/ 28 January 2011

Making a good start

A new year means the possibility of a fresh start. As you start 2011, you will have a new class with children you have not taught before. They don’t know what to expect from you and you don’t know much about each child and how he or she is coping with mathematics. The fresh year gives you the opportunity to establish the classroom culture anew and the opportunity to get to know each child in your class.

You need to learn the names of the children, their special needs and, most importantly, their developmental stages with respect to mathematics. A well thought through start will pay handsome dividends for the rest of year.

The culture of the mathematics classroom defines the way in which maths is learned and experienced. Your sense of mathematics also shapes the culture. Is yours a classroom in which mathematics is regarded as the memorisation of the facts, rules, formulas and procedures needed to determine the answers to questions? Or is yours a classroom in which mathematics is regarded as a meaningful, sense-making, problem-solving activity?

If you regard mathematics as only the memorisation of the facts, rules, formulas and procedures needed to determine the answers to questions, then you need to create a classroom culture in which children expect you to demonstrate how things are done, and they see their role as remembering how to do what you have demonstrated. In this kind of classroom, the teacher decides whether answers are correct and children accept the verdict of the teacher. Children learn that they don’t have to understand how or why things work — only that they do work. Children also learn that what is most valued in the class is the correct answer.

In this kind of classroom, you would do well to start the year with a test using the kinds of question you expect children coming into this grade to be able to answer. You should mark the test and rank the children from weakest to strongest and then set about covering the curriculum.

If you read my columns in 2010, you would know that the kind of classroom I have just described is not one in which I believe children can learn mathematics effectively. It is in the kind of classroom I have just described that children develop anxieties and fears about mathematics. My concern with the classroom described above is that it stifles the child. The classroom culture robs the child of being able to think for himself or herself and shows no regard for a natural ability for and interest in solving problems.

My ideal mathematics classrooms are those in which children experience mathematics as a meaningful, sense-making, problem-solving activity; classrooms in which children have opinions and their opinions are valued; classrooms in which mathematics can be understood, appreciated and enjoyed.

So how does a teacher go about establishing such a classroom? The book Adding it Up (Helping Children Learn Mathematics) provides some useful suggestions and answers to this question. It can be downloaded at no charge or via the link in the downloads page at: www.brombacher.co.za.

In Adding it Up (Helping Children Learn Mathematics) the authors describe what they call “mathematical proficiency”. In particular, they describe five different strands or aspects to being mathematically proficient. If you are concerned about creating a mathematics classroom that supports mathematical proficiency then you may want to use these strands as a guide to creating the classroom culture.

As you create the classroom culture you should be concerned with children understanding what they are doing. In your class it is not enough that children choose the correct operation and produce the correct answer. You also want them to know what they are doing and why they are doing it. You will ask questions such as: “Can you explain why you did what you did?” and/or “Why do you think that will work?”

You should be concerned with children being able to apply what they have learned to solving unfamiliar problems. In your class, you pose problems expecting that children can make a range of plans to solve them. You have faith in children’s ability to develop a strategy and give them time to do so.

This last point is important. In many classrooms teachers pose problems and get impatient or want to “help” the children and so they give the children hints or even solve the problem for the children. Children quickly learn that the teacher will solve the problem anyway, so they stop trying.

You should expect children to reason about what they have done. In your class, children are expected to reflect on their solution, they are expected to decide whether the answer is reasonable or not and to explore ways of testing the validity of their solutions. They are expected to be able to convince others about their solutions, justifying their explanation using logical argument. They are expected to make mistakes and to learn from them. Mistakes are valued as lessons and are not regarded as signs of weakness or incompetence.

It goes without saying that you expect children to be able to compute. You expect them to know the number facts and calculation strategies that are appropriate to their grade and you expect them to know them fluently. The point is not that children shouldn’t have strong mental arithmetic skills and be able to perform the calculations that are expected at their grade level; it is simply that this alone is not enough.

Finally, you need to create an environment in which children want to engage, an environment in which children expect to work hard, to take risks and to struggle. You create this environment by modelling such behaviour yourself and by encouraging such behaviour in the children.

The start of the new year offers you a wonderful opportunity to begin with the establishment of the culture described above. How better to do so than by means of a well-chosen problem. Maybe even an investigation. Investigations provide you with a window into the minds of the children in your class. They reveal their problem-solving strategies, their attitudes and even their mathematical knowledge and skills.

In concluding today’s column, I would like to suggest an investigation that you might like to try in your class. I have suggested it with great success to teachers participating in courses I have presented who are dealing with children from the Intermediate Phase.

What is the greatest product that can be made from the numbers that add up to a given number?
Let me get you started. Consider the number 8.
8 = 1 + 2 + 3 + 2 and 1 + 2 + 3 x 2 = 12
8 = 5 + 3 and 5 x 3 = 15

Now that you understand the question, we can ask:

  • Is there a bigger product?
  • What about the numbers 9 and 10?
  • In my next column I will discuss a solution to this question and talk about how different children of different ages might respond.

    Aarnout Brombacher is a private maths consultant. For more information go to www.brombacher.co.za