Help for Those Who Hate Math

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 ones, 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!"

"What!"

"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?

"Sure."

"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?"

"Two"

"So, if we had those pieces, what kind of fraction would we have?"

"Sixths."

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."

"Really?"

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)

Clear Thinking about Math Part 1

Sometimes, the best plans fall apart. Pamela seemed to track the review of fraction addition I did early last month. When we went back to area and perimeter problems, something got lost in translation. Fortunately, a friend posted about how she made tiles with numbers and symbols to use on a magnetic board with her grandson. Having to write added enough challenge to chip away at his working memory and make it difficult to a new concept to gel. Pamela writes and draws very well, but I thought it worthwhile to remove any potential memory drain while she was trying to visualize adding fractions. I suspect Pamela lost her thread of thought every time she had to stop and draw her thinking.

I also heeded Mary Boole's advice in an old Parents' Review article,

"Beware of writing, in play-lessons, anything which does not represent some process actually going on in the child's mind."

I created a set of pie charts in a spreadsheet, some representing wholes, some representing wholes split into fractions, and fractions. I stayed simple by limiting it to halves, thirds, fourths, and sixths. I cut out all the shapes, covered them with clear contact paper, and cut them again to make them more durable. Before we worked on a problem, we sorted between wholes and fractions to help Pamela familiarize herself with these homemade manipulatives. You can see the first step in our first lesson in the video below. Since Pamela understands fractions, I am using very declarative language as we collaborate.

Then, we started working on her problem, adding 4 3/4 and 4 1/2. Before writing, she set up a model for each addend so that she could represent her mental process visually and spotlight what adding fractions and simplifying meant. You can watch how we worked through the problem together: first, she made both denominators alike. Because I didn't build any models for eighths, she had to think through another option: fourths.


Then, she added them and ended up with an improper fraction 5/4.


Using the models helped her see what she was doing when converting to a mixed fraction and adding the wholes again.



The video below shows how we collaborated step by step. We wanted to show what we were doing physically and write it on paper.

Working together like this cleared up other glitches. Pamela had a habit of forgetting to write the wholes until she needed them again. While she usually remembered to pick the wholes back up when she needed it, that mathematically incorrect habit could lead to disaster in algebra. When finding a common denominator, she tended to multiply the denominators (2 x 4 = 8) rather than going for the least common multiple (4). The lack of eighths forced her to think of a smaller denominator, which turned out to be the LCM. We worked on similar problems together for about a week. I made a set of twelfths for more challenging ones. Then, I faded myself out of the picture and she did well flying solo without anymore issues.

"Let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring." ~ Charlotte Mason (page 261)

Grapes of Math

Last September, a fellow blogger turned me onto Greg Tang books for dynamic thinking about math, and I blogged an update on our progress in November. So far, we have slowly worked our way through the following books: Best of Times, Math Appeal, and Math for All Seasons. We are looking forward to working through the five books we have not read yet.

Right now, we are starting Grapes Of Math and I thought it high-time to share what we are doing with these deceptively simple, elementary-school-level books. Basically, you read the poem and use the clue to count items in a picture. In this counting exercise, the clue is "Never fear, I have a hunch there is a match for every bunch." When you match a bunch of seven purple grapes with three green grapes, you get ten. Five matches of ten grapes yields the answer ten.


Even though Pamela is near the end of sixth grade math, these books are valuable because she is learning to group and count items in unique ways that make them easier to count. They guide you to count dynamically! Rather than get wrapped up in the poem in a static way, we adopt various strategies for counting the items. Pamela and I take turns showing each other a new way to count. We usually do three patterns each. Today, two interesting things happened. Pamela made a mistake in mentally adding and ended up with the wrong answer: 52. Rather than telling her she was wrong, I counted and got the right answer: 50. Then, we observed our answers were different and kept trying various counting strategies. From that point on our answers were always the same: 50. Pamela surprised me in her last strategy: she counted them individually and silently! The video clip shows a typical session with the book.


Best of Times gave us a fresh look at multiplication tables. There was nothing new about x 0, x 1, and x 10, but the way the author explored other numbers was original for this old dog (you know, ME). These unique ways of multiplying captured Pamela's imagination, and I plan to apply them when we are doing mental math while we shop at our favorite grocery stores!

x 2 - Double the number by adding it to itself:
37 x 2 = 37 + 37 = 74

x 3 - Triple the number by doubling it first and adding the double to itself:
45 x 3 = (45 + 45) + 45 = 90 + 45 = 135

x 4 - Double the double of the number (and I bet you can guess times eight now):
42 x 4 = (42 + 42) + (42 + 42) = 84 + 84 = 168

x 5 - Find x 10 and cut in half:
57 x 5 = (57 x 10) / 2 = 570 / 2 = 285

Here's a dynamic newsflash, not in the book: add the double to the triple:
57 x 5 = (57 + 57) + [(57 + 57) + 57]
= 114 + (114 + 57) = 114 + 171 = 285

Aside, not in the book: why not double the double and add it to itself?
57 x 5 = (57 + 57) + (57 + 57) + 57
= (114 + 114) + 57 = 228 + 57 = 285

x 6 - Triple the double (or vice versa):
28 x 6 = [(28 + 28) + (28 + 28)] + (28 + 28)
= (56 + 56) + 56 = 112 + 56 = 168

How about try x 5 and add that to itself:
28 x 6 = [(28 x 10) / 2] + 28
= (280 / 2 ) + 28 = 140 + 28 = 168

x 7 - Add x 5 and x 2 (or figure out your own variation):
82 x 7 = [(82 x 10) / 2] + (82 + 82)
= (820 / 2) + 164 = 410 + 164 = 574

x 8 - Double the double of double of the number:
65 x 8 = [(65 + 65) + (65 + 65)] + [(65 + 65) + (65 + 65)]
= (130 + 130) + (130 + 130) = 260 + 260 = 520

x 9 - Figure out x 10 and subtract the number:
19 x 9 = 19 x 10 - 19 = 190 - 19 = 171

Working on Dynamic Thinking in Math

A fellow blogger recommended
Greg Tang's series of living math books that target dynamic thinking. I got a copy of Math Appeal: Mind-Stretching Math Riddles (for free at Paperbackswap--if you sign up, my email address is tammyglaser798 at earthlink dot net). Pamela blew me away today with how quickly she figured out that she could count the peas in groups of eleven! While she has an eye for patterns, she cannot count a box of toothpicks spilled onto the floor a la Rainman. I love how quickly her mind works as you can see in the video.


Pamela surprised me greatly today. She handed me this postcard mailed to us to generate donations for victims of Hurricane Gustav. She said to me in a meaningful way, "It's the Gulf Coast."

I was curious if she knew where the Gulf Coast was because it is not something I have explicitly taught her. I asked, "What states are on the Gulf Coast?"

She quickly answered, "Louisiana" (where many of our relatives live).

Household Ways

Charlotte Mason viewed the natural conditions of the home and household ways as valuable opportunities for learning.

We all know the natural conditions under which a child should live; how he shares household ways with his mother, romps with his father, is teased by his brothers and petted by his sisters; is taught by his tumbles; learns self-denial by the baby's needs, the delightfulness of furniture by playing at battle and siege with sofa and table; learns veneration for the old by the visits of his great-grandmother; how to live with his equals by the chums he gathers round him; learns intimacy with animals from his dog and cat; delight in the fields where the buttercups grow and greater delight in the blackberry hedges (page 96).

One reason why I love RDI is because it focuses parents on lifestyle as a target-rich environment for framing objectives. Before lunch yesterday, Pamela and I sat on the rocking chairs on the back porch, working on her math. We watched my dad trim the pecan tree over the garage. Then, Pamela and I carried the limbs to the curb and cleaned up for him. I thought it would be a great opportunity to work on upper body because some the limbs were awkward to carry. She helped me haul all of that debris to the curb for trash pickup!


We are continuing to do our daily walks, timing them for the coolest time of the day on oppressively hot ones! The thermometer nearly hit 100 yesterday, and it sure felt like it! We have learned so many interesting things about our neighborhood and met Treebeard. David told me that the hardware store a block from the house sells these things. Pamela calls it the talking tree and you can bet her imagination is flying back to Middle Earth and Narnia (Pamela loved Prince Caspian by the way). The day after our walk, I write up a math problem sheet so Pamela can see how math is part of every day life (click the picture to see it enlarged).


Since she is doing different operations with fractions, I have been writing up sheets that focus on how we use fractions in real life, too.

Putting Pamela in the Driver's Seat

Because most of Pamela's gestures are imperative, our first objective is to teach Pamela how to use gestures declaratively. Since verbal language is such a challenge because of her aphasia, we decided to scaffold this by focusing on nonverbals first. We hope Pamela will transfer the discovery of declarative gestures to declarative language. We focused on three common gestures:

• Pointing - Draws my attention to whatever she is sharing with me.
• Nodding - Tells me I am getting what she is seeing.
• Shaking her head - Tells me I am on the wrong track in what she is seeing.

We wanted to come up with an activity in which she would be the active person driving the conversation. We decided to go on data collection walks in which I am the recorder and Pamela is the observer. One side benefit is that I can turn the data into a math lesson!

Before our walk to a nearby playground, I told Pamela we were going to do a project about house colors in our neighborhood. I needed help figuring out what colors to use. Then, she told me a bunch of colors, and I typed them into a spreadsheet. I printed out the tally sheet, and we left.

At each house, I stopped and looked down at my feet as if I had shut down. It took a bit for Pamela to realize she had to activate me. She poked me! Then, I looked at her face, waiting for her to take action. Again, it took a bit for her to realize she had to point to draw my attention to the house she was describing. Then, I overemphasized turning and looking at the house, and I turned back to her to tell her a color. Sometimes, I would say the right color so she could nod her head. Sometimes, I would say the wrong color so she could shake her head. Then, I recorded the data. Occasionally, I looked at a building, like a church, library, or office. I left it for her to decide whether or not to count them in our study--she shook her head.

That activity could become very static, so, on the way home, we expanded it dynamically. I heard a siren and exclaimed, "An airplane!" I looked at Pamela, and she shook her head. I went through several guesses before I got it right. I waited to see if she would direct my attention to something; if not, I would point out something else, either correctly or incorrectly, so she could give her observation by nods or head shakes.

Because RDI is about lifestyle and not activities, I am finding ways to incorporate it into our daily life. For example, she can get my attention in several ways: shoulder-touch, poke, move in front of me, etc. During the day, I found times to "shut down" so she would have to do something active to "reboot" me.

The language is not the only component of our plan to put Pamela in the driver's seat. Some of Pamela's passive nature might be due to sensory under-responsivity, which I will address tomorrow. By the way, the math lesson I wrote based on our walk is pictured below!

Why Not?

When did your children start asking why? Pamela has not quite reached the stage of asking why questions, but she is starting with "Why not?" Today, during math, I decided to review adding and subtracting fractions because we have been working so hard on multiplying and decided. She excelled at everything except when she had to borrow for subtraction. She fell back on carrying the one as a ten, not as the fraction indicated by the denominator.

First, I tried showing the chain of logic involved and ask Pamela along the way what I should write next. She had a very hard time and clearly had forgotten the concrete meaning of these steps.

Second, I tried drawing pictures but the steps were not clear-cut enough to demonstrate the chain of logic. Pamela has a very sequential, organized mind, and I knew this was within reach. But, she told me, "It's too hard!"

Third, I tried color-coding, pink for whole and green for fraction. I would do the color-coded diagram, two pink boxes for whole and 3/4 for fraction, and she would have to tell me what to write, 2 + 3/4. Then, I would make the next set, one pink box for whole and one green box and 3/4 for fraction, and she would have to tell me what to write, 1 + 1 + 3/4. Then, I would make the next set, one pink box for whole and 4/4 and 3/4 for fraction, and she would have to tell me what to write, 1 + 4/4 + 3/4. I had to give Pamela lots and lots of guidance. I even wrote the numbers in pink and green. She improved a little bit on the second problem, but not much!

By the second sheet, I continued to ask her as we went along because I was trying to figure out if she understood. Everytime, I had to correct her, she would ask in a semi-whiny voice, "Why not?" I was so glad that she asked that question because usually she just gets fussy or argues. But, this time, she asked why! Suddenly, something clicked. I can always tell because she starts to giggle and flash a smile. I can almost see the light bulb flick on in a cartoon buble over her head. She did the last two problems without help or correction. I like to end on a positive, so we will see what she remembers tomorrow.

Pamela decided to hang up this Bronco calender in the kitchen. Her Uncle Randy would be proud!

Wordless Wednesday: More Self-Assigned Math Problems!

Panel: Schools Should Focus on Fractions

Introducing Dividing by Fractions

This week, we introduced multiplying and dividing by fractions. I thought Making Math Meaningful Level 6 plowed through the first page too quickly. We have been spending more time exploring concrete versions of this activity. Yesterday, we focused on scooping measured cups of water into a bowl and scooping the water back with fractional measuring cups (1/2-cup, 1/3-cup, 1/4-cup).

The first couple of clips may seem dull to you, but there is a payoff! If you can believe it, Pamela picked up the pattern for division problems like this (a ÷ 1/D) by the time we reached the second one. If you do not have time for all of these clips, watch the clip for Problem 2 in which I do a double-take when she solves the second problem in her head!

Here is the first problem we explored yesterday:
Problem 1
Put 6 cups of water in a bowl. Scoop out 1/2-cup at a time. How many scoops did you have?
The math equation for this problem is 6 ÷ 1/2 = _______.

The clip below shows me introducing the problem. I wanted to see how well scaffolding works with academics. First, I asked her which measuring cup she should use to make six cups, and she wisely and logically chose the two-cup measurer. She is not very accurate with measuring liquids, so I put masking tape to show the exact line. I was very particular about getting eye-level to measure. I am doing a great deal of thinking out loud, so Pamela sees me solve problems out loud.


In this step, we scoop 6 cups of water, back into the pot, by 1/2 cups to demonstrate dividing by a fraction. She was a little distracted thinking about You-Tube videos but quickly got on track. Pamela has a little trouble verbalizing the count because of the pause in the action when I dump the full cup back into the bowl Either that, or she was getting ahead of herself).


Pamela invented a game called Fly or Drive last week. She tells me two states, and I have to say whether or not I'd fly or drive. In this clip, I redirect her to get back on track with the fraction lesson: figure out how many 1/2-cup scoops equals 6 cups. She easily gets back to business. We finished the first problem.


Problem 2
Put 5 cups of water in a bowl. Scoop out 1/3-cup at a time. How many scoops did you have?
The math equation for this problem is 5 ÷ 1/3 = _______.

Pamela astounded me!!!! On the second problem, Pamela solved 5 divided by 1/3 in her head!!! Before we did any measuring, she knew the answer was 15. She had already spotted the pattern to division problems with this format: a ÷ 1/D. I am pretty well-versed in her abilities, but today she made me do a mega double-take! This clip is the million-dollar, must-see clip if you are pressed for time today!


Here are the rest of the problems. By the time we reached Problem 5, it was obvious that Pamela knew the pattern. So, I let her tell me the answers for the rest and we only needed to do concrete measuring for the first three problems.

Problem 3
Put 2 cups of water in a bowl. Scoop out 1/4-cup at a time. How many scoops did you have?
The math equation for this problem is 2 ÷ 1/4 = _______.

Problem 4
Put 1 cup of water in a bowl. Scoop out 1/4-cup at a time. How many scoops did you have?
The math equation for this problem is 1 ÷ 1/4 = _______.

Problem 5
Put 4 cups of water in a bowl. Scoop out 1/2-cup at a time. How many scoops did you have?
The math equation for this problem is 4 ÷ 1/2 = _______.

Problem 6
Put 4 cups of water in a bowl. Scoop out 1/3-cup at a time. How many scoops did you have?
The math equation for this problem is 4 ÷ 1/3 = _______.

Problem 7
Put 4 cups of water in a bowl. Scoop out 1/4-cup at a time. How many scoops did you have?
The math equation for this problem is 4 ÷ 1/4 = _______.

After three problems, I did a quick review of the first three which we did by hand by scooping water. In this clip, you see us review the answers and then she correctly calculates the answers to Problems 5 through 7 in her head.


On Monday, we will do this with cutting up strips of paper and see if she can come up with a predictable rule written algebraically with letters. In Making Math Meaningful, children work with letters from the earliest books, and Pamela understands that letters can mean number. I am sure, with a tiny bit of guidance she will be able to come up with something like:

a ÷ 1/D = a x D

Wordless Wednesday: Self-Created Math Problems! and P.S. For Jamberry

P.S.

How Not to Be Frazzled by Fractions!

Last week, Pamela was supposed to be wrapping up the final lesson on subtracting and adding fractions, including complex problems like finding a common denominator. She usually finds it easy to keep pace with the logic required in Making Math Meaningful Level 6. She could do everything except borrow one to make the fraction in the minuend (positive term) larger than the fraction in the subtrahend (negative term). The question I pondered all weekend was how to apply scaffolding to this!

At first, I fell back on my visual-kinesthetic worksheet-ish sort of thing because I thought, once she visualized it, she would understand it. I set up all of the colored boxes and let her fill in the numbers.

Great idea?????

NO!!!! It tanked!

So, I decided to try the scaffolding approach:

(1) Shared Understanding: I recalled how Pamela worked these problems to figure out what she could do independently:

• Subtract the whole numbers.
• Know when to find a common denominator.
• Give the fractions common denominators.
• Subtract the fractions IF the first one is big enough.
• Know whether or not to borrow.
• Write the whole number 1 as the fraction number.
  

(2) Zone of Proximal Development: Then I figured out what Pamela needed to learn to do:

• Reduce the whole number by one.
• Add the fraction version of the whole number 1 to the original fraction.

I considered whether or not these skills are in the Zone of Proximal Development. Then, I recalled that she has no problems borrowing for numbers with more than two digits. And then, I realized that building upon what she already knows might be the best way to nail it! Clearly, this skill is in the zone.

(3) Scaffolding: Scaffolding involves three elements: warm encouragement, self-regulation, and joint problem solving. Self-regulation is the ability to let thought guide behavior. According to Dr. Laura Berk in Awakening Children's Minds, "when adults ask children questions and make suggestions that permit them to participate in the discovery of solutions, then transfer of useful strategies to the child is maximized" (page 49). Charlotte Mason's book Formation of Character provides many examples of how parents can teach children to self-regulate. The key is to let language help the child reflect on alternatives, which is what she put in her twenty principles:

Children must learn the difference between "I want" and "I will." They must learn to distract their thoughts when tempted to do what they may want but know is not right, and think of something else, or do something else, interesting enough to occupy their mind. After a short diversion, their mind will be refreshed and able to will with renewed strength. Principle 17 translated into modern English by Leslie Laurio

Pamela and I have been doing joint problem solving for years! Scaffolding requires the parent to observe the child's level of competence. When the child falters, the parent provides more support. When the child is sailing, the parent simply observes and makes little encouraging remarks. A nearly mastered task, like this one, only needs indirect hints, while tasks at the outer edge of the zone require more direct, pointed support. Laura Berk gives a great example of how a father and his daughter put a puzzle together with these joint problem solving techniques in Chapter 2 of her book.

Warm parenting is a no-brainer for me. I had already determined how to help Pamela self-regulate by comparing borrowing with fractions to what she already knows. All I needed to do was to watch for moments of confusion and guide her thinking with an insightful question or statement.

The first thing I did was to remind her how we borrow for a two-digit number. Charlotte Mason loved this technique of taking what is already known and relating it to the unknown.

Then, I pointed out to her how we can do the same thing with fractions. "What do you think we could do with the 4? . . . What fraction do you need? . . . What does the whole number 1 equal? . . . What can you do with the fractions I circled? . . . Do you see how we can rewrite the problem?"















Here is her first attempt. She did pause a couple of times and reference me when uncertain. She was quite excited to see that what puzzled her yesterday became clear as day today.

The exciting thing was Pamela did most of these problems without hesitation in the many variations you can have. You would think that mastering this within the first five minutes of applying this strategy would have made my day. But, no, I have even more excitement to share once you scroll past the pictures!











Emotion Sharing Highlights

• Yesterday, Pamela told me she had a stomach-ache. She never tells me her stomach hurts except right before she throws up. Later last night, I figured out she had PMS! That was the first time in five years she ever told me about cramps!
• About half way through working on fractions, she turned to me and said, "Steve will be proud!" You can bet that, when I showed him the sheet, he told her how proud he was and then made a copy of it to take with him to the office!
• She INVENTED a new game: drive or fly. She names two states and asks whether or not you should drive or fly. She even corrected me when she disagreed. She laughed at my play on words: highway or flyway.