Latest from ORIGO

Two Rising Country Musicians Take Maths to a New Level

April 2013

Pop singer/songwriter, Coles Whalen who describes her style as a little bit of folk and country, collaborated with Loren Ellis, lead guitarist for Easton Corbin; and ORIGO Education to create a catchy new album to help children learn maths. The record is a soundtrack for the ORIGO Big Books series, which helps teach a variety of mathematical concepts from patterns to skip counting to division. Whalen’s enchanting voice and high-energy music bring the characters in the books to life. The Big Book Tunes is the third children’s album that Whalen & Ellis have produced under CLEW Media, the production company the pair co-founded. Their mission is to enhance young minds while encouraging kids and parents to have fun.

These engaging new Big Book Tunes really bring your mathematics classroom alive

“I had such a great time making the ORIGO Big Book Tunes and I think you can hear it in the record, it transfers through to our music,” says Whalen “Loren is the right mix of musicianship and whimsicalness, he doesn’t take himself too seriously.” The book series has several different illustrators so each title has its own individuality. Whalen says, “We tried to express each book’s unique personality in our songs.” Her favourite song to create was Pirates' Gold, a book about division and sharing. The tale of six greedy pirates learning to share and dividing up a gold chest is animated by the duo’s voices complete with pirate sounds and a wise cracking parrot.

Whalen originally began writing children’s music as an experiment to raise awareness for a good cause. She found that it challenged her in a way that she did not expect, but she truly enjoyed it. “I’m used to writing adult music and know how to express those ideas. It was very different to trying to translate an idea a kid might grab onto and put it to music.”

The ORIGO Big Book Tunes was the result of a chance meeting. Coles was travelling in Australia with a friend shortly after her music video had gained circulation on Country Music Channel in the US. Her companion introduced her to James Burnett, the President of ORIGO Education. He had been looking for someone to create music for the ORIGO Big Books. Whalen was enamoured with the books from the start. “I wanted to read every single one. I love large format concept for promoting interactivity in the classroom.” Young students have a natural love for learning, for stories, and for numbers. The ORIGO Big Books are beautifully illustrated, large format storybooks that build on this enthusiasm to help teachers introduce key mathematical concepts in Foundation – Year 2. Each of the 36 ORIGO Big Books introduces and reinforces the essential mathematical language. Click here for samples from the new album.

About Coles Whalen: Coles Whalen is an Americana Pop singer-songwriter from Denver. She started performing at a young age, touring internationally with The Colorado Children's Chorale in the early 1990’s. She graduated from The University of Southern California with a BA in 2003. Whalen has released five independent records. Her most recent, I Wrote This for You (2012) has been heard on La Galere (Paper Airplane) and seen on CMC (Call on Me, music video). She continues to play hundreds of shows each year including opening for Pat Benatar, Joan Jett, Rufus Wainwright, Paula Cole and Kellie Pickler. She is planning to release another record late in 2013.

About Loren Ellis: Loren Ellis is the lead guitarist for country music artist Easton Corbin. He was previously a member of the Drew Davis Band. Born and raised in Agoura Hills, CA, Loren’s talent was apparent at the ripe old age of 4 when he began playing the piano. Shortly thereafter, he was writing his own music. At 12 he discovered the guitar, or a guitar. Specifically, his father’s prized Telecaster. Loren learned to play by sneaking into his father’s closet to practice. Throughout his youth, Loren performed in various bands and earned a BM in Music at the University of Southern California.

Maths Games

Article by Dr Calvin Irons

As published in the Courier Mail (4 September 2012)

Games have long been used to motivate students and to help them enjoy mathematics, but there are other reasons for using instructional games in the mathematics classroom. Read more.

Patterns Appear Everywhere!

Article by Dr Calvin Irons

As published in the Courier Mail (28 August 2012)Most patterns are visual and are analysed for the beauty they display. The pretty visual patterns can usually be described with numbers. Number patterns often lead to some powerful applications of mathematics. This article will highlight just a few of the contemporary applications of mathematics that involve patterns. Read more.

Geometry: It's Everywhere

Article by Dr Calvin IronsAs published in the Courier Mail (27 March 2012)

In school, geometry seems to receive less attention than other aspects of mathematics. But it certainly shouldn't! We live in a three-dimensional world full of natural and man-made objects. We also interact with science and technology, where geometry plays a large role. This article highlights a few examples of the geometry that surrounds us – and some of the key understandings from early primary school upon which all geometry is based. Read full article.

Maths Made Easy:Thinking Strategies in Primary Mathematics

Article by Dr Calvin IronsAs published in the Courier Mail (13 March 2012).

As of this year, there is a new curriculum for English, Science and Mathematics. The main aim across all three subjects in the new curriculum is to promote thinking. For mathematics, that means thinking strategies, mental mathematics, reasoning through complex situations and solving problems. Read full article.

10 Tips for Teaching with an Interactive Whiteboard

17 December 2008

Interactive whiteboard technology can enrich the learning and teaching experience. However, using this technology – and using it well – may seem a little overwhelming. These research-based tips will help make its introduction and ongoing use in your classroom easy and effective.

PractisePractise using the board and the software before you use them in a lesson.Let the students explore the board and software and experiment with the functions.PreparePrepare and save activity files before class to reduce in-lesson preparation time. Keep the files where you can easily find and open them.Check that colours and writing size can be seen from the back of the classroom. Ensure that light (natural and artificial) is not affecting visibility. Clean any smudges and dust from the screen and data projector lens.Recalibrate the board before each lesson. Recalibration aligns the touch screen with the image. If the calibration is out, your writing, for example, will appear in a different position to where you touch the screen.Check that students can comfortably reach most of the board.ProceedUse the touch screen, not the computer, to navigate around the software. When you sit behind the computer you are detached from the students. You can better engage with them when you use the touch screen.Make sure there is a purpose to all student–software interaction. Students really see the mathematics when they manipulate the software tools. However, interaction for the sake of interaction distracts from the learning.Pace yourself. The technology increases efficiency, which means you can present the next step in the lesson more quickly (e.g. a comparison number line). Use this time to focus on the teaching and discussion.Use the board and software to their full potential. Think of them as resources for developing questioning and interactive learning, rather than just as tools for presenting information (e.g. leave “gaps” in activity files for the students to discuss and fill in).Bibliography

Actis Ltd. (n.d.). Interactive whiteboards: An approach to an effective methodology. Retrieved January 18, 2007, from http://www.virtuallearning.org.uk/whiteboards/An_approach_to_an_effective_methodology.pdf

Armstrong, V., Barnes, S., Sutherland, R., Curran, S., Mills, S., & Thompson, I. (2005). Collaborative research methodology for investigating teaching and learning: The use of interactive whiteboard technology. Educational Review, 57, 457–469.

Beauchamp, G. (2004). Teacher use of the interactive whiteboard in primary schools: Towards an effective transition framework. Technology, Pedagogy and Education, 13, 327–348.

Hall, I., & Higgins, S. (2005). Primary school students’ perceptions of interactive whiteboards. Journal of Computer Assisted Learning, 21, 102–117

Higgins, S., Beauchamp, G., & Miller, D. (2007). Reviewing the literature on interactive whiteboards. Learning, Media and Technology, 32, 213–225.

Kent, P. (2006). Using interactive whiteboards to enhance maths teaching. Australian Primary Mathematics Classroom, 11, 23–26.

Levy, P. (2002). Interactive whiteboards in learning and teaching in two Sheffield schools: A developmental study. Retrieved January 17, 2007, from http://dis.shef.ac.uk/eirg/projects/wboards.htm

Smith, H.J., Higgins, S., Wall, K., & Miller, J. (2005). Interactive whiteboards: Boon or bandwagon? A critical review of the literature. Journal of Computer Assisted Learning, 21, 91–101.

Wall, K., Higgins, S., & Smith, H. (2005). “The visual helps me understand the complicated things”: Pupil views of teaching and learning with interactive whiteboards. British Journal of Educational Technology, 36, 851–867.

Developing Essential Strategies for Computation

Mathematics education leaders are often asked about strategies for teaching students to calculate both mentally and with paper and pencil. The first significant step for calculating involves learning the basic number facts associated with each operation. These facts form the basis for all future number work.

Thinking Strategies

The most effective way for students to learn the basic facts is to arrange the facts into clusters (Fuson 2003; Thornton, 1990). For example, the idea of using a double, or numbers close to a double, forms the basis of the strategy in the “use doubles” addition cluster. The other addition clusters involve “counting on” and “bridging to ten”. Between these three clusters, students can master all 100 basic addition facts. “Using addition” is the most effective thinking strategy for helping students to learn the basic subtraction facts.

The idea of using a known multiplication fact involving ten, forms the basis of the strategy in the “use tens” multiplication cluster for the fives facts. The other multiplication clusters involve “doubling” for the twos, fours, and eights facts; “using a rule” to multiply by one or zero; and “building up or down” from a known fact for the sixes and nines facts. The majority of the remaining facts are covered by the turnarounds of the above. Just as subtraction is the inverse operation of addition, division is the inverse operation of multiplication. Because of this relationship between the two operations, “using multiplication” is the most effective thinking strategy for helping students to learn the basic division facts.

Teaching Sequence and Stages

The most widely accepted sequence for teaching the addition facts is as follows:• Count on (1, 2, 3, and 0) with their turnarounds• Use doubles (double, double plus 1, and double plus 2) with their turnarounds• Bridge to 10 with their turnarounds

The sequence for teaching the multiplication facts is:• Use tens (fives facts) with their turnarounds• Doubling (twos, fours, and eights facts) with their turnarounds• Use a rule (ones and zeros facts) with their turnarounds• Build up and build down (sixes and nines facts) with their turnarounds

Along with the above sequences for teaching the strategies, the activities for each strategy are also sequenced according to the following four stages of teaching and learning: introduce the strategy, reinforce the strategy, practice the facts, extend the strategy. The use tens multiplication strategy for the fives facts will be used to explain and illustrate each of these stages.

Introduce the Strategy

Hands-on materials, real-world stories, pictures, discussions, and familiar visual aids are typical of the activities that are used to model and introduce each thinking strategy in the first stage. For multiplication, a rectangular array of dots is often used to show facts and their turnarounds. For example six rows of ten dots show 6 x 10 or 10 x 6. A card such as the one below, can then be used to demonstrate the thinking required to figure out a fives fact. After determining the total number of dots, fold or cover up half the array and have the student verbalise what they see and how they can figure out the total number of dots.

Reinforce the Strategy

The activities in this stage are designed to make links between the concrete/pictorial and symbolic representations of the facts being examined. Students also reflect on how the strategy works and the numbers to which it applies. Given pictures such as the one below, students can be asked to write the answer to a tens fact they see, then colour half the array and write a number fact to match what is now coloured. Experiences such as this serve to reinforce the relationship between the two facts and the thinking strategy that is involved.

Practice the Facts

At this stage, games, flashcards, worksheets, and other activities provide students with opportunities to apply and demonstrate their knowledge of the facts. The students should use mental computation only and fast recall is stressed.

The following game for two or three players is a simple way to practice the fives facts.

Materials:• About ten counters of one colour for each student.• Two blank cubes with the numerals 2, 4, 3, 3, 5, and 5 written on the faces of one cube and 6, 8, 7, 7, 9, and 9 on the faces of the other. Each 6 and 9 should be underlined to avoid confusion.• A game board. Have the students sketch the game board below onto a grid or sheet of paper.

To play the game:• The first player rolls both cubes.• The player then chooses one of the numbers and multiplies it by five. Encourage the player to verbalize their thinking, for example, “Four tens are forty, so four fives are twenty.”• The player finds the product and places a counter on the game board.• The next player has a turn.• As the game continues, a player misses a turn if they cannot cover a number on the board (only one counter can be placed on any one space).• The player who has the greatest number of counters on the board at the end is the winner.

Extend the Strategy

Once mastered, students are encouraged to apply the strategy to numbers beyond the range of the basic number facts. The activities in this stage are designed to further strengthen students’ number sense. To extend the Use-Tens strategy, have the students explain how they can use the first answer in each number sentence below to figure out the second answer.

10 x 18 = ____ so 5 x 18 = ____

10 x 26 = ____ so 5 x 26 = ____

10 x 22 = ____ so 5 x 22 = ____

10 x 15 = ____ so 5 x 15 = ____

Later, students will learn to vary this strategy by doubling and halving the two factors rather than doubling one factor and halving the product. For example, by doubling and halving, students can quickly calculate that 14 x 35 has the same answer as 7 x 70, which is easy to calculate mentally.

Conclusion

It has long been known that students should have instant and accurate recall of the basic number facts. But now, more so than ever before, with the increased emphasis on mental computation in contemporary curricula, it is essential that these facts be taught using strategies that serve as the foundation of mental computation.

As co-founder of ORIGO Education, James Burnett strives to lift the profile of mathematics through dynamic professional learning and the development of quality, research-based classroom materials. James frequently presents workshops and speaks at conferences, and has authored more than 200 mathematics books for teachers and primary students.

For more information about ORIGO Education professional learning and our unique resources, such as the Box of Facts kits shown in this article (images), please contact us at: info@origo.com.au or 1300 674 461.

References

Burnett, J., Irons, C. & Turton, A. (2007). The Book of Facts. (Series title) Brisbane: ORIGO Education.Fuson, K. C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards to school mathematics (pp. 68-94). Reston, VA: National Council of Teachers of Mathematics.Irons, C., Burnett, J. & Irons, R. (2006). The Box of Facts (Kits) Brisbane: ORIGO Education. Thornton, C. (1990). Strategies for the basic facts. In J. N. Payne (Ed.), Mathematics for the young child (pp. 131-151). Reston, VA: National Council of Teachers of Mathematics.

Algebra: 4 Big Ideas

For many, algebra is a confusing jumble of symbols and equations. Until recently, this abstract area of mathematics was first introduced to students in high school, where it was often approached as a set of procedural rules for solving equations (Kilpatrick & Izsák, 2008). Developing algebraic thinking in the early school years provides a solid foundation for later algebra symbol work (Warren, 2008). Teaching children the big ideas (key concepts) of early algebra (Warren, 2008; explained below) through real-world problems helps them understand its rules and applications.

1 Equivalence and Equations

“Equals” means equivalent sets rather than a place to write an answer. Simple real-world problems with unknowns can be represented as equations. Equations remain true (balanced) if the same change occurs to each side. Unknowns can be found using the balance strategy.

2 Patterns and Functions

Operations almost always change an original number to a new number. Simple real-world problems with variables can be represented as “change situations”. “Backtracking” or reversing a change can be used to find unknowns.

3 Properties

Arithmetic properties apply. The commutative law and associative law apply to addition and multiplication but not to subtraction and division. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. Adding or subtracting zero, and multiplying or dividing by 1, leaves the original number unchanged. In certain circumstances, multiplication and division distribute over addition and subtraction.

4 Representations

Different representations (e.g. graphs, tables of values, equations, drawings, everyday language) help with identifying trends and finding and interpreting solutions to real-world problems.

References

Kilpatrick, J., & Izsák, A. (2008). A history of algebra in the school curriculum. In Greenes, C.E. & Rubenstein, R. (Eds.), Algebra and algebraic thinking in school mathematics: Seventieth yearbook (pp. 3–18). Reston, VA: The National Council of Teachers of Mathematics.

Warren, E. (2008). Algebra for all. Brisbane, Australia: ORIGO Education.

Composite Versus Multi-Age Classes

Combining multiple ages or year levels in a class is not a new move by schools. You may have taught such a class, or even been part of one as a child. What is relatively new is a development in the reasons for forming combined classes, and therefore the teaching philosophy used.

The traditional composite class, also called combination, double, split, or mixed-age (Russell, Rowe, & Hill, 1998), is generally formed for administrative reasons (Naylor, 2000; Sydney Morning Herald, 2003); for example, if too many (Sydney Morning Herald, 2003) or too few (Naylor, 2000) students enrol in a year level. Therefore, the composite class usually exists only for that one year. To illustrate, if too few Year 4 students enrol one year, the school may need to create a 4/5 composite class. The following year, enrolment numbers for Year 4 may be up, and additional students may join Year 5, so a 4/5 composite class is not needed. If, however, the numbers in Year 5 remain the same, a new 5/6 composite class may be formed. In composite classes, students learn separately within their year level – the teacher works with each group in turn while the other group works independently (Mulcahy, 2000).

The newer multi-age class and its teaching philosophy allow for the reality that all children in a year level do not progress at the same pace (Ladd, 2007). Some students learn more slowly than their peers, others more quickly. In multi-age classes, students may work in groups according to their ability, not their age or year level (Ladd, 2007), or they may work together as a mixed-ability group. Unlike composite classes, the multi-age class continues to exist from one school year to the next, and students and teacher remain together for the year levels the class spans (Ladd, 2007; Mulcahy, 2000).

References

Ladd, K. (2007, August). Classroom collectives. Retrieved September 22, 2008, from http://www.australianassociationofmultiageeducation.org/documents/classroom-collectives.pdf

Mulcahy, D.M. (2000). Multiage and multi-grade: Similarities and differences. Retrieved September 22, 2008, from http://www.mun.ca/educ/faculty/mwatch/win2000/mulcahy.html

Naylor, C. (2000). Split-grade and multi-age classes: A review of the research and a consideration of the B.C. context. BCTF Research Report. Vancouver, BC: British Columbia Teachers’ Federation.

Russell, V.J., Rowe, K.J., & Hill, P.W. (1998). Effects of multigrade classes on student progress in literacy and numeracy: Quantitative evidence and perceptions of teachers and school leaders. Paper presented at the 1998 Annual Conference of the Australian Association for Research in Education, Adelaide. Retrieved September 22, 2008, from http://www.aare.edu.au/98pap/rus98154.htm

A class of their own. (2003, December 5). The Sydney Morning Herald.

Dominoes: More Than a Game

Have you ever played with dominoes? As well as being fun, domino games involve calculation and thinking strategies, which makes dominoes an ideal resource for teaching various aspects of mathematics. They can be used to develop number concepts, computation strategies, patterns and algebra, and chance and data. This sample activity shows how double-nine dot dominoes can be used to investigate addition involving three addends.

Sum Snakes

Activity

Have students work in groups. Each student in the group selects nine dominoes. Place one domino in the middle of a table. The students take turns to place their dominoes at either end of the “snake” as shown below (three ends in a section). The sum of the dots in each three-end section must always equal 12. A player who is unable to place a domino to make 12 misses a turn. The winner is the first player to use all of their tiles or have the fewest number of tiles left over in the event that no-one can play.

Variation

Play the game with different target numbers from 8 to 14.

Activity from A Little Book of Big Ideas: Double-Nine Dot Dominoes, 2007, ORIGO Education Bibliography

Irons, C. (2007). A little book of big ideas: Double-nine dot dominoes. Brisbane, Australia: ORIGO Education.

Fingers: A Powerful Representation of Ten

Representations of ten are used for teaching place value, number sense, and computation. However, many of the resources used for representing ten (frames, sticks, cubes, rods, and blocks) do not clearly show the individual ones within the number or effectively demonstrate three- and four-digit numbers. For example, all one thousand smaller cubes are not visible within a one thousand base-10 cube.

Fingers are a ready and powerful resource for representing ten that students already use naturally in mathematics (e.g. for counting). They readily show ten as 10 ones or 1 ten, and so are ideal for further developing understanding of two-, three-, and four-digit numbers. This property also makes them an effective resource for developing mental computation strategies such as bridging to 10 and splitting numbers into places.

Representing 0–10 with fingers

Have each student hold both their hands up palms forward and make fists, then raise their fingers one at a time, starting from the little finger of their right hand (the audience’s left).

Presenting the numbers in this way shows left-to-right reading for the audience. The raised and unraised fingers also represent the two parts that total 10 (e.g. “I’m showing 7 fingers and I need 3 more to make 10.”).

Representing numbers beyond 10 with fingers

Have more than one student show the tens and ones. For example, three students will show 28 like this:

As well as showing the 28 ones, it also shows that there are 2 tens and 8 ones, and that 2 more ones are needed to make 30. Showing the tens and ones simultaneously helps students develop a deep understanding of the place-value system.

Bibliography

Tickle, B. (2007). DecaCards: A real hands-on approach to teaching place value. Brisbane, Australia: ORIGO Education.

Mental Computation Strategies: Addition

It is essential that all students are able to calculate mentally as well as with paper and pencil. Students can make mental calculations quickly and easily with the help of strategies. There are mental computation strategies for each of the four operations: addition, subtraction, multiplication, and division. Strategies for addition are explained here.

Adding the places

Start with one addend, then add the value of the digits of the other addend(s).

See 35 + 23

Think 35 + 20 + 3

Bridging to ten

Start with one addend, count up to the next multiple of 10 (100, 1000 etc.), then add the balance of the second addend.

See 17 + 8

Think 17 + 3 + 5

Synonyms: bridge the decades; bridge to ten; make a ten; make to ten; use ten

Compensating

Round one or both addends before adding. Then adjust the answer to compensate for the rounding.

See 28 + 36

Think (30 + 36) – 2

Synonyms: compensation; round and adjust; round or adjust; use a nearby number

Counting on

Start with one addend, then count on parts (not places) of the other addend.

See 58 + 24

Think 58 + 10 + 10 + 4

Synonym: jump

Sub-strategies: count on 1; count on 2; count on 3; count on 0

Using compatible addends

Choose pairs of addends to make the calculation more manageable. This strategy applies only when there are three or more addends.

See 14 + 23 + 16

Think 14 + 16 + 23

Synonym: use compatible pairs

Using doubles

Use a known nearby double.

See 7 + 8 = ___

Think 7 + 8 = 15 because 7 + 7 = 14

Synonym: near doubles

Sub-strategies: double plus 1; double plus 2

Using place value

Expand the addends into places before adding the value of the digits in each place.

See 56 + 17

Think 50 + 10 + 6 + 7 or 6 + 7 + 50 + 10

Synonym: split

(For doubling)

Split one of the addends into places, double the value of the digits in each place, then add the doubled values.

See 12 + 12

Think double 10 + double 2

From The ORIGO Handbook of Mathematics Education, 2007, ORIGO Education

These addition strategies, as well as strategies for subtraction, multiplication, and division, are explained in The ORIGO Handbook of Mathematics Education.

Bibliography

Burnett, J. (2001). Mastering mental maths: Addition. Brisbane, Australia: Prime Education.

Burnett, J., & Irons, C. (2005). GO maths teacher sourcebook: Level 2B. Brisbane, Australia: ORIGO Education.

Irons, C., Burnett, J., & Turton, A. (2007). The book of facts: Addition. Brisbane, Australia: ORIGO Education.

The ORIGO handbook of mathematics education. (2007). Brisbane, Australia: ORIGO Education.