You Know You're a Geek When You're IM'ing about Factoring Polynomials

Two posts in two days? Pass the smelling salts!

I've been busy writing what amounts to a fifty-page research paper on the teaching of mathematics. It has been eating up much of my time and distracting me from my blog, and I do it gladly. On top of that, Google has decided to delete old videos next month, forcing me to transfer my blog videos to You-Tube and download my private ones. It's a good excuse to attach labels and revamp the blog. The timing stinks!

Wednesday, I tutored an intelligent young woman taking college algebra. She was struggling with factoring polynomials, so we spent over an hour working through problems. My primary goal in tutoring math students is to put them on a search for meaning. If they understand why they are doing what they are doing, they will be more likely to remember it. Even if they forget, meaning helps them reason their way through a problem. If I sniff any hint of wavering, I will ask them why something is true.

My friend had to solve 9x⁵ - 9x³ = 0. She had no problems with step one, dividing both sides of the equation by 9 to get x⁵ - x³ = 0 and knew to pull out x³. When asked what x⁵ divided by x³ was, she made the fatal mistake. She raised her eyebrows and questioned, "X squared?"

Me: "Are you sure about that?"

Her: "Our teacher said you subtract."

Me: "Did she explain why?"

Her: "No. She's old school. She just tells us how to do it. She doesn't have time to explain why."

Sigh. She is smart. She is perfectly capable of understanding why. People who disrespect motivated students enough not to explain why bug me. So, we headed down the path of meaning, peeling back her uncertainty until we reached something solid.

Me: "What does x⁵ mean?"

Her: "You times x by itself 5 times. You know, x times x times x times x times x."

Me: "Good. Any time I'm unsure about a procedure, I start thinking about meaning. If you freeze on a test and forget whether to add, subtract, multiply or divide, you can always fall back on meaning. Write it out the long way."

So, she wrote x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x). Then I showed her how that is just like saying x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x). Then her face lit up, "Oh! Then you can cancel and get 0."

Believe it or not, that is a misunderstanding because we throw around words without meaning and precision and end up confusing students. I responded, "No. Lots of students do that. Let's go back go meaning. What does x divided by x mean?"

Her face went blank. Yes, I know I'm a pain, but this is important! Math makes sense when taught properly. So, I peeled the onion back further. I said, "Sometimes, it is easier to think about numbers. What does 5 divided by 5 mean?"

Another blank stare. The way we teach math focuses on doing, not thinking deeply. I explained, "Dividing means putting objects into equal groups. Suppose you had to share 5 cookies with 5 people. How many cookies would each person get?"

Her: "Oh, 1!"

Me: "What if you shared 10 cookies equally with 10 people?"

Her: "They'd each get 1."

Me: "What if you shared a million cookies equally with a million people?

Her: "They'd get 1!"

Me: "Now, let's get back to x divided by x. What does x mean?"

Her: "I don't know."

My friend answered correctly without realizing it. I explained to her that we use x to represent a number we don't know. It's a placeholder that means a number that we don't know. Having placeholders allows us to set up relationships between known and unknown numbers and manipulate them to figure out the unknowns or refine those relationships. I added, "We have a number of objects and the same number of people. We'll call that number x. If we have x pencils to give to x people, how many pencils would each person get?"

Her: "It's 1."

Me: "What is x divided by x?"

Her: "It has to be 1."

Then, everything fell into place, and she understood:

x⁵ ÷ x³ = (x ∙ x ∙ x ∙ x ∙ x) ÷ (x ∙ x ∙ x)

= x ∙ x ∙ (x ÷ x) ∙ (x ÷ x) ∙ (x ÷ x)

= x² ∙ 1 ∙ 1 ∙ 1

= x²

We had to peel the onion for only a few more glitches. My friend said she had a much better understanding. It disappoints me to know how much rote, meaningless instruction is happening in the math world.

Training Habits without Losing Your Mind!

If you are expecting charts, stickers, timeouts, schedules, programs, organizational tips-- (which seem to work best for the organized)--please find another blog. You will not see any of that here!

Back in February, a friend started a local Charlotte Mason study group. This is the first time I have ever gotten to read her books and swap stories with skin friends. Last week we covered habit training, which we can easily look like dog training with velvet gloves if we get all legalistic. When I applied the ideas our group shared to guide kids in our church's after-school program in relational ways, class went more smoothly. Another friend who teaches a college class on disabilities set up a video call with me, so her students could talk to a family living what they are learning. They asked my opinion of behaviorism and received the flip side of raising autistic children: through relationships, not rules! Last week, our church Bible study focused on living Christ's principles (by staying connected to Him Colossians 2:19-23) and to avoid becoming enslaved by worldly principles ("Do not handle! Do not taste! Do not touch!"). Then, a cyber friend asked about helping her teenager keep a cleaner room in a Mason way. God tossed all of these circumstance into my life in the past two days, so I got the hint and decided to blog it!

Relationship Development Intervention (RDI) (our favorite autism therapy) and Charlotte Mason (our favorite method of education) have similar ways of guiding children in their thinking. When our thoughts change, changed behavior will follow. Both begin with parents: the way we think affects the way we parent. We need to start with the head and heart for change to occur: the head and heart of the family changing first, which then flows into the head and heart of the child. Mason recommended parents cultivating three habits when training their children: tact, watchfulness, and persistence. These three habits dovetail very nicely with RDI.

Tact - Tact means saying, "Dave, can you come here for a minute?" instead of "DAVID JOSEPH GLASER!!" Tone of voice alone can mean the difference between a strong-willed child showing up with an open hand or closed fists. It means using a hopeful and expectant look rather than a frown and glaring eyes. It is giving a few brief words and having a conversation, not a dreary monologue, to help your child understand why a habit is important to you and will benefit them. Half the battle is won when you and your child agree about why a habit is needed. Tact means knowing your child well enough to find the right idea to inspire him to reform his ways.

Watchfulness - Watchfulness means observing your child and figuring out the best way to approach a new habit based on her nature. It means being aware of triggers for poor behavior and preventing them. It means knowing what contrary habit might work best in helping her succeed. Watchfulness means setting up a situation (timing, environment, control, brainstorming) conducive to forming the habit and keeping it in the long term. It means altering the plan if the outcome isn't quite what you envisioned. Watchfulness means never allowing friction creating an ever-widening gulf within your family.

Persistence - Persistence means being vigilant until the habit sticks and not relaxing, undoing weeks of effort. It means staying hopeful, not cranky, when progress is slow. It means realizing that building one habit at a time in the marathon of childhood gets you farther than short bursts of programs that rob you of your energy and joy. Persistence means knowing that habits are not going to change overnight and require patience and consistency. Persistence does not mean helicopter parenting.

Before describing habit training, keep in mind these points:

• Habit training is hard and requires vigilance. Focus on one habit at a time or you may lose your mind.
• Try indirect cues. Commands rob children of the chance to think for themselves. Use hopeful and expectant looks. Speak with a calm and friendly voice. Simply stating their name may be enough to check their behavior.
• Avoid a "running fire of Do and Don't" (page 134). Consistent habit training will prevent that in the long run.
• Allow natural consequences to dictate the reward: washing the dishes means Mom has time to bake brownies, controlling a nasty temper lets the whole family enjoy life, paying attention during lessons leaves more time for play!
• Let divine grace rule your interactions. "But the little fellow has really not much power to recollect, and the mother will have to adopt various little devices to remind him... she never lets the matter be a cause of friction between herself and the child, taking the line of his friendly ally to help him against that bad memory of his" (page 123).

If you are habit training all the live-long day, something needs to change. Mason's analogy provides a sense of proportion once you have gotten into the habit of habit training:

Let me say that the education of habit is successful in so far as it enables the mother to let her children alone, not teasing them with perpetual commands and directions... but letting them go their own way and grow, having first secured that they will go the right way, and grow to fruitful purpose. The gardener, it is true, 'digs about and dungs,' prunes and trains, his peach tree; but that occupies a small fraction of the tree's life: all the rest of the time the sweet airs and sunshine, the rains and dews, play about it and breathe upon it, get into its substance, and the result is––peaches. But let the gardener neglect his part, and the peaches will be no better than sloes (page 134).

So, how does it look in action? Mason provides several examples, the first of a little girl who wastes time lacing her boots (pages 120-121). Instead of going for the surface issue (lacing), she zeroes on the real problem: daydreaming. (In fact, if you have a dawdler on your hands, I suggest you read Inconstant Kitty while you're at it.) Mom gives her daughter a few words about how much more time she'll have to play if she can dress for outdoors in five minutes. She watches her without a word, with expectant, warm looks, and only a slight touch if the girl lapses into a reverie. The mother makes a point to join her for all those instances where the girl dawdles. As the child becomes reliable after a few weeks, mother fades out of the picture.

The next example is about shutting the door of a room (pages 122-123). Mother gives her son the request and explains why—for the comfort of others (living in a drafty, old house, I spent the winter training Pamela on this habit). She promises to remind him if he forgets, and she does with a pleasant voice for she knows crying out in exasperation will only encourage him to increase the distance between himself and the open door. She reminds him indirectly because he needs to think for himself: she glances at the door and says, "I said I should try to remind you." Each time, she varies her gentle cues to transfer responsibility for remembering to him.

How would I work on a messy room? First, I would plant the idea of why her room should be cleaner and then I would think of a contrary habit to replace the messy one. There are plenty of good reasons to clean a room: fire hazard, tripping over stuff, not being able to find stuff, relatives coming for a visit, etc. Whatever your reasoning, it needs to be brief and presented in a way that the teenager gets. Avoid the temptation to kill interest with a monologue. On the contrary habit, you might try brainstorming possibilities that have a relational twist to it that makes you an ally. Pick just one habit and work on that. When that becomes automatic, pick another habit. Here are some ideas:

• Every day, set the timer for ten minutes and the two of you clean up together. Have you ever seen the ten-second tidy on Big Comfy Couch? It can be silly, fun, fast, whatever. Turn on some upbeat music. Laugh at the disgusting things you find. Let the teenager pick the time of day: it needs to happen once a day before the deadline you set. Some kids are happier having some control.
• Let the teenager pick the tasks hated least. Let the kid who hates dusting vacuum the messy room and perhaps others while you dust. In real life, people bargain.
• Before watching TV, pour the stuff from one drawer into a clothes basket. Take it to the TV room and have a trash can nearby. In two weeks of watching television, the drawers will probably show improvement. Buy any organizer things and start consolidating the scattered stuff.
• Brainstorm something completely different than what I have here. With some kids, having an ally to bounce off ideas is half the battle. Having a say in the matter, bargaining, and feeling like everyone's opinion matters makes a difference. It needs to be a daily habit that is short and palatable. Over time, you'll see progress.

Sometimes, family dynamics make the situation worse. The parent who is frustrated about the mess needs to back off and assess progress in a month. It helps to reframe words that may be truthful but fan the flames. "Your room is complete disaster. WHY did YOU leave it that way?" will cause the strong-willed child to fire right back. Asking why for rhetorical reasons invites conflict. It may require experimentation until the least divisive words are found. Asking "Have you had a chance to do the ten-minute tidy?" with pleasant, relaxed nonverbals may be all that is necessary. If not, asking the parent with the stronger relationship to handle may be a better way.

The Bounty of the Outdoor Life

A variety of commitments have kept me from blogging. I have a little breathing space and yesterday's episode of the outdoor life gave me something worth sharing. Even though we've not homeschooled much in the past two weeks, we've made time for nature study. One can't help it with all the beautiful things blooming and the unpredictable weather.

This week, my heart has been turning over Carroll and Andy Smith's post on nature study and how to live it.

Education, including Nature Study, as Mason told the young lady whom she interviewed to attend her college, is about living. I have thought about it and I have asked myself the question, “Are these (mentioned above) ways of teaching Nature Study more about “doing” Nature Study weekly or are they about “living” Nature Study. We are to develop the habit of living fully and part of that living is relating to nature and knowing the places where we live, not just doing activities, even Mason inspired ones!"

Ouch! Even the best of intentions can end up focused on doing rather than being, especially during busy times like this past month!

I told Pamela I was heading out to do nature study, but I didn't tell her to join me. I wanted to see if she would come on her own initiative. While she was wrapping up whatever she was doing on the computer, I was outside, painting wisteria shoots. While we planted seven seeds at the same time, three have sprouted and emerged, one by one, and they illustrate stages of shooting up. I wanted to capture the differences a few days makes. The thought occurred to me that even seeds remind us of something about child development: it happens at their own pace and in their own time, no matter how much we yearn to rush it. By the time I began my notes, Pamela was sitting on the brick steps with me, admiring the seedlings.

Last night, a storm blasted our town, knocking out power for five hours, which killed our elderly algae-eating fish. Today, we found pieces of pecan branches on the ground. I picked them up and noticed a jelly-like fungus that I later learned goes by many names Auricularia auricula, wood ear, cloud ear mushroom, Judas ear, and a few others I bet. The fungus intrigued Pamela, and she began picking it off the branch. It almost looked like a cross between a golden raisin and a dark raisin after being soaked in water for a few hours. Pamela didn't seem to mind the squishy texture one bit!



For the past few weeks, I've been trying more "sight-seeing" with Pamela. By "sight-seeing", I mean looking around for something outside that is small and within reach and studying it carefully enough to supply a detailed description (pages 45-48). This activity trains observation skills and expressive language and builds vocabulary. Just like picture study, Pamela spent a few minutes observing the attributes of the fungus, knowing a narration will follow.



When she was ready, I asked her to turn around and narrate a description from memory, which I recorded in the video below.

I noticed a couple of interesting things.

• She illustrated nominal aphasia quite well: Pamela knows the word branch but what came out of her mouth was bench. Word retrieval glitches are part of her language challenges and thankfully she doesn't feel self-concious about it. We value communication over perfect speech. I understood what she meant and I respect her efforts.
• Her eye movements shift to the right most of the time when she is speaking. She is concentrating so hard on verbalizing what she saw that she can't look at me while figuring out what to say. Lateral eye movements are a sign of answering difficult questions. Just talking requires her to overcome the neurological barriers we feel when asked a challenging question.
• She processed well enough to look up at the tree when I pointed up. RDI gave her another channel of communication and understanding when language fails her. Tapping into nonverbals must be such a relief for her!
• When we had more of a conversation, going back and forth, she attempts to shift attention to me and then quickly looks away as she processes what she is going to say. When she realizes I am going to give her the name of what she described, she is better able to share eye contact because she doesn't have to talk. Once she starts repeating the word fungus, her eye contact becomes uneven again. It reminds me of how far she has come. Ten years ago, listening was just as challenging as talking for her!

Pamela sat down to draw the wisteria, and then she began writing. My jaw dropped when she asked me how to spell fungus, a word that isn't a regular part of her vocabulary. Her easy recall of a fairly unfamiliar word pleasantly surprised me--aphasia gets in way of even well-known words.

A thought struck me that the outdoor life’s bounty provided therapy for her as I alluded to in my ChildLightUSA blogpost called Rethinking the Culture of Therapy. Picking off the fungus worked on her fine motor skills, the domain of occupational therapy. Learning a new word in a contextualized way and expressing what she saw verbally was speech therapy. Touching this squishy gooey stuff—something she could not do fifteen years ago because of her tactile defensiveness—falls under sensory integration therapy. Anyone who has purchased materials for a sensory diet knows how expensive that stuff can be, and the outdoor life offered it for free! Keeping a nature notebook is an artful blend of science, language arts, and art.

If the child is left to himself he will think more and better, if less showily. Let him go and come freely, let him touch real things, and combine his impressions for himself, instead of sitting indoors at a little round table, while a sweet-voiced teacher suggests that he build a stone wall with his wooden blocks, or make a rainbow out of strips of coloured paper, plant straw trees in bead flower-pots. Such teaching fills the mind with artificial associations that must be got rid of before the child can develop independent ideas out of actual experiences.
~ Anne Sullivan, May 8, 1887