Last July, I had the pleasure of meeting Richele Baburina, author of Mathematics, An Instrument for Living Teaching
, at the Living Education Retreat
. Her insights have helped me enormously with a student in the autism spectrum who has declared loudly to us at the school
: "I HATE MATH!" "IT'S TOO HARD!" "IT'S BORING!" "YOU JUST WANT TO ME WRITE PAGES AND PAGES OF STUFF!"
I empathize with his views of typical math curricula. As Richele points out in her first blogpost on mathematics
, "Though its use in daily life was important, it was the beauty and truth of mathematics, that awakening of a sense of awe in God’s fixed laws of the universe, that afforded its study a rightful place in Charlotte’s curriculum."
Does typical math curricula inspire awe over God's fixed laws of the universe? Does it point to the beauty of mathematics?
When he sees a worksheet full of equations, my young friend shuts down or melts down! As Richele states in her second post on mathematics
, such worksheets are not CM-friendly. They are convenient for moms and teachers because they give us a break from individualized instruction.
Rather than haul out workbooks, I assessed his addition facts orally with different manipulatives: dominoes, dice, etc. He seemed to know them, so the headmaster of our school and I assessed him in two-digit addition. Rather than pass out a worksheet, Angie pulled out her 5" x 8" notepad to emphasize the shortness of the lesson! She asked him how many problems he could do. He told her six. So, she gave him a couple of problems that did not require carrying. He made no errors.
When she wrote down one that required carrying, he struggled. Rather than disrupt the flow by pulling out manipulatives, she appealed to his sense of reason. She wrote above the two columns of the problem, tens
, and explained that this number is like a house. It has two rooms, the tens
room and the ones
room. Only numbers that are 9 or less can fit in the room. She asked him where he thought then ten part of 13 should go. He answered, "The tens room. Is that why they do that
?" (carry the ten). From that day, he always knew when to carry and when not to carry. That week, he gave correct answers for tricky two-digit addition problems: some with a three-digit answer or with 0 in the ones place of the solution. He sailed through three-digit problems!
His math book offered addition problems with decimals next, so I asked his mother what he understood. Not much. I asked her about his understanding of fractions because they lead to decimals. She stated that he knows the basics, so, this week, I shifted to assessing him in fractions.
My friend bores easily, so variety is the name of the game. Because I am mindful of shared experiences
(the joy that comes from collaboration—a challenge for those in the autism spectrum), I seek situations that invite him to work with me. Richele calls this living teaching
- Teach math concepts in a hands-on, life-related way that assures understanding.
- Encourage daily mental effort from your students with oral work.
- Cultivate and reinforce good habits in your math lessons.
- Awaken a sense of awe in God’s fixed laws of the universe.
On the first day of our foray into fractions, he explored fraction overlays. To pique his interest, I asked, "Guess what I made!"
"I made this basket."
"You did? What's in it?"
"Some fraction overlays.
"What are those?"
"Take them out and see!"
Eman pulled out all of the overlays and made circles with the fraction slices. As he put them away in the way he found them, we talked about the names of the denominators for halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths. He knew them all, so I took notes on what he did and what he knew. This task covered more than fractions: taking out and storing the pieces exactly as he found them required fine motor skills and practiced the habits of attentiveness and order. He spent at least twenty minutes doing math and enjoyed it!
Knowing that Eman loathes repetition, I found another hands-on task the next day. I asked, "Guess what the kids in your class are making!"
"I don't know. What?"
"Leonardo da Vinci's parachute
"Really? Can I try?
"But I want to work outside!"
"We can do that."
We headed outdoors with pencil, ruler, and four pieces of paper. Together we folded each paper in half, drew diagonals, and cut along the diagonals until we had four triangles. We talked about the shapes we noticed (rectangles and triangles). Then, we put them together as shown in the picture and I said, "It reminds me of the fraction overlays from yesterday." He agreed, so I probed.
"It looks like there are pieces missing. How many do you think are missing?"
"So, if we had those pieces, what kind of fraction would we have?"
After that exchange, I began to wonder if boredom might be the culprit. This hands-on, meaningful task revealed a keen understanding of fractions that rows and rows of problems might not have uncovered. Then, we taped the triangles together and I showed him how he could make a tent. I asked, "Do you know what this shape is called?" "A pyramid." Yes, he really is bright.
Eman's mother loved the parachute project and told me that his visual-spatial awareness is keen. The next day, I printed out a model of a dodecahedron
. Before we started, we had a little chat about the pentagons. Then, he wanted to know what a ten-sided shape was called and then a twelve-sided shape. I made a grid with twelve squares for him to color to represent each side: 3 red, 3 green, 3 yellow, and 3 blue. My printer was running out of cyan, so only one pentagon was true blue. Eman insisted that the other two were purple, which made for a more interesting problem. While coloring the grid, he said, "I remember doing these in school. I liked it." Then, he wrote fractions for all five colors.
He seemed worried that cutting would be too hard. When asked what he could not do, he said, "The black lines."
"I can cut the white tabs. What can you do?"
"The colored ones."
We took turns cutting, and then he folded all the sides without any help. Then, we took turns taping it all together. When finished, he just had to roll the dodecahedron like a dice!
To avoid boring him, I chose six red and six blue buttons the next day. Eman had to sort them by color, size, and number of button holes and make fraction grids. After doing the color count, he told me he wanted to try his own problem. He insisted.
I asked for a topic. He chose cats and dogs, and I selected a much more challenging problem: conduct a survey about liking cats or dogs. When Eman had a hard time choosing cats or dogs, he created a new category: both. He polled students, teachers, parents, and even the ladies painting the new elementary classroom. He interviewed 27 people and checked their preferences. He asked how to spell their names and wrote them down! This math problem encouraged writing, communication, social interaction, attentiveness, and patience (we had to wait for kids in the primary class to come out for bathroom breaks and lunch). Moreover, this problem inspired Eman beyond the length of a typical math lesson.
I made a printout to show his data and apply equivalent fractions. Tasks were picking a color scheme, coloring a grid that had bars the same size as labels for him to convert thirtieths to fifths, and making a bar graph as well as a pie chart in both denominations of fractions.
Because of the trust we have built, his first reaction was not complaining
about it being too hard. He studied it for few seconds and asked, "Did you make this?"
"Yes. I did. I learned how to make these in college."
He enjoyed picking out the color scheme, counting up the responses, and coloring in the grid. At one point, he told me, "I like this!" He figured out the fraction in thirtieths and had no problem seeing that 6/30 was the same as one bar and that he needed three brown bars to make 18/30. He has not fussed about math in over a week.
Tomorrow, we will make the connection to fifths, color the bar graph, and make the pie charts. In time, I hope he will learn to love math for its sake because he has encountered enough inspiring ideas to endure the repetition required to learn those facts that must be learned.
Education should be a science of proportion, and any one subject that assumes undue importance does so at the expense of other subjects which a child's mind should deal with. ~ Charlotte Mason (page 231)
Mathematics depend upon the teacher rather than upon the text-book and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas, what Coleridge calls, the 'Captain' ideas, which should quicken imagination. ~ Charlotte Mason (page 233)