Last year, *Education Review* published my opinion piece on the need to provide greater detail in the Australian mathematics standards to give teachers better insight into what they need to teach and how to teach it. In that article I compared how the current Australian Curriculum: Mathematics addressed the standards for teaching common fractions in primary school against those detailed in Singapore's primary mathematics syllabus.

Common fractions was chosen due to the considerable research (Siegler, 2012, 2015, 2016) showing that success in high school mathematics is closely correlated with a student's confidence with common fractions. The Singapore syllabus was chosen as it is the first of four curricula that ACARA has benchmarked the Australian mathematics curriculum against (see the link to the review on the ACARA homepage). Singapore is also an English-speaking country that consistently ranks Number 1 or 2 on the TIMMS or PISA international tests.

With a total of 28 standards relating to common fractions, requiring 880 words of elaboration, compared to ACARA’s 12 standards (360 words), there is no doubt that Singapore gives their teachers better guidance on what and how to teach at each year level. The current Australian mathematics curriculum is scant at best, which allows it to be far too open to interpretation. As they continue their review, they must find the right balance between decluttering and better informing teachers on what is required. Until the results of the review are officially released, we wait in eager anticipation that common sense will prevail.

Until then, we support the federal education minister’s review of the teacher education sector. While a quality national curriculum is essential, we will continue to regress if we don’t have quality upskilling, cross-skilling and re-skilling courses offered to educators.

By way of example, we were recently told that a ‘numeracy coach’ for teachers did not know that zero has no value. It seems the coach was insisting that zero be shown on a counting board, which had the effect of giving zero a value that is equal to the other counting numbers on the board. Practices such as this will lead to major misconceptions for students throughout their schooling years and significant errors later in life. (I am sure we all recall the drama around the Y2K confusion.)

We also often hear teachers say generalisations such as, “When you multiply, the answer is always bigger!” While this might seem true in the lower grades where students are dealing solely with numbers greater than one, comments such as this – made by well-meaning teachers – cause major problems when students start multiplying by numbers less than one (e.g. 4 x 0.5 is not greater than 4). It’s like the notion that subtracting always makes my answer smaller. Is this really the case? If I subtract a negative amount (debt) off my bank balance, then my balance actually goes up!

Examples such as these provide strong evidence of gaps in understanding. More worrisome are the comments that we hear from mathematics coaches, influencers, publishing representatives and others that verge on what can only be labelled as education malpractice. It is only a matter of time before the life-long damage caused by such ‘advisors’ leads to some kind of legal action.

Universities certainly have a big responsibility to ensure teachers are given the right initial training, but the private and public education sectors must also carry their share of responsibility to ensure teachers continue their re-skilling while in service.

As Australia’s largest mathematics education company we have decided to support mathematics leaders through a professional learning ‘academy’ where, over six days, they will be armed with the knowledge and skills required to become primary mathematics learning facilitators.

**Dr Calvin Irons is the co-founder of ORIGO Education**

**James Burnett is the co-founder and executive chairperson of ****ORIGO Education**

For many of us, the lessons happened long ago and we’re now having a quick refresher course.

We live in a world where mathematics is important: for public health, for family budgeting, for making good decisions in many ways. We also live in a world where many believe mathematics is difficult, or that only some of us are good at maths and that it is okay to say, “I don’t do maths.”

Our mission is to change those beliefs and lay a foundation for mathematics to be meaningful, enjoyable and accessible for all.

We are not alone in the mission. The National Council for Teaching Mathematics’ *Catalyzing Change *initiative also focuses on broadening the purpose of school mathematics, dismantling obstacles faced when learning mathematics, implementing equitable instructional practices, and organising mathematics to provide a strong foundation of deep mathematical understanding for each and every child. (NCTM, 2020)

Teachers, too, are working hard to change the narrative of mathematics from a painful time of memorisation and worksheets to an enjoyable sensemaking and problem-solving experience.

As school moves home, we also want to see these conversations being shared at home. Doing mathematics at home should not be the land of “death by worksheet” and endless skill practice.

Students should be learning by explaining their thinking. Yes, there is some skill practice, but much of that can be in the form of games and applications, not just constantly completing worksheets.

Today, parents join the fight. But this is not without challenge or sacrifice. Families are finding a new normal where everyone is together all the time, which can be overwhelming. Learning with your child should never become a burden.

Our undertaking is to open the doors to teachable moments, so adults and children enjoy learning mathematics together.

**If you’re a teacher**, we encourage you to consider resources that can be implemented at home, by caregivers, and understand that parents may be uncomfortable with maths.

You might need to provide background or guidance on how to introduce a concept or strategy, for example, the count-on strategy. The assignments and resources you share could be an opportunity to change adult perceptions of mathematics, while helping children learn.

Be realistic in your expectations and look for opportunities to integrate mathematics with other subjects.

**If you’re an administrator**, we encourage you to support your school community, both teachers and families, in finding opportunities to experience mathematics as meaningful and enjoyable.

**If you’re a parent**, we encourage you to talk with your child about mathematics. Open their eyes to the everyday applications of mathematics by including them in the day to day decisions you make.

Ask an older child to do the maths of the grocery budget or scaling a recipe. Ask a younger child to help match socks by colour or count the eggs in the carton.

(Reference: National Council of Teachers of Mathematics (2020). *Catalyzing Change in Early Childhood and Elementary Mathematics: Initiating Critical Conversations*. (Huinker, D., Editor) Reston, VA: Author )

We have put together a series of open access resources designed to encourage mathematics learning at home, and for the long term when schools do reopen.

Written for parents and caregivers in grades PreK-6, teachers can share ORIGO at Home with their school community to reinforce maths concepts and skills along a research-based, developmental scope & sequence. To find out more go to www.origoeducation.com/athome.

**Dr** **Sara Delano Moore is director of professional learning and mathematics advisory board chair, and Peter Stowasser is lead writer and mathematics advisory board member at ORIGO Education.**

No detail was provided about what these skills might be. No mention was made about the skills the future workforce currently studying in primary and secondary schools should be receiving now.

So, what skills from the discipline of mathematics are important today?

The Australian curriculum (ACARA) for mathematics focuses on four key aspects (called proficiency strands) for every content area at every grade level. The proficiency strands are **understanding, fluency, problem-solving** and **reasoning**.

This delineation helps stress the fact that there is much more to mathematics than meaningless out-of-date procedures such as long multiplication or long division.

The following extract from the ACARA document indicates what is expected for each of the proficiency strands at Year 6 when teaching decimals, fractions and percentages:

**understanding**: representing fractions and decimals in various ways and describing connections between them, and making reasonable estimations**fluency**: includes converting between fractions and decimals, using operations with fractions, decimals and percentages**problem-solving**: includes formulating and solving authentic problems using fractions, decimals and percentages**reasoning**: includes explaining mental strategies for performing calculations

It is no accident that these four strands appear in this order. You need understanding before you become fluent with related skills, which then enables you to solve problems and to reason mathematically.

Interestingly, though, a study of textbooks shows that many publishers have chosen to swap the order of the understanding and fluency. This reflects the traditional sequence from the past where skills were often stressed before any understanding of the ideas was developed.

Without professional training, this rearrangement of the order to teach skills first is usually supported by teachers. Unfortunately, when fluency is the first focus of the teaching, procedures are taught in a rote manner which requires much more valuable time. This leaves most students floundering when it comes to assessment that measures problem solving and reasoning skills.

Practical activities provide pictures that help students build **understanding**. A ruler is a good tool to help students see the connections between fractions and decimals.

For example, the dimensions of the table can be reported in whole units (1125mm), as a decimal (1.125m), or a fraction (1 1/8m) for the lengths. Of course, the ruler has many other practical everyday applications. **Fluency** is the stage of reinforcing the understanding by making the conversion between the decimals and fractions without the ruler. The move from one stage to the other requires careful teaching.

Once students know both dimensions, they can use **problem solving** and **reasoning** to work with the length/width data. This could involve the operations of addition (find the perimeter), subtraction (find the difference between the dimensions), multiplication (find the area of the surface) or division (describe a ratio).

There are of course more sophisticated tasks that might involve deciding the dimensions and area of a table cloth that needs to drop down 40cm all the way around the table edge.

All of these tasks require the necessary pre-requisites such as understanding what is meant by perimeter, area and ratio and are examples of some of the most important mathematical ‘skills’ for today. Each of these ideas also require careful development that begins with understanding.

In addition to the everyday applications described in the previous paragraphs, most future jobs, and certainly all future university students, will require knowledge of some previously underemphasised topics. Many of these important future skills are collectively described as quantitative studies.

Specifically, this includes knowledge of a widening range of statistical techniques, applications involving discrete methods, and the capability to search data (sometimes called data mining). All of these are becoming important because they involve routine mathematical tasks that are completed by a digital device.

Universities have revamped courses so that some of the first core subjects that students must complete in many faculties are titled **quantitative methods**. These are the tools that the treasurer mentioned in his speech in early May 2020.

Statistical techniques have always been important, but technology now makes it possible and necessary to study many new mathematical methods in greater depth. With these technologies more problem-solving activities involving descriptive techniques such as the mean, standard deviation and finding a line-of-best-fit can be included in school curricula.

This knowledge can then be applied to methods in inferential statistics. This is an important topic which helps provide forecasts and make predictions, but is currently under-represented in secondary school mathematics. These two aspects of statistics are now very important as they help produce (mathematical) models of real-life situations.

Most mathematical solutions to a real-world problem involve discrete methods and provide discrete solutions. Time is a continuous variable. A graph of an event such as temperature over an interval of time is a continuous smooth curve.

When time is measured, the readings are taken at distinct instances of time and should therefore be shown as separate, albeit very close points. This type of graph is a set of points that represents a discrete variable. The readings (time, temperature) can be taken to any level of precision, but are still discrete entities. Applied mathematics in the real world (as opposed to pure theoretical mathematics) is nearly always discrete.

Time along one dimension of a graph is about the only safe example of a continuous variable. However, the construction of this type of representation would require the continuous collection of data – think images of your heart beat on a hospital monitor or the rolling collection of seismic activity during an earthquake.

Computers are binary beasts (using 0 and 1 to do the work) and therefore discrete entities that are ever increasingly being used to solve real-world problems. The number of numbers involved in a computer calculation is massive, but never-the-less isolated from one another; so discrete. This makes computers ideal for mining data which involves elements of both of the two aspects mentioned above.

When they are programmed appropriately, computers can pour through the massive amount of known discrete data to build a statistical picture that is as close as possible to real world. These results are then used to make the inferences that are necessary for forward planning, decision making, and overall modelling. In the current literature, the word modelling is used with ever increasing frequency, but this writer formally studied mathematical modelling in 1972.

In order for a student to be successful in the three topics mentioned above, they need to have a broad grounding in algebra with experience to build and solve a system of equations for a real-world situation. Using the earlier discussion, this means understanding, being fluent, being successful at solving problems and completing reasoning tasks that involve algebra.

Working back, this implies prior success with the prerequisites to algebra: all aspects of numbers (whole numbers, common and decimal fractions), all properties involving the four basic operations and exponents; and the facility to manipulate expressions around an equal sign, for example knowing that adding 2 to both sides of an = sign means the expressions on both sides are still equal – an idea that can be shown with hands-on materials as early as Foundation (Kindergarten).

Mathematics is a highly structured subject that is built on interrelated ideas. Topics at the top of that structure might be new, but there is a foundation that still uses many of the same understandings that have always been included.

In the future, students might not need long multiplication or long division, but they will benefit knowing that multiplication distributes across an addition expression as illustrated in this sequence of pictures to find the unknown partial lengths in the garden plot.

This understanding with whole numbers will transfer to situations where they need to simplify algebraic equations.

In summary a mathematics curriculum for primary schools today should, as a minimum, include a sound understanding of the four operations and properties of those operations for whole numbers, decimal fractions and particularly common fractions, all measurement attributes, balancing simple equations with two operations and a good introduction to statistical representations.

In the secondary years, students should use the knowledge from the primary school to gain confidence with basic aspects of algebra as it relates to discrete mathematics, the fundamental of descriptive and inferential statistics, and creating coding algorithms that can search data.

Overall, the emphasis in teaching the four proficiency strands in mathematics should move from what is currently typical (shown in the first diagram) to a better balance for the applied nature of mathematics in an eco-commerce digital world (shown in the second diagram).

**Dr Calvin J Irons BA, MA, PhD in mathematics and mathematics education is co-founder of ORIGO Education**.