Math: When A Picture Is Worth 1,000 Words
by The Link staff with contributions from Guillermo Mendieta
It is well known that too many of our students struggle with gaining a meaningful understanding of key mathematical concepts. From place-value and fractions, to variables and equations, too many of our students find themselves lost in a sea of abstraction.
The solution is to re-conceptualize the teaching and learning of math. The “pictorial mathematics” approach, fathered by teacher and author, Guillermo Mendieta accomplishes this task.
Pictorial mathematics focuses on helping students, and those who help them, into developing a deeper conceptual understanding of every major school mathematics concept in grades 2 through 9.
The Mr. Mendieta developed meaningful and engaging visual representations for each concept and used them as a bridge between concrete and more abstract representations of each mathematical idea.
According to Mr. Mendieta, the underlying idea of pictorial mathematics is particularly well-suited to homeschoolers because their parents often recognize the importance of gaining a meaningful conceptual understanding of mathematics and are looking for ways to meet the individual learning styles of their children. Guillermo’s ideas are set forth in his new book, Pictorial Mathematics.
To illustrate the approach used in the book, consider the concept of multiplying mixed fractions.
Try to recall the way your math teacher taught you how to multiply two mixed fractions, such as 1 ½ x 4 ½. My teacher, like most other teachers, taught us to multiply 4 ½ x 1 ½ by following the following rules:
• Change the 1 and ½ to an improper fraction. To do so, multiply the whole number (1) by the denominator (2) and add it to the numerator (1). In our example, this gives us 3 x 2 + 1 = 9. This is the new numerator (3) of your first fraction. Keep the same denominator (2). Thus, the new fraction is 3/2.
• Now change the 4 ½ to an improper fraction. Again, the process is as follows: multiply the whole number (4) by the denominator (2) and add it to its numerator (1). This gives us 1 x 2 + 1 = 9. Thus, (9) is the new numerator of the second fraction. Keep the same denominator (2). Thus, the new fraction is 9/2.
• Multiply the numerators, then multiply the denominators. Your new fraction is 27/4.
• Is the numerator greater than the denominator? If yes, continue with the next step, otherwise, simply reduce the fraction. In our example, the numerator (27) is larger than the denominator (4), so we go to the next step.
• Since 4 does go evenly into 27, when you divide 27 by 4 your answer will give you mixed fraction; you must keep in mind that the quotient (6) is the whole number part of your answer. The remainder is the numerator (3), and the dividend (4) is the denominator. Thus your final answer is 6 and ¾.
This procedural, symbolic or abstract representation of the multiplication of mixed fractions made very little sense to me and to my classmates. It did not help us understand what multiplying two fractions meant. You either memorized the steps in the right order, or you did not get the right answer. But whether you got the right or the wrong answer, this way of representing the process of multiplying mixed fractions did little to promote conceptual understanding.
Compare this procedural and rote approach to the conceptual pictorial approach used in Pictorial Mathematics: To teach multiplication of fractions conceptually, 1 ½ x 4 ½ should be read in one of two ways:
1 ½ x 4 ½ should be read as 1 and ½ groups of 4 and ½ or 4 ½ repeated 1 ½ times.
Students should be introduced to the pictorial representation of 1 x 4 ½ first: The following drawing represents 1 x 4 ½ (or 1 group of 4 ½)
The original multiplication was 1 ½ x 4 ½. How is this multiplication different than 1 x 4 ½? The difference is that 1 x 4 ½ is only group of 4 ½ but 1 ½ x 4 ½ is one AND a Half groups of 4 ½. The drawing below shows two rows of squares. The first one represents 1 group of 4 ½. The second row represents half a group of 4 ½.
[Note about the drawing
Notice that the second row is exactly half of the first row. The first row shows 4 ½, the second row shows half of 4 ½. ]
Teachers and parents who have not had experiences with pictorial representations of this type might at first struggle to see that the picture above is the pictorial representation of 1 ½ x 4 ½. The author introduces this model developmentally, starting with simple multiplications of whole numbers such as 2x3, and slowly helping the learner create a visual representation framework for more complex problems, including algebraic concepts such as factoring. The reader is encouraged to see that by combining the pieces above, we get 6 ¾, which is the answer to 1 ½ x 4 ½.
Pictorial mathematics is a great resource for homeschooling families. It includes:
• Master templates to create an unlimited number of problems using the ones provided by the author as a guide
• 38 Pictorial templates that can be used to teach every major concept in grades 2 through 9
• More than 1,000 visually engaging models and exercises
• Two sets of reproducible gaming cards, one for fractions, place value and percent, and one for variables and expressions, along with 4 different games for each.
• Access to web downloadable updates that include additional templates, exercises and additional pictorial games for other concepts.
This is what K.C. Cole, award winning L.A. Times Journal and Science author had to say about Pictorial Mathematics:
“You’ve done math teachers (and students) a great service with this book. It shows parents and teachers ways to convey deep (and cool) concepts even while working under the “tyranny of tests.” It’s lively, concise, reader-friendly and fun. I’m very impressed.”
Pictorial Mathematics: An Engaging Visual Approach To The Teaching And Learning Of Mathematics, published by Meaningful Learning Press in Feburary, 2006, has already been widely acclaimed as a breakthrough educational resource. It has been adopted by The Los Angeles County Office of Education to support the meaningful teaching of mathematics and they are offering a series of workshops on how to use the 400-page book most effectively with students.
Mr. Mendieta will be presenting at The Link Homeschool Conference in Woodland Hills, June 8-11. He is also a featured speaker at the California Math Conference in Palm Springs, the largest math conference in the state. To see a preview of his book or to place an order, visit the book’s website at www.PictorialMath.com. The book is $34.95, but a special price of $32.50 has been arranged for the readers of The Link. ■