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AGR-TEAC61-04 · Lesson 4 · 14 clock hours · demo

Mathematics Instruction Support: Concept Development and Problem Solving

Objective. Students will facilitate a concrete-pictorial-abstract (CPA) math lesson segment using manipulatives and worked examples, accurately representing the mathematical concept being taught.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by a licensed professional in the field or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

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What Is the CPA Approach?

The Concrete-Pictorial-Abstract (CPA) approach is a research-supported instructional sequence used widely in K–12 mathematics education. It moves students through three stages of understanding:

As a paraprofessional, your role is to support the certified teacher by guiding small groups or individual students through these stages accurately. If you introduce a mathematical error at any stage, that misunderstanding can persist and grow. Accuracy is not optional — it is your primary responsibility.

Stage 1: Concrete

Manipulatives give students something to touch and manipulate. For multi-digit addition, base-ten blocks are the standard tool. A small cube represents one unit, a rod of ten cubes represents ten, and a flat represents one hundred.

To model 143 + 28 with base-ten blocks, set out 1 hundred flat, 4 ten-rods, and 3 unit cubes for the first number. Then set out 2 ten-rods and 8 unit cubes for the second number. Combine the units: 3 + 8 = 11 units. Trade 10 unit cubes for 1 ten-rod (this is regrouping, or "carrying"). You now have 1 ten-rod from the trade, plus 4 + 2 = 6 ten-rods, for a total of 7 ten-rods, plus the 1 hundred flat, giving 171.

Always use accurate mathematical language. Say "regroup" rather than "borrow." Say "ten ones equal one ten" rather than vague language like "put it over there." Precise vocabulary prevents misconceptions.

Stage 2: Pictorial

Once students show understanding with physical blocks, move to a place-value chart drawing. Students draw or label columns (Hundreds | Tens | Ones) and use tally marks, dots, or simple sketches to represent each group. The drawing mirrors exactly what the blocks showed. When regrouping occurs, students cross out ten marks in the ones column and draw one mark in the tens column.

The pictorial stage bridges the gap between handling objects and reading symbols. Students who skip this stage often struggle with the abstract algorithm because they have no mental image to fall back on.

Stage 3: Abstract

At the abstract stage, students use the standard written algorithm with numerals and operation symbols only. For 143 + 28, they write the numbers aligned by place value and apply regrouping rules with digits. Your job is to confirm that the student's written work matches what they demonstrated with blocks and drawings. If the abstract answer differs from the concrete result, that is a signal to return to an earlier stage — do not simply tell the student the correct answer. Instead, ask guiding questions: "How many ones did we have altogether?" or "What did we do when we had more than nine in one column?"

Supporting Concept Development Without Introducing Errors

The most common paraprofessional errors in math support are:

1. Using imprecise language (e.g., calling the denominator "the bottom number" without explaining its meaning).

2. Skipping the pictorial stage because it feels slow.

3. Giving the answer instead of asking guiding questions.

4. Misrepresenting the concept at the concrete stage, which then carries forward.

Before any support session, review the concept yourself using the teacher's lesson plan and your ParaPro Mathematics skill area reference sheet. If you are uncertain about a concept, ask the supervising teacher before working with students — never guess.

Error Analysis: Identifying and Responding to Misconceptions

Error analysis is a professional skill. When you review a student's work, first categorize the error:

Respond with questions that redirect thinking rather than simply correcting. For a student who writes 1/2 + 1/3 = 2/5, you might say: "Use the fraction tiles. Show me one-half and one-third together. Is that the same as two-fifths?" This lets the student discover the error rather than simply receiving a correction, which builds deeper understanding.

Co-Planning a CPA Mini-Lesson

When you co-plan a mini-lesson with a peer, confirm three things before you rehearse:

1. Is the concrete model mathematically accurate for this concept?

2. Does the pictorial representation show exactly the same relationships as the concrete model?

3. Does the abstract notation match both earlier stages?

Use the content-accuracy checklist during rehearsal. A critical error — any step that would teach students an incorrect mathematical relationship — must be corrected before you deliver the lesson to actual students.

Safety Considerations

Manipulatives such as small unit cubes are choking hazards. When working with young learners, supervise materials at all times and follow the supervising teacher's protocols for managing and storing manipulatives. Report any broken or missing pieces immediately. Keep the workspace clear of clutter so materials do not become trip hazards.

Key terms8

Concrete-Pictorial-Abstract (CPA)An instructional sequence that moves from physical manipulatives, to diagrams or drawings, to numerals and symbols, building mathematical understanding at each stage.
ManipulativeA physical object — such as a base-ten block or fraction tile — that students handle to explore a mathematical concept.
RegroupingThe process of exchanging a set of units for the equivalent in the next place value (e.g., trading 10 ones for 1 ten) during addition or subtraction.
Place valueThe value of a digit determined by its position in a number; for example, the digit 3 in 143 represents 3 ones, while the digit 4 represents 4 tens.
Conceptual errorA student mistake that reveals a misunderstanding of the underlying mathematical idea, not just a procedural slip.
Procedural errorA student mistake made during the steps of a calculation even though the student understands the concept.
Guiding questionA question a paraprofessional asks to redirect a student's thinking toward the correct idea without directly giving the answer.
Content-accuracy checklistA structured tool used to verify that every stage of a lesson segment represents the mathematical concept correctly and introduces no errors.

Worked examples4

1. Model 136 + 47 using the full CPA sequence and find the sum.
  1. Concrete — Place 1 hundred flat, 3 ten-rods, and 6 unit cubes on the workspace for 136. Add 4 ten-rods and 7 unit cubes for 47.
  2. Concrete — Combine the unit cubes: 6 + 7 = 13 units. Trade 10 unit cubes for 1 ten-rod (regrouping). You now have 3 unit cubes remaining and 1 new ten-rod.
  3. Concrete — Combine the ten-rods: 3 + 4 + 1 (regrouped) = 8 ten-rods. Combine with the 1 hundred flat. Result: 1 hundred, 8 tens, 3 ones.
  4. Pictorial — Draw a place-value chart with columns H | T | O. Mark 1 in H, 3 in T, 6 in O for the first number; 0 in H, 4 in T, 7 in O for the second. Cross out 10 marks in the O column, write 1 above the T column, and record 3 in O. Add the T column: 3 + 4 + 1 = 8. Record 8 in T and 1 in H.
  5. Abstract — Write the standard algorithm: 136 + 47, aligned by place value. Add ones: 6 + 7 = 13; write 3, carry 1. Add tens: 3 + 4 + 1 = 8; write 8. Add hundreds: 1 + 0 = 1; write 1. Result: 183.
  6. Verify: the concrete, pictorial, and abstract stages all produce the same answer, confirming accuracy.
Answer: 183
2. A student's paper shows 1/2 + 1/4 = 2/6. Identify the error type and write one guiding question.
  1. Examine the work: the student added numerators (1 + 1 = 2) and denominators (2 + 4 = 6) separately.
  2. Classify the error: this is a conceptual error. The student does not understand that denominators represent the size of each part; they cannot be added the same way numerators are.
  3. Check the correct process: to add fractions, the denominators must be the same. The least common denominator of 2 and 4 is 4. Rewrite 1/2 as 2/4. Then 2/4 + 1/4 = 3/4.
  4. Write a guiding question: 'Use your fraction tiles — show me what one-half looks like and what one-fourth looks like. How many fourth-pieces fit together to equal one-half? Now put all those pieces together — what fraction does that make?'
Answer: The error type is conceptual (adding denominators as if they were independent counts). The correct sum is 3/4. One guiding question: 'Use your fraction tiles to show one-half and one-fourth together, then count how many fourth-sized pieces you have in all.'
3. During a CPA lesson on basic algebra, a paraprofessional uses tiles to model solving x + 3 = 7. Walk through each stage.
  1. Concrete — Use algebra tiles or counters. Place an unknown tile (representing x) and 3 positive unit tiles on one side of a balance mat. Place 7 positive unit tiles on the other side.
  2. Concrete — Remove (subtract) 3 unit tiles from each side to keep the balance equal. The unknown tile alone remains on the left; 4 unit tiles remain on the right. Therefore x = 4.
  3. Pictorial — Draw the balance: a box (representing x) and 3 circles on the left; 7 circles on the right. Cross out 3 circles on each side. The box equals 4 remaining circles.
  4. Abstract — Write the equation: x + 3 = 7. Subtract 3 from both sides: x + 3 − 3 = 7 − 3. Simplify: x = 4.
  5. Confirm all three stages show x = 4, verifying no error was introduced.
Answer: x = 4
4. Use base-ten blocks to model 204 − 67 and find the difference.
  1. Concrete — Represent 204 with 2 hundred flats, 0 ten-rods, and 4 unit cubes.
  2. Concrete — Try to subtract 7 ones: only 4 unit cubes are available. Trade 1 hundred flat for 10 ten-rods, then trade 1 ten-rod for 10 unit cubes. Now have 1 hundred flat, 9 ten-rods, and 14 unit cubes.
  3. Concrete — Subtract 7 unit cubes from 14: 14 − 7 = 7 unit cubes remain. Subtract 6 ten-rods from 9: 9 − 6 = 3 ten-rods remain. 1 hundred flat remains. Result: 1 hundred, 3 tens, 7 ones.
  4. Abstract — Write 204 − 67 aligned by place value. Regroup as described: 204 becomes 1 hundred, 9 tens, 14 ones. Subtract: ones 14 − 7 = 7; tens 9 − 6 = 3; hundreds 1 − 0 = 1. Result: 137.
  5. Verify: concrete and abstract both yield 137.
Answer: 137
In class: Instructor demonstrates a CPA sequence for multi-digit addition using base-ten blocks (concrete), a place-value chart drawing (pictorial), and the standard algorithm (abstract); students replicate each stage at their own workspace. · Students work in pairs to co-plan a 10-minute CPA mini-lesson for one of three assigned topics (fractions, geometry vocabulary, or basic algebra concepts); they rehearse the lesson and peer-evaluate using a content-accuracy checklist. · Error analysis activity: students receive five student work samples with common math errors; they identify the error type, describe the likely misconception, and write a script of what they would say to guide the student without simply giving the answer.
Materials: Base-ten blocks, fraction tiles, geometric shape sets, place-value charts, error-analysis work samples, ParaPro Assessment Mathematics skill area reference sheet, content-accuracy checklist
Check: Instructor observes each student's CPA mini-lesson; student must correctly represent the mathematical concept at all three levels and avoid introducing mathematical errors; content-accuracy checklist must be passed with no critical errors. Maps to ParaPro Mathematics skill area.

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

ParaPro Mathematics — Assisting with Instruction
1. In the CPA sequence, what is the primary purpose of the pictorial stage?
ParaPro Mathematics — Mathematical Skills
2. A student adds 2/3 + 1/5 and writes 3/8 as the answer. What type of error has the student made?
ParaPro Mathematics — Mathematical Skills
3. When using base-ten blocks to model addition and the ones column totals more than nine, the correct action is to:
ParaPro Mathematics — Assisting with Instruction
4. A paraprofessional is guiding a student who solved x + 5 = 12 and wrote x = 17. Which guiding question best addresses the error without giving away the answer?
ParaPro Mathematics — Assisting with Instruction
5. Before supporting a small group in a CPA lesson on geometry vocabulary, a paraprofessional realizes she is unsure of the definition of a 'parallelogram.' What should she do?
ParaPro Mathematics — Mathematical Skills
6. Which of the following best describes a procedural error in student math work?
ParaPro Mathematics — Assisting with Instruction
7. A content-accuracy checklist flags a 'critical error' in a peer's planned mini-lesson. What is the appropriate next step before that mini-lesson is delivered to students?
ParaPro — Instructional Support and Classroom Safety
8. A paraprofessional working with first graders notices that several small base-ten unit cubes have fallen on the floor. What is the most important immediate action?
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