Take genuine delight in the learning of your student. This is your most powerful tool. There is much that is delightful in good thinking, a new solution, or an insightful comment. There is much danger for us adults when we come to think that we have seen it all, particularly when working on seemingly simple, 'childish' problem. Often so intent are we on getting the student to see things our way that we miss the new perspective uttered by the child seeing the world for the first time. Take delight in those new perspectives...they are a gift to you. Often these perspectives will be simple and obvious yet profound in their truth. Be careful that you don't cast them aside with a 'but of course.'
Be aware enough to separate a good thought path from the correctness of an answer. A child [in the process of] explaining will often find a good path but follow it poorly. The original idea may be a good one even if it does not produce the correct solution. Instead of saying 'No, that is not right,' honor the attempt. 'That was a good idea; I understand what you think even though it doesn't work here. It was a good try.' Or ask questions that will test the hypothesis in other ways. 'Are there other cases where this will work? Can you think of problems where it won't work?'
Be aware also that a child in a comfortable learning situation rarely says nonsensical things. Even an answer which may seem absurd to you makes sense to the child. Try to understand the model the child is using, the model that will make sense of what the child says. When you listen carefully to what the child is struggling to articulate, you show that you value his thoughts by your behavior. Your act of valuing and delighting in your child's thoughts will go many miles further than any reprimand you could give. You will, by your praise, cultivate a desire to search and test, and this will strengthen them against the frustration of error. The child will learn that thinking is fun.
If delight is your greatest asset, then impatience is your greatest liability. Impatient teaching is humiliating to the learner. One aspect of impatience is expecting children to give snap answers to questions. The better the question the more time a child will need to mull it over . You must allow the child time to think. Silent times in my teaching have been as long as three to five minutes. Give children time to think."
From Guidelines to the Lessons Patterns in Arithmetic
Free Exploration or How to Keep from Being a Nag
by Alysia Krafel Top of Page
I recall an incident years ago when my six year old nephew was learning to set the table. Dinner was almost ready and there he was popping his fists down onto fork tines flipping them into the air. The stage was set for an unpleasant confrontation. I see now that the problem is one of time frames. We needed him to get the job done, he needed to explore. When we press a child to use a tool seriously and with skill before we have provided fiddle time, we set him up to be seen as a behavior problem and ourselves to be seen as nags. If a child can do his necessary playing and exploring with materials when he is not under pressure or expectations, before he needs to use them as we wish him to use them, not only will he learn the new skill faster and with more understanding, he will learn it with more delight (or at least with less complaining). The first association will be of joy and competence rather than frustration and forced anxiety.
Learning to use math manipulatives is like learning to use other tools. An adult will look at a screwdriver and use it to drive screws. A child will roll it on the floor, listening intently as the grooves in the plastic handle thump. She will roll it over her tongue, poke it into cracks, bang it on the table, spin it or balance it on her finger. When finally satisfied with the investigation, she will watch you screw in the screws and then try it herself. And so it is with all of our doodads.
Play, unhindered exploration, is very important to children. At the Farm School, we came to realize that this self-directed learning style was not an impediment but a boon to our aims. We found that children who had enough time to fiddle with the plexi-glass tiles came to their first fractions lessons knowing that the smaller the pieces there were, the more of them they had; that you could cover the black (1/2) tile* with the yellow (1/4) tile or orange (1/6) pieces but you couldn't with the clear ones (1/3). When they began to work with the formal fractions lessons, they brought into action all that their senses already knew about these tiles. As soon as the labels "halves" and "fourths" were attached to the colored tiles, they knew that 1/2 = 2/4. They found the patterns very quickly and mastered the materials faster than the previous children who had not been allowed fiddle time.
The problem the teachers at the Farm School had encountered was that we wanted to teach x amount of material in a certain time frame. When the children needed three weeks of fiddle time with the fraction tiles, we experienced that as a delay, a delay that made us nervous. We just had to teach and have the children master fractions by the end of the school term. The solution was to introduce the materials long, long before (in many cases years before) we intended to use them as teaching tools. After ample free exploration time, not only did the children not resist the use of the materials in the specific way that the formal lessons demanded, they eagerly attended to the new way to interact with the now familiar materials. The new work was for them a continuation of the old play. The experience did not feel all that different to them. Since the struggle between teacher and child had been eased, the adversarial relationship became a co-worker relationship. The instruction then proceeded easily. The children and the teachers learned together to truly understand the world of numbers and enjoyed doing it.
My life suddenly changed when I had a baby. When giving her a bath one day, I had the idea to give her as a toy one thing she would later use as a tool, a cup. In the tub, she could pour, drink, (or sputter), spill, bang and splash to her heart's content without undoing whatever housework managed to get done. She would talk into the cup and put it on her head. Doing these things at the table with a cup filled with milk would have caused an uproar just as fiddling with colored tiles during a fraction lesson did. She had had the cup as a toy many months before we ever put milk into it for her to drink at the table.
Interestingly enough, when given milk in the cup at the table, she reacted in the same way as the children in the fractions lesson did. Since pouring liquid from the cup was old hat and drinking milk from it like Mommy and Daddy was new, she tried very hard to do it right because that's what she was ready to do, that was the new fun. She wouldn't pour milk out at the table and say "oh no" when she spilled it. Later, when she showed an interest, spoons and bowls were added to the toy shelf and to the bath set. While in the tub recently, she filled her bowl with water, bent down and began to blow bubbles into it. She dumped out the water and put the bowl on her head and said, "Hat." I thought to myself, "I'm glad that was warm water on her naked body instead of hot tomato soup on her white sweater!" An unintended side effect of all of this was that she became competent with a cup at 14 months and with a spoon and bowl by 16.
Having done all of that, I was surprised recently to find myself remaking that same mistake of not allowing exploration as a parent as I had as a beginning teacher. When my daughter put eggs and Cheerios in her cup of milk I said, "No put food in cup! No put eggs in milk!" I was expecting her to be like me, to use tools as tools, food as food, instead of tools and food as toys. Resisting my constant scolding, she persisted in this behavior. Obviously, she needed to explore how things, like scrambled eggs and Cherrios, would interact with milk in her cup. My choice was either to press her to "act properly at the table," triggering tears and tantrums, or to allow her to explore with the knowledge (or at least great hope) that when she was satisfied, she would stop. This is always a difficult decision for me to make. One doesn't want to be so permissive as to develop a totally undisciplined child. On the other hand, one doesn't want to be a repressive nag either. Since food couldn't be a toy anywhere else but at the table, I gave in on unconventional mixing, squashing food through fingers and food painting but drew the line on throwing. So with a certain amount of plaintive sighing, I watched her put my lovingly prepared, warm eggs into her milk, got out the dish towels, and added them to her toy box. To my delight, she ate the cold, milky eggs out of her cup with the spoon without as much mess as one would expect! My husband realized that the current fascination was food and liquid together. Now we put the liquid in the bowl with the food to begin with, and everyone is happy. She certainly is not like me.
I wonder how many things children must learn, like setting the table, feeding the cats, making the beds and so on, would move more easily if we parents allowed the child to play with the new materials long before we or society demanded that they use them properly and with skill.
*Fraction Tiles by Lee Jenkins and Peggy McLean