M3 - Planning and designing lessons

Introduction

In this third module of the Teaching series we investigate the planning and design of classes. Professor Eamon Murphy of the Department of Social Sciences at Curtin University devotes an entire chapter of Lecturing at University (1998) to emphasise his view that careful planning is the most important aspect of lecturing.

"the success of the lecture is largely determined before you actually get up to speak. The most successful lecturers develop a systematic approach to lecturing. Most have a planned step by step approach which in the long run saves time and energy. ... . Planning helps put you in control." (Murphy, 1998, p. 5)

This module explores a range of processes from which you as an educator are encouraged to fashion your own approach to designing and planning lessons to achieve your desired student learning outcomes. The next module (Module 4) then goes on to address conducting mathematics lessons.

Learning Outcomes

By end of this module, you will be able to design lessons using an outcomes-oriented, student centred lesson planning process. You will be able to:

Explain the importance of consolidating background knowledge as the foundation for generating new understanding of mathematical concepts, and to develop techniques for higher order mathematical thinking.
Identify relevant mathematical concepts and both mathematical and general skills in proposed lesson content, and be able to signal and enhance these during learning activities.
Place new mathematical knowledge in both a broader context and a student-meaningful real world context.

Module Structure

This module outlines the following steps to designing a lesson:

Create a planning framework
Select topic and determine the goal of the lesson
Determine prior learning and skills
Determine learning outcomes and progress indicators
Select and organise resources
Determine a sequencing for the development of knowledge and skills
Select appropriate teaching strategies and assessment tasks
Reflect on and evaluate the lesson.

Create a planning framework

A typical mathematics lecture or tutorial is a fifty minute learning opportunity for a large group of students. During that fifty minutes students encounter and explore specific knowledge or concepts that have been identified by the unit co-ordinator during the unit planning process, and which are documented in the unit outline.

The teacher's actions, activities and importantly enthusiasm play a significant role in achieving the desired student outcomes - through the development of understandings, connections and skills. The careful planning of these activities is a significant contributor to achieving student learning.

Brown (2006) emphasises that student needs are "central to the planning process". In practice, this means that the planning and design process for a given session starts with an identification of the student learning outcomes for the session. The other processes then proceed backwards from the outcomes. Seven stages of the planning process are set out below and expanded in the table below.

Select topic and determine the goal of the lesson
Determine prior learning and skills
Decide on student learning outcomes and indicators of students' progress
Select and organise resources
Determine a sequencing for the development of knowledge and skills
Select appropriate teaching strategies and assessment tasks
Reflect on and evaluate the lesson

Key questions and resources for each of the stages are outlined in the table below:

Select topic and determine the goal of the lesson	What are the key concepts, ideas and theories? Why are these important?
Determine prior learning and skills	What understanding do the students already have? What are their and your preconceptions and misconceptions?
Decide on student learning outcomes and indicators of students' progress	What will students know, and be able to do, by the end of the session? What indicators will you use to determine if students have achieved these outcomes? One useful approach is to write lesson outcomes, expressed using verbs to indicate what the students will achieve.
Select and organise resources	What resources are available to design and use as part of the session? Some resources you might find helpful are textbooks, colleagues' notes, online resources, applicable teaching articles.
Determine a sequencing for the development of knowledge and skills	What is the optimal ordering of the material to consolidate and extend students' knowledge? At what stage should background material and notation be introduced? How will the current theory be linked to previous work?
Select appropriate teaching strategies and assessment tasks	What experiences will consolidate students' understanding and allow them to demonstrate their achievement of the lesson outcomes?

Reflect on and evaluate the lesson

How can you use feedback from students to respond to the experience and characteristics of your student cohort?

We now go through the above planning approach giving examples and tasks associated with each of the seven points.

If you are a tutor in a unit, then you may feel that you do not have full control over what or how you teach. However, understanding the intended outcomes for your particular class, and how they will be assessed, and seeing that ideas and tasks are sequenced to build on prior knowledge are relevant to the effectiveness of your classes, which support the lectures. If you consider these issues, then you will be a great source of feedback to the subject lecturer as they reflect on and evaluate the teaching sequence.

Select topic and determine the goal of the lesson

The selection of the session topic will often be predetermined by the subject outline and this will be discussed further in a later module (Module 8). However, goals for each session should be established and used to guide the session planning. Note that the goal is not focused on the tasks of the teacher and the delivery of the lesson, but the student learning process. So for instance, "explaining differentiation from first principles", should not be the goal, whereas "students understand how to differentiate from first principles" might be.

Cartoon by Nick D Kim, strange-matter.net. Used by permission.

We will continue to use this thread of preparing a lesson on differentiation from first principles to illustrate the remaining steps in the planning approach. In parallel, you will be asked to carry out the planning process for a lesson on simple differential equations.

Task 3.1

Write down a set of goals for a lesson on techniques for solving linear first-order differential equations.

Determine prior learning and skills

The acquisition of mathematical knowledge is hierarchical by nature and so it is imperative that new knowledge and skills build on existing knowledge and skills. Hence once the topic and goals have been set, it is important to check that the new material is being presented at the appropriate level. This is especially important for service teaching.

For instance, if students have never worked with the concept of graphing linear functions and evaluating gradients they will not be able to grasp the concept of the gradient of the tangent line as a means of calculating instantaneous rate of change. If they are algebraically challenged then they may not be able to cope with the algebraic manipulation needed to obtain solutions without support. On the other hand, if students have seen the derivation of the formula for differentiation from first principles then little is to be gained from repeating this explanation; more might be gained by proving from first principles that the derivative of a sum is the sum of the derivatives, or deriving the product rule, for example. You can ascertain this sort of information by checking the National Secondary Curriculum, earlier level unit outlines for the appropriate degree programs, looking at exam papers for these earlier level units, checking textbooks and talking to students and colleagues. Getting regular feedback via devices such as ‘clickers' and from students' own work may also help.

Another point to consider at this stage is whether your intended lesson involves any threshold concepts (see Module 2) that may relate to prior knowledge in a transformative or integrative way.

Task 3.2

What are the knowledge and skills which students will require to understand a lesson on techniques for solving simple differential equations? Are any threshold concepts going to be encountered? Explain your reasoning.

Determine learning outcomes and progress indicators

In determining learning outcomes, it is useful to use verbs to indicate what the students will achieve, for instance in terms of recollection of knowledge, advancement in performance and changes in thinking processes. The choice of an appropriate verb can indicate where on a taxonomy (such as the Anderson and Krathwohl’s (2001) version of Bloom’s taxonomy, introduced in Module 2) you intend your students’ understanding to be. The learning outcomes you propose for your lesson should also convey (again possibly by choice of verb) whether you intend students to achieve an instrumental or relational understanding (in the language of Skemp, from Module 2). You will need to decide how to gauge this achievement.

Task 3.3

Post your responses to these three questions on the discussion board:

Below is a list of verbs that describe some mathematics-related activity. Which of the following could be used to describe or indicate higher order mathematical thinking? Explain your reasoning.
1. identify
2. explain
3. solve
4. analyse
5. prove
6. reason
7. generalise
Do we use these verbs in any special way in mathematics or statistics? What about the verbs “evaluate” and “integrate”? They are generally considered “high level” relational verbs in educational literature, but what about in mathematics? Can you think of other verbs that we use in a special way and which you may use in writing lesson learning outcomes?
Can you add any other verbs that may be used to describe when students are engaging in higher order mathematical thinking?

To illustrate this point further we give a possible set of learning outcomes for our mathematics lesson.

Example learning outcomes

At the end of this session on differentiation from first principles students will be able to:

Analyse and explain the concept of instantaneous rate of change of a function of one variable, and generalise ideas and adapt concepts to relate the gradient of a function in one variable to the gradient of its tangent line.
Use critical reasoning to derive a formula for the calculation of the instantaneous rate of change of these functions, understanding the importance of the limit that appears there.
Collect, analyse and organise information when evaluating the derivative of a given (simple) function from first principles.
Demonstrate a comprehensive and well-founded knowledge of differentiation and the solution of applied problems involving derivatives.
Apply critical reasoning and independent thought to master the concept that every differentiable function is continuous but that the reverse doesn't necessarily hold.

It is relatively easy to see how we could use problem sheets and submitted work to ascertain indicators of student achievement for Outcomes 3 and 4.

However, it is harder to see how we can gauge students' performance on Outcomes 1, 2 and 5. One possible approach might be to set students an exercise where they are given a function, its graph and the graph of the derivative. Then ask them to connect this information to the concept of a derivative and relate this back to first principles.

Alternatively, if students are involved in small group work, they could be asked to present to the class an explanation of the derivative, maybe by getting one student to start and then others to take over at various points. Another approach might be to get the students to present a poster or an essay on Newton and Leibniz and the relevance of their work to the current topic.

Task 3.4

For a lesson on the solution of linear first-order differential equations, write a list of student outcomes that use some of the verbs in task 3.3.
Now write a list of indicators you could use to determine if the students have achieved these outcomes.
Is it harder to suggest indicators when the learning outcome concerns Skemp’s relational rather than instrumental understanding (see Module 2)? Why is it still important to include them?

Select and organise resources

You should gather together resources that will help you plan your lesson. You are looking for well-designed resources that will allow you to quickly identify insightful examples and relevant applications. These should include:

an outline of the curriculum
textbooks on the topic
publishers' support resource materials
colleagues' notes, if these are available
online resources
teaching notes in professional publications, such as the Gazette of the Australian Mathematical Society and The American Mathematical Monthly
an applicable teaching article; these can be sourced from journals, but it is often advisable to start with a search using Google Scholar or a similar tool.

In planning your lesson you may choose not to stick exactly to the presentation given in the above documents, but you should use these as references as they will help you clarify your own ideas on how to present the material.

As part of your planning process you should also consider how to utilise resources such as tutorial assistance, computer laboratory classes, software such as Maple or MATLAB, and online simulation. For instance, there are many simulations on the web that demonstrate the change in the gradient of a tangent line and its dependence on the concavity of the function.

Determine a sequencing for the development of knowledge and skills

In many cases, the new knowledge and skills that you wish the students to acquire will be built on prior knowledge (which the students may or may not have), and might be impaired by existing knowledge that is either incorrect or limited. If threshold concepts are involved, you will need to both build on existing knowledge and possibly transform it. You need to consider how interest will be developed and prior knowledge activated.

This problem can, in part, be addressed by the design and delivery of logically sequenced material that is contextualised - especially in service teaching - and which recognises the broader mathematical framework, providing real world applications where possible. This process is summarised in Figure 1:

Figure 1. Process of determining a sequencing for the development of knowledge and skills

It is fundamental and indeed imperative to the preparation of material for a mathematics lesson, be it a lecture or tutorial, that students are exposed to tasks based on a logical sequencing of material that builds the solid basis for their mathematical understanding.

Note that at this stage, we are still considering the sequencing of ideas, and not specifying what the “tasks” might be. These will include activities done by the teacher for the benefit of the students and activities for the students to do themselves.

The careful choice of graduated examples will help students attain a range of skills and understand key mathematical concepts. These examples should be chosen in such a way as to avoid excluding sections of the class. The careful choice of graduated examples will help students attain a range of skills and understand key mathematical concepts. These examples should also form a basis for student activities that enhance skills such as analytical thinking, logical presentation, mathematical communication and problem solving. Make sure your planning includes an allocation of time to each task, thus ensuring that students fully grasp the topic, but at the same time not swamping them with too much detail. Returning to the example of a lesson on differentiation from first principles, in a lecture one might decide to sequence the ideas as follows:

Revise function notation and introduce any other required notation.
Revise the concept of the gradient of a straight line.
Connect the concept of the gradient of a straight line to the average rate of change of a function over a given interval.
Develop a formula for the average rate of change over a given interval.
Introduce the concept of taking a limit to approximate the instantaneous rate of change of a function.
Develop a formula for the concept of a derivative using the above derivation.
Implement the above for linear functions.
Implement the above for other polynomials.
Emphasise the importance of the continuity and smoothness of the function.
Apply knowledge to real world problems.

Task 3.5

Read these notes on how "Solving linear first-order ordinary differential equations" could be taught. Make a list of the key concepts that are being presented and identify the sequencing of material. Discuss how this sequencing will support student learning.

In sequencing concepts and activities to enhance student learning relating directly to the current topic, it is important to take the opportunity to develop general mathematical skills, including:

logical presentation,
critical and analytical thinking,
mathematical communication,
problem-solving.

Knowledge of mathematical concepts and the development of mathematical skills are an integral part of mature mathematical thinking and cannot be taken in isolation; they should be interwoven in any lesson planning. Moreover, such skills, though here being developed in the mathematics context, have value for students and should transfer to their whole education and their development as professionals, as was noted at the start of Module 1. Many universities now expect that disciplines can describe and document the development of general skills or graduate capabilities in their units and courses; this development begins by building them into your lessons.

In relation to a lesson on differentiation from first principles one could develop skills as follows:

Logical presentation and mathematical communication can be enhanced through a clear explanation of the steps involved such as the writing down of the definition, the substitution of the specific function into the definition, the simplification techniques and the taking of limits to obtain a solution. Remember that graphical aids can be used to facilitate understanding.
Analytical thinking can be facilitated by assembling known information both generic (definition) and specific (given function), by substituting the specific information into the generic information, by manipulating and applying known techniques to approximate a solution (algebraic manipulation including concepts such as multiplying by a conjugate).
Problem solving can be enabled by collecting and organising relevant information, abstracting into the theory especially for complicated functions, by identifying difficult or complex elements of the problem (the step at which you cannot easily go forward) and using known techniques to simplify and reorganise material, moving closer to the final solution.

Task 3.6

Return to the notes on "Solving linear first-order ordinary differential equations" and identify points in the discussion that can be used as opportunities to develop students’ general skills.

Select appropriate teaching strategies and assessment tasks

Assessment tasks are addressed in Module 6 and Module 10, and we refer you to these modules. However, it should be noted that assessment tasks should be aligned with the student learning outcomes, and that we have already observed that considering how achievement of an outcome could be demonstrated by the students and measured in assessment is important in the planning stage.

Selection of teaching strategies is an important part of the planning process and once again the focus should be put on student learning. The literature contains many articles on various teaching strategies, and these will vary depending on the type of class, the number of students and the cohort of students. See, for example, resources for teaching large classes developed by The University of Queensland (Teaching and Educational Development Institute, 2006). It is important not to be locked into one type of teaching mode, but to be flexible and adapt according to students needs both at the class and individual level. For instance, there may be students with special learning needs such as visual impairments or hearing difficulties, so the teacher may need to organise for accessible resources to be made available. There will be more discussion of teaching strategies in the next module (Module 4), but here it is relevant to completing the planning and design of the lesson.

To illustrate this point we return to the list of concepts to be addressed in a lecture on differentiation from first principles as set out above under Determine a sequencing for the development of knowledge and skills , and give some activities which might attract student's interest and activate their learning process, and at the same time develop their skills in critical and analytical thinking.

You might begin the lecture by splitting the students into groups and giving them problems which revise their understanding of function notation and equation and gradient of a straight line. Follow this with an applied problem on finding the velocity of a particle for which the distance from the origin is increasing at a constant rate. Encourage each group to write a solution to this last problem on a sheet of paper, illustrating their ideas using graphs. Vary this last problem, so that the distance from the origin is increasing quadratically and get them to find the average velocity over a given interval.
Use "clickers" or otherwise to check solutions to simple problems and collect a number of written solutions to applied problems and project these using the visualiser. Use the one group's solution to extend the concept of average rate of change to instantaneous rate of change.
Then develop a formula based on this extension.
Implement this formula for a simple function, then a slightly more complicated function.
Get the students to return to their groups and implement the formula for finding the derivative for a more complex function such as . Encourage the students to identify points of difficulty and strategies for moving past these points.
Give the students a sheet containing a sketch of a general polynomial function and a blank set of axes immediately under it. Ask the students to draw the tangent lines at various points on the function. Then ask them to sketch the function for the derivative.
Repeat the last step for a discontinuous function.
Give the students applied problems so they may put the material in the context of their disciplines.

In implementing the above ideas you may not choose to get the students to work individually or in groups, but instead to demonstrate these ideas yourself. Alternatively you may have a small class, or a tutorial, and can get the students to present their ideas to the class. Another alternative is to include these types of questions on problem sheets or as part of the assessment process, as they will gauge if students have achieved the learning outcomes.

Task 3.7

Return to the teaching notes "Solving linear first-order ordinary differential equations" and suggest a series of activities which can be used to facilitate student learning in this area. Incorporate activities, including some collaborative ones, that can help students enhance their general skills in the mathematics context. You can draw on the suggestions above, the resources below, ideas you already use, and ideas you would like to try.

Read this lesson plan on differentiation which takes the above ideas and develops them into a formal lesson planning document. Some of these ideas have also been implemented in this screencast which takes the student through the derivation of the formula for differentiation from first principles.

Screencast rate of change of a linear function

Task 3.8

Take the ideas you have developed in the preceding tasks for a lesson on "techniques for solving linear first-order ordinary differential equations" and use them to compile a formal lesson planning document.

Reflect on and evaluate the lessons

It is important to evaluate your teaching as it enables you to respond to the needs of your particular student cohort and to adjust your approach for future delivery. Module 7 is dedicated to this topic.

Relation to Assessment Task Two

In assessment task two, you will be asked to plan a sequence of classes. For full details of this task, the submission date and the marking rubric, please consult the unit outline.

Review and conclusion

Always remember: "Careful planning will ensure that most of your lectures will be effective - some will be very good and there will be the odd magic occasion when an occasional lecture will be brilliant. The odd brilliant lecture where everything clicks will be a bonus. Students do not expect you to give brilliant lectures. They are very grateful if you get the basics correct: a lecture that is well organised and clearly presented." (Murphy, 1998, p. 5)

In the next module we address conducting mathematics lessons.

References

Brown, N. (2006). Planning for flexible approaches to tertiary courses. Australian Association for Research Education Conference. Retrieved from http://www.aare.edu.au/06pap/bro06609.pdf
Murphy, E. (1998). Lecturing at university. Bentley: Curtin University.
Teaching and Educational Development Institute. (2006). Teaching large classes: AUTC project. Retrieved 18 February, 2011 from http://www.tedi.uq.edu.au/largeclasses/.
Race, P., & Pickford, R. (2007). Making teaching work; teaching smarter in post-compulsory education. London: Sage Publications.