I was recently reading a teacher’s manual for fourth
grade. It suggested if students
were struggling with a particular concept that the teacher should “make the
font bigger.” I don’t think this
strategy is going to help for most students but it does bring up the
question, “How can I help students
who are struggling?”
Development of math concepts proceeds along a continuum from
concrete to abstract. When
students are first learning, they should start with a concrete representation
of the concept. As their skill
grows, they can move to a more abstract understanding. For instance, when working on a
subtraction problem such as 32-19, students should first work with unifix cubes
or other objects so that they can count out 32 cubes and then take away 19 of
them. As they develop more
proficiency, they can move on to using more abstract manipulatives such as base
ten blocks. These blocks include a
stick with marks on it to show that it is a stick of ten. Unlike the cubes, however, one cannot
break it apart and take some off.
To solve 32-19, one can take away one stick of ten. Then to subtract the nine, one needs to
take away another stick of ten and trade it in for ten units in order to be
able to take away nine units. Once
students have worked with the blocks enough, physically trading in the ten
stick for ten units, they will then understand why we “cross out the 3 and make
it a two” when using the algorithm.
Often when students struggle it is because they haven’t
spent enough time at the concrete stage or they have been shown the algorithm
directly and haven’t spent any time
at the concrete stage. They have
memorized a procedure but there is no understanding about why it works. If they are presented with a problem
that doesn’t fit their memorized procedure exactly, they cannot solve it.
To support these students in building their understanding,
you need to go back to a more concrete representation, such as
manipulatives. These will provide
a visual model of the concept.
Students can work through problems by creating models. They will physically move the materials
and be able to see what happens to the numbers as they work. If they do this they will be using
their kinesthetic and visual modes of learning. If they then discuss what they are doing they will be using
their auditory sense as well. By
approaching the problem using so many pathways, they will develop a deeper
understanding. They will see that
math makes sense.
Consider an example from middle school. When I have worked with middle school
students and brought up the concept of trapezoids, I am often met with many
blank stares. It is difficult for some
students to remember what a trapezoid is and how it is different from other
quadrilaterals. On the other hand,
students who have used pattern blocks a lot in their earlier years know exactly
what I am talking about. When they
were young, they had the opportunity to hold trapezoids often. They built designs with them. They put two trapezoids together to
build a hexagon. They fit the
acute angle of a trapezoid between two equilateral triangles to make an
interesting design. (Though they
had no idea about this terminology when they did it.) When I mention trapezoids to middle school students who have
had these concrete experiences they have a physical reaction to the word. They often exclaim, “the red one!” (In
a set of pattern blocks the trapezoid is red.) Many hold out their hand as if they are remembering what a
trapezoid feels like. They can
also tell me that it has two sides, two sides are parallel, two angles are
acute, two angles are obtuse. They
can work from this specific information to generalize a definition of a
trapezoid. They are building on a
solid foundation of experiences with the shape and expanding their
understanding with new experiences.
The students’ early activities with the manipulatives have an effect on
their math understanding, even many years later.
Often students begin to struggle with math in fifth grade. Up until this point these students may
have been good at memorizing algorithms and teacher instructions. Their math experiences did not include
early experiences with manipulatives and problem solving that helped them make
sense of the math. Fifth grade is
when students need to start applying algorithms, often in sequence. If they do not have a firm
understanding of what they are doing, the process falls apart for them because
they do not have a firm enough understanding of what is happening to be able to
follow the numbers and make sense of the work. It is like building a house on sand. The foundation is not firm enough to
support the weight of the heavier math.
More often than not, this results in a frustrated student who says, “I
hate math.” This student often
starts to turn away from math at this point and is unlikely to pursue math
challenges and courses in the future.
We need to go back and shore up the foundation for these
students by helping them build their understanding from concrete to abstract. By using manipulatives and helping
students make sense of the math then they will be able to move forward with a
more positive attitude. They will
be more likely to stick with math and gain skills that are needed for the jobs
of the 21st century.
