mav.vic.edu.au - Mathematical Association of Victoria

Foundation Maths is a course designed for students who have had difficulty with mathematics in earlier years of the secondary education. The aims of the course are to provide these students with basic competencies and the confidence to use mathematics in their everyday lives.

The students who take this course typically move on to TAFE courses and trades, or to employment upon finishing their V.C.E. The types of mathematics that are useful to them are then mathematics of trade, business and personal interests.

As a corollary to this point, today’s students need to be familiar with and proficient with a variety of equipment and technologies. While this brings to mind visions of computers, spreadsheets and CAD programs, it is important not to forget about rulers, compasses, hammers and handsaws.

These students work well in environments where the activities are practical, concrete and often physical in nature. Some perform tasks in a step by step sequential order so breaking larger tasks into their components is important.

The approach that I advocate for Foundation classes is two pronged:

 

1. The use of thematic projects that are real and practical in nature. These themes will produce an end result, whether it be a budget or a construction or a plan of a room. The will incorporate within them mathematics from several of the "traditional" concept areas.

 

 

2. Identification and "remediation" of weaknesses. The class sizes are small. This allows for a higher degree of teacher student interaction on a one to one level. When weaknesses are identified, or as a result of pro-active skill building, time is to be spent on overcoming them.

 

The Oxford University Press text that has been developed incorporates this approach. Two of every three chapters are thematic in nature and use mathematics that is typically useful for the majority of the population. Every other chapter is a Skills Practice unit, focussing on a particular related group of concepts like fractions, interest, area, volume etc. These chapters are very similar to standard text chapters presently in wide use.

Teachers can alternate between these two different chapter styles, or work exclusively with one or the other type, whichever they feel more comfortable with.

The single most important question to be answered for the students in Foundation Mathematics classes is "When are we ever going to use this?"

The second most important question is "How are we going to use this?"

This second approach starts with skill development and practice in a particular area of study and moves toward the use of these skills in an application. The skills practice lessons may continue for one to two weeks before moving on to an application for another two weeks.

A teacher may, for example, have the class investigate area and volume of common two and three-dimensional shapes and then undertake an application task examining areas and volumes contained within different buildings.

The application tasks for this model would cover fewer concepts than in the thematic approach, particularly in the early stages of the course. It is important to note that even though the skills practiced at the beginning of any topic may be specific to one area of study, the Application Task and Assessment should cover mathematical content from all areas when appropriately used.

In both approaches the thematic or application periods should be centred on ideas that are of interest or need to the students in the class. The course is intended to equip them with the mathematics that they will need for daily life: work, further study, recreation and the home. The next page lists several ideas themes that could be explored. There is also a list of Skills Practice Areas that tie into one or more of the themes.

Example Topics (from Foundation Mathematics, Oxford University Press)

Either in the design stage of the theme or during its delivery in class with the students, it may become apparent that several members of the class are having difficulty with a particular skill or certain knowledge. At such times it is appropriate to introduce skill development material for the students to use as practice before continuing. Such material may be found from existing textbooks or worksheets. Refer to the list of resources at the end of this advice material.

In this example Budgeting has again been used. The same content is covered as in the Thematic topic but in a different order. The style of delivery is the decision of the teacher.

Foundation Mathematics is structured around three Outcomes. These outcomes can be separated into two criteria each to assist in assessing student work as follows.

Outcome 1: (Content) Students should confidently and competently use mathematical skills and concepts from the areas of study ‘Space and Shape’, ‘Patterns in Number’, ‘Handling Data’ and ‘Measurement and Design’.

It is not necessary to check individual dot points within each of the areas of study. A well-balanced course with a variety of Themes or Skill foci will naturally cover most if not all of the dot points. See Sample Course outlines for examples.

Outcome 2: (Context) Students should be able to apply and discuss basic mathematical procedures in contexts relating to familiar situations, personal work and study.

Outcome 3: (Technology) Students should be able to select and use technology to apply mathematics to a range of practical contexts.

 

1. Short Tests and Worksheets (15% of semester) – These can be used effectively to assess Outcome 1, the content of the course and to some extent Outcome 3 where the use of calculators is involved. While the questions in such tests may be worded in a particular context (Outcome 2) many students who undertake this course have a limited ability to reason out what skills and rules to apply under test conditions. Tests and worksheets should assess a student’s ability to use skills and knowledge in the course, not necessarily choose the most appropriate skill under these conditions.

 

 

2. Applications Tasks (40% of semester) – These can be used to assess all three outcomes, and are to be contextualised. Written material, presentations and finished products can all be included as part of the assessment. The sample assessment sheets at the end of this course advice give two options for Applications Tasks:

 

 

1. Technology – where the use of software plays an integral part in the Task. Students may need to use Calculators, Spreadsheets, PowerPoint, Dynamic Geometry packages to complete the work. Budgets, Renovation schedules and Survey Maps may all be assessed in this way.

 

 

2. Materials – where a finished product is to be assessed. This may take the form of a Plan Drawing, a Toolbox or a Recipe. The use of appropriate tools mathematical or otherwise can be assessed in Outcome 3.

 

The difference between these two types of Application Task is in what is used to achieve them. Software is not the only form of technology that students need to be familiar with in their workplace and home environments. Students should undertake at least one Technology Application Task per semester.

 

1. Reports (Mathematics in Context) (20% of semester) – For assessing Outcomes 2 and 3. In reports students are to choose an issue or theme that makes use of mathematics. The student is then to research this topic and prepare a report of presentation on how mathematics is used. Topics that can be assessed in this way are Mathematics in sport, Mathematics in the Media, Advertising, Alcohol and its Effects, and Gambling. While students need to be able explain what mathematics is used and how it is used, they may not need to undertake any calculations of their own for a report.

 

 

2. Examinations (25% of semester) – As with tests, examinations can be used to assess Outcomes 1 and 3. Some use of short answer questions, with adequate direction of students, may also enable an examination to assess some components of Outcome 2. In all cases questions should be broken down clearly into parts. Where formulas or rules are to be used these should be written into the question as and when needed.

 

Technology should be incorporated into the teaching and learning process wherever applicable. Decisions of this nature will depend on the type of topics selected and on the teacher’s familiarity with various software packages.

Technology that may be used successfully in Foundation Mathematics include:

scientific calculators, graphics calculators (where available, for example as a class set kept by the school), Excel, PowerPoint, Cabri, Geometer’s Sketchpad, CAD and the Internet.

It is worth noting that measuring instruments such as tape measures and compasses, and tools like those found in a workshop, home, restaurant, or office can also be classed as technology. This course can cater for such diverse interests and needs. This is also an avenue for linking Foundation Mathematics to other V.C.E. and V.E.T. courses.

In addition to electronic technology, the use of other materials and resources is worthwhile. The students who undertake the Foundation Mathematics course are often visual and kinaesthetic learners. Hands-on, practical activities can be very successful.

Foundation Mathematics, E. Flittman: Longman, 2000

Foundation Mathematics, O’Connor, Gaton, Haese & Haese: Oxford, 2000

Mathematics at Work, Ian Lowe,

Investigations in Space and Number, B Hodgson, et al, Jacaranda, 1991

Roads and Traffic Authority publications
Consumer Maths Success Kit, Hawker Brownlow Education, 1993

Hands-on Maths in Practice, Forster et al, Longman, 1997

Trade & Business Maths, Thompson et al, Longman, 1998

General Maths Applications, Daly & Cody, McGraw Hill, 1994

Geometry Everywhere, Ministry of Education and Training, 1991

Chance & Data Exploring Real Data, Finlay & Lowe, Curriculum Corporation, 1993

Dollars & Sense, Sadler & Swan, 1988

Maths Projects & Investigations, Ferguson et al, Nelson, 1990

Environment Counts, MAV

Most General Mathematics texts contain material that can be adapted.

Some of the early Space & Number projects and problem solving can provide ideas

The MAV is exploring the possibility of using its website for hosting themes and application material submitted by teachers in PDF format.

 

1. Find six examples of graphs from the Herald Sun.

 

2. Stick these graphs on to the A3 sheet of paper provided.

 

3. Work out the mean, median and modal values for each of these graphs.
4. For the Vote Line Graph and one of the other graphs convert the information from a bar chart into a pie chart. Draw the pie chart underneath the bar graph of the same information.
5. For the Pie graph titled "Distribution of Classes" convert the percentages quoted into angles. Check these angles against the ones used in the graph using a protractor. Are they close enough to exact given the size of the graph.