An Activity for Upper Level Students: Making Connections among Mathematical Concepts
Too often, students are taught to solve mathematical problems in only
one way - the algorithmic approach. This strategy requires students to
follow a standardized procedure to reach an answer. Often, algorithms
are learned by repeated "drill and practice" approaches. Drill
and practice can be extremely effective for teaching short-lived procedural
skills, but for real understanding to occur, students must be engaged
in
a deeper exploration of what the problem means and how to resolve it.
Liping Ma, in her book Knowing and Teaching Elementary Mathematics,
talks to the importance of helping students find multiple solutions to
a problem: "Being able to and tending to solve a problem in more
than one way, therefore, reveals the ability and the predilection to make
connections between and among mathematical areas and topics" (112).
Encouraging the exploration of a wide range of approaches and examples
with the same problem can help students value and use powerful problem
solving strategies based on a deeper understanding of fundamental mathematical
concepts. Naturally, when you try to express something, you understand
it better ("Wired for Mathematics: A Conversation with Brian Butterworth,"
Educational Leadership, 59 (3), p. 18-19).
One problem with teaching mathematics strictly at a procedural level
is that students have no framework of what the multiplication of mixed
numbers really means or why it works the way it does. Taught in isolation,
none of these approaches alone enables students to attain a clear picture
of what it means to multiply two mixed numbers. They are forced to carry
around a large set of disconnected procedures for each mathematical problem.
However, when students look at many different ways to approach a mathematical
problem, it is easier to make critical connections that enable them to
construct and remember the "big picture."
Making Connections among Mathematical Concepts challenges students
to create different ways or manipulatives to solve one problem. Then,
they are encouraged to make connections between each of these manipulatives
so that they develop an understanding of what it means to create and solve
an algorithm. At the end of the exercise, they will have the confidence
and base knowledge they need to tackle similar mathematical problems from
many perspectives.
How many different ways can you find to solve a multiplication problem?
Multiple Multiplication Perspectives
For this activity, challenge teams of students to find as many solutions
to a word problem as they can. One way to encourage their creativity is
to look at it from different mathematical systems. Though the methods
to solve the problem each depend on a different knowledge base, they all
lead to the same "correct" solution.
Divide students into pairs, taking fullest advantage of the diverse abilities
of the classroom. Present the word problem in a way that is situated in
the local knowledge of your classroom and community that reflects 3 1-2
times 2 1-2 . Perhaps offering problems in a variety of contexts to strike
chords with your diverse student population might be ideal. Have fun and
be creative and flexible. Here is an example:
While your parents were away, your little sister decided to go bowling
on your parents' kitchen floor, destroying the tiles in the corner section
of the room. The tiles that must be replaced cover a 3 1/2 by 2 1/2
foot area. If each of the tiles is one foot by one foot, how many will
you need to repair the destroyed area?
Challenge each pair to find two unique ways to solve the problem. This
may take some prompting from you. Suggest that they consider the problem
visually or algebraically to stimulate additional solutions. Allow the
students to make arguments about which method works or does not work and
why. What is equivalent? What is different?
Here are some perspectives that may come out of the group work.
Number
Sense
What is multiplication? In
its simplest form, multiplication is the repeated addition of equal
sized groups. For example, 3 x 2 = 2 + 2 + 2.
If 3 x 2 by definition is three groups of
two in each group, then 3 1/2 x 2 1/2 means three and a half groups
of two and a half in each group.
This number sense method (three and a half
groups of two and a half units each) can be expanded like this:
Students may have difficulty with the final
term 2 1/2 / 2.
They might figure it out if it was rewritten
2/2 + 1/2 / 2, but even in this expanded form, they need to understand
how to divide fractions. It could help if they think of 2/2 as 1/2
of 2, which everyone knows equals 1. They still may struggle with
1/2 of 1/2, but they should be able to comprehend that using money.
Another way to solve this is by laying out
the problem geometrically, using a different approach so that division
does not have to take place. (See the graphic below.)
The
Geometric or Measurement Perspective Using Area
This perspective encourages the student to
create a picture to help visualize the problem. In this case, the
tool used to solve the problem is a rectangular grid using the idea
of area and square units. Drawn to scale, the area of each rectangle
reflects the size of the fraction it represents. Students can solve
this without actually doing any "multiplication." Instead,
they can add like terms (ones, halves, quarters), then combine them.
(six individual whole units)
(five one-half units - three vertical and two horizontal)
(one half of a half unit)
sum of the rectangular units, "unit" being a 1 x 1 square
Algebraic
Perspective Using the Distributive Property
Students may be unaware that they have been
using the distributive property since they started multiplying multi-digit
numbers. They may need to see whole numbers such as 23 x 54, for
example, decomposed and then multiplied horizontally as opposed
to the vertical method more commonly used.
Using the above example and substituting
the problem's values of 3 1/2 and 2 1/2, students' work might look
like this.
Ask students to compare their algebraic solution
with the number sense and geometrical approaches discussed earlier.
There are similarities. All three perspectives partition out the
problem so that addition can be used to solve the multiplication
problem.
Using
Decimals
Decimals are not often considered fractions,
but they are. In fact, parts of wholes are more commonly represented
by decimals in the real world than by fractions. Restated in decimals,
the problem looks like this.
3.5 x 2.5 =
Once restated in decimal notation, the problem
can be solved using any of the approaches already described. Take
the distributive property, for example:
Seeing the similarities between the four
approaches can be powerful for helping students understand visually,
conceptually, and, finally, procedurally how the multiplication
of mixed numbers works.
What is taught in grade school
as arithmetic is, for the most part, not ideas about numbers but automatic
procedures for performing operations on numerals - procedures that
give consistent and stable results. Being able to carry out such operations
does not mean that you have learned meaningful content about the nature
of numbers, even if you always get the right answers!
-George Lakoff and Rafael E. Nuñez, Where Mathematics Come From: How the Embodied Mind Brings Mathematics
into Being, p. 86
The
Standard Algorithm Perspective
The algorithm is the most efficient way to
do the mathematics but is not the best way to understand it. The
algorithm provides a process for solving the problem that relies
on breaking down the problem down into easily calculated steps.
Algorithms are so simple that calculators and computers rely on
them heavily to solve complex problems, but they depend heavily
on rote meaning of procedural rules and contribute little to students'
conceptual understanding. They should be used only after the student
understands the underlying mathematics that go into solving the
problem.
3 1/2 x 2 1/2 =
7/2 X 5/2 = 35/4 = 8 3/4
Students change the mixed numbers to improper
fractions, do the necessary multiplication, then simplify the fraction
and/or convert the answer to a mixed number. The standard algorithm
statement looks very close to the algebraic problem without the
addition: (3 1/2 ) (2 1/2). When creating the problem from the top
row (2 1/2) and the left column (3 1/2) of the geometric perspective,
the standard algorithm is reflected. Of course, when the fractions
in the algorithm are turned to decimals ( 1/2 = .5), the decimal
problem is created.
How many children leave school with good grades in mathematics
but no understanding of what they were doing? Surely a lot, judging
from the large numbers of perfectly intelligent adults who cannot
add fractions. If only they understood what was going on, they would
never forget how to do it. Without such understanding, however,
few can remember such a complicated procedure for long once the
final exam has ended.
-Keith Devlin, The Math Gene: How Mathematical Thinking Evolved and Why Numbers
Are Like Gossip, p. 67
Making
Connections
Solely
teaching the algorithm shortchanges the development of students'
understanding of how a problem works. According to Ma, "Being
able to calculate in multiple ways means that one has transcended
the formality of an algorithm and reached the essence of the numerical
operations - the underlying mathematical ideas and principles"
(112). Based on students' understanding of multiple strategies,
they can make connections between the parts of the problem. During
this exercise, students have used geometry, algebra, decimals, and
a visual diagram to solve one multiplication problem. Challenging
students to solve mathematical problems a number of ways allows
them to relate underlying mathematical relationships that they will
remember when using the more efficient algorithm method.
This lesson is an adaptation of a SCIMAST teacher professional development
training module. Special thanks to Concepcion "Como" Molina,
SCIMAST program specialist, for his assistance with this lesson.
Solely
teaching the algorithm shortchanges the development of students'
understanding of how a problem works. According to Ma, "Being
able to calculate in multiple ways means that one has transcended
the formality of an algorithm and reached the essence of the numerical
operations - the underlying mathematical ideas and principles"
(112). Based on students' understanding of multiple strategies,
they can make connections between the parts of the problem. During
this exercise, students have used geometry, algebra, decimals, and
a visual diagram to solve one multiplication problem. Challenging
students to solve mathematical problems a number of ways allows
them to relate underlying mathematical relationships that they will
remember when using the more efficient algorithm method.