The concrete–pictorial–abstract (CPA) approach teaches a Primary 2 Maths idea in three connected forms: handle or arrange objects, draw a picture that preserves the relationship, then express it with numbers and symbols. For four groups of five, a child can first arrange twenty counters into four equal groups, next draw four circles with five dots each, and finally write 4 × 5 = 20. These are not three unrelated tricks—they describe the same quantity.
This guide shows parents and tutors exactly how to use CPA for P2 place value, exchanging across zero, multiplication/division, fractions, money and data. MOE’s current October 2025 Primary Mathematics syllabus emphasises representations, mathematical relationships and clear communication; earlier MOE material explicitly discusses the CPA approach. CPA is a teaching method, not a promised SEAB P2 examination format. Formal P1/P2 exams and weighted assessments have been removed since 2019.
- Concrete: use counters, bundled tens, fraction strips, price cards or real shapes to make a relationship visible.
- Pictorial: draw a labelled diagram retaining the same group count, units or whole.
- Abstract: write an equation, compare values or apply a mathematical rule.
- For independence: gradually remove prompts, then test with changed numbers rather than the same memorised question.
- Included: 20 complete CPA examples, a 24-item parent observation checklist and troubleshooting guidance.
What CPA means and how it relates to official Singapore curriculum
The concrete–pictorial–abstract approach is associated with Singapore Mathematics pedagogy and appears explicitly in earlier MOE primary syllabus material. The current Primary Mathematics Syllabus, updated October 2025, explains mathematical representations, equivalence, diagrams, communication and metacognition. We use CPA here as a practical way of connecting those ideas, not as a claim that the 2025 content table requires three drawings for every question.
Concrete does not mean “easy work for weak students”. It means that a child can see, handle or act out a mathematical relationship. A physical model provides evidence about the quantity. Pictorial means turning that evidence into an illustration or diagram. Abstract means the relationship can be communicated through symbols or mental reasoning without a full object model.
The stages should connect. If a pupil has twenty counters in four equal groups, the drawing should show the same four groups, and the number sentence should express the same total. If the child accidentally draws five groups of four, the numerical product happens to remain twenty, but the representation no longer matches the exact story. Ask which number counts groups and which counts each group.
Teaching need not proceed in a rigid one-way staircase. A confident pupil may begin with symbols and return to counters to check an unfamiliar mistake. Another may move between a diagram and number sentence several times. The goal is flexibility and independence, not a rule that the concrete phase must be abandoned permanently.
Why P2 is a useful year for concrete representations
P2 extends the number range from P1’s 100 to 1,000, introduces an explicit hundreds place and develops algorithms up to three digits. The child is required to understand several units of different sizes and how ten of one unit can be exchanged for one of the next. Bundled counters or place-value blocks show this far better than unsupported slogans about “carrying”.
Other P2 strands create a similar need. Multiplication means repeated equal groups. Fractions involve equal parts of a fixed whole. Dollars and cents are linked through a hundred-to-one unit relationship. Picture graphs use a symbol that stands for a given number of objects. Each idea is fundamentally about representation and equivalence.
For example, a star in a picture graph may stand for two pupils. Physically place two counters below each star, then draw the same connection, and finally calculate five stars times two pupils each. The symbols convey meaning only because the key defines the relationship.
Adults sometimes stop at numerical accuracy. A child may know that 4 × 5 = 20 while failing to describe twenty divided into four equal groups. Returning to concrete groups can reveal that the memorised fact is not yet a flexible concept. Conversely, a child who already explains the grouping should not be forced to move every counter for every new product.
Stage one: concrete objects without expensive teaching equipment
Household materials can be sufficient: paper dots, buttons, beans kept safely out of reach of younger children, cubes, pencils, folded strips or numbered cards. The materials should be easy to group, move and count. For hundreds, use labelled cards representing ten tens rather than insisting on counting hundreds of individual items.
Ask the child to state the quantity before moving it. “There are four groups with five in each” means twenty objects. The adult should not arrange everything in advance if the learning goal is for the child to create equal groups independently. The physical action is part of the reasoning.
For subtraction 500 − 176, a base-ten block representation or labelled hundred/tens/ones cards is more efficient than five hundred loose counters. The pupil can exchange one hundred for ten tens and one ten for ten ones, showing that the total value remains constant. This is the foundation of the written algorithm.
Use an object only while it helps clarify the relationship. If a student becomes distracted by toys, simpler paper circles or plain dots may be better. The concrete stage is not a reward game that must be lengthy; it is a mathematical demonstration.
Stage two: pictorial diagrams should preserve the exact quantities
Draw what the counters showed. Four physical plates with five objects each can become four simple circles with five dots in each. A number bond can show a whole of 125 and parts of 48 and an unknown. Two aligned bars can show the difference between 163 and 135. The drawing must retain the group sizes, labels and unknown quantity.
Pictorial does not mean an artistic illustration. A row of small dots or two plain rectangles is sufficient. Many children waste time drawing elaborate objects while the underlying numbers remain unlabelled. Encourage quick, clear models that a second reader can interpret.
A picture can reveal a misread story. If a pupil draws Mira’s bar shorter when the problem says she has more than Ali, the diagram contradicts the wording. Repair the representation before calculating. This is more efficient than correcting an arithmetic answer that came from a mistaken picture.
The pupil should be able to translate back: point to each part of the drawing and explain which physical objects or quantities it represents. If they cannot, the diagram may have been copied by rote. Ask for a smaller number and rebuild the connection.
Stage three: abstract notation and independent calculation
The abstract stage is not simply writing a final answer. It means that symbols accurately express the relationship: 4 × 5 = 20, 125 − 48 = 77, or 163 − 135 = 28. Each equation should be read in words and linked to the objects or diagram that gave it meaning.
For a number bond, both 48 + 77 = 125 and 125 − 48 = 77 are valid related facts. Understanding the whole and parts allows the child to choose one equation when the unknown moves. This is stronger than remembering that every empty-box problem has a particular sign attached.
When the child can choose a method without the physical model, try a changed case to test transfer. If 398 + 7 was taught with seven counters on a number line, change to 397 + 6 and ask for the bridge to 400. The learner should partition six into three and three, producing 403.
A learner may still choose to sketch a short bar for an unfamiliar word problem. That is not necessarily a failure to reach abstraction. Mature mathematical reasoning involves choosing a useful representation, including returning to a drawing when it clarifies the question.
Do not mistake a three-stage routine for a guaranteed mark scheme
Singapore Primary Two has no formal examination and no weighted assessment under MOE’s current framework. Neither the 2025 MOE syllabus nor the policy requires a child to earn marks for showing all three CPA stages in every question. Classroom teachers may give task-specific instructions on working and representation; follow those instructions in school.
The educational purpose of CPA is understanding. A child who writes 20 ÷ 5 = 4 and can explain four per child may not need to draw twenty pencils. Another child may know the division fact but still be unable to identify whether the quotient counts bags or objects per bag; a concrete demonstration will help.
Use concrete objects when a concept is new or unstable, pictorial forms when the relationship needs to be made visible, and abstract notation for efficient communication and generalisation. Make room for the child to move backward when checking an unexpected result.
Do not invent a national paper requirement or a “CPA formula” for every problem. The aim is transferable mathematical thinking, not performing a fixed sequence of teaching aids for approval.
Teaching place value from ones to hundreds
Begin with bundles: ten individual items become one bundle of ten; ten bundles of ten become one hundred. Make 352 using three hundred cards, five ten cards and two one counters. Ask the learner to explain 300 + 50 + 2 = 352. The same amount is represented concretely, pictorially and symbolically.
Now show 305 and 350. The first has three hundreds, zero tens and five ones; the second has three hundreds, five tens and zero ones. Their values differ by 45. A place-value table is the pictorial form, and the number comparison is the abstract form. The zero indicates an empty place without changing other digits’ positions.
Do not tell pupils to “move the zero” as the main explanation. Zeros serve as placeholders in positional notation; moving digits between places changes the value. A changed-case comparison such as 407 and 470 tests the same relationship with new quantities.
When the learner explains all three forms independently, the concrete blocks can be put away for that problem. Retain the place-value table as a quick diagnostic tool for later arithmetic difficulties.
Teaching regrouping across zero with exchange
For 500 − 176, begin with five hundred blocks. To remove six ones and seven tens, exchange one hundred for ten tens, then one ten for ten ones. The amount represented is still five hundred, now expressed as four hundreds, nine tens and ten ones. Remove one hundred, seven tens and six ones to leave three hundreds, two tens and four ones: 324.
Draw a hundreds–tens–ones table to show the exchange, then write the vertical subtraction. The written crossing-out marks should correspond to the changed count of unit blocks. If the child writes 476, ask which digits were subtracted and whether the amounts were conserved.
The inverse provides an independent abstract check: 324 + 176 = 500. If the sum fails to return to five hundred, at least one operation or representation is wrong. That is more useful feedback than a bare cross next to the answer.
Change the original to 600 − 176. The remainder becomes 424. The extra hundred in the starting number produces one additional hundred in the final amount, while the same exchange pattern is used. This changed case tests transfer.
Teaching equal groups and both meanings of division
Place twenty counters in five circles with four in each. This concrete representation supports 5 × 4 = 20. Draw the circles and dots as the pictorial stage, then write 20 ÷ 5 = 4 when the question asks how many items each of five children receives.
Now repack the same twenty counters into groups of five. Four groups are formed, so 20 ÷ 5 = 4 when the question asks the number of bags that each hold five items. The mathematical sentence is identical, but the quotient counts different things. The story’s units determine the interpretation.
For a different divisor, 20 ÷ 4 = 5. Ask the child to identify whether four describes group count or group size in each version. This prevents the child from interpreting every division question as one universal sharing routine.
As fluency grows, ask for fact families from one arrangement. Four groups of six give 24, so 24 ÷ 4 = 6 and 24 ÷ 6 = 4. The four-times table is in P2’s specified set, making this appropriate foundational reasoning.
Teaching fraction notation from equal paper strips
Take a paper strip representing one whole. Fold or mark it into eight equal sections and shade three. The concrete or physical representation is one whole split into eighths; the pictorial representation is a drawing of eight equal boxes with three coloured; the abstract notation is 3/8.
Ask what the denominator eight describes: the number and size of equal parts making the whole. Ask what the numerator three counts: the selected parts. If the child writes 8/3, they have reversed the roles, and the model exposes the error.
For 3/8 + 2/8, shade two more eighths on the same strip. Five eighths are shaded; the denominator remains eight. The model shows why the unit-sized pieces did not become sixteenths. The same reasoning works for subtracting like fractions within one whole.
To compare 1/4 and 1/8, use two equal-sized wholes. Without common-sized wholes, a physical comparison of the pieces may be misleading. The part–whole relationship must be defined before symbols can be compared fairly.
Teaching dollars and cents through value equivalence
Use cards worth one hundred cents for dollars and smaller cards for tens and ones of cents. Build $2.05 with two hundred-cent cards and five single-cent units. A drawing can show 200 + 5, and the abstract notation is 205 cents = $2.05.
Build $2.50 with two hundred-cent cards and five ten-cent cards, yielding 250 cents. Both amounts begin with two whole dollars, but the cents differ. Ask the learner to compare the two values: 250 − 205 = 45 cents.
The zero in $2.05 is not optional decoration; it indicates no tens of cents. A child who writes $2.50 for 205 cents needs to connect the drawn units to the notation again, not to memorise a rule about moving a decimal point.
For independent abstraction, give 407 cents without cards and ask the child to write $4.07, then explain that the 7 counts ones of cents. Reverse it to 470 cents = $4.70. This is a place-value lesson expressed through money.
Teaching clock time and duration with a visible model
A clock with movable hands can help a child distinguish the hour hand from the minute hand. Show 8:43 and ask what the smaller and larger hands represent. Move the minute hand to show the next minute and explain the change of the written time.
For duration conversion, place sixty minute marks on a circular clock or a straight timeline, then add twenty-five more marks. One hour and twenty-five minutes equal 85 minutes. The pictorial model is a sixty-minute segment followed by twenty-five minutes; the abstract equation is 60 + 25 = 85 minutes.
Do not teach 1 h 25 min as 125 minutes. The relationship between hours and minutes is sixty-to-one, not one hundred-to-one. The visual model shows the value of a complete hour rather than relying on a misleading analogy with dollar notation.
Formal starting/finishing-time word problems with varied clock formats are developed further in later syllabus work. Parents can use simple age-appropriate clock games as enrichment while keeping the P2 named outcomes clear.
Teaching picture graphs with a visible scale
Draw five stars in a row and write the key “★ = 2 pupils”. Place two counters under each star, making ten counters altogether. Next draw the ten represented pupils as grouped pairs. The abstract number sentence is 5 × 2 = 10 pupils.
Change the key to three pupils per star while keeping five stars. Now the represented value is fifteen pupils, even though the visible icon count remains five. This is a strong test of whether the student reads the legend before computing.
Ask what the answer should count: pupils, not stars. If the child writes five pupils, they have counted the symbols but ignored the representation rule. Returning to the counters is a precise correction for that gap.
As the student progresses, compare categories by calculating each represented value, then subtracting to find a difference. The underlying skills are multiplication facts, units and interpretation of a picture graph with scales—the named P2 statistics topic.
How to tell when to fade the concrete stage
Do not remove objects simply because a number of calendar days has passed. Look for independent explanation. Can the child arrange four groups of five without the adult placing the groups first? Can they draw the corresponding circles with the correct group count? Can they write 4 × 5 = 20 and reverse it as a division fact?
If all three representations are accurate, offer a fresh problem without announcing the method. The learner may start abstractly. When a mistake occurs, ask them to choose a picture or object model to check it. Self-selecting a representation is an important step toward independence.
One child may need counters for division but solve place-value addition mentally. Another may have secure multiplication facts but still require paper strips to understand fractions. There is no mathematical reason to insist that the child must be wholly “at the abstract stage” across every P2 topic at once.
Use spaced changed-case checks. After 398 + 7, ask 397 + 6 two days later. A pupil who can explain the new bridge without the adult counting counters aloud has demonstrated transferable understanding.
Twenty complete CPA lesson examples with correct answers
These are original classroom-style demonstrations. Each starts with a visible relationship, then a suggested sketch, then a concise number sentence or statement. They are not official SEAB P2 questions, and no national mark allocation for CPA stages is implied.
A 24-point CPA teaching and observation checklist
Use the checklist as a practical conversation guide, not a test grade. A learner may need help in the pictorial representation of a fraction while already being independent with a number line. The result tells the adult which representation to strengthen next.
CPA example 1: Make 352
Concept: three hundreds, five tens and two ones.
Concrete: three hundred-squares + five ten-bars + two counters. Pictorial: H/T/O table showing 3, 5 and 2. Abstract: 300 + 50 + 2 = 352.
Changed-case transfer: Move five tens to the ones column only if exchanging each ten for ten ones. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 2: Compare 407 and 470
Concept: four hundreds and seven units placed differently.
Concrete: four hundred cards plus seven one-counters versus seven ten-bars. Pictorial: two aligned H/T/O tables with a zero in a different column. Abstract: 470 > 407; difference 63.
Changed-case transfer: Explain why zero changes the position of the 7. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 3: Add 368 and 245
Concept: hundreds, tens and ones combined with exchanges.
Concrete: make both values using labelled blocks and exchange ten units. Pictorial: place-value columns with carried ten and hundred. Abstract: 368 + 245 = 613.
Changed-case transfer: Change 245 to 235; the total is 603. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 4: Subtract 500 − 176
Concept: regroup five hundreds before removing smaller units.
Concrete: exchange one hundred for ten tens then one ten for ten ones. Pictorial: show 4 H, 9 T, 10 O before subtraction. Abstract: 500 − 176 = 324.
Changed-case transfer: Add 324 + 176 to recover 500. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 5: Mentally add seven to 398
Concept: seven single units crossing a hundred boundary.
Concrete: place two counters to reach 400 then another five. Pictorial: number line 398 → 400 → 405. Abstract: 398 + 7 = 405.
Changed-case transfer: Try 397 + 6 = 403. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 6: Mental subtraction 403 − 20
Concept: two tens removed from a number with zero tens.
Concrete: move backward twice by ten using labelled cards. Pictorial: number-line jumps 403 → 393 → 383. Abstract: 403 − 20 = 383.
Changed-case transfer: Check 383 + 20 = 403. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 7: Four groups of five
Concept: four equal packets with five in each.
Concrete: four circles each containing five counters. Pictorial: four circles with five dots in each. Abstract: 4 × 5 = 20.
Changed-case transfer: Rotate array to see five groups of four without changing total. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 8: Sharing twenty among five
Concept: twenty counters shared evenly between five groups.
Concrete: distribute to five cups until none remain. Pictorial: five circles each containing four dots. Abstract: 20 ÷ 5 = 4 per group.
Changed-case transfer: Compare packing twenty four to a bag. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 9: Packing twenty five per bag
Concept: total twenty and group size five.
Concrete: fill bags of five until all twenty are packed. Pictorial: four separate drawn groups of five. Abstract: 20 ÷ 5 = 4 bags.
Changed-case transfer: Ask why the quotient counts bags, not items per bag. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 10: Number bond 125
Concept: a whole 125 and known part 48.
Concrete: build with labelled quantity cards and cover missing part. Pictorial: one whole bar divided into known 48 and unknown section. Abstract: 125 − 48 = 77.
Changed-case transfer: Check 48 + 77 = 125. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 11: Comparison 163 versus 135
Concept: two quantities with one greater by an extra amount.
Concrete: align two rows of labelled counters or cards. Pictorial: two aligned bars showing gap. Abstract: 163 − 135 = 28.
Changed-case transfer: Swap unknown to smaller amount: 163 − 28 = 135. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 12: One quarter versus eighth
Concept: one common whole partitioned into differently sized equal pieces.
Concrete: fold identical strips into four and eight sections. Pictorial: two equal-length bars with one section shaded. Abstract: 1/4 > 1/8 for equal wholes.
Changed-case transfer: Compare one third and one sixth. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 13: Represent three eighths
Concept: one whole and three selected of eight equal parts.
Concrete: fold one strip into eight equal sections; shade three. Pictorial: eight equal boxes, three shaded. Abstract: 3/8.
Changed-case transfer: Explain numerator and denominator in words. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 14: Add like fractions
Concept: three eighth-sized parts combined with two eighth-sized parts.
Concrete: shade three plus two of one eight-part strip. Pictorial: five of eight sections shown in a drawing. Abstract: 3/8 + 2/8 = 5/8.
Changed-case transfer: Check that the denominator remains eight. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 15: Money $2.05
Concept: two dollars and five cents.
Concrete: two 100-cent cards and five one-cent counters. Pictorial: place-value columns for hundreds/tens/ones of cents. Abstract: $2.05 = 205 cents.
Changed-case transfer: Contrast with $2.50 = 250 cents. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 16: Money $3.70
Concept: three dollars and seventy cents.
Concrete: three hundred-cent cards and seven ten-cent cards. Pictorial: H/T/O cents table showing 3,7,0. Abstract: $3.70 = 370 cents.
Changed-case transfer: Compare with $3.07 = 307 cents. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 17: Hours and minutes
Concept: one complete hour plus twenty-five minutes.
Concrete: one sixty-minute time strip and twenty-five counters. Pictorial: number line 0 → 60 → 85 minutes. Abstract: 1 h 25 min = 85 min.
Changed-case transfer: Explain why it is not 125 minutes. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 18: Read the metre scale
Concept: same length represented by repeated standard metre units.
Concrete: place a metre rule end-to-end along safe floor. Pictorial: line segment with three 1 m gaps. Abstract: 3 metre units = 3 m.
Changed-case transfer: Begin at the actual zero mark and count intervals. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 19: Identify a cylinder
Concept: a solid with two flat circles and a curved side.
Concrete: turn a safe can-shaped model on its side. Pictorial: sketch two circles joined by a curved side. Abstract: solid name: cylinder.
Changed-case transfer: Compare with sphere, which has no flat faces. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
CPA example 20: Scale graph symbols
Concept: each star represents two pupils.
Concrete: five paper stars with two counters under each. Pictorial: five rows or icons paired with a key. Abstract: 5 × 2 = 10 pupils.
Changed-case transfer: Change the key to three; five stars then mean fifteen. The student should be able to say what remains equivalent across the three forms and why the chosen operation matches the story. If one representation disagrees with another, repair the first mismatch before advancing.
Common problems when CPA is used badly
| What went wrong | Why it is unhelpful | Better response |
|---|---|---|
| Adult moves every object while child watches | Student may imitate rather than reason | Ask the learner to arrange the next changed case |
| Diagram does not match the story | Numbers appear correct but units or ownership are wrong | Label each group and confirm its meaning |
| Symbol introduced without meaning | Child memorises a manipulation rule | Rebuild the operation with familiar objects |
| Child must draw models for every easy fact | Unnecessary work may obscure independent fluency | Use representations when they clarify or check |
| Concrete model remains forever with no fading | Child does not practise selecting methods independently | Move gradually to diagrams and symbols |
| Different topics treated as one fixed stage | Understanding develops unevenly across strands | Assess representation needs separately by concept |
A good teaching approach adapts the amount of support to the actual response. If a child already understands a concept, ask for reasoning or a changed example; do not insist on moving every counter. If the child cannot explain an exchange, return to physical or drawn place-value units before repeating the written procedure.
- Can name which quantity is the whole before modelling its parts. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can show the same number with objects and a place-value chart. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can explain what zero tens or zero ones mean in a numeral. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can exchange ten ones for one ten without changing the total. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can exchange ten tens for one hundred and explain its value. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can connect the crossing-out marks in subtraction to a real exchange. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can draw the same number of groups that appear in the story. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can distinguish group count from size per group. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can use an array to explain a multiplication fact. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can use the same array for a related division sentence. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can show a missing addend as a missing part of a whole. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can place the labels larger/smaller/difference on two comparison bars. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can give a complete answer with the unit requested. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can create equal parts of the same-sized whole. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can say what numerator and denominator each mean. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can show why like-fraction addition preserves denominator. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can convert dollars and cents into consistent cent units. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can explain the zero in $2.05 versus $2.50. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can relate one hour to sixty minutes rather than one hundred. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can read a picture graph’s key before counting represented items. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can identify a rotated solid by properties instead of orientation. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can choose among counters, a diagram or direct notation independently. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can check a correct answer using a second representation. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
- Can solve a changed-case question after the adult removes prompts. Record “independent”, “prompted” or “still developing” and request a new example when uncertain.
The checklist does not produce an official grade, diagnosis or pass/fail outcome. A learner’s representational needs may differ substantially by topic. Use these observations to guide the smallest next teaching task.
A seven-session family teaching sequence using CPA
Session 1: build two numbers with tens and hundreds blocks, then write them in a place-value table. Session 2: model a regrouping subtraction and check by addition. Session 3: make and draw equal groups, then write a multiplication/division fact family. Session 4: fold fraction strips and compare equal parts of a common whole.
Session 5: use money cards to match $2.05 with 205 cents and $2.50 with 250 cents. Session 6: read a clock or picture graph with a clear key. Session 7: mix three unfamiliar questions and let the child choose whether an object, picture or direct number sentence will help.
This is an optional family plan, not a prescribed school timetable. A child who is unsure about a single stage may spend several sessions there, while a child who is independent can move quickly to changed cases. The learner’s explanation and teacher feedback should set the pace.
After each session, record one observation: “Can build 407 physically but sometimes writes 470” or “Can draw equal groups and now explains 20 ÷ 5.” Those notes point to a specific next lesson more clearly than a made-up national P2 exam grade.
Frequently asked questions about CPA for P2
What does concrete–pictorial–abstract mean?
It means moving among objects or physical actions, a matching drawing or diagram, and numbers or symbols expressing the same mathematical relationship.
Does MOE require children to draw all three stages for each question?
No such universal P2 examination rule applies. CPA is a teaching approach compatible with the MOE emphasis on representations and understanding, not an official mark scheme for every calculation.
What if my child already knows the answer mentally?
Allow an efficient correct method. Ask for a model when a concept is unfamiliar or an answer needs checking; representations support reasoning rather than being permanent compulsory steps.
Are concrete objects only for struggling students?
No. They can introduce new concepts and help check surprising results. Even confident students may use models to explore a new relationship.
When should we remove counters?
Gradually, when the child can draw, explain and solve changed cases independently. Return to counters when a new concept needs a visible model.
Does CPA guarantee a higher Maths score?
No teaching method guarantees a particular future grade. Use actual independent progress and teacher observations rather than promises about an official P2 examination that does not exist.
Sources and the rest of the P2 learning pathway
Current curriculum: MOE Primary Mathematics Syllabus, updated October 2025, framework and P2 pages 33–34. Pedagogical background: Earlier MOE Primary Mathematics syllabus discussing the concrete–pictorial–abstract approach. Policy: MOE Education Conversations, removal of P1/P2 weighted assessments and exams since 2019.
For related examples see P1-to-P2 Maths transition, regrouping within 1,000, equal groups and fact families, fractions with equal parts and clear working and checking.
Last reviewed 13 November 2026. CPA stages are a teaching aid; questions, illustrations and classroom activities are original, not official P2 examination paper specifications.