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What do we mean by rich and balanced?

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Many authors have attempted to define the term rich in relation to mathematics tasks or lessons (see, e.g., Downton, Clarke, Knight, & Lewis, 2006; Flewelling, & Higginson, 2001). Although it is very rare for a single task or lesson to have all of these features, the following are often mentioned:

• It draws on a range of important mathematical content;
• It is engaging for students;
• All students are able to make a start, as it caters for a range of levels of understanding;
• It can be successfully undertaken using a range of methods or approaches;
• It provides a measure of choice or openness, leading to a sense of student ownership;
• It involves students actively in their own learning;
• It shows the way in which mathematics can help to make sense of the world;
• It makes appropriate and effective use of technology;
• It allows students to show connections they are able to make between the concepts they have learned;
• It draws the attention of students to important aspects of mathematical activity; and
• It helps teachers to decide what specific help students may require in the relevant content areas, or ways in which students might be extended.
  

There are many such lists in mathematics education articles, and in working with teachers, we have encouraged them to put the lessons we share with them and the ones they develop themselves under the microscope, and to ask the questions, “Is this lesson rich? What are the features which make it so?” These features therefore provide a kind of checklist, and thus form our design elements in developing lessons. In the following discussion of three particular lessons, we will summarise the features which teachers identify in relation to the lessons.

When we refer to balanced, we mean that the features above work “in harmony”, are mutually self-supportive and not over or under weight in any aspect. Many traditional text-book type lessons are arguably overweight in the rush to skill development, while ignoring a range of other equally important teaching and learning principles. The balancing act is needed within any lesson, and by implication, across any set of lessons. We will explore the way in which the three lessons we discuss can be regarded as balanced.

The three lessons which we encourage the reader to place under the microscope in this article are titled:

• Temperature graphs
• Maths in motion
• Mathematics of Lotto

In each, we will present a description of the lesson followed by a critique designed to identify the elements of the design profile, and also compare and contrast the lesson with an equivalent stereotypical textbook version.

Our context

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In the 1980s, there were major concerns with the teaching of mathematics in Australia, particularly among teachers in the middle years of schooling (Grades 5-8): We are confident that the following list generated at that time will come as no surprise to readers:

• Maths was seen as boring and irrelevant;
• Not enough thinking nor genuine understanding was required of students;
• The topic was too abstract;
• Fear of failure and poor attitudes were evident;
• Maths was seen as elitist, and designed for tertiary bound students only;
• There was too much content to address and not enough time to do so;
• Assessment approaches were too narrow;
• Catering for a wide ability range was very challenging; and
• Teachers were struggling with problematic parent and community expectations.

The authors first worked together jointly running a national program, titled the Mathematics Curriculum and Teaching Program (MCTP, Lovitt, Stephens, Clarke & Romberg, 1990). In this project, we had the opportunity to consider how best to address the above issues, in conjunction with teachers, mathematics consultants and education systems.

Our response

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Our focus was, and in many ways still is, to use “the classroom lesson” as the basis of professional learning conversations. We attempted to “capture” examples of good practice from around Australia by documenting lessons, sharing these as “works in progress”, and encouraging teachers in supportive professional learning environments, to consider how they could both enhance the lessons, and more particularly expand their personal comfort zone or repertoire of teaching strategies. This personal repertoire of features can be called their design profile—the personal collection from which a teacher draws in creating (designing) learning experiences.

The model shown in Figure 1 was used as a basis for these conversations (Owen, Johnson, Clarke, Lovitt, & Morony, 1988, p. 4.)

Each teacher comes to any learning situation with their personal current practice or ‘comfort zone’. As teachers worked through lessons in the role of the student in professional learning settings, and subsequently trialled them in their own classrooms, they were encouraged to seriously scrutinise the lessons for the features they contained, particularly those new to themselves, with a view to ultimately expanding their comfort zone or teaching repertoire.

In our documentation for a given lesson, which always went through many iterations, and, in one sense, was never finished, we attempted to tell the story of the lesson, which was always an “amalgam” of the ways in which the lesson has played out in a variety of settings.

We used photographs, sketches, samples of student work, comments from teachers, and insights from trialling to paint the picture of a lesson, in a way that the reader could hopefully visualise.

MCTP also focused significantly on the professional development (PD) settings in which these “lesson conversations’” could occur. We analysed existing PD models from around Australia, developed a set of criteria for effective PD (see Clarke, 1994) summarised the various PD offerings into eight generic models (Owen, et al., 1998), and supported consultants and curriculum leaders to use these as a basis for generating their own local PD settings.

Our work at the time was clearly informed by prevailing national policy and our understandings of the literature, but was also largely intuitive and arguably more of a grassroots approach. One memorable day, a colleague came to us with an article and said “This is what you guys are on about!” He shared with us the seminal work of Lee Shulman (1986, 1987) on “pedagogical content knowledge”, or “capturing the wisdom of practice”.

Among the statements which resonated particularly with us and the work we were undertaking at the time, were the following:

“Teaching is characterized by a collective and individual amnesia,… the consistency with which the best creations of its practitioners are constantly being lost to both current and future peers… It is devoid of a history of practice”
(Shulman, 1987, p. 11).

“Pedagogical Content Knowledge is …an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of the most frequently taught topics and lessons … if the preconceptions are misconceptions, which they so often are, teachers need knowledge of the strategies most likely to be fruitful in reorganising the understanding of learners. … the most powerful analogies, illustrations, examples, explanations, and demonstrations related to the topic one teaches. … the ways of representing and formulating the subject that make it comprehensible for others.”
(Shulman, 1986, p. 9).

We realised that in attempting to document the best lessons from around Australia, we were both highlighting the wisdom of practice, and encouraging teachers to explore those aspects of pedagogical content knowledge which could enable a teacher to take a basic idea for a lesson and make it both rich and balanced.

Our response, which started with MCTP has largely continued in several follow up projects through to the present. One such current project using classroom lessons as the basis for professional learning conversations is Maths300. This is managed by a government agency, Education Services Australia. Its ambition is to find the 300 most ‘interesting’ (i.e. worthy of professional discussions) lessons from across Australia and mount these on a website (www.maths300.esa.edu.au) to be used by teachers and curriculum leaders in various PD settings. All three lessons discussed here are available to teachers via Maths300.

The three lessons

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As has been mentioned, our approach has been to introduce the lessons to teachers in a workshop setting, and then to discuss the features of the lessons which are evident to them, and which may or may not be part of their current teaching repertoire.

The lessons are now discussed in some detail. The lessons have been chosen to reflect a range of design features, different mathematical content, and different grade levels. For each lesson, we describe how the lesson might play out (with teachers and with students), and then discuss the nature of the debrief which occurs with teachers in professional learning settings.

Teachers recognised that the purpose of such questions target just one outcome - namely the skill of correctly reading a scale. In the alternative version, it was easily noted that this skill has not been abandoned; it is still there. But now the deliberate addition of some extra pedagogical features leads to other additional worthwhile learning outcomes. The additional features which teachers often identify during discussions include the following:

• Context based. The choice of cities puts a focus on links with geography and ideas of weather in relation to location. The context is important and students can learn more about the world they live in.
• Challenge. Having a challenge is seen to be engaging. Teachers reported much higher levels of ‘participation’. The challenge has elements of the intrigue of a puzzle.
• Thinking, reasoning, problem solving. The challenge requires students to think, argue and justify desirable outcomes, features missing from the text book equivalent.
• Group work. The higher order outcomes of reasoning and communication are seen to be developed better by setting a group challenge rather that an individual one.
• Personal. Teachers reported that students know something about many of the cities and were keen to share this personal knowledge. For example, student often ask, “where is Anchorage?”. If students have personal knowledge to contribute, then their commitment to the learning is enhanced.
• Technology. The worksheet used initially is printed from the software drawn from a provided data base of cities. The capacity to produce multiple worksheets for any combination of cities, the quick feedback, the opportunity for students to choose their own questions, the opportunity for research on additional cities are all aspects provided by the structure of the software. All were seen to add value to the lesson in ways not available in the more traditional textbook version.
• Ownership. This was seen as a particularly valuable feature. In the text book question, the decision on what work is to be tackled is likely to be taken by some textbook author writing questions in their home office, possibly 10 years ago. It is this decision which is largely dictating the journey of both teacher and student. The software elegantly hands over ownership and control to the teacher and students. Teachers commented strongly that when students were selecting cities for their challenges they selected places of interest to themselves. They had a degree of control over their learning and were more committed to that learning.
• Skill. As students debate in small groups, it was clearly seen that the desirable skill of reading graph scales was still prominent and being developed and enhanced.
• Mixed ability – multiple levels of success. Some teachers initially commented, “my students might struggle with this; they don’t know enough about these cities.” But on the software there is the option to choose just two cities, say Singapore and Beijing. What do they need to know for success? Singapore is near the equator and is tropical, Beijing is up north and gets cold.
• Open-ended. An important aspect of being able to go further and in more depth (if the teachers wants to do so). For example, “Our school is in Mt. Isa (country Queensland, Australia) and it is not in the database on the software. But it could be.” Go to a search engine, put in “Mt. Isa Temperature” and in just a few seconds, the data will be available. Enter that into the data base as a new city (and the data will stay there). Then we could put in other regional cities and produce a worksheet for the class just for the State of Queensland.

The number of deliberate design features began to look impressive and as one teacher commented:

“This lesson is now profoundly different, richer and more balanced that the typical textbook”.

Up until now, the comments and insights from teachers are deliberately intuitive and informal. Further reinforcement for the features they have recognised can come from considering formal curriculum policy documents. One such example is the much respected International Baccalaureate program (International Baccalaureate Organisation, 2008)

Teachers commented:

“This design structure of the lesson exhibits all four of these – the text book version exhibits virtually none of them.”

Summary

As indicated earlier, one of the main purposes of the lessons and the discussion which surrounds them is for teachers to reflect on particular features of a given lesson, and to consider expanding their pedagogical repertoire. So, in the case of Temperature Graphs, for example, teachers might ponder whether they can give their students more ownership using a piece of software, or by other means. The whole lesson is also a transferable ‘template’ – the lesson structure, complete with all its features can be repeated using a different context such as rainfall, or population growth, or rates of deforestation, or indeed any measurable commodity that might be presented graphically.

Lesson 2: Maths in Motion

• Creativity. This was seen as a highlight feature – given the chance to invent challenges caused many students to peer into the task differently and come up with unexpected yet profitable directions. Teachers involved in the professional development also were able to add creative twists and turns.
• Ownership. When tasks were created by the students and the teachers, the sense of owning the direction of learning clearly increased involvement and commitment.
• Multiple interconnected content. Some teachers reported starting with a narrow focus such as ordering numbers from 1 to 10, but finding themselves in other mathematical territory such as finding a strategy to find all the different numbers that can be created by students ‘3’, ‘2’ and ‘8.’ This led to insights about place value and also combination counting methods.

Teachers’ informal and intuitive judgments can be backed up by research on effective teaching (e.g., Askew, Brown, Rhodes, Johnson, & Wiliam, 1997; ), and emphases in policy documents. This can give teachers added confidence that their observations are valid and justifiable. For example, the emerging (currently in draft form) Australian National Mathematics Curriculum (ACARA 2010, p. 5) states in a section titled Implications for teaching and Learning:

In Mathematics, challenging problems can be posed using basic content, and content acceleration may not be the best way to extend students. Choosing engaging experiences as contexts for a variety of tasks assists in making mathematics inclusive, differentiating both for students experiencing difficulty and those who complete tasks easily.

Summary

Teachers in professional learning settings readily agreed that each of the features listed above, deliberately designed into the lesson, added to the overall quality, richness and balance of the experience. Yet every single feature could have been ignored or eliminated, as often happens in some resources, where rote learning is more often the focus. Importantly, teachers agreed that the list of features by themselves would not be empowering. It is the feature backed up by the imagery of its use that gives meaning.

Lesson Three: Maths of Lotto

This lesson, currently in development, arose spontaneously because a mathematics teacher noticed newspaper reports about the social issues behind citizens over-spending on such things as Lotto Games to the detriment of their family and financial circumstances. One hypothesis is that such victims seriously do not understand their real chances of success and are lured by propaganda and mathematical half-truth advertising in a desperate attempt to improve their circumstances. Unfortunately, usually the opposite happens.

So how might a social context like this become a component of a rich and balanced mathematics program?

The teacher has a repertoire of lesson design features and considers systematically their use as a lesson structure evolves. The process is deliberate and systematic.

• Social Issue context. The social context of gambling and the problems this brings combines obvious opportunities to link mathematics with moral and ethical challenges. It was seen by teachers as a very relevant context. Importantly, the major outcome of the lesson is in understanding the context and hopefully being lass exploitable. The mathematics is not the major outcome, but a pathway to social empowerment. This is a reversal of text book approaches where the context is merely the vehicle to learn more mathematics.
• A class game. The 6:2 class game has been a huge highlight; very engaging and personally involving for students and a powerful first step towards the investigation that follows.
• Estimation. An important aspect of the lesson is to find out students’ current perceptions of success. The investigation is premised on the hypothesis that students will see the game as easier than it really is. Also, if students make a commitment to an initial guess, their motivation to find out if their ‘guess’ is accurate is greater.

Many of the features of this lesson are advocated within various policy documents. Once again, referring to the draft Australian National Curriculum (ACARA 2010, p. 4-5):

The curriculum lists fewer detailed topics with the intention to encourage the development of ideas in more depth and promote the interconnectedness of the mathematical concepts…

The … general capabilities of self-management, teamwork, intercultural understanding, ethical behavior and social competence are all relevant to the pedagogy used by teachers of mathematics.

Summary

This last of the three lessons we believe illustrates our general argument. If teachers wish to create rich and balanced mathematics experiences for their students, they need command of a pool of design features. These features may be expressed intuitively, but are backed up by the imagery of their successful use in other lessons as well as supported by current research on effective teaching and prevailing educational policies.

Our design features and their relationship to Pedagogical Content Knowledge (PCK)

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When we asked teachers in workshop settings to identify features of interest within the three lessons described above, and which in their opinion contributed to the quality of the learning experience, the descriptors they chose were as follows: The features chosen are expressed intuitively, informally and are born out of teacher conversations. We wanted the words used to be ‘teacher words’, generated by their observations and experiences and not laden with jargon. When used in discussion, the practical classroom meaning of the (design) features are illuminated by the context (i.e., lessons) in which they appear and later mediated through a background learning theory lens or a system policy lens.

The three lessons and features identified by teachers

Lesson 1	Lesson 2	Lesson 3
Temperature Graphs	Maths in Motion	Maths of Lotto
Context based Challenge Thinking, reasoning, problem solving Group work Personal Technology Ownership Skill Mixed ability – multiple levels of success Open-ended	Active / Kinesthetic Group work Storyshells Challenge, Problem solving Creativity Ownership Multiple interconnected content	Social Issue context (community gambling) Mathematical Investigative process A class game context Strategy (solve a simpler problem) Estimation First principles approach Technology First hand data for analysis, Ownership Multiple content Mixed ability

All the elements from these and other ‘lighthouse’ lessons aggregate into a single portfolio or personal design repertoire. A teacher’s confidence in selecting from these to enrich new lessons of their own design is a function of personal experience, in seeing them used successfully within a pool or library of lighthouse classroom lessons.

The teacher is truly an educational designer.

References

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