The term rich in relation to Mathematics lesson usually have these following features: draws on a range of important mathematical content; engaging for students; all students are able to make a start, as it caters for a range of levels of understanding; successfully undertaken using a range of methods or approaches; provides a measure of choice or openness, leading to a sense of student ownership; involves students actively in their own learning; shows the way in which mathematics can help to make sense of the world; makes appropriate and effective use of technology; allows students to show connections they are able to make between the concepts they have learned; draws the attention of students to important aspects of mathematical activity; and helps teachers to decide what specific help students may require in the relevant content areas, or ways in which students might be extended. Then, balanced means that those feature work in harmony, mutually self-supportive and not over or under weight in any aspect.
Furthermore, there were major concerns with the teaching of mathematics in Australia, particularly among teachers in the middle years of schooling (Grades 5-8) in the 19880s, such as: Mathematics 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; Mathematics 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.
One of ways as the basis of professional learning conversations is the using of classroom lesson. Each teacher comes to any learning situation with their personal current practice or ‘comfort zone’, such as: group investigations and problem solving, story-shell frameworks, mental arithmetic, application, computers, estimation, equity, visual imagery, iteration and numerical methods, concrete materials, social issues, concept learning, physical/outdoors, Mathematical modeling, writing mathematics, and provide non-threatening learning environment.
More importantly, there are three lessons explained in this journal, they are: Temperature Graphs, Math in Motions, and Maths of Lotto.
- Temperature Graphs
This lesson is recommended for students in grade 5 to 10. The additional features which teachers often identify during discussion include the following:
- Context based. 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.
- Technology. It can be 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.
- 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. On the software there is the option to choose just two cities, then the teacher can ask to the students suitable with their horizon.
- Open-ended. An important aspect of being able to go further and in more depth.
2. Math in Motions
This lesson targets children in grade level Kindergarten. The additional features which teachers often identify during discussion include the following:
- Active / physical / personal. All of the lesson could be done with 10 small numbers cut from cardboard, with all students working individually at their desk, but the teachers reported it was far more memorable and involving with students physically ‘becoming the numbers’.
- Group work. Working as a group to solve the challenges increased both cooperation and communication.
- Storyshell contexts. The ‘fantasy’ stories about ducklings and other scenarios greatly increased student interest and involvement, and led to greater engagement with the mathematical content ideas.
- Challenge, problem solving. The challenges clearly needed the students to think and reason as well as learn number skills and facts.
- 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.
3. Maths of Lotto
- Social Issue context. The social context of gambling and the problem this brings combines obvious opportunities to link Mathematics with moral and ethical challenges.
- A class game. 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.
- Mathematical investigative process. A mathematician go about investigating a context, such as: find an interesting, meaningful or worthwhile problem; informally explore, experiment, collect data; create conjecture, hypotheses, theories from pattern; use toolbox of problem solving strategies to prove or disapprove theories; use toolbox of basic skills; extend or generalize what else that can be learned; then communicate it.
- Strategy. There are some problem solving strategies, such as: working backwards, act it out, see a similar problem, make a model, guess and check, make a table, seek exception, test all possible, break problem into manageable part, solve a simpler problem, and draw a diagram.
- First principles approach. When working out the chances of winning game, the teacher reported greater understanding is developed by writing down all the pairs from a first principles approach than using algorithmic formula from combination theory.
- Technology. The software supports the investigation in important ways, and allows the transfer from the simple game towards the more commonly used community context.
- First hand data for analysis. Playing the game produces data and hence is owned by the students.
- Multiple content. There are many interconnected big ideas of probability and statistics all being developed concurrently.
- Mixed ability. Teachers strongly stated that all students were engaged and involved and all felt they had learned something worthwhile, even though this is different for different students.
Lovitt, C., Clarke, D. 2011. A Designer Speaks. The Features of a Rich and Balanced Mathematics Lesson: Teacher as Designer. Journal of the International Society for Design and Development in Education.