A Practical Approach to CPA Instruction

March 11, 2020

Concrete, Pictorial, Abstract (CPA) instruction follows the following rationale:

• Students gain understanding of mathematical concepts through concrete experiences.

• Once this understanding is achieved, they segue into pictorial representations of the concrete.

• Later, students represent the pictures through abstractions.

Within these three pedagogical stages, there are many intermediary levels.  Observing a teacher handle a manipulative is less concrete than doing so individually.  Ten popsicle sticks bound together by a rubber band is more concrete than a ten-stick rod, which is more concrete than a foam number disk with the number ten written atop it.  Drawing two 10-disks is more abstract than handling two 10-disks, and marking two dots in the tens column of a place value chart is more abstract than both.  Writing the digit 2 in the tens column of a place value chart is more abstract than any of the aforementioned but less so than writing 20 on a blank surface.

The greatest math teachers blend these mini-phases into a fluid, coherent continuum, gradually increasing complexity through abstraction (see first grade example on right).

T:  How many pens do you see?

S:  5.

T:  How many pieces of paper do you see?

S:  7.

T:  Are there more pens or pieces of paper?

S:  Pieces of paper.

T:  Are their fewer pens or pieces of paper?

S:  Pens.

T:  How could we find out how many more pieces of paper than pens there are?  Turn and talk to your partner.

S:  Place a pen on top of each piece of paper and see how many pieces of paper are left over.

T:  (Places one pen on each piece of paper.)

T:  How many pieces of paper don’t have a pen?

S:  2.

T:  How many more pieces of paper than pens are there?

S:  2.

T:  Repeat after me…There are 2 more pieces of paper than pens.

S:  There are 2 more pieces of paper than pens.

T:  7 is 2 more than 5.

S:  7 is 2 more than 5.

In this example, the position of the manipulatives centralizes student focus.  Because they’re not distracted by handling objects themselves, children likely see the more/fewer relationship with greater clarity than if they were trying to follow teacher directives and solve problems using their own set of manipulatives.  This concrete teaching method is less tiring for teachers and often leads to greater student comprehension.  If all children were handling seven pens and five pieces of paper, the instructor might find them self constantly redirecting individuals, who would undoubtedly be fumbling their materials and struggling to follow directions.

After repeating the above process with different examples, the teacher might feel that student comprehension is strong enough to segue into the pictorial learning phase.  They would then adapt the script, using pictures instead of objects, before directing students to practice pictorial problems on their own (see example on right).

When facilitating lessons in which all students handle manipulatives, the teacher carefully selects the items students work with.  Recognizing that novelty often leads to distraction, they are not seduced by a marketing industry that prioritizes flare over learning.  They are wary of incorporating bright colors, pleasurable textures, and tasty foods into their instruction, because each has a tendency to distract student attention away from lesson content.

Understanding the beauty of mathematics and its wondrous discoveries, they see manipulative uniformity as exciting rather than bland.  Identical black and white linker cube sets produce more lesson comprehension than multi-colored collections that lead students to admire and/or envy their neighbor’s manipulatives.  Immersed in mathematics rather than visual distraction, the child enters euphoric understanding and recognizes unexplored vistas of learning.

The most effective math manipulative is also the most ancient.  When taught and used well, student fingers serve as better learning tools than any other object.  They are the foundation of mathematics’ base-ten system and, barring a catastrophe, never leave the child.  Fingers create representations of one-to-one correspondence and communicate students’ comprehension or lack thereof.  They also help children efficiently count, compute, and organize information.

The next best math manipulatives are durable items that students regularly see outside of school.  Pennies, beans, popsicle sticks, and paper strips are more effective (and cheaper) than most marketed items. Linker cubes and number disks are exceptions to this rule.  Although neither are commonly found in students’ homes, both are powerful early grade learning tools.  To maximize their effectiveness, teachers invest significant time and energy familiarizing their students with them.  This is best achieved by consistently incorporating the manipulative into different sections of the lesson.

During a single math class, kindergarteners can use linker cubes to:

• Perform One more than, One less than fluency.

• Represent part-whole relationships in a story problem.

• Demonstrate put-together problems during an Addition within 5 lesson.

Within the span of 25 instructional minutes, second graders might use number disks to:

• Participate in place value fluency activities.

• Compute numbers while solving an Adding with Renaming word problem.

• Practice problems during and/or after a Subtract with Renaming lesson.

With repeated exposure to a small set of manipulatives, the learning tools’ novelty wears off.  When this happens, students recognize and use them in the same efficient, utilitarian manner as pencils and paper.

The wise math teacher rarely – if ever – uses students as manipulatives.  This is because a child acting out a mathematical concept doesn’t experience the activity through an observatory lens. A novice actor cannot see their stage or scenes holistically when they are focused on playing their roles, and a child immersed in a new lesson context lacks perspective on the math they’re supposed to be interpreting.  If the teacher uses children as learning manipulatives they are careful to select students who are able to participate while still comprehending the concept being taught.

Regardless of their choices, the educator values concrete learning experiences, but constantly pits its worth against efficiency.  Distributing materials and facilitating learning while students use them is time consuming and often leads to disciplinary problems and/or student distraction.  A math manipulative ceases to be a learning tool if children don’t recognize and treat it as such.  When this happens, they conclude that utilizing pictures while maintaining student concentration is better practice than teaching with manipulatives devoid of children’s focus.  Although they’re not commonly recognized as concrete learning instruments, pencils, paper, graphic organizers, markers, and white boards are also manipulatives that help students gain understanding.

The best math teachers minimize the number of manipulatives they have their students work with, and the best schools standardize the manipulatives used in each grade.  When students see the same learning tools from year to year, new concepts are easier to comprehend and their mathematical experience is enriched (see sample list on right).

Management and distractibility challenges aside, the teacher never uses manipulatives superfluously.  They will begin instruction at the pictorial or even abstract level if it will lead to greater teaching efficiency without sacrificing student understanding.  Fifth graders don’t need to fill containers with water to understand volume properties.  In the decade that they’ve lived, these students have had enough concrete experiences to understand basic physical laws, e.g. a rock dropping into a tank will raise its water level.  In these instances, the instructor begins teaching at the pictorial level.  For example, a sketch of a tank half-filled with water is tangible enough to ensure student comprehension (see graphic below).

The higher the grade, the more critical these decisions become, and the more concessions teachers need to make.  For many upper elementary topics, instructors will never be able to address grade level content, if they always begin teaching topics on the concrete level.  Writing the digit 2 in a place value chart is an abstraction that fourth graders must work with if they are to master the base ten system on their grade level.  If they need two seashells, bundles of popsicle sticks, or number disks to reach conceptual understanding, their teacher might justifiably conclude that they can’t backtrack this far in the CPA continuum and still hope to teach required content.

When this happens, they petition administrators to carve out intervention time.  If it’s not provided, they accept that they occupy an imperfect job within an imperfect profession that exists inside an imperfect world and sacrifice conceptual understanding of a few to advance the mass.

***

Expert teachers pass their students through the CPA continuum, starting at the most abstract point of collective student understanding.  Therefore, they don’t use manipulatives if understanding can be gained using a picture.  Likewise, they teach through abstraction when neither concrete or pictorial teaching are needed for students to comprehend.  They also recognize writing tools and surfaces as manipulatives. 

Student learning, not a pedagogical philosophy, is their aim.