Students solve mathematical problems in a variety of ways. Some students need to touch and manipulate cubes or tiles to help them visualize the problem. Some may use software to investigate and explore different models. Some may use symbols to represent their solution algebraically.
But how do we get those students who are at the concrete stage to represent their models algebraically? Do students who represent their solutions symbolically really understand what their algebraic expressions mean or have they just learned to manipulate symbols?
Intermediate mathematics should emphasize making connections between the concrete and algebraic models. If we rush to teach rules or algebraic manipulation without first imparting an understanding of what these rules or expressions mean, then what are students really learning? The purpose of making connection is to understand the development of the rules, the formulas and the algebraic expressions.
In grade 7, students are expected to learn how to add and subtract integers. When they are presented with a question like: “What is the value of 7 – (-3) ?“, we may get a solution that looks like this…
Although the solution is correct, what does this student really know? If, instead of paper and pencil work, we use tiles and an integer mat, we can focus on and assess students’ understanding of the zero principle.
By going through many examples and seeing the pattern of subtracting a negative, students begin to understand what the rule “two negatives = a positive” means.
We can also use visual or concrete models as tools to aide in the understanding of problems in areas such as fractions. In the following problem, the grid on the left mirrors the calculations on the right. The grid is used to not only generate the solution (1/8) but to add meaning and intuition to each step of the algebraic derivation on the right.
Simaya’s dad cut her birthday cake into 12 equal parts and Simaya ate one of the pieces. Later on, there were only a few pieces left, so her dad cut each of the original pieces in half. Simaya then ate 1 of those smaller pieces. What fraction of the cake did she eat in total? Explain your reasoning.
In this Patterning & Algebra question, students are asked to complete the chart and find the 20th term in this growing pattern.
Some students may ignore the pattern of dots and focus on developing a rule which generates the sequence of numbers 2,6,12,20,… at the bottom of the chart. While their rule may work, the connection between the numeric sequence and the dots sequence is lost. They would have difficulty justifying their rule. By focusing on both the pattern of numbers and the pattern of dots at the same time, a student connects a dot rectangle with a number in the sequence and can potentially derive and justify a general expression for the nth term in both sequences. Below are two different models which help students visualize and justify two ‘different’ equations for the nth term in the sequence.
Paper folding is a great way to concretely demonstrate mathematical concepts. It’s inexpensive and requires minimal preparation. The following exercise motivates and develops the formulas for the area of a trapezoid and parallelogram. Notice how the exercise implicitly includes concepts such as conservation of area under transformations.
Making connections naturally leads to better mathematical intuition and an understanding of the derivation of rules, formula and algebraic manipulations. A focus on the final answer to a problem, without a motivating model to connect the solutions steps is like seeing only the last ten minutes of a movie. The end doesn’t make nearly as much sense compared to watching the movie from start to finish. Besides, watching the whole movie is more fun, usually something interesting happens along the way.
- Area Formulas by Paperfolding – http://www.mathnstuff.com/math/spoken/here/2class/150/area.htm
- Mathstar – http://mathstar.lacoe.edu/newmedia/integers/welcome.html