Follow Through: Four important levels when using

Follow Through:
Four important levels when using
Alfred Ojelel, PhD
([email protected])
Alpha Secondary School
BCAMT Conference, October 21, 2011
Consider this sequence . . .
Head down
Knees bent
Keep your arms straight
Follow through
We’ve all heard it
but it is doing it
that matters!
The objective of this sequence in a golf swing of any shape,
clearly, is to consistently get the golf ball airborne.
• Any tangible object, tool, model, or mechanism
that may be used to clearly demonstrate
(introduce, practice, or remediate) a depth of
understanding, while problem solving, about a
specified mathematical concept or concepts.
• We find manipulatives in most classrooms now:
buckets of pattern blocks;
trays of tiles and cubes; and
collections of geoboards, tangrams, counters,
spinners . . .
• They are being touted as a way to help students
learn math more easily.
But many teachers still ask . . .
• Are manipulatives a fad, a craze, a rage?
• How do I fit them into my instruction?
• How often should I use manipulatives?
• How do I make sure students see them as
learning tools, not toys?
• Are they useful for upper-grade students, too?
Why are manipulatives important?
• Research based studies show that, with
manipulatives, students:
– develop more precise and more comprehensive
mental representations,
– often show more motivation and on-task behaviour,
– understand mathematical ideas, and better apply
these ideas to life situations.
• According to learning theory (based on Jean Piaget's
research), children are active learners who master
concepts by progressing through three levels of
knowledge :
– concrete,
– pictorial, and
– symbolic.
• Piaget (1965) approached the construction of
knowledge through questioning and building on
learner’s answers while they constructed
• Vygotsky (1962) felt that learners could be guided to
stronger mathematical understandings as they
progressively analyzed complex skills on their own
with the teacher nearby to scaffold or facilitate as
We can deduce then that . . .
• Manipulatives enable students to explore
concepts at the concrete level of understanding.
• When students manipulate objects, they are
taking the necessary first steps toward building
understanding and internalizing math processes
and procedures.
• For a learner to understand and handle a
concept at the abstract or symbolic level
successfully, he or she must first understand
the concept at the other two levels, in the
order given - concrete then pictorial.
• When students demonstrate understanding at
the concrete level then they are ready to move
to the next level.
• They can then apply their knowledge using
representations of the objects in place of the
objects themselves.
Concrete to Concrete exploration
• Instruction proceeds through a sequence with
each concept first modeled with concrete
materials e.g. red and yellow chips.
• The materials by themselves are not enough.
Teacher guidance is essential.
– How did you make this model?
– What did you show?
• We should try to "get inside students' heads" as they
work with concrete materials to solve a problem by
asking questions that elicit their thinking and
– To can get a better sense of what students know
and don't know,
– To identify alternate conceptions, thereby
– To develop a basis for intervention strategies.
Concrete to Abstract
Abstract to Concrete
• The concept is modeled at the semi-concrete level
which may involve drawing pictures that represent
concrete objects e.g. circles, dots, tallies, . . .
– Could you draw this model?
– How did you do that?
– What did you draw to show groups?
– How did you group these?
• This representational level affords learners
multiple opportunities to construct mathematical
knowledge while making reasonable connections
to everyday tasks.
• This is the essence of tool- and problem-based
teaching and learning.
• This level is the real bridge between the concrete
and abstract.
Abstract to Abstract
• Here the concept is modeled at the abstract
(symbolic) level using only numbers, notation,
and mathematical symbols
– What numbers, operations, expressions . . . did you
use to show this problem?
– Is this answer a reasonable number?
Follow the sequence through . . .
How can we make this happen?
1. Clearly set and maintain behaviour standards
for manipulatives.
2. Clearly state and set the purpose of the
manipulative within the mathematics lesson.
3. Facilitate cooperative and partner work to
enhance mathematics language development.
4. Allow students an introductory timeframe for free
5. Model manipulatives clearly and often.
6. Incorporate a variety of ways to use each
7. Support and respect manipulative use by all
8. Make manipulatives available and accessible.
9. Support risk-taking and inventiveness in both
students and colleagues.
10.Establish a performance-based assessment
Example of exploration sequence
Thank you.
Comments, suggestions, questions may be directed to
[email protected]

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