OTRNet: Background on Using Investigations for Mathematics Education: Math Investigations

An investigation may be defined as "a situation originating in mathematics or the real world which lends itself to inquiry". This definition allow for the inclusion of the many different types of investigations discussed further on in this information paper.

A mathematics investigation allows students to examine this situation using various techniques and in the process of their exploration develop skills that can be applied to other problems. The type of skills normally associated with investigations are generally higher order skills or processes. These processes fall under the broad heading of Working Mathematically in the Curriculum Frameworks.

Are All Maths Investigations the Same Format?

Although all maths investigations aim at developing higher order processes the form that they take can vary significantly. In some cases an investigation may be a very open exploration starting from a single statement such as " Investigate the ratios of volume to surface area in 3D objects". This same situation could also be presented as a series of tasks leading students along certain lines of thought. An example of this is in the sample investigation of Category A: Investigations 1 of our Subscription Service.

What Type of Maths Investigations Will I Find in OTRNet's Subscription Service

The investigations in Categories A and B of OTRNet's Subscription Service are presented in the form of a series of tasks. These introduce some situation and then prompt students to examine various aspects of the situation, generalising wherever possible. They tend to become less prescriptive towards the end often leaving open avenues for further exploration by better students. Our investigations are presented in this format so as to provide a full resource to teachers, their online format allows for modification to make them more or less open ended as the user sees fit. You can see examples of this format in either of our samples for Category A or Category B.

What Can I Achieve by Using Investigations?

Investigations require students to use mathematical processes to understand the problem. The types of processes developed by work on investigations include: data collection, symbolizing, classifying, simplifying, abstracting, following and extending patterns, conjecturing, communicating, generalising, justifying, proving, hypothesising and predicting.

Having developed these thinking skills through various investigations students become more able to apply and transfer this knowledge to new, non-routine situations as they arise.

How Can I Select Maths Investigations Suitable for My Students?

The selection of what type is appropriate for a given class depends on what the aim of the work is, what time constraints are applied, whether it is being used for an assessment item, what experience the students have and a number of other factors. It is possible to select particular investigations with the aim of developing (or assessing) certain processes.

An important part of the selection process should be completing the investigation yourself before deciding on its use. In doing this you can accurately assess its suitability and begin to build up an understanding of what is involved. This also help to decide on what is expected from students and how the item will be assessed (if this is required).

How Do I Implement Maths Investigations Into My Courses?

If you intend to use investigations as part of the learning experiences of students then they will need to be written into your programme of work.

Other planning that can take place at the programming stage is; planning the number of investigations, pre-selecting investigations to suit your objectives and deciding on the part they play in your assessment structure.

How Do I Implement Maths Investigations Into My Classroom?

Investigations should be presented in an interesting and suitable format. When students have had time to read through an investigation it may be necessary to go over it and clarify any restrictions etc. Make available any types of resource materials required such as graph paper, dotty paper, cubes, matchsticks or whatever will help their exploration.

It may be necessary after the students have had some time to look at the investigation to discuss their progress and ensure that they are following any assumptions or restrictions that it involves.

Helping students probably should be limited to encouragement and stimulation rather than hints. It is the 'mental leaps' that we are often trying to get kids to take, giving them these leaps removes this benefit. It may be necessary to go over any content that is hindering their progress or to remind them of suitable investigative techniques such as simplifying the problem, being systematic, tabulating results etc.

You will need to monitor or set the level of cooperation occurring between students. I would strongly suggest that students are not allowed to share the 'hackwork' as it is during this hackwork that students often gain the insight into the investigation.

In the end the level of help given to students depends on what you deem appropriate for their experience. In early investigations a lot of help might be useful but ultimately we want students to be able to 'conduct' the investigation on their own with only encouragement and stimulation from us.

Once students have some experience with investigations my usual procedure is;

give out the investigation and allow students an introductory look at what it involves
a few days later briefly discuss progress, assumptions etc
after one week of students exploring have them submit their work and validate their personal understanding using an assessment task designed for that investigation

How Do I Use the Maths Investigations in OTRNet's Subscription Service?

Browse through the directory looking for an investigation at a suitable level and with the required purpose.
Open the investigation by following the link and look at it in detail to determine its suitability.
Download your choice of investigation to your computer in Word, Word Perfect or PDF. (Remember - if you wish to edit it you will need the software - i.e. Word etc)
Make any changes you require. This may be just adding a header and due date or it could involve adding or deleting a particular question. The choice is yours.
Print a copy from which you can duplicate the students copies.
Each investigation comes with an assessment task that you can also download and edit if necessary before duplicating for students.

Providing Feedback to Students on Maths Investigations

Allow time for discussion of investigations when you return them to students. Provide written comments on student work wherever possible. Provide students with a copy of the marking guide so that they can look at their performance on each part of the investigation. It is often worthwhile to show students your attempt at the investigation or use some of the best student work as a model.

Discuss items of general interest with the class and also go through the marking guide and how it was used. These things set the scene for the next investigation and make the whole process more of a learning experience.

Improving Student Performance on Maths Investigations

Perhaps the simplest and most significant step in improving students' performance on investigations is to allow them to attempt as many investigations as possible. By implementing and practising the processes involved students will be more likely to apply these processes in other problems. I have seen this steady improvement in students over the long term to the point where they will apply techniques learned in investigations to solve other problems.

Considerable benefit can be gained by providing a model for your students. Displaying copies of and discussing your attempts at investigations is worthwhile. Perhaps even more useful is working on an investigation with the students. Thinking aloud, discussing ideas and strategies, and demonstrating techniques will allow students to follow your model and will also help to give them more confidence with their own exploring.

Make students well aware of what is expected from each investigation. If it to be assessed outline the assessment process that will be used. It may also help to go over the objectives of each piece of work before and after its use.

Allow for discussion between students after they have completed their work. Vary the types of investigations you give so that students experiences are varied. While marking each item keep a list of 'good' and 'bad' strategies to discuss when the work is returned to students.

Assessing Maths Investigations

I will outline three possible structures for assessing students performance on investigative items.

1. It is possible to set up a model solution or marking key for each investigation and to mark students work comparatively against this model. This method is reliable but suggests that a unique way of solving a problem exists.

2. Another method is to outline a series of processes that should be used in an investigation. This might include headings such as: Comprehension, Representation, Generalisations, Application and Communication. Each could be weighted for specific investigations. This assessment strategy allows for student work to be quite divergent while still gaining credit for use of high level processes. It is less reliable than the previous method but has advantages when assessing work produced on very open initial problems.

3. A third method is to devise a series of tasks that attempt to measure students understanding and retention of concepts covered in their investigative work. It can also include a task that requires students to apply processes to investigating previously unseen aspects of the topic. This is the format of the assessment tasks we provide.

Each of these structures has advantages and disadvantages. It is also important to remember that Investigations are an excellent learning experience and may not involve any form of assessment at all.

The Assessment Tasks Provided with OTRNet's Maths Investigations

Each investigation in OTRNet's Subscription Service comes with an assessment task based on validating each students work. These tasks follow the format in point 3 above. They are designed to allow students to demonstrate the knowledge they have gained by completing the investigation, and to apply investigative techniques to further explore the topic.

These assessment tasks can be used after the students have completed their work on the investigation. If given to students individually without reference to the written work they have completed it allows teachers to assess the level of understanding and retention of key concepts from their investigative work.

They can be used as the sole method of assessment, in combination with students submitted work, or simply to validate the mark allocated for submitted work

Problems to Overcome in Implementing Maths Investigations

There are a number of problems associated with implementing maths investigations as part of a mathematics course.

If they are new to your students then there may be a problem with acceptance of investigations as a different form of learning. This can be alleviated by a clear explanation to students, and perhaps parents, of the purpose and value of investigations. This explanation should include reference to the type of outcomes desired which relate to mathematical processes or "Working Mathematically". Outcomes of the form below are significantly developed by work on maths investigations:

Call on a repertoire of general problem-solving techniques, appropriate technology and personal and collaborative management strategies when working mathematically.

Choose mathematical ideas and tools to fit the constraints in a practical situation, interpret and make sense of the results within the context and evaluate the appropriateness of the methods used.

Investigate, generalise and reason about patterns in number, space and data, explaining and justifying conclusions reached.

Including the assessment of investigations as part of the student's assessment will also help to show the level of importance you place on them.

A second problem can be the increased workload placed on students and teachers. To help teachers, a ready source of prepared investigations needs to be available, and the assessment method needs to be easy to understand and use. To help students there is a need to give adequate time on investigative tasks, give advance warning if they are to be partially completed at home and also in this case reduce the volume of other homework.

If investigations are to be a part of assessment, and time is allowed to work on them out of class, then measures will need to be taken to ensure the validity of these assessments. This is true of any "out of class" work. These measures need to include a method of validating the out of class work and of ensuring that students spend an adequate amount of time working on the investigation.

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If you would like to find out more about maths investigations send an e-mail to mark@otrnet.com.au and we'll send you more information.

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