Sunday, November 3, 2013

Multiplication using Charlotte Mason's Methods

"There is no royal road to the multiplication table; it must be learnt by heart.  This is a fact which faces every teacher of elementary arithmetic, and which each must prepare for in the best way possible" (Irene Stephens, The Teaching of Mathematics to Young Children. p. 10).

Irene Stephens* was a lecturer in mathematics at Ambleside and author of the handbook Charlotte recommended for use both alone and alongside arithmetic books in Forms I and II (approximately our Years 1-6). Of course, most of us would readily agree with Miss Stephens’ play on Euclid’s statement that, “There is no royal road to Geometry.” Indeed, a few months ago I was reviewing a course in Introductory Algebra when, in the first lesson, the teacher instructed the students to take a week or two to learn the multiplication tables if they had never properly done so before continuing on with Algebra.

Let’s take a look at the second part of Miss Stephens’ assertion and how students were prepared for learning the multiplication table by heart “in the best way possible.”  To begin with, children were given a full year to thoroughly examine and become comfortable with the numbers 1 to 100. As the numbers that the child worked with grew larger, the combinations grew more plentiful and a natural overlap with multiplication occurred such as 'two fours make eight' and 'three twos make six' (for a brief overview of elementary arithmetic see my posts Charlotte Mason and Math and Math and the Write Stuff or for a thorough exploration of elementary arithmetic see Mathematics, An Instrument for Living Teaching).

Now we'll fast-forward to the point in our scope & sequence when multiplication is formally introduced.  Using Charlotte Mason's methods, I've scripted a lesson so you are able to see how it would have looked in Charlotte's classrooms and how it could look in your own. You will need a blackboard or dry erase board for yourself as well as a personal slate or dry erase board for each student. The student will also be using his or her coin bag full of pennies and dimes.  To begin:

Introduce multiplication as repeated addition through simple, interesting problems using coins or other manipulatives. The student's answer is in parenthesis and you will be writing on the board until they are instructed to do so. Thus, begin by saying:

John had 2¢ and a friend gave him 2¢ more. How many cents had he then? (4¢)
How many times do we have 2 cents? (2)

If 3 children had 2¢ each, how much had they altogether? (6¢)
How many times do we have 2 cents? (3)

Then you write on the board:

2 + 2 + 2 = 6.

You bought five gumballs at 2¢ apiece, how much did they cost altogether?

Write it up on the board:

2¢ + 2¢ + 2¢ + 2¢ + 2¢ = 10¢  (or 1 dime).

Several examples are given before suggesting that it may be written down more shortly:

We can write this more shortly. Let’s take a look at our last problem. 

You bought five gumballs at 2¢ apiece, how much did they cost altogether? (10¢ or 1 dime)

How many times do you have 2¢? (5)

Write it up on the board: 

2¢ × 5 = 10¢.

So the  "× 5" means multiplied by 5 - That is, each of the quantities is to be taken 5 times, so that 2¢ × 5 means five 2¢.

The sign “×” - read multiplied by, means the first is to be multiplied by the second; so ”2 × 5 = 10" shows that 2 multiplied by 5 = 10.

The “×” sign may be read times.  2 x 5 shows 2 is taken five times.

So 4¢ × 3 would mean, 4¢ multiplied by 3 = 12 pennies,or  4¢ times 3 and so on.

Let’s work a few problems on your dry erase board using the multiplication sign.

If 4 children had 5¢ each, how much had they altogether?

5¢ (multiplied by or taken how many times) × 4 = 20¢      (2 dimes).

Emma has 3 ribbons and her sister has 3 times as many. How many ribbons does Emma's sister have? (3 x 3 = 9)

Now let's take our coins and make a multiplication table for 2. Let's make 12 rows of coins with 2 coins in each row. Let your students answer:

2 and 2 are (4), and 2 are (6), and 2 are (8), and 2 are (10), and 2 are (12), etc. 

How many 2s are in 10? (5)

So it is right to say 2 x 5 = 10

If Lego bricks cost 2¢ apiece, how much would 6 Lego bricks cost? (12¢ or 1 dime and 2 pennies).

2 x 6 = 
2 x 3 =
2 x 7 =

How many 2s in 14? How many 2s in 6? How many 2s in 18? etc.

This concludes the introduction to multiplication and the lesson ends with five minutes of rapid mental work, incorporating everything studied in their arithmetic lessons up to this point. If all has gone well, your child or student will be ready to construct a written multiplication table for their next lesson.  The "how-to" will be in my next post.

*Do you love hearing people's stories as much as I do? Irene Stephens was actually a resident of Madras, India, traveling to Ambleside in 1911 (happily, coinciding with the census) where she lectured in mathematics while her handbook was published.  She stayed at Greenbank Cottage, Ambleside (perhaps like the one pictured above) with a house painter, George Alldis and his wife, Susannah, whom had no children. Hats off to the Cumbria Family History Society for helping me with my research.

Tuesday, October 22, 2013

Step Inside Our Story

Autumn rates right up there with spring for wanting to just close the books and head outside. Since we are spending most of our days outdoors in both work and play, I thought you might enjoy a tour around our grounds and then I'll catch you up on our doings.

Our home started out as a storefront for Berkshire Glass Works in the mid-1800's. The exceptional sand of our county (97-99% silica) makes for very fine glass and it is reported that Tiffany himself visited the Glass Works in order to pick the tints he incorporated into his designs. Above is some of the glass slag our youngest pulled out of the ground near a creek bed. 

In the 1930's, New England artist Leo B. Blake lived here with his family. He added a studio with a vaulted ceiling and celestory to the home, carving his initials on the outside of the addition that is now our living room. He also added a wall of windows to the north side of the barn which served as his summer studio.

Our son, Luca, was thrilled that his initials are the same as Leo Blake's. Here he sketches at a favorite spot we call "reading rock."

The original outhouse now functions as a garden shed. A hand-lettered sign by Mr. Blake still hangs on the inside though, warning of the dangers of improper cigarette disposal.

 A view of the garden shed and one of the former barns which is now my husband's tool shed. The back is filling up nicely with wood for winter heating.

 Another view of reading rock with a path that heads west along the brook.

This is the view heading east along the south side of the brook. It is home to calendula, flowering raspberries and jewelweed to name a few species. The path leads into the woods where the original rock fence remains.

The barn looking from the south side of the brook which houses my writing studio along with our chickens. Our hope is to put an alternative print-process photo studio in the barn for my husband next year as well. 

Obligatory picture of our 5-month-old Standard Poodle puppy, Aquila, who has become an important part of our family.  Watching him see the wonders of God's creation for the first time has renewed our own appreciation for the beauty in it all.  Now for some current events:

Currently attending: A puppy training class with Aquila. If you've ever seen Cesar Millan, known as the Dog Whisperer, you'll know what we are going through. The class is actually more human training than dog training and, with five families and five very cute puppies, would make for great reality TV.

Currently reading: Galatians. If this letter Paul wrote to the Galatians has been as influential to your life as it has to mine, you may enjoy John Sheasby's Introduction to Galatians. Put the Sheasby's podcasts on while doing the dishes and you will love doing dishes.

Currently planning: A live-action role-play fete at our home this Friday. A large group of intrepid kids will spend the day free-ranging here for a game of "Town," which Max describes as a "glorified game of house...only outdoors." Think "Shire of Middle-earth" with Nerf-weapons. I'll try to get pics.

Until then.

Saturday, October 5, 2013

A Mother's Debt

Whilst cleaning up the computer, I found this draft that had never been posted. Though it was begun long ago, happily the sentiments remain fresh and true.

Mushroom hunting!

Is it any wonder it's called Tanglewood?

Living in the past a bit as I searched for pictures to share with our Compassion child so thought I'd share them with you as well.

I have been living in the past as I write, realising how much happiness I owe to the vision of one woman. My case no doubt is similar to many others, scattered all over the world. Others will write of Miss Mason's work from the point of view of the trained teacher, but how much greater is the debt of the mother who without any training at all, could teach her children through the method that Miss Mason had worked out. It was she who made the impossible possible, who shewed us term by term what books to use and how to use them, who taught us to take the children straight to the fountain head and let them learn from the books themselves. It was she who realised what home education might become, who changed the whole atmosphere of the home schoolroom, who inspired us for our work and gave us the power to carry it out; a pioneer who blazed the trail that many of us followed with keen enjoyment and grateful hearts. -E. M. CAPRON

Monday, July 8, 2013

Our Barefoot World Atlas app is free to download this week!

Personally, I don't keep a lot of apps on my phone but among those you will find is the Barefoot World Atlas. As an ambassador for Barefoot Books, I'm thrilled to let you know that the Barefoot World Atlas has been named by Apple one of the top ten apps from among the nearly million available in their App Store. To celebrate, you can Download the Barefoot World Atlas for free this week! The Barefoot World Atlas app is based on the Barefoot Books World Atlas, written by Nick Crane and illustrated by David Dean. 

If you are really into celebrating, select Travel the World titles are 25% off now until July 14th using the code APPLE. Among them are some of our family-favorites, including world-renowned flautist Guo Yue's Little Leap Forward, a semi-autobiographical story about a boy growing up during China's Cultural Revolution.

 Isn't it good to know living books are still being published?

Thursday, June 13, 2013

Math and the Write Stuff - Charlotte Mason-style (Part 1)

 Would you be surprised by the suggestion to place Handwriting or Handcraft directly after the arithmetic lesson? After all, in a Charlotte Mason education variety in the order of lessons is important so neither the brain nor body tires. 

This makes sense when arithmetic lessons are primarily oral. Rather than being worksheet driven, with a child working a page of exercises such as the one pictured below, the actual writing of sums is used very sparingly in the early years.

Not CM-friendly.

Let's take a look at the first few lessons in Elementary Arithmetic and how the reading and writing of digits are to be taught. We'll use Charlotte's writings as well as the booklet "The Teaching of Mathematics to Young Children" by Irene Stephens, Lecturer in Mathematics at Ambleside, as our guide.

In Charlotte’s schools, elementary arithmetic lessons were referred to as Numbers or Sums, referring to the child’s investigation of each number by working out addition and subtraction sums involving its use. For the first lesson, the number one is taken and the child is asked to point out one of something in the room, that is, anything that exists singly. One door, one nose, one pencil, etc.; following this, the symbol for one is learnt. Whenever we see a stroke “1” we know that it stands for one of something” (Stephens, 1911, p. A2).

The children pick out the ones from among a group of figures, then, at last, they have the chance to learn to write the symbol one. Here you will notice a striking similarity between Charlotte’s initial lessons in Numbers and her first reading lessons.

The child should be taught from the first to regard the printed word as he already regards the spoken word, as the symbol of fact or idea full of interest (CM, Vol. 1, p. 216).

Interest at once; he knows the thing…and the written symbol is pleasant in his eyes because it is associated with an existing idea in his mind (CM, Vol. 1, p. 218).

The initial writing of numbers would have been done on the child’s slate with chalk but if you prefer a small dry-erase board, feel free to use that. Just as in Charlotte’s early hand-writing lessons, neatness and accuracy are desired, with the child getting their written symbol for one as straight and perfect as they are able.

Next, the numbers two through nine are taken systematically in the same way, with little sums involving each number given orally.  Numbers are first written out on the child’s slate and then a gridded notebook is introduced with each number occupying a square.

Just as you wouldn't begin handwriting lessons with a college-ruled notebook, your child will need a larger grid than the 4sq/inch or .5cm square most readily available for purchase at your local office supply but the right size can be found at certain Waldorf supply stores. A Toy Garden carries these 9x12 math notebooks which are 2.5 sq/inch, while Paper, Scissors, Stone carries a slightly smaller size math notebook (8x10) which is 3 sq/inch. Both notebooks are small-hand-friendly.

Can you imagine, rather than being bent over a worksheet for 20 minutes, the opportunity to write in a math notebook being considered a real privilege by a child? In the next post, we'll take a further look at the act of writing in these early stages of elementary arithmetic and how it fits into the later years of mathematics.

Friday, June 7, 2013

Charlotte Mason and Mathematics

Through a series of steps Max is led to discover the rule for finding area.

Hello, best beloveds. We have been humming along with two new bee hives, final exams and a succession of enchanting house guests (perhaps entertaining angels unaware) whilst also working out how to best relay the joy of Charlotte Mason Mathematics in an hour at the Living Education Retreat in July. All endeavors which require tending but result in much pleasure.

Being thoroughly embarrassed by the number of visits my first post on CM and arithmetic still receives, I've resolved to include you in what I have learned since then. We'll start with an introductory overview which first appeared over at Simply Charlotte Mason's blog then, in the following weeks, I'll share some of what I consider the most fascinating practical aspects of her methods, such as the introduction of writing in elementary arithmetic and its place in CM math, money as manipulatives, how to learn the multiplication tables without hating them and what handicraft and geography have to do with math.

The study of mathematics falls within Charlotte’s definition that “education is an atmosphere, a discipline, and a life.” Are you surprised? Its study was not an afterthought nor a "follow my methods in every subject then tack on a math curriculum."

Charlotte valued the study of arithmetic primarily for its use in training mental and moral habits, including accuracy, attention, careful execution, neatness, and truthfulness. 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. More on beauty and truth in a future post.

Now let’s take a brief look at how mathematics are taught in a CM education—because without living teaching, that sense of wonder might not be awakened nor the desired habit training take place.
The Early Years—Before the age of six, a child’s education is by means of his senses, natural environment, and unstudied games. Direct preparation for mathematics in these years is considered not only undesirable but detrimental. Yikes! Detrimental? That's right. Charlotte said it.
Elementary Arithmetic—The formal study of arithmetic begins at about six years of age and is characterized by thorough, careful work in which the children make discoveries for themselves. Its study follows Charlotte’s basic principles of short lessons with concentrated attention.
Manipulatives—Though the term math manipulative did not exist in Charlotte’s time, the use of concrete objects as aids in conveying ideas is significant in her method of teaching arithmetic.
Some important points to remember:

  • All the manipulatives you need can be found in your own home—beads, buttons, and craft sticks to name just a few. A variety of simple objects should be used rather than a single specially-designed manipulative so the child doesn’t form a hard-and-fast connection between the math facts and the manipulative.
  • Manipulatives are only a tool to the presentation or investigation of an idea. If a manipulative’s use requires too much teaching, it becomes more important than the idea it is to represent.
  • Arithmetic tables should not be memorized until the child proves the facts first through the use of manipulatives.
  • Allow your child enough time to work with the manipulatives but then progress to working with imaginary objects. Once the child can mentally picture the number, or has grasped the abstract, put away the manipulative until the introduction of a new concept.
“A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to ‘do sums’ on his slate” (Vol. 1, p. 256).
Luca works out double-digit addition using popsicle sticks in units and 'ten bundles.'
Mental Arithmetic and Oral Work—In Charlotte’s methods of teaching mathematics, written work is used sparingly. Mental arithmetic and oral work help reinforce math facts and vocabulary, plus they are instrumental in the training of good habits.
“Give him short sums, in words rather than in figures, and excite in him the enthusiasm which produces concentrated attention and rapid work. 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” (Vol. 1, p. 261).
While children advance in their understanding, the oral questions should always remain within their ability.
“Engage the child upon little problems within his comprehension from the first, rather than upon set sums” (Vol. 1, p. 254).
“Now he is ready for more ambitious problems: thus, ‘A boy had twice ten apples; how many heaps of 4 could he make?’ ” (Vol. 1, p. 257).
Some points to consider:
  • The oral questions we give our children should be engaging. For example, “How old will you be when your sister is four” will be more apt to fix your child’s attention than the same question given as, “Add 4 + 5.” 
  • Require your child to give fully worded answers in complete sentences for the most benefit. 
  • Along with oral work throughout the math lesson, consider following Charlotte’s schedule of five minutes of rapid drill at the end of the lesson or ten minutes for older children at another time in the daily schedule.
Careful Teaching vs. Careless Teaching—Charlotte felt that careless teaching—which includes offering crutches and failing to pronounce sums wrong—fosters habits of carelessness in children. In contrast, carefully graduated lessons, along with Charlotte’s methods already mentioned, foster the training of good habits.
“Arithmetic is valuable as a means of training children in habits of strict accuracy, but the ingenuity which makes this exact science tend to foster slipshod habits of mind, a disregard of truth and common honesty, is worthy of admiration! The copying, prompting, telling, helping over difficulties, working with an eye to the answer which he knows, that are allowed in the arithmetic lesson, under an inferior teacher, are enough to vitiate any child; and quite as bad as these is the habit of allowing that a sum is nearly right, two figures wrong, and so on, and letting the child work it over again. Pronounce a sum wrong, or right—it cannot be something between the two.” (Vol. 1, p. 260).
“Therefore his progress must be carefully graduated; but there is no subject in which the teacher has a more delightful consciousness of drawing out from day to day new power in the child. Do not offer him a crutch; it is in his own power he must go” (Vol. 1, p. 261).
Living Math Books—Charlotte believed mathematics fell outside her rule of literary presentations. She stated:
“…mathematics, like music, is a speech in itself, a speech irrefragibly logical, of exquisite clarity, meeting the requirements of mind” (Vol. 6, pp. 333, 334).
Charlotte did not employ the modern notion of “living math books” to teach mathematical concepts. She advocated acquainting the children with the “captain” ideas of math by introducing the different branches or their great thinkers through an interesting or exciting history.
Advanced Mathematics—The methods we’ve discussed today are not just for the teaching of elementary arithmetic; they also apply to more advanced arithmetic: geometry, algebra, and beyond. Whether you are comfortable teaching the higher levels of mathematics or rely more heavily on textbooks, a curriculum, or a tutor, be sure to ensure a living treatment of math for your older child as well.
  • Guide your older child in discovery, allowing her to think for herself. Be patient and advance slowly. Allowing your older child to wonder, discover and permit ideas to germinate.
  • Practical exercises should continue along deductive exercises in geometry, and the practical side of algebra should be introduced as early as possible.
  • Provide a slow, steady approach with lots of practice.
  • Exclude long or tedious examples for calculation.
There you have it, our general overview with the fun stuff to come!

Thursday, March 21, 2013

Working Outside the Home and Homeschooling

I missed my rays of sunshine whilst at work.

Probably the single-most question I am asked is how we home schooled during this year "abroad" while running a restaurant. This was no easy feat and I had to constantly remind myself to not try to get through the subjects but into them and enjoy the time, no matter how little, with my boys. The logistics, along with apologies if this is a bit dry:

Alternating day shift and afternoon/evening shift at the cafe' meant an alternating school regimen. On the days I was at the cafe', the boys came with me and we brought our book bags with us to the restaurant. Breakfast was ordered (yes, that was a plus) and I read our literature and poetry selections while the boys narrated - never speaking with their mouths full, of course.

Max had independent work while Luca and I had math and reading together. Max rejoined for handwriting and then we had history and geography in the restaurant's dining room where a map collection is hung. Luca then had play time and Max and I did math together. At this point, we went upstairs to an apartment we'd turned into a hotel suite and the boys were set up for free reading/drawing/handwriting/clay modeling/Liberty's Kids, etc. while I dashed down to the cafe' during the lunch rush. Mind you, this all took place while I managed the front of house of a restaurant that seats 140.

After the lunch crowd left, it was our turn to have lunch. Luca, our younger, then usually stayed with me and Max would go to the library or youth group until I was finished for the day.  We read copiously in the evenings, helping to make up for snuggle-time and "school time" lost.

On the mornings I didn't have to go into the cafe' we would spend time out of doors in rebellion of all the time spent in the restaurant. Living with five additional children in the house is joyfully noisy so we retreated to our Airstream camper for our lessons until the weather turned too cold.

"What of poetry, and hymns, and Shakespeare, and handicraft, and...and...?!" comes the outcry of those who embrace Charlotte's generous curriculum.  Aah, one of the highlights of our year was the wonderful opportunity of being a part of a real-life, fine arts Charlotte Mason homeschool co-op. That delicacy will be another post.