A Caterpillar Convention, or Deer Rubs Don't Run

For the third week in a row, we took a turn, a rather long and meandering turn, at the Santee National Wildlife Refuge. The ideas of Charlotte Mason (how this family learns and lives) and Relationship Development Intervention (how this family guides persons with autism) began criss-crossing in my mind like the muscadine vines along the path.

Take this pair of mushrooms for example. Our state has a long list of things that ought to be taught and when they ought to be taught. I don't have the time to read through it, but I began to wonder when they recommend children should begin to learn about fungi. Will their first brush with the fungus among us be as something to pick off pizza and toss out? Will they ever see mushrooms in the wild before they meet them in books? Will they have ever tried poking small puff balls to release spores and notice that the white ones are duds, and only the brown ones work sometimes? Will they have seen the many beautiful colors that toadstool mushrooms show off for the world? Will they have ever seen shelf fungi growing at the bottom of a trunk or a dead stump covered in all sorts of mushroomy looking things? What creates the most awe and wonder: reading about them in a unit on fungus or finding a yet another sort of fungal friend in one of many long walks?

Will they ever see a toad hanging out in a chestnut tree crotch before they study the life cycle of a frog in books? Find a dozen frogs dotting the swamp grass? Catch a toad and wonder why their palms feel wet?


Will they ever see for themselves that The Very Hungry Caterpillar and The Very Busy Spider both spin webs? They will if they ever see this caterpillar house in the woods or if they find web worm nests in trees.


Speaking of said caterpillar, will they find eggs before they meet the book? A wee, whispery voice in me wonders if seeing a spotty leaf in the woods one week, then finding it overrun with wee caterpillars the next, and seeing the leaf transformed into holy lace another week inspires more delight.

Will they feel the thrill of naming a cluster of critters—"caterpillar convention" coined by one child naturalist—before hearing a very proper term army? Will they know on their own that a caterpillar army hugging a limb might relate to autumn's chill?

Will they see all sorts of camouflage like the little critter on the left who cannot change colors but hides well in autumn spectrum colors? Or will they think animals that change color and shape is only to amuse little children?

What, you don't see the little amphibian yet?

That's a clue!

Will they ever plunge clean hands into a pond to grab a water hyacinth? Will they dirty their nails to rip open its bulb and find styrofoam that God hid inside? Do textbook writers find this idea worthy of making the cut?

Can children whose time is ground up by busywork, homework, worksheets, testing, and standardized exam preparation ever find out if one really can make ink by grinding up oak galls? How many galls will it take?


"But, wait!" exclaim my RDI friends. What does this have to do with autism and experience sharing? What does walking through the woods have to do with developing relationships with people in the autism spectrum?

My friend, the child naturalist, knows the answer. When looking at a deer rub, he shared, "Deer rubs don't run!" He knows that the woods are full of wonderful things to see and know. It takes time to hunt crickets and catch them in a bug glass. Deer rubs take less time to catch because they don't run. Two years ago, when my friend's days were busy with traditional schooling, time was precious. He never had enough time to know all the things he longed to know. Sometimes, his parents took him and his siblings out of school to go on raptor rescue adventures. But, he still never had his fill of time.

It takes lots of quiet time and long experience in the woods learning how to provoke the antlion larva just enough to come out of its home to fight off intruders. Explaining too much too soon destroys the pleasure of seeing it unfold after long hours of exploring. Pumping buzz words into a child and attempting to extract said words steals joy and mystery from the experience. Pushing through the trail quickly while an expert explains all the important stuff on a field trip pushes out time for rabbit trails. Experience sharing is not about getting the job done and all the stuff known. It is about meandering and musing like the endless weaving of muscadine vines.

It also takes someone more experienced in the ways of the woods to guide. My friend spent many long hours learning things as a child naturalist. She carefully chooses when to share her knowledge (how to turn a ginormous beetle on its back and hear if its a clicker beetle) and when to let her children discover for themselves. To know what time is ripe for exploring and discovering a new thing without a lot of words. To know what time is best for supplying a well-timed word, for holding back an answer to foster wondering and yearning to know, for wandering off the path, for returning back to it. Too many words, too fast a pace, too few visits, too many prompts destroy the wonder of the experience.

I watched this little one pet his first frog and caterpillar today. We had to slow down our pace of interaction for him to keep up. When we did, his joint attention was lovely. Rather than flood his mind with blitz of correct jargon explaining everything a preschooler should know about frogs, his mama elaborated on a familiar play he enjoys. "What does a frog say? Ribbet!" Only then, after carefully studying the amphibian, after his mama quietly coaxed him, after slowly, but steadily reaching his hand toward the frog, did he finally pet the critter.

What life lessons would falter if she had goaded him to touch it, forced compliance, pushed out his time to wonder and ponder?

Even the most nontraditional homeschoolers know when to move on. We came across a garter snake stretched out along the road in a patch of sun. When enough big, loud, wiggly folks ventured too close, the garter snake suddenly sprang into action and sidled to a safe place to hide from the big bad world (just as a person with autism does when we rush too often). We watched the snake wind itself along a muscadine vine, moving onward and upward to blend into the security blanket of camouflage. We all longed to stay for a half hour and see how far it would go. The stomachs of the littlest ones rumbled. The minds of the adult ones knew time was short for we had a hard deadline in the afternoon (the feast of the mind to come). The more experience guides ushered the children along the trail because the time was right to transition.

We know that we will come back to our beloved trail and further our relationship with it. Another day may reveal how high garter snakes climb. Charlotte Mason often said, "Sow a thought, reap an action; sow an action, reap a habit; sow a habit, reap a character." To reap a relationship, we must sow experiences slowly, gently, humbly. When we do that consistently and carefully over the years with our children with autism, then relationships will weave in and out of their lives like the muscadine vines that bear the sweet fruit warmed by the patient sun.

Wish Granted . . . Abundantly and Lavishly

If you have ever read The Five Children and It, then you know the dangers of wishes come true. If you have never read that gem of a book, step away from the computer, find that book now, and enjoy!

Little wishes aren't so dangerous. On Thursday, Paperbackswap granted my wish for a science book that has been sitting on my wish list for two years. It is the kind of book that is truly worth only $15 new but, because it is out-of-print and sought-after in homeschooling circles, is going for $9,999 new paperback, $313 used paperback, and  only $82 new hardcover at Amazon. To think that I'm getting mine for the cost of media mail isn't really mind-blowing because the 48-page book truly isn't worth the inflated-pricing schemes. Perhaps, the tulip mania chapter in Hans Brinker and the Silver Skates isn't so far-fetched after all.

A friend of mine posted a photograph of the Io moth caterpillar about a month ago. This creature is so amazing that I wished to see it in real life. Yesterday, when we headed out to Santee National Wildlife Refuge, my request was granted. The children spied one soaking up sunshine, while resting on the railing of the boardwalk. My cellphone doesn't do it justice, but the picture was clear enough to earn a confirmed sighting of the Io over at BAMONA. Apparently, I'm not as weird as I thought because my sister-in-law, who is so much more hip than I ever was or ever will be, had a cool shot of a caterpillar from her yard on her cell phone too!

I've had another wish, a very long-term wish, really a prayer that finally came true yesterday. In our third year of homeschooling (back in 1997), we joined a homeschooling co-op and loved it. Two things happened in 2000 that changed everything: we moved to another state (and have moved several times since) and we shifted to a Charlotte Mason style of homeschooling. Our paradigm change meant that, while we still occasionally attended co-ops, they never really meshed with how we homeschooled.

Until yesterday!

Backing up a bit, since moving to Carolina, I have tried to start local Charlotte Mason discussion groups, but nobody seemed interest. One spring, a group of us starting meeting in Columbia, but it was such a long drive and life was so busy that it fizzled out. Often, my timing isn't God's timing, nor my plan His plan.

Then, in Spring 2011, God dropped a study group in my lap, or Phase I of my Heavenly Father checking off an item on my bucket list. Long story short, a friend came to me when she first started thinking of homeschooling and I let her borrow For the Children's Sake. Six months later, she plunged into homeschooling and never brought up Mason's ideas, so I assumed that our unusual ways weren't meant for her family. Six months after that, she began telling me that she and some of her homeschooling friends had been reading Mason and they wanted to get together so they could learn more. The first couple of meetings I did some "show and tell" so they could see the sorts of things Pamela and I did and the sorts of books we read. Then, we began plowing through Home Education and moved onto Towards a Philosophy of Education at the beginning of 2012, talking about habit training, exams, mother culture books, and other things CM. Pamela has show an interest in getting to know the babies of the families. During the past eighteen months, we have grown together in our friendship and understanding of Mason's ideas. One friend delighted me with a gift from the British Museum (her husband's favorite haunt in London when he isn't busy flying airplanes).

Last summer, we began hatching a new plan, Phase II of progress toward a bucket list. We all felt brave enough to launch an afternoon of enrichment, which we now call "The Feast". That may sound like an odd name for a homeschooling gathering. We are trying to avoid the "c" word because we are on the road less traveled. We hope to alert fellow homeschoolers we aren't quite what one expects in a typical co-op.

One of my friends said recently that the hardest part of homeschooling and living this way is that you feel sorry for the children who are cooped up in rooms, doing worksheets, reading textbooks, regurgitating facts, too busy to think, imagine, or dream. At that moment, I could finally confide that I felt the same way two years ago when we headed off on our raptor adventure together. I kept the thought to myself that they would probably enjoy the freedom of homeschooling. When she started homeschooling last fall, I kept the thought to myself that they would probably enjoy the Charlotte Mason homeschooling life. Now, her family as well as others in our area are traveling the same road and building relationships as we learn together and share a feast that is both abundant and lavish.

I will hold off on posting on how the day went. Instead, I will share how we spent the past two Friday mornings: walking the trail at the Santee National Wildlife Refuge. The first time we went, Pamela didn't know exactly what to do, surrounded by so many children. She joined in to see what we all found but didn't seemed to find anything herself. Yesterday, she collected sticks, which made me smile.

Juggling TWO Chatterboxes and NINE Wee Turtles

Last fall, a friend brought baby slider turtles to Pamela's watercolor class. At the time, I had wished we could have observed them from the very beginning when my friend had first started posting pictures of the nest on Facebook. Having learned my lesson, I contacted Pat right away when she announced the laying of more eggs in last summer.

On July 20, the mama came to my friend's yard in a golf course community, dug up some dirt, and laid some eggs. Pat heard crows in the neighborhood telling everyone of a pending meal. She grabbed her protective gear and covered the nest with two cages spiked into the ground and nestled in bricks strapped together. Without protection, crows, foxes, and raccoons might help themselves to an easy dinner.



A month later, my friend and her granddaughter opened up the nest to count the eggs and see how they were doing. Slider turtles usually lay eggs in May or June, so this mama was a bit off schedule. Pat found eleven eggs, and all looked well. We paid our first visit on September 7, the day we met the box turtle. Pat removed the protective gear but decided not to dig into the nest. Slider turtle eggs incubate for eighty to ninety days, so they were due to hatch between September 8 and September 18. We saw no signs of hatching (disturbed dirt), so, after taking pictures, we covered up the nest.



A week later, Pat dug around the nest to check the progress of the eggs. We were very nervous, acting like expecting parents. Were they all duds? Were they okay? What were we going to find? She carefully dug and dug, and then we saw four turtles. Since they were not strong enough to dig themselves out, we left them in the ground. We covered them up with dirt and protected them as before.



Pat had to go out of town, so she asked a neighbor to keep a close eye on the nest. She was hoping that the turtles would sit tight until her return on October 1. They did not and dug out of the nest while she was gone. On September 30, the neighbor had to retrieve the babies and bring them indoors, where Pat will raise them until they are large enough to make it on their own. Pamela and I headed over to her house and the first thing we did was to locate the duds. We wanted to make sure Pat's friend hadn't accidentally missed a live turtle. We found the remnants of the last two eggs.



Then, the fun began. Pat wanted to clean out a smaller bin with rocks where they hang out during the day. We helped her carry the nine turtles to a large bin, which Pamela called the "swimming pool." We watched them swim and measured them for their first well-baby checkup. They were 1 3/8 inches in diameter.

Pat, Pamela, and I watched the turtles swim, which I recorded. I was so excited at first that I could not help myself and chattered away, as did Pat. From a Relationship Development Intervention point of view, I did everything exactly opposite, talking quickly and barely giving Pamela time to process.

And yet, she kept up!



This video illustrates how well she engages in a shared experience. Her comments connect directly to the situations. She shares what she thinks. She manages to engage with nine turtles and add her part to a conversation with two chatterboxes. She pays particular attention to one aspect of animals that has fascinated her for a long time: the claws (which is why she thought of her dogs Loa and Arwen).

Awe and Wonder in Bible Study

Children should have the joy of living in far lands, in other persons, in other times—a delightful double existence ~ Charlotte Mason

Since January 2011, I have had the pleasure, yes pleasure, of teaching young children a Bible lesson in my church's afterschool program every Wednesday. I shared my first reflection in June 2011 and my second in November 2011. Our class spent the first few months of 2011 studying Moses and the school year of 2012 reading most of the book of Mark. We ended the year three chapters short of our goal. This year, we plan to finish one whole book of the Bible: Daniel. When I looked up June's post, I noticed a new comment in which Courtney asked the following question about coordinating Sunday school curricula:

We started from scratch and I have attempted to implement Charlotte Mason principles from the beginning. I have done a fair share of the leading, but others have definitely caught on to some of the principles. However, I need a break from creating the curriculum. My hope is to find a curriculum, tweak it to free it from twaddle, competition, teacher pleasing techniques, etc and equip our leaders to lead the children with this method. We have about 20+ children in one room ages 3-8. It’s been a challenge to get in a rhythm. If you have any words of counsel on where to find a base curriculum that would free me from some labor – I would be so appreciative.


Unfortunately, I am rarely happy with children's curricula, so I stick with finding high quality pictures to immerse them in a time period, maps, and the Bible itself. Our class rhythm continues to be the same: play time, snack, song (I downloaded a version of Something about That Name that doesn't drag), prayers, Bible lesson, activity (sometimes related, sometimes not), homework time, and play time. After 2.5 hours, the children leave physically, mentally, emotionally, and spiritually fed!

We continue to use The Illustrated Children's Bible for the Bible lesson (for free samples, go here). I downloaded a copy of of the Old Testament, but you can order it in a binder with CDs with the PDF files. Every week, I scour the Internet and download a couple of high-resolution pictures of artifacts from the time of Daniel to give the children background knowledge and to help immerse them in all things Babylonia. It typically takes a half hour to find suitable images. For example, to go with the first week, they studied and talked quite animatedly about soldiers attacking the Jews in Lachish and another of soldiers carrying Jews into exile. (I didn't go into depth on the fact that the Assyrians are the masters here, instead of the Babylonians, because of the quick change of hands during this period of history.) Seeing men, women, and children heading into exile personalized Daniel to them: they were astonished to see children their age having to walk five hundred miles in the desert to live in a new land with a new language and new customs. They noticed all kinds of details: the differences in clothing styles, vehicles, weapons, hair styles, etc.

I introduced them to King Nebuchadnezzar and showed them what a document looked like in his day. We talked about the clay tablets and how they were baked in ovens. One girl was amazed that you couldn't simply close a book. I told them about how the king had build an amazing palace that had fallen apart and had been covered in sand over the past two-thousand years. I showed them a picture of modern mud-brick palace built by the late Saddam Hussein next to the ruins of Nebuchadnezzar's palace. I told them that a hundred years ago some German archaelogists started finding these beautiful blue bricks that are very much like the tile we use in fireplaces and bathrooms today. They realized it was the gorgeous Ishtar gate and reassembled it in a museum. We talked about the animals on the walls and how many represented Babylonian gods, which were animals, planets, the sun, and moon. I showed them a second picture to give them an idea of the scale of the gate, and they were impressed!

I used to worry about how children unaccustomed to a Charlotte Mason style of education would react. Looking at those pictures and talking about them together incorporates the study of architecture and sculpture, which I am quite sure is neglected in their education. I offer it in a familiar medium, television, but done in an active, not passive way. I created two slideshows into a DVD (which one could also do in powerpoint): one has the pictures just described and the other has pictures of the pages of the Bible we are reading, cut and paste from the PDF file). One benefit of using a DVD is that, if a child misses a week, I can send home the DVD for them to catch up on what they missed the previous week. I simply show a picture and pause it while we have a grand conversation. They love it! The children learn so much just by seeing the artifacts of Daniel's world and noticing little details. It takes me a half hour to prepare the weekly DVD.

Last week was a wonderful example of how our afternoon goes. Children arrive in stages because different schools have different dismissal times. They go to the gym or youth room to play. Once everyone has arrived, we go to the fellowship hall for a snack. Then, we go to the classroom, sing a hymn or two, and pray. Then, we have a conversation about last week's lesson. Typically, I have a student who missed last week. So, I said, "Laura missed the lesson. Who would like to tell her what happened to Daniel last week?" Hands flew up. I picked her friend, sitting next to her, who gave a narration that went something like this

There was this guy. I don't remember his name. [Someone pops up with, "Daniel!"] Yeah, Daniel. This king with a really long name came to his city and chose only young men who were healthy and good-looking. He took them to the palace to be a servant. First, Daniel had to learn a new language and learn to read and write. When he got to the palace, this guard told him he had to eat the king's food, which was really bad for you. But, Daniel only wanted to eat vegetables and water, but the guard wasn't happy. He might get in trouble with the king, and, when the king got mad, he cut off people's heads. Daniel asked for a test of ten days. If he and his friends looked bad after ten days, then they would eat the king's food. We don't know what happened because Mrs. Tammy stopped reading at that part.

Hands flew up after her narration, and I picked a boy who was eager to share his part, which went something like this,

Daniel used to live in a city that was surrounded by walls. When the king's army came to Daniel's town, soldiers were everywhere. Nobody could leave the city. They didn't have guns. They had bows and arrows and, if anyone tried to leave the city, the guards would shoot like this. [He demonstrates shooting an arrow with a bow.] Daniel's city didn't want to run out of food, so they gave up.

Another boy added something like this,

We wondered why Daniel didn't run away when he was in the palace. I thought he should have waited until everyone was sleeping and then sneak out. But, Mrs. Tammy said that they had soldiers guarding them even in the middle of the night. Somebody was always watching them.

Reminded of another reason why, I added, "The other problem, Laura, is that the king's palace was five-hundred miles away from David's old home. He would have to survive in the desert to get home."

You might think that narrating comes easy to these students because they have been in my class for almost two years. This year, we have three new boys. They eagerly added smaller parts. One said something like,

Hey, do you remember the soldiers had pointy hats? Daniel's people had round hats, so that's how you can tell them apart!

Another added,

Oh, yeah! And, the king didn't take just young men. There were children, boys, and girls. Some were so little they rode the carts!

Another remembered,

Mrs. Tammy showed us pictures of the king's city. They had a wall with pictures of animals that were their gods. They had all kinds of gods, not at all like God.

My class is an even mix of public and private school children, so they  have never been in a Mason setting outside of my class. I teared up watching them narrate so happily and so beautifully. They have their own way of expressing their thoughts, and they enjoy sharing what inspired them the most. They were so excited to talk that I had to help them transition to the Bible lesson.

First, we studied the pictures on the television screen and discussed new artifacts from Babylon and a map of the area. Then, we read Daniel 1:15-2:12. I still read all the parts by the narrator and hand out the dialog to volunteers. We pause from time to time and the children tell what they notice or ask questions. Several gasped when I read Daniel 2:2, "So the king called for his fortune-tellers, magicians, wizards and wise men." They cried out, "Wizards? Magicians?" So, I asked them a question to encourage them to think it through. "What if you could bring King Nebuchadnezzar to our time and let him watch television? What would he think about it?" One answered, "He would think we were magicians." I added, "We know that televisions are based on science, but Nebuchadnezzar wouldn't." I stopped at a cliffhanger to leave them begging for more: Daniel 2:12, "When the king heard that, he became very angry. He gave an order for all the wise men of Babylon to be killed." The kids were properly horrified because they realized that Daniel and his friends were on the hit list.

After the lesson, we did an activity: they painted the clay sculptures they made last week. In the background, Gershwin's Rhapsody in Blue played. Pamela and I are studying it right now, and David's marching band is including it in their half-time show. Sometimes, our activities have nothing to do with the Bible lesson. This week, I'm brought in our temporary pet snail for nature study. Then, the children did their homework and had free play after that.

I want them to enjoy wondering about the Bible. The big question they asked me on the first day concerned Daniel 1:2, "The Lord allowed Nebuchadnezzar to capture Jehoiakim king of Judah. Nebuchadnezzar also took some of the things from the Temple of God. He carried them to Babylonia and put them in the temple of his gods." One of them popped up with, "Why did God let that happen?" Another was angry about the stuff he stole from the temple for, when we studied Moses, we built the tabernacle out of cardstock. I never answered the question about why God let it happen. I told them that, by Christmas, they might figure it out. Every time we come across the word "wise men" I hold my tongue because they will experience more awe once they make the connection themselves.

A Turtle's Eye View of the World: A Metaphor for Autism

Last fall, Pamela and I made a new friend who loves turtles. The other day, Pat found an eastern box turtle (Terrapene carolina carolina) stranded in the road, closed up tight in her shell, too frightened to head out of traffic. She took the turtle home to make sure all was well. We paid our first visit on the day our friend found it (September 7). We only observed the shell for the turtle was not hungry enough to risk peeking out of her refuge. We took pictures, answered some of the questions from the Comstock book (our stock in trade for nature study), and later made an entry in our nature notebooks at home.



Seeing the turtle shut up tight in its shell reminded me of what happens to children with autism. Something in the environment (a sound, a small change, too much information, a fast pace) makes them feeling like a turtle trying to cross the road with cars racing by. If they cannot retreat into a shell (quiet place, repeated words, rocking), they meltdown and tantrum or they completely zone out. If we try to make the world too predictable and safe, as my friend Di described in her recent blogpost, then the whole family ends up boxed up into a very small world.

Many people wonder how to help children in the spectrum find the world less frightening. Our second visit (September 14) to the turtle illustrates how to make this happen. After a week of living in an alien world, she—her brown eyes, flat bottom shell, and high upper shell are clues to her gender—learned to come out of her shell. Because she could see us through the clear, blue bin, Pamela and I created a little bit of anxiety. She slowly crawled into one corner of the bin and turned her tail to us. She tolerated us as long as we sat very still. If we moved too much, the turtle would move away and, at one point, she retreated into her shell.

Di posted a graphic with guidelines for encouraging meaningful interaction. We applied these ideas in our observation of the turtle: stay calm, be aware of sensory issues, stop the action, and speak less. When Pamela feels anxious, I force myself to remain calm and quiet, even if a storm is blowing and the power has been out for awhile. When she seems inexplicably upset, I pay attention to the environment to search for a cause, so we can deal with it. When she blusters because I am not giving into an unrealistic demand of hers, I stop the action. Any action I take causes her to increase her intensity. Doing nothing helps her realize the ineffectiveness of storming. When she is ready to process so we can figure out a realistic option, I match my verbal communication to hers so that she feels like an equal partner. I communicate with my face, hands, and body because she can understand and express herself in like manner (Relationship Development Intervention helped us achieve that miracle). I speak with declarative language (rather than issuing commands) because she needs to think for herself.



Slowly, the turtle emerged. She was hungry for she hadn't eaten the day before (typical, for turtles). Pat dug up four juicy earthworms and plopped dinner into the bin. The first worm wiggled fiercely but the turtle ignored it. We sat quietly and waited and waited and waited and waited. Ants crawled across the pavement. And we waited. Hummingbirds buzzed overhead. And we waited. Pat dropped a juicy worm right in front of the turtle, who sat on the wiggly thing. Mosquitos sucked our blood. And we waited. During this downtime, we reminded Pamela to sit still and avoid swatting the bugs. We reminded ourselves to sit still. The turtle took forever to find her courage.

Finally, we noticed her eyes tracking a worm. She slowly shifted her body in the right direction. A spider walked near Pamela's hand. And we waited. The turtle's head stretched out slowly. Then, very suddenly, very snakelike, she struck! She snapped open her jaws, picked up the worm, and gummed it to death for turtles lack teeth. She was a messy eater, leaving pieces behind. One bit of worm hung out of her mouth like a cigar. She was full after two worms.

People with autism process at slower speeds. While many can react to prompts quickly, you have to give them plenty of time to observe, process, decide, and respond. My friend Di had to wait 45 seconds for her son to respond when she first changed her communication style to foster interactions. His processing speed has increased dramatically now that they have been working on this for awhile! Pamela needs about 10 seconds to think through simple tasks. If language is involved, it takes much longer. When you give our kids the time they need to process and think, they develop the habit of processing and thinking and learn to do it more efficiently.

Pat decided to release her turtle into the woods near her home. Pamela watches and wishes her well.

Helping Others Find Joy in Math

Last spring, a friend of mine from church came to me a few weeks before finals for help in high school mathematics. Jay is a bright student who has earned passing grades in math by memorizing formulas without understanding and cramming for tests. I usually prefer to peel back the layers of knowledge to what a student understands and build up from there. With so little time, I had to focus on major habits and enough understanding to pass. I prioritized teaching what would give her the most bang for the buck in the little time she had before finals. She passed but we both knew she could have done better.

Even though she is not taking precalculus until next semester, Jay came to me for help this week. She has set a goal of earning a 90 or above in all her classes in her last semester of high school. We both know she has gaps in understanding, relies too much on the calculator, and mixes up formulas. Rather than revisiting difficult topics while under the pressure of mastering new material, she is going to build a solid base now when she can relax and learn.

I asked Jay what concept she would like to study first. "Negative numbers! I know what to do when you multiply and divide. I get confused about what happens when you add a negative and a positive. I never know what the sign should be."

I drew a number line, and we did some simple problems. I said, "Explain to me what happens when you add two positive numbers like +5 and +3." I illustrated the problem on the number line.



She told me, "When you add two positives, you head in a positive direction. They stay positive."

We did the same thing for adding two negative numbers like -5 and -3. Jay gave a similar explanation of heading in a negative direction and staying positive.



Then, we worked on the piece of the puzzle that had long mystified Jay. I plotted adding +5 and -3. Jay looked at it for a moment and said, "Oh, I see. The negative number is not large enough to cross zero. So, the answer is positive."



We then studied what would yield a negative answer: adding +3 and -5. Jay smiled, "I get it! Since 5 is larger than 3, it is going to cross zero and the answer is negative! Wow!!"



We continued pursuing this line thought by adding -5 and +3 and then -3 and +5. I gave her a couple of problems to make sure she could apply what she understood.





I like to give students other ways to understand a problem. Jacob's Algebra offered an alternative view of adding positive and negative numbers. The book depicts a number as a set of circles. A positive number has circles with no filling (white), and a negative number has filled-in circles. The picture below shows two numbers: +6 is the first row of circles and -6 is the second row. I drew a picture like this for Jay, and she saw immediately how adding them together results in an answer of 0.



I drew another picture with -2 in the first row of circles and +2 in the second row. Again, she saw immediately how adding them together results in an answer of 0.



The picture below illustrates -5 plus +16. Jay had no trouble explaining that the answer had to be positive since the number of positive circles is greater than the number of negative circles. Since there are 11 circles left, the answer must be 11.



The picture below illustrates +3 plus -8. Jay easily explained that the answer had to be negative since the number of negative circles is greater than the number of positive circles. Since there are 5 circles left, the answer must be -5.



I assigned several addition problems for her to solve. Jay got them all correct. Even better, she smiled the whole time because she didn't have to guess the sign of the resulting sum. Then I picked a more difficult problem from the book:

1 + -3 + 5 + -7 + 9 + -11

She drew a picture like the one below and said the answer had to be -6.



Since Jay needs practice with mental math, I processed the problem aloud in a different way for her, "Do you see that 1 plus 5 plus 9 is 6 plus 9, and that is 15?" Then, I wrote down the number 15. I said, "And, -3 plus -7 plus -11 is -10 plus -11 and that is 21." I wrote "+ -21 = -6" to finish the solution. To practice both ways of adding positive and negative numbers, I assigned the next problem in the book.

-1 + 3 + -5 + 7 + -9 + 11

She drew the following picture and told me the answer was 6. Then, she did some mental math and wrote down the equation, "21 + -15 = 6."



Then, she paused and said, "That's funny. You get the same answer only the signs are reversed. Oh, wait! All the signs are opposite. Had I thought about it, I wouldn't have had to do all that work. That's cool." She warmed my heart when she said, "I want to keep this paper! It finally makes sense!"

Before closing, I will summarize some of the important ideas that help people who may not care for math to experience joy.

• Figure out what the student knows. I usually do this by asking them to explain things to me. When they get, I ask a question about a more foundational concept until I find the student on solid ground.
• Take small steps of logic by demonstrating the process with objects or in pictures. Too many students think in memorized procedures and formulas but cannot explain the rationale behind them. Once they understand why, procedures and formulas become effortless.
• Ask the student to explain the rationale behind what they are doing. If they cannot explain it, then they lack a solid foundation.
• Show how to solve a problem in different ways. It helps students see how math can be a creative process.
• If a student struggles with mental math, assign problems with simple numbers and avoid using the calculator. I find that students who rely on calculators for simple problems usually lack number sense.

Breathing the Rarified Air of a Mountainous Land

David is a student at Charleston Southern University, taking 16 credits in your average freshman schedule. The main reason why he dipped back into math last spring was that his ACT math scores were less than ideal. I will refrain from sharing the actual number, but the folks at CSU felt he needed remedial math. I agreed. I suggested he retake the ACT to boost his math score and avoid a class in which he would study for long hours but not receive any credit. He pointed out that his college of choice had already accepted him. I did some digging and let him know that CSU offered a math placement exam. If he passed the test, he could take a regular math class. He saw wisdom in that idea and spent the spring and summer diligently studying algebra.

Why would the son of an engineer (my husband has three engineering degrees) and a mathematician (I have two math degrees) fare so poorly in math? For a long time, I have suspected that, while David has the intelligence to do well in STEM subjects, he prefers more creative outlets. He tackled math by memorizing and cramming, even though we had always encouraged him to think. Since he always seemed to struggle with math, I could never prove my hypothesis.

Long story short, David studied Jacob's Algebra from cover to cover. He was surprised to find the book, both interesting and readable. David told me that Jacobs teaches you how to look at problems logically and to understand algebra. For years, he focused on cramming formulas, which he mixed-up on comprehensive tests. Rather than feeling anxious when he couldn't think of a formula, he learned to relax, write down ideas, and reason his way through problems. He also realized that asking for help is not a sign of weakness and reached out rather than stare at a page until his brain froze.

I think Jacob's Algebra is a fabulous book for many reasons. The author has not updated the book since the late 1970s. Unlike textbooks of today, it does not have ADHD, distracting students with irrelevant graphics, sidebars, etc. When I peeked at David's college math text, the glossy pictures of movie scenes in which mathematics played a role nauseated me. The money wasted on pop culture would be better spent showing students how to reason. I personally think all the eye candy in the world will not attract students to this much maligned subject. Force-feeding formulas to children without understanding and encouraging them to cram to get ready for standardized tests make them loathe math. Jacobs carefully unfolds algebraic thinking through brain teasers and puzzles before getting abstract. Learning to reason and apply logic changed David's attitude toward math.

He also struggled with fluency and accuracy. The month before taking the placement test, I wrote algebra quizzes based upon ACT problems. I had him run through problems at Khan Academy to practice fluency and accuracy. Because he understood more, he had to memorize less. He could focus his attention on practicing and improving these habits.

The end result is that he earned an 81 on the placement test and is taking freshman math. But, even more important, he enjoys mathematics now. When asked what about his favorite classes, he put English Composition right at the top followed by math: "Actually for some reason I'm preferring the math problems over reading... Pretty odd, huh?" He dislikes reading because he is on a steady diet of typical textbooks (yawn). He even added, "I might take some extra math classes as soon as I get all my required math courses out of the way."

This fall, I have another opportunity to win another student over to mathematics. I will share the beginning of her story in my next post.

Another realm open to Intellect has an uninviting name, and travelling therein is difficult, what with steep faces of rock to climb and deep ravines to cross. The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving, though some people find it too rare for their breathing. It differs from most mountainous countries in this, that you cannot lose your way, and that every step taken is on firm ground. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth. ~ Charlotte Mason