What are the ingredients for an effective mathematics lesson? Teachers are continually faced with a range of advice or ideas to improve their mathematics lessons and often this just creates confusion. It’s a little bit like being a cook. New recipes appear online and in cookbooks on bookstore shelves, but often they’re just adaptations of classic recipes that have been around before, their foundation ingredients are tried and tested, and often evidence based. There are always the staple ingredients and methods that are required for the meal to be successful.
The following is a list of what I consider to be important ingredients when planning and teaching an effective mathematics lesson. The list (or recipe) is split into two parts: lesson planning and lesson structure.
- Be clear about your goal. What exactly do you want your students to learn in this lesson? How are you going to integrate mathematical content with mathematical processes? (The proficiencies or Working Mathematically components) Will you consider the General Capabilities in your planning?
- Know the mathematics. If you don’t have a deep understanding of the mathematics or how students learn that aspect of mathematics, how can you teach it effectively? Where does the mathematics link across the various strands within the mathematics curriculum?
- Choose good resources. Whether they are digital or concrete materials, make sure they are the right ones for the job. Are they going to enhance students’ learning, or will they cause confusion? Be very critical about the resources you use, and don’t use them just because you have them available to you!
- Select appropriate and purposeful tasks. Is it better to have one or two rich tasks or problems, or pages of worksheets that involve lots of repetition? Hopefully you’ve selected the first option – it is better to have fewer, high quality tasks rather than the traditional worksheet or text book page. You also need to select tasks that are going to promote lots of thinking and discussion.
- Less is more. We often overestimate what students will be able to do in the length one lesson. We need to make sure students have time to think, so don’t cram in too many activities.
- You don’t have to start and finish a task in one lesson. Don’t feel that every lesson needs to be self-contained. Children (and adults) often need time to work on complex problems and tasks – asking students to begin and end a task within a short period of time often doesn’t give them time to become deeply engaged in the mathematics. Mathematics is not a race!
- Begin with a hook. How are you going to engage your students to ensure their brains are switched on and ready to think mathematically from the start of each lesson? There are lots of ways to get students hooked into the lesson, and it’s a good idea to change the type of hook you use to avoid boredom. Things like mathematically interesting photographs, YouTube clips, problems, newspaper articles or even a strategy such as number busting are all good strategies.
- Introduction: Make links to prior learning. Ensure you make some links to mathematics content or processes from prior learning – this will make the lesson more meaningful for students and will reassure anxious students. Use this time to find out what students recall about the particular topic – avoid being the focus of attention and share the lesson with students. Talk about why the topic of the lesson is important – where else does it link within the curriculum, and beyond, into real life?
- Make your intentions clear. Let students know what they’re doing why they’re doing it. How and where is knowing this mathematics going to help them?
- Body: This is a good time for some collaboration, problem solving and mathematical investigation. It’s a time to get students to apply what they know, and make links to prior learning and across the mathematics curriculum. This is also a time to be providing differentiation to ensure all student needs are addressed.
- Closure: This is probably the most important time in any mathematics lesson. You must always include reflection. This provides an opportunity for students to think deeply about what they have learned, to make connections, and to pose questions. It’s also a powerful way for you, the teacher, to collect important evidence of learning. Reflection can be individual, in groups, and can be oral or written. It doesn’t matter, as long as it happens every single lesson.
There are many variables to the ingredients for a good mathematics lesson, but most importantly, know what and how you are teaching, provide opportunities for all students to achieve success, and be enthusiastic and passionate about mathematics!
In several of my previous posts I discussed the importance of promoting critical thinking in mathematics teaching and learning. I’ve also discussed at length various ways to contextualise mathematics to provide opportunities for students to apply prior learning, build on concepts, and recognise the relevance of mathematics in our world. In addition, investigations provide excellent assessment material – usually when we assess in mathematics we ask for specific answers. In investigations, students can show us a range of mathematics, often beyond our expectations. They are also a great way to integrate other subjects areas such as literacy and science.
In this blog post I am going to share some ideas for open ended and inquiry-based mathematical tasks based on two items that most students would be familiar with – beach towels and pencil cases!
Let’s start with pencil cases. It’s the start of the 2018 school year next week and many children begin each school year with brand new stationery, in brand new pencil cases. Even if they’re not brand new, most children have a pencil case. I came across an interesting article relating to pencil cases a few days ago, and I think this could be used to spark interest and curiosity. The article can be found here:
- Who has the heaviest pencil case? Compare the mass of your pencil case with the pencil cases of your group members. Who has the lightest? Estimate the mass, then use scales to test your estimations. How close were the estimations?
- Estimate, then calculate the surface area of your pencil case. What units are the most appropriate to use? Explain how you measured the surface area.
- Faber Castell is a famous brand of pencils. Investigate the history of Faber Castell and illustrate this on a timeline.
- According to the Faber Castell website, it takes one ‘pinus caribaea’ tree 14 years to be ready to be used to manufacture pencils. Each tree can produce 2500 pencils. If one tree was allocated to each school, how many pencils do you think each child in your school might receive? How did you work this out?
- If each of the 2,500 pencils were sold for $1.50, how much do you think the entire tree be worth in pencil sales?
- At the beginning of each school year many children get brand new pens and pencils to take to school. Investigate how much it would cost to buy your stationary. Which shop offers the best value for money?
- Some pencil cases like the one in the photo and in the Missing Letter article have small clear plastic pockets to put your name in. If a pencil case has only eight pockets, is this enough for your name? Investigate the length of names in your class. What would be the average length name in your class? What else could you explore about names?
- The pencil case in the picture came with some pre-printed letters for the clear pockets. There are more of some letters than others. Investigate the most common letter occurring in students’ Christian names. Do you think it would be the same in all countries?
- Design and make a pencil case to suit your individual stationery needs. Write about the mathematics you use to do this.
- Design a new and improved pencil and explain the changes you have made.
- Design, justify, and create a marketing campaign for a new, ‘miracle’ pen.
- Research and discuss the following statement: “To save the environment, wooden pencils will no longer be manufactured”.
Promoting Curiosity and Wonder
Mathematical investigations should promote curiosity and wonder. The pencil case questions and investigations are open, yet provide some structure and support. They give enough detail to communicate the type of mathematics required to complete the task or investigation. Students should eventually be able to feel confident enough to come up with their own questions and follow their own path in terms of the mathematics they access and apply, just like mathematicians do.
Round Beach Towels?
In the last year or two a new beach towel has emerged onto the beach towel scene. It’s round. Now this idea immediately caused some concern for my mathematical brain. I had questions.
- Is there more fabric in a round beach towel than a regular, rectangular beach towel?
- Is there more fringe, and wouldn’t this make the towel more expensive?
- How does one fold a round beach towel?
- Could you wrap a round beach towel around you the way you wrap a rectangular beach towel?
- How much more area on the beach gets taken up by people spreading round beach towels?
- Does this mean less people get to lay on the sand?
- Could you design a round beach towel that has a tessellating pattern?
All of the questions above can be explored using a range of mathematics…I wonder how many more questions your students could come up with?