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Articles:

Memorization and Understanding in Mathematics
Mere Memorization Cannot Lead to Understanding
Vital to our understanding of language is relevant tactile & other sensory experiences associated with each word or concept that denotes them. From birth we learn language cues in concert with the sight, feel, taste, smell or sound of an experience when the word is presented to us. The case of Helen Keller demonstrates of how difficult learning language is without sight & hearing. more..

The Psychological Principles at work in Teaching Math With Manipulatives

Galileo's method in science owed much to his ability to devise 'thought' experiments (like Einstein's), to visualize things as they might be. In 'The Assayer' he wrote, "the book of nature is written in the language of mathematics. In MM we use the rectangle to develop the powers of visualization in children. I think Galileo would be pleased.

Mental Images - If you use plastic bears to teach counting, when you say the word 'six' to the child will they see a half dozen bears?
But use a generic shape like a unit square, when you say 6, the child will readily see a rectangle 6 over 1 up - the way we present 6 when counting. In the MM case the child can easily see that 6 is bigger than 5. Not so certain when using bears. :)

"Acquiring Meaning"
Your average ten-year-old boy would rather be outside playing baseball than memorize times tables, which has as much meaning to him as reciting poetry in ancient Greek (think Homer). It is obvious that we must present the opportunity for the students to experience feelings when learning math. One feeling we all would like them to experience is the joy that comes with success.
http://www.geoffwhite.ws/meaning.html

"Understanding is the Goal if you want children to learn language you must provide a language-rich environment. If you want children to learn math concepts you must provide a math-rich environment.

Comparing "Mortensen More Than Math" Method with traditional

math

Mathematical Language Our job as teachers is to decode this mathematical language into a spatial reality. The keys to doing this are the manipulatives and the rectangle model to show the relationship between two factors, over and up.

Mathematical language is any string of symbols, eg. 123 x 45 = or (4x)(2x) = or 768/12= and any ordinary language describing mathematical relationships, for example story problems.

How does "6" become meaningful?
To put it bluntly, for there to be meaning, there must be feeling. Could there be a rote recall of an equivalent meaning as in vocabulary lists? Yes, of course, but for there to be a visceral understanding of a word or concept, there must be a feeling associated with it. A corollary of this is, that if there are no feelings associated with an experience then it is meaningless.

Math isn't about just manipulating symbols: +, 3, x, =, etc. according to rules. That is, it doesn't have to be. It could be about 'seeing' the values two-dimensionally, about visualizing rerlationships. See '6' as a rectangle that is 2 over and 3 up. See See 156 as a rectangle 13 over, 12 up, as in the figure at upper left on this page

Using the pieces at upper left: unit, ten bar, hundred square, try this exercise. Imagine the construction for all the squares from 2 - 100. 2 squared is a rectangle 2 over, 2 up = 4. 3 squared is a rectangle 3 over, 3 up = 9, and so on. MM is not a "faster" way to solve math problems, it is a path to understanding and visualization is key. So, when teacher says, How many is 24, - several rectangles may be imagined, then she says, "divided by.. 6." Suddenly, a single rectangle 6 over is "seen." Then the down dimension of 4 is realized

Visualization is an important part of the MM method. This means seeing all numbers as rectangles, even fractions. There are many rectangles to represent a given number, say 24. It could be a rectangle 3 over, 8 up. Or 4 over, 6 up. Or 12 over, 2 up, etc. Those familiar with doing this will hold the vision as a fluid form until additional info is known. So, when teacher says, How many is 24, - several rectangles may be envisioned.

Building rectangles with manipulatives facilitates rapid, accurate counting - which builds confidence and improves self-esteem. If you build it, they will comprehend.

The best tools are Mortensen's Combo blocks. Next best would be base ten blocks of contrasting colours, then base ten blocks - even if of uniform colour Then tiles - which can be made from coloured card if the budget is tight. The method is key, but the manipulatives are easily 25% of the effectiveness of the program

Lesson #2: See all numbers as rectangles. One is a rectangle one unit over & one unit up. When you think of six, think of a rectangle that is one unit over and six units up. Similarly, think of nine as 3 over & 3 up. NB. numbers can be re-shaped.

Lesson #1.: All we do in math is count. We cannot count until we know what 'one' is, that is, what the unit reference is. Hold up a little green cube and say, "This is one. It is one over, and it is one up. I have one.

MM adds 2 stages to the traditional math education experience: construction with manipulatives & drawing - in all learning experiences we seek to provide the child with as many sensory experiences as possible. Thank Maria

Montessori for that innovation. Jerry Mortensen provided the rectilinear model. Explore and Discover is the theme. Understanding is the goal. Confidence and Competence are the intended outcomes.

Any M.M. lesson involves 4 parts: build it with manipulatives, draw it to reinforce the experiences, do the notation, record the answer. Traditional teaching only employs the latter two stages. What MM adds, the tactile-motor-kinaesthetic and visual experiences, are precisely what the child needs to achieve understanding.

Is your first impression that math consists merely of numbers and symbols requiring a set of remembered rules? That's the way math has been taught for more than a hundred years. It doesn't have to be that way.

Math could be about developing an understanding of the world using visual images accessible to everyone, including dyslexics & learning challenged, using intuitive operations.

Relevance of Success at Math Questioned
Wash. Post recently: Prof. Ramanathan says, "Math is less relevant to daily life than literature, history, politics, music and communication skills. Your average person doesn't need "higher math." That's fine. Is 'average' all you want your kid to be?

Today, I am shipping copies of my Teacher's Handbook for Teaching Math with manipulatives, using the Mortensen method to Singapore and to Manila. There are Montessori schools in Manila using the Mortensen "more than math" method. www.geoffwhite.ws

"Math is the study of numbers and even the smallest child knows what you do with numbers - you count them." - Jerry Mortensen.

The easiest way to evoke feelings with a math problem is to tell a story. Thus story problems ought to be welcomed by the student rather than feared. Start early, say, age 2 or 3 and dramatize it with animals with names, personalize it by using their name and those of their peers & family. see my essay at http://www.geoffwhite.ws/meaning.html

We must evoke feelings in association with a concept in order that students acquire meaning viscerally. if there are no feelings associated with an experience then it is meaningless. When learning math, meaning is often lacking and learning is sometimes tedious, students need to experience strong feelings related to the topic. One feeling we would like them to experience is the joy that comes with success.

Let's use our imagination. See all numbers as rectangles, so many 'over' and so many 'up' Now counting and doing all operations +, -, x, / is building rectangles. 156 is a rectangle 12 over and 13 up.

Visual perception is so acute that with one glance we can take in, and retain, many times more information than by reading and memorizing. With brightly coloured manipulatives MM achieves exactly this. One glance at a construction, as at left, and you can recover info from it such as the relationship: 12x13=156

Visual perception is so acute that with one glance we can take in, and retain, many times more information than by reading and memorizing. With brightly coloured manipulatives MM achieves exactly this. One glance at a construction, as at left, and you can recover info from it such as the relationship: 12x13=156