Research Background

Implications of recent brain research for teaching mathematics

The quantity of current brain research is formidable and has incredible implications for instruction. Still, it is important to understand that the nature of change is so fast that anything said may in fact need to be adjusted tomorrow. We are in a tremendous state of flux. Still, it is worth outlining a few things, which seem to be supported by past practice and by common sense.

Mind and body are one, so we need to provide lots of opportunities for movement. When you move the brain re-engages; we need to move often or we become distracted. We can only sit for about the same number of minutes as our age without drifting off. Most of us never get over age twenty in this respect. Movement prepares the body for upcoming learning and helps boost self-confidence because movement helps create an attitude that learning is fun.

A good example of a simple movement activity is the Every Pupil Response Technique mentioned earlier. Sometimes we just need a physical break to help us refocus. There are many brain gym activities such as "Simon Says" or the "Cook’s Kitchen" (cross over your arms and join hands reversed, then bring your hands toward your face, then reverse the motion) which require movement and cause the brain to cross over the corpus callosum thus stimulating us into using both sides of our brain. We each move differently and have different movement experiences. Each brain is unique. Likewise, students must make connections to their own life and express ideas in their own words. Teachers should always ask if students can word an answer in another way or if anyone worded their answer differently. When the teacher stops after one correct response they give the impression that the method used is the only correct method when in fact there many be many correct methods. Always ask, "Can anyone do it (or say it) another way?"

A good way to make something your own is to talk about it. Students can turn to a partner and have a one-minute discussion about their understanding of a concept, and then it becomes their own learning. This is an excellent tool for problem solving. Reinhart (2000, p. 482) refers to this as the "think-pair-share" strategy. It is very important to work independently first or some students simply wait for someone else to do the work or take the risks. Sharing in pairs, then quads and even octets can expose learners to new strategies in less threatening, more open environments. This is especially true as children get older and is crucial in adolescence, where an idea from a peer is exponentially more acceptable than an idea from an adult. Peer feedback is very important because it usually poses less threat.

Feedback is crucial to learning. Most teacher feedback is too late, too vague, too little and lacks emotion. More importantly, teachers cannot provide enough feedback so they need to create opportunities for peer feedback and personal feedback. This can be done through the "think-pair share strategy" mentioned above and also through reflection journals, making up your own questions, peer tutoring, role playing, using parent volunteers, utilizing partners, creating a buddy class, using the thumbs up, thumbs down feedback systems, instituting a 1-2-3-4-5 voting systems, marking your own work, implementing student-led conferences, creating memorable bulletin boards which are changed regularly, teaching paraphrasing skills, and encouraging brainstorming.

We not only need to work with different people in different ways, we need to do different things in different parts of the room. This creates several locations for the episodic memory thus making it easier to remember where we learned a concept or process. For example, if we do the weekly graph on the side board instead of the front board then children will be able to visualize the material better because it won’t be meshed with everything we learn at the front board. If we teach certain multiplication tables in the gym or outside on the playground then the memory of those facts will be easier to retrieve. The teacher can say, "Go down to the gym in your head if you are trying to remember the three times table. What were we doing? Bouncing balls. That is right. Now do you remember how to do the threes?"

Challenge is crucial to learning so there must be opportunities for choice. The teacher or the parent simply cannot always know what is best for the learner. Choice, control and commitment often go together and can include ways of representing the learning, choosing the amount of time and the time of day.

Much of our learning is non-conscious (up to ninety-nine percent). Ritual is important because it creates a sense of emotional comfort and a feeling of predictability, thus a sense of control. Novelty is just as important, if everything is taught in the same way or the same place, we are lulled into not noticing or remembering the significant parts of the learning experience. Teaching everything in the same part of the room such as the front board causes different concepts to run together and be indistinguishable. A good example of this is teaching "two, to and too" or "there, their, and they’re" in the same lesson. The mathematical equivalent is to start a lesson by defining all the terms, many of which are totally outside the learner’s experience or often have different meanings in everyday, non-mathematical life. Examples include words like edge and side, acute and obtuse, parallel and perpendicular, or odd and even. These words have non-mathematical meanings that interfere with their use in math class and when they are all defined at the beginning of the lessons students get them confused with their everyday meanings, which are the first connections that form in their minds.

Attention is a function of motivation and novelty. The brain loves novelty and motion, which is why our eyes often move to the flickering television screen. Motivation and concentration come when we make meaning. When we see that what we are doing is relevant and we can relate it to a context we are better able to extract the patterns. If students can maintain attention let them work for longer periods of time. We need to spend longer on fewer, more complex and thus more interesting projects.

Emotion is crucial to learning because it is integral to the decision-making process. When we are making a decision we are using and connecting many different parts of the brain. Emotion makes the learning personally real, triggers our values and activates long-term memory. It helps us make faster decisions and better long-term decisions. We facilitate this through tone, non-threatening posture, literature, music, visual peripherals, role playing, pictures, photographs, smells, appropriate deadlines, modeling, storytelling, sports, movement, feedback, touching, writing, making connections to family, celebrations, good lighting, drama, costumes, props, and setting appropriate priorities.

Music increases cortical firing, it boosts learning because it can act as a carrier for words and other messages. Some Asian languages make rhymes and songs of the multiplication facts making them easier to learn. Remember the alphabet song. Different kinds of music, especially baroque music, seem to prepare the mind for learning. It is important to experiment with the use of music during math class – my experience is that it has a positive effect on attention with some classes and not with others.

The brain likes complexity and often works better from the whole to the part but most mathematics learning in traditional textbooks is presented from the simple to the complex. Some students see the parts only after they see the whole. Sometimes students can learn the whole complex process or concept in one sitting when it is presented holistically, which is usually from a context of meaning. The whole is associated with meaning and is often what is motivating the student to learn. When students are motivated to learn something then they are more likely to understand why it is important to learn the process in smaller steps.

Introducing a concept through a problem can be initially frustrating but it is often more interesting and motivating. When children are interested in the problem because it is relevant or when they see the problem as something to play with, then they are not as stressed and are less likely to downshift. (Caine and Caine, 1991, p.63) Students need to learn the strategies for overcoming frustration and they need to learn perseverance – these two things are much more learnable if the students are not threatened. Practice in overcoming frustration leads to a sense of control and subsequently to an ability to handle ambiguity and uncertainty. This leads to ability to delay gratification – a skill much needed in today’s society. Part to whole teaching doesn’t do this very effectively because it often lacks relevance and appropriate levels of challenge. Part to whole teaching in traditional textbooks gives the unwritten message that the student has permission to forget the material as soon as she is finished the chapter review and chapter test. The problem is that many children demand this kind of teaching because they appreciate the feeling of adult control and prefer to avoid ambiguity. By giving it to them we solve the short-term problem and increase the long-term dependence.

Some students become very threatened by mathematics. Timed tests are threatening if the students have poor fine motor control or if the students do not already know the facts being tested. Students can feel threatened because they are moving through a concept too quickly or because they think the question or problem is too challenging. Some students shut down and demonstrate a form of learned helplessness; they simply give up. The solution is not to give more timed tests but to stop timing all together until the student understands the material and/or chooses to be timed.

Every brain is unique so our evaluation should include choice wherever possible. Students who participate in the creation of criteria for evaluation are more committed to the product and understand the evaluation, making it doubly effective. Choice also implies variety; this could include tests, portfolios, journals, interviews, and observations. Evaluation should include different forms of representing learning such as oral, written, and visual. Learning is also social so some assessment of group processes and products is necessary.

Evaluation of problem solving must include evaluation of the process. Students need practice and feedback on getting unstuck, doing the problem in more than one way and developing their intuition about when they have made an error. Self-evaluation can be very effective for increasing student awareness of processes. Journals work well for this.

Memory is often episodic so students will generally do better on tests if they are being tested in the same place they learned the material. This is very true of high stakes testing such as government exams where the tension can be very high and students very prone to downshifting and learned helplessness. If it is not possible to write the test where the material was learned then students need to practice writing the test in the place where the test will occur. Students also need practice in the format of the test or the test does not measure their knowledge of the material but rather measures their familiarity with the process and the format. Brains operate in cycles so it is better to rotate times when tests or subjects are taught keeping in mind that routine also reduces stress.

(The above excerpt is taken from Mathematics as a Teachable Moment, by Trevor Calkins.)