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.


Melissa said...
This comment has been removed by a blog administrator.
erin kate said...

Well I am most eager for the next post! This post has been wonderfully clarifying for me. Many thanks. Warmly, Erin

Anonymous said...

This post is like a little present just for me. Thanks, R!
Bobby Jo

Nancy said...

I am thankful that you share your expertise and experience with math the CM way, Richele. These posts may be influencing the atmosphere in so many homes where math has been less-than-pleasant. I know that I refer people to your book and blog often.


Melissa said...

Brilliant. Thorough explanation and practical.

walking said...

Thank you for sharing how Mason would have introduced students to multiplication. My experience in tutoring students today is that there is no royal road to addition and that journey must happen before multiplication! I was so glad to have such practical and carefully described advice on hand for the carnival.