MATH PERSPECTIVES INSTRUCTOR/CONSULTANT BLOGS
Marilynn Magnani
Name:
Marilynn Magnani
Location:
Newark, DE
Marilynn Magnani is Director of Education and a classroom teacher at the Newark Center for Creative Learning. She is one of the founders of this private, non-graded, parent-cooperative school for students aged five to 14 years old.
She has been teaching for over thirty-seven years and has served in a variety of leadership roles in the education community. Marilynn is dedicated to helping classroom teacher rethink how they teach mathematics and helping them become confident and competent in the mathematics they teach. She is the developer of the Math Perspectives Understanding Fractions workshop.
News from Marilynn:
I believe all children can and should learn math with understanding. Below is a story from my classroom that I want to share.
A Classroom Lesson: From. Fractions to Formulas on the Geoboard by Marilynn Magnani
As I walked around the classroom helping my students hammer in the last pins on their self-made geoboards, I wondered where the next several days' investigations would take us.
My class, a multi-age group of fourth and fifth graders, had been studying fractions and was beginning a unit on plane geometry. I expected that the geoboard was a good tool for modeling both fractions and geometry and would be a way to connect the two units. In addition, it would give me a chance to assess how the students generalized what they know about fractions to a new format.
Having taught for over twenty years, I was well aware that merely memorizing area formulas as isolated facts did little to develop a sense of geometric relationships. My main goal was to have the students build an understanding of such relationships as they discovered formulas themselves. I had done my homework and had an overall plan but was also aware that plans are one thing, reality is another. Would the students really develop a firm understanding? Would they be able to discover the formulas?
I decided to have the students make geoboards larger than the ones commercially available. This would allow for a greater number of models to be built. Each student was given an 11.5 inch square piece of wood and asked to figure out how to space the pins evenly in seven rows and seven columns. They were to make a paper pattern before any hammer was put to nail. This was a challenge and many questions arose. I answered clarifying questions but controlled the temptation to solve the problem for the students. By the end of the class most of the students had drawn up an acceptable plan and I had a chance to assess real life problem solving.
Two days, and several bruised fingers later, the geoboards were done and the students began making intricate designs with rubber bands. Of course, rules for acceptable use of the rubber bands needed to be established.
It was agreed that any student who broke the rules would have to use dot paper rather than a geoboard for two class periods. The students enforced this consequence themselves and in no time everyone was going by the rules.
As the students made designs I walked around and began to ask questions about the sizes of parts in their designs.
"What part is that triangle in relation to the whole?" I queried. "How much of the geoboard does that octagon take up?"
These questions began a search for how to define the whole. Halves and fourths were easy to distinguish but smaller parts were difficult. I let the students struggle with this difficulty, sure that figuring out these questions was what mathematics was about. It would require applying information the students had learned previously about fractions to a new format in order to solve a problem. I asked guiding questions.
"Sara, why did you say that triangle took up one-fourth of the board? What does that mean?"
"It's one-fourth because it is one of four equal parts of the whole:' she answered.
She had made a connection from previous learning. I continued, "If each of those triangles is one of four equal parts, what about the small triangle? What is it one part of?"
I kept circulating and directing the students' thinking by asking questions. Certainly, some students were not making the connections as Sara had. I supported their thinking by finding out what they understood and moving on from there.
Toward the end of class, several students had identified shapes that took up various fractional parts. I called the class together. It was time to share what was discovered and define the units of measurement we would use for our investigations on the geoboards.
I asked what part of the whole was the smallest square, one bound by only four pins. With very little hesitation the students identified it as one thirty-sixth.
"Why?" I asked.
"Because there are thirty-six of them on the geoboard," came the reply. Exactly what I had hoped. They saw the relationship of part to whole. This was an essential concept for making sense of fractions.
We then defined the measurements on the board. The distance between adjacent pins in a row or column would be one unit and the square with a perimeter of four units would be one square unit.
As the class ended several students took their geoboards to recess. I heard one student pose the question, "If we put bands across the small square we can make even smaller parts.
I wonder what's the smallest part we can make on this thing?"
For the next three classes and for homework, we continued making shapes and determining their areas by counting square units. Students began to automatically know the fractional parts. One-half of a square unit was one seventy-second of the board; a few of the ways one-fourth of the board could be referred to were two-eighths, three-twelfths, and nine thirty-sixths. Not only were they getting comfortable with measurement on the geoboard, they were reinforcing their understanding of equivalent fractions.
At this point, I felt the students were ready to focus on finding formulas. They were very comfortable with the terms length and width but they needed to learn the terms base and height as it applied to triangles. We spent a class period making triangles and finding the bases and heights. To aid in describing the vertices of specific triangles, I had the students mark their boards with Cartesian coordinates using the lower left corner pin as the origin, thus making the first left-hand column the Y axis and the bottom row the X axis.
I made a chart on the marker board with columns headed: Triangle, Base, Height, and Area. Under the triangle column, students were to enter the coordinates of the vertices of their triangles and continue to fill in the other columns with the measurements for that particular triangle. They were to make this chart in their journals, then make and record the information for at least twelve different triangles. Each student was to enter information from one of his or her triangles on the chart on the marker board.
Again I circulated around and observed how the students went about their work. Area was easy to determine on some triangles, not so easy on others. I realized that I would need to direct the students' focus to get them to articulate what they knew about the relationship of the area of a triangle and the area of a rectangle. I asked them to make several different rectangles, divide each into two equal triangles, and then compare the area of the rectangle to the area of one of its triangles. In very little time the word was out. The area of each triangle circumscribed by a rectangle was one-half the area of that rectangle. The first hint of the formula.
"See if this helps you find the area of some of your problem triangles," I stated.
It did for many, but not all the students. This was acceptable. Students who could, recorded information on difficult triangles, those who could not, stuck with easy triangles.
When work was done and the chart on the marker board filled, I asked the students to see if they could find any relationships between the area and base and height. As students saw relationships, I asked them to state the relationship and give an example. Then I upped the ante by asking them to find relationships which were true for all the triangles recorded. In other words, the relationship had to work for all the triangles not just specific ones.
As relationships were stated we tested them by trying them on several different triangles.
"I think there is a pattern," Matthew said.
"Look, if you multiply the base times the height and divide it in half, you get the area.
I think it works for all of them," he said as his eyes moved down the chart.
"Mmm," I said, "can anyone find a triangle where Matthew's relationship doesn't work?"
The rest of the class time was spent trying to find a triangle that would disprove the relationship. None could be found.
Now was the time to formalize their findings. I asked if anyone could tell me the formula for finding the area of a rectangle and illustrate it on the geoboard. I knew that this was knowledge they already had.
"Area is equal to length times width," a student said.
Others agreed.
Tell me how to write that with symbols. "A = L x W" came the response.
Now how can we put Matthew's relationship into a formula?" I asked. The students began turning the words of the relationship into symbols. There were several variations proposed, all of them very much like the standard formula, A = 1/2 B x H.
"You did it," I shouted. "You have found the formula for finding the area of any possible triangle.” The students congratulated each other.
Then Tav said, “Hey, that's like saying the area of a triangle is equal to one-half the length times the width of the rectangle made around it.”
I thought I would burst with joy.
“Tav has made a conjecture that the area of a triangle is equal to one-half the area of a rectangle drawn around it. 'Does that connect with any work we've “ done?'” I asked. “That's your homework for tonight. Take your geoboards home and make different size triangles. Make a rectangle around each one and compare the area of the rectangle with the area of the triangle. See what you can discover.”
I was exhausted and exhilarated at the same time. I knew there would need to be many more investigations, but my goal was in sight. The students were finding formulas and discovering relationships.
As we walked out of the classroom Daniel asked, “How long is the diagonal across one square unit on a geoboard?” He stated that it was longer than one unit but he could not figure out exactly how much longer.
“Great question,” I replied. “Long ago a man named Pythagoras had that very same question. Let's see if we can answer it in the next few days.”
Strong partnerships and communication with parents is an important part of supporting student learning. Below is a letter I sent to the parents of the students in my classroom that I believe was an effective communication between home and school and helped parents understand the mathematics I am teaching their children. Marilynn
Dear Parents,
I want to share some observations I have made while listening to your children talk about their math. By listening to them explain their work I can better understand what sense they are making of our number system and how it works. I particularly want to talk about how your children make sense of computation and what supports sense making and what interferes with it.
Many of us think that children aren't able to add, subtract, multiply or divide unless they are taught the standard algorithm or procedure for those operations. Therefore, we try to teach the procedures first before we let the children compute. Faced with a problem like 6 x $4.95, we feel compelled to teach the standard algorithm for multiplication. How else will a child be able to find the answer?
In fact, children can carry out all kinds of computation without formal procedures and in so doing learn crucial information about our number system, information which allows them to make great sense out of their number work.
An algorithm can be very useful as a short cut for recording computation on certain problems. But, because it is a short cut, an abbreviated or coded form of the operations, it is very abstract. Think about learning the algorithm for long division or where to put the zeroes as place holders in double digit multiplication. In most cases, the algorithms make very little sense to children and so they struggle to memorize them without thinking about the actual numbers they are computing.
When algorithms are taught too soon, they interfere with how children make sense of numbers. The importance is shifted from number sense to memorization of seemingly unrelated procedures. Not surprisingly, many children lose what understanding they have because of the belief that the algorithm is the only right way to get the answer. There develops an over reliance on the procedure and thus an inability to discern when other strategies would be more efficient.
Rather than develop mathematical flexibility we do the opposite, and, to add insult to injury, we have children practice these procedures over and over in the name of mastery. My question is, mastery of what? Certainly we don't want them to master something which interferes with understanding.
Students need to be given many experiences which help them develop their own understanding of numbers and operations. They need to use what they truly understand about numbers to help them solve new, more complicated problems. And we have to give them the opportunity to do this.
This doesn't mean that we cast them out with no support or guidance. It does mean that we listen carefully to their reasoning, help them clarify their thoughts and lead them towards greater understanding and efficiency.
I'd like to describe two examples from the classroom. These are not isolated examples. First, let's go back to the multiplication problem above, 6 x $4.95. One child, who relied on the algorithm to get the answer, often became confused as to which place to put the numbers and then what to do with the decimal point. When I asked her if she had an estimate of what the answer might be she told me that she couldn't tell until she was done. However, I knew that this child had decent number sense and in other cases, without relying on the algorithm, she would have been able to figure out the answer with ease.
Another child changed the $4.95 to $5.00 multiplied to get $30.00 and then took away the extra 30 cents. She did this in a minute's time. She was able to do this because she was thinking about what those numbers meant. Right away she knew that the answer was going to be a little less than $30.00.
The next example concerns the "long division" algorithm, second only to the algorithm for dividing fractions, in its ability to obscure number sense. In the traditional algorithm students are taught to Divide, Multiply, Subtract and Bring Down (There are countless mnemonics used to help remember the order, one being, Does Mac Donald's Serve Burgers? I'm not even going to go down that road.)
One student set up the problem, 126 divided by 15, in the long division form. Before he began, I asked him if he had an estimate of what the answer might be to which he replied, "No." I then asked him what it meant to divide and he said that it meant to do what he was doing, and he indicated the problem he was working on. He then began the procedure by saying, "Fifteen goes into 12, zero times, I think. But no, I'm supposed to divide first. So 15 divides into 12 once and now I'm supposed to either subtract or multiply." I stopped him at this point and asked if he could tell me in words what the problem was asking and he said he wasn't sure but he thought it meant 15 divided by 126.
I could go on, but I hope I have made my point. This child was very confused and I believe much of it had to do with his determination to remember the steps of the procedure rather than trying to understand what was being asked.
Another child in the class approached this problem by thinking about how many groups of 15 could be made from 126. Right away he said it couldn't be 10 because you would need 150 for that. He then used the multiples of 15 he knew, and wrote down, 4 15's = 60, so 8 15's = 120. The answer is 8 groups of 15 with 6 left over.
When students have been given time to develop a solid understanding of numbers and operations using their own strategies, algorithms can be learned and applied when useful. They can be looked at from a different perspective, one which includes and understanding of why they work. Then, and only then, is the student master of the procedure rather than the other way around.
If we teach our children tricks such as, when you multiply by 10, add a zero, or add two zeros when you multiply by 100, we are inadvertently supporting the notion that math is done without meaning. If we want our children to be powerful mathematically, they deserve more than tricks. They deserve the time to build and understanding about our base 10 number system so that they know what it means to multiply by 10, or a hundred or .1 (one tenth). This is truly powerful.
Building a solid understanding of numbers, takes time. Often we feel rushed and want to push the process along. We want to show our children the "quick" way of getting an answer. Our children become masters of nodding their heads when we say, "Do you get that? Now do you see? Just put the zero over here because now you're multiplying in the 10's place. Understand?" as we quickly jot down our solution.
Allow your children to solve problems their way even if it seems terribly inefficient. Ask them to explain their reasoning and listen to their explanations. They are learning about how to work with numbers and most important, they are doing it in a way that makes sense to them. As they do this, their understanding grows and they are able to use this understanding in a very powerful way.
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