Two years ago now we began to question the relevance of paper and pencil, worksheets and drill and kill methods in the development of foundational basic number facts and understandings. We began by testing a few mental activities that involved the whole class simultaneously in fun and relevant activities and as we did so we observed that when children are engaged in mental activities certain conditions need to be present to make them have maximum benefits. These are as follows:

1. Concrete materials used as tools by children.
2. Feedback is immediate and involves sharing and discussing strategies as well as showing equal respect for them all.
3. Errors are seen as learning opportunities for all.
4. Questions provide success for all as well as challenges for others.
5. All children need to be engaged at their own level during the process.
6. Children see themselves as a community of learners where everyone has a role to play in the development of thinking and learning.

It was with these criteria in mind that we began to explore the potential of mental routines, as we have chosen to call them.

About Mental Routines

The purpose of mental routines is to develop useful strategies that will lead to mastery and a solid foundation in basic maths concepts. Mental strategies as far as possible should relate to the methods that children develop intuitively and within their own culture. They should also relate as far as possible to the ways in which those strategies are applied in the real world.

"This means that mathematics instruction must use contexts and pedagogies that allow students to use their own cultural, ethnic, and gender preferences and approaches".

(Ladson-Billings, 1994)

When we refer to the conditions that need to be present for the effective development of mental strategies we see that this view is clearly reflected.

Concrete Materials

It may seem like an oxymoron to say that hands on materials could be used as part of mental routines but let's explore this idea a little further. It is our belief that mental maths can involve the use of concrete materials and does not have to be totally abstract. For instance when children are first becoming familiar with doubles and near doubles (doubles plus or minus one) the use of materials such as Unifix cubes can become a mental routine. The children can be asked closed questions such as show me a near double that makes seven. The children may hold up a stack of three and a stack of four to show double three plus one or they may hold up two stacks of four and then snap off one Unifix cube to show double four take one. The beauty of this is that the children can actually see the blocks and show how they match the strategies. This means that language and visual imagery are combined to chunk the information into a meaningful whole.

"So there are two points to keep in mind as we discuss the uses of tools: First, meaning is not inherent in the tool; students construct meaning for it. Second, meaning developed for tools and meaning developed with tools both result from actively using tools. Teachers don't need to provide long demonstrations before allowing students to use tools; teachers just need to be aware that when students are using tools they are working on two fronts simultaneously: what the tool means and how it can be used effectively to understand something else."

(Heibert 1997)

Feedback

Feedback should be immediate and useful and should create a win-win situation for all rather than the competitive win-lose situation that so many students are familiar with. By this we mean that there is no place for the over learning of number facts or for the stressful learning and testing practices that often typify mental arithmetic.

The intention is to replace this with a situation where children share their solutions and strategies, where they consider the benefits of different approaches to something as simple as 4 plus 7 and in so doing  receive valuable feedback as they hear comments such as the following:

"I did my rainbow fact to 10 and then added 1 on"

"I took 1 from the 7and put it with the 4 and knew I had to double 5 plus 1."

"I did a turnaround and counted on from 7 until I got to 11".

"I just knew that fact".

"I thought it was 10 because I took 1 from 7 and gave it to the 4 but then I forgot the extra 1 that was left over".

Teacher feedback is barely necessary, is it? The last example shows that the child, while listening to the others realised that an error had been made and wondered how and where. There was no embarrassment about the error just a willingness to share with others a common error in such methods. The final questions from the teacher enhances this feedback by requiring the children to reflect on what was said as they answer questions such as:

"Which strategy did you think worked best in this example and why?"

"Which strategy would you like to explore a little further?"

"When we do another one will you stick with the strategy you just used or try a different one?"

Through this process the children have been engaged in the feedback process and have been asked to self-reflect and use that reflection as a planning tool ready for next time.

Errors as Learning Opportunities

It appears that there is little value in participating in mental activities which are too easy or which are already well developed. Activities, then, need to be just on the edge of children's' comfort zones, scaffolding them to the next level. If this is the case then obviously errors in computation are going to occur from time to time. We found at first when we worked in this way children would be derisive and snicker at errors. We also found that some children would not have a go for fear of failure. It was interesting however to see how quickly this mindset turned around. The children seemed to embrace the idea of using errors as learning opportunities and were often heard saying, "Oh good a learning opportunity". Very often as a child is explaining their strategy they notice their own errors and are keen to fix them up on the spot. Other times though the error is not noticed, for instance, recently a 6-year old who had responded to a reading of the Very Hungry Caterpillar by drawing the foods eaten during the week, was showing how she had drawn two rows of things that the Hungry Caterpillar had eaten during the week. She counted in twos but only touched one thing at a time. She was unperturbed by her error: she had counted by ones earlier and written the number 26 on her paper. One of the other children was very impressed by her ability to count so fast and so far by twos, but another asked her why she only touched one thing at a time. She stood perplexed looking at her page and asked the other child to come up and show what she meant. She laughed out loud when she saw what she should have done and proceeded to repeat her counting correctly this time.

"We recall when children have wanted to share mistakes; after presenting their solutions, they explain their errors. The children also become very supportive of one another and understand that errors are a natural part of doing mathematics,. Errors often can lead to new understandings about the concept".

(Trafton, P. and Thiessen, D. 1999).

Questions for Success

There are children at a variety of levels in every classroom and they all need to have some success and some challenge. When planning mental activities it is essential to build in closed, open and reflective questions at three levels as will be explained in more detail later. What this means is that questions can target certain children with specific needs. We also use strategies, outlined later, whereby the children can show their answers in a variety of ways rather than having to give them orally. This keeps the children engaged without drawing attention to individuals unnecessarily.

Engagement: The ten or so minutes set aside for mental routines are fast and pacey. They may involve concrete materials, number cards, 100 squares, dice, bottle tops etc. The children engage with the activities because they are different to the rest of the lesson. When we first began exploring the mental activities that we suggest here we had no idea how much fun and of course learning would flow from them. We soon realised that we didn't need to make up a new mental activity every day because the nature of the tasks and the children's interest in them meant that they could be used and easily changed for several days, hence the term mental routines. We now use the routines for several consecutive days, all the time watching to see the level of engagement and of course we switch to a new routine if we think the interest is dwindling.

Language: As we introduce each routine we use the meta-language of the strategies or process that go with it. At first it was our intention to simply immerse the children in the meta-language but the children were so captivated by words, such as, subitize and unitise, that they soon wanted to use them too. Watching the children engage with the activities has been rewarding for us and for them too. When the children are having fun and are engaged they seem to be hungry for more. We have seen even the switched off learners reengage through the mental routines.

Community of Learners

 To gain the most from these activities the students do need to become a community of learners. They need to really listen to the ideas of others. To give positive feedback, ask questions, make suggestions and comparisons and finally to evaluate the strategies presented by others. They need to feel safe to take a risk, present their ideas and to comment on the ideas of others. They need to learn to justify their viewpoints and stick with them. For instance, if after hearing how near doubles can be used for an addition a child still prefers a count on strategy then they should be able to explain why they prefer it. And at the end of the day if the response is, "Because I know it always gives me a correct answer". Then that justification has to be seen by all as valid for that child at that time and as such should be respected.

"Learning to be a member of a mathematical community means taking ownership of the goals and accepting the norms of social interaction. Why is it important that classrooms become mathematical communities and that all students participate? Because such communities provide rich environments for developing deep understandings of mathematics."

 (Heibert, J. et al 1997).

Bibliography

Ladson-Billings, Gloria. The Dreamkeepers: Successful Teachers of Afro American Children. San Francisco: Josey-Bass, 1994.

Heibert, James et al. Making Sense: Teaching and Learning mathematics with Understanding: Heinemann, Portsmouth, NH. 1997

Trafton, Paul, R. and Thiessen, Diane. Learning Through Problems: Number Sense and Computational Strategies. Heinemann, NH, 1999