The grade fours have been working on multiplication and the fives on division. All the ideas below apply to both grades as the fives are already expected to understand the multiplication (they need to also understand 2 digit by 2 digit multiplication) and the fours will be moving on to division later.
With all concepts in mathematics there is a progression from a concrete understanding to an abstract understanding. There are phases of development that all students go through, some move from concrete to abstract quickly and others need to remain concrete for longer before they are developmentally ready to move to abstract thinking about the concept. Concrete means needing manipulative and/or pictures to solve problems.
This developmental process is why I do not teach them the algorithm right away and when I do teach it, I insist they understand what they are doing. A child can memorize a series of steps to do for multiplying or dividing without understanding the what or the why. This can lead to problems in math in later years, particularly in algebra. (as an aside, preschoolers can do pre-algebra easily, it seems the steps and formulas are what mess up students!)
I use a chart that shows movement through the phases for understanding the operations (+ – x \ ). It was developed for Nelson by Dr. Marion Small. The phases of operations development are across the top of the chart and the concepts and skills are down the left side of the chart.
The five phases are:
Phase 1 – Beginners, focus on counting to solve problems.
Phase 2 – Concrete – formal operations with numbers to 20; Concrete operations with numbers to 100
Phase 3 – Whole Number Comfort – formal operations with whole numbers; concrete operations with decimals
Phase 4 – More Abstract – Fluency (can use multiple methods to solve) with whole number operations; formal operations with decimals
Phase 5 – Flexible – Fluency with whole number and decimal operations; Concrete operations with integers and fractions
There are also 3 concepts and 3 skills on the chart but for this post I am only looking at:
Concept 2 – Multiplication and division are extensions of addition and subtraction. Multiplication and division are intricately related.
Concept 3 – There are many algorithms for performing a given operation with multi-digit numbers.
Skill 3 – Computes with multi-digit whole numbers and decimals using pencil and paper without the aid of a calculator.
I approach multiplication and division from a problem solving perspective. This means I pose problems that the children can relate to and then I have them solve the problems in pairs using whatever method they would like to use. I ask them to check their work using a different strategy. The groups solve their problems on chart paper and then they share their solutions with the class. The children are exposed to many different ways to solve the same problem and they see that there are many ways of thinking about and solving problems. Every year I learn something new from the students, there are ways to solve grade 4 & 5 problems that I have not thought of yet after 12 years of teaching math! That is awesome!
Here are a few examples of the grade 4s solving a multiplication problem:
After we share our examples we talk about how efficient the methods are. In this case, 23 x 3, drawing a picture is doable but not very efficient. Repeated adding is fairly efficient. But what if it were 233 x 23? Would drawing a picture be efficient? Repeated adding? From here I get into algorithms (procedures for calculating). I try to direct the children to come up with their own algorithms because they really remember those the best! Naturally, none of the children has come up with the traditional algorithm on their own so I do have to teach those, but I want them to fully understand what they are doing. But before I do that, I let them discover the break apart method!
The purpose of this post is to explain the development of students’ developmental learning in math. Some students go through the phases quickly – some can multiply and divide quite well using several methods. Others take longer and they are still not quite there. This is normal and is one of the challenges of organizing children by age into grades. All kids develop at their own rate (of course there are learning skills such as attention, work ethic, etc., that also come into play). It is important for ALL kids to understand how to multiply and divide concretely before they move into the more abstract algorithms such as the ones we learned.
The next post will describe the break apart method of multiplying and the alternative long division algorithm.