able to develop a better understanding of concepts, principles and processes which can then be interpreted through numeracy as it is applied to real life situations. Numeracy is practical as it can be used in different situations in life. Some mathematical concepts include patterns, relationships and measurement whereas numeracy is being able to read the time and timetables for classes, following a recipe, ordering a group of people by height etc.
When teaching children about the practicability of numeracy, teachers should relate the situations to the children’s lives at present. For example teachers should use classroom timetables rather than bus timetables when teaching children as they are more likely to see and use class timetables during primary school rather than bus timetables. Teachers can then talk about the mathematical concept in reading timetables is the numbers and determining the patterns of the classes. Through these patterns the students will learn that every class runs for one hour (numeracy).
Numeracy and mathematics are closely interrelated. All numeracy is underpinned by some mathematics; this is why mathematics at school plays an important role in the development of children’s numeracy. Teachers have a responsibility to develop mathematics that underpins numeracy. Without the knowledge of numbers, mathematical language, symbols and abbreviations, the interpretation and use of tables and visual images people would not be able to be numerate. Students need to understand the mathematics before they can use it in their lives. E.g. if you didn’t understand fractions and volume, you wouldn’t be able to cook and measure ingredients.
Being numerate involves using mathematical concepts and skills, mathematical thinking and strategies and general thinking skills and all teachers are responsible in helping and showing children how they can use their mathematical skills in every day tasks. “A person’s disposition to use mathematics is also critical in numeracy. This includes personal confidence, comfort and willingness to ‘have-a-go’ through the use of mathematical or quantitative means” (Department of Employment, Education, Training and Youth Affairs, 1997). Therefore as teachers, we must ensure we are positively encouraging students to continue to attempt mathematics through the use of positive reinforcement, encouragement and giving students different ways to solve problems so they can choose their own way of getting to the correct answer.
An area of the Australian Curriculum is fractions and decimals. In order to teach students how to use their mathematics in the real world and how numeracy is interrelated with mathematics, I would prepare a yummy and healthy learning experience by using some of the children’s favorite fruits in the class. I would then cut the fruits into different portions and demonstrate fractions to them this way. To incorporate something a little different into the lesson, i will play the fruit fraction video on the next slide. After the healthy lesson on fractions, I would then have a lesson using blocks of chocolate. This time I will get the children to show me their fractions using the small blocks of chocolate.
After both lessons, we will convert the fractions we made suing the fruits and chocolate, by writing them down on paper as some children learn visually, some through manipulating objects. Through these lessons, the children are learning how to use the math’s concept of fractions through converting this concept into a numerical process by making fractions with their fruit or chocolate. An extension from this activity would be a cooking experience where the children would learn to measure half cups of flour, a quarter cups of sugar etc.
In this cooking lesson, the students will be learning the math concept of measuring as they measure the ingredients. They will then put this maths into context through numeracy when they following the recipe.
There are different ways in which math’s can be taught to students in order to see how it is used in the real world. One way this can be done is through role play. I have organized an activity for grade 4 students as it involves solving problems involving purchases and the calculation of change (ACMNA080). In order to teach students about Australian currency and its worth I would begin by giving the children an opportunity to make their own coins and notes out of paper (teaching them about size and proportion and enabling the students to recognize and describe the Australian coins and their value). They will then use this money to buy things from a pretend shop full of empty boxes, containers and play food. I will put up price tags and give the children purses and wallets to put their money in. The students will get opportunities to be both the customer and the shop keeper. For younger students, this activity will help with their simple addition and subtraction, as the students get older (or according to their ability) it will be a matter of multiplication, such as multiplying the cost of 3 cartoons of milk. It can also be used to teach students the value of money e.g. 10 cents is more than 5 cents and $1 is more than 50 cents.
In order to make learning about time tables more enjoyable for children, I would relate school classes with movies they may have seen. Such as:
• Assembly with solders from Toy Story (30 minutes)
• Sport with King Fu Panda ( 30 minutes)
• Swimming with Finding Nemo (30 minutes)
• Maths at Monsters University (1 hour)
• Music with The Muppets (1 hour)
• Geography with Blu and Linda from Rio (1 hour)
• History with Knomeo and Juliet (1 hour)
• Science with Flint (cloudy with a change of meatballs) (1 hour)
I will then get the children to put the above information into a timetable, adding their lunch and recess into the table.
Through this lesson, the children are using math (numbers) and transferring these numbers and lessons into a graph in order to see how long their school day is (numeracy). This relates to the Australian Curriculum area ACMSP050 as the children are learning to create displays of data using a table and interpret this information.
An extension from this lesson would be for the children to collect data and then represent this in a table. This could include going on an excursion to the local park and counting the different coloured cars that drive by. The children would then represent this information in a table, which could then be transferred into a graph (ACHGS021).
Mathematics is the study of concepts, principles and processes; whereas numeracy is the process of applying these mathematical concepts into everyday life situation. Both mathematics and numeracy have a number of similarities and differences; however in the end they are both interrelated.
Numeracy is a big part of people’s lives, it is important that children feel confident and comfortable when learning numeracy, so that they are able to apply this knowledge in their own situations, both when young and older. The Australian Curriculum has noticed this importance and emphasises the important role teacher’s play in students' learning through their teaching. Teachers are expected to positively encourage students to continue to attempt mathematics through the use of positive reinforcement, encouragement and giving students different ways to solve problems so they can choose their own way of getting to the correct answer.
Teachers have a duty to make their lessons interesting and interactive, using different teaching techniques to cater for individual children's learning styles. This can be done through a variety of different learning demonstrated by Howard Gardiner as the 7 different learning styles: visual-spatial, bodily-kinesthetics, musical, interpersonal, intrapersonal, linguistic and logical-mathematical.
Mathematics plays an important part in our environment as it is shown throughout nature. We can see this through the spots and stripes on animals, patterns throughout the sand, symmetry of people’s faces, the shapes of volcanoes, the planet and bubbles, spirals in a shell, sunflower, and acorns and from the angle in which branches and leaves grow on trees.
Mathematics is all around us and used collectively with numeracy in every day life. It plays an important role as without mathematics we would not be able to perform simple everyday tasks such as shopping, preparing meals and planning our day.