Schema-based instruction for word problems: how to teach it
By the IEP Atlas editorial teamLast updated
Short answer
Schema-based instruction teaches students to sort a word problem into a type, or schema, by its structure, then solve it with that type's diagram. Teach one schema at a time with stories that have no question yet, add the diagram, practice a short solving routine, then mix types so students must tell them apart. Probe weekly with mixed problem sets.
Word problems defeat many students because they read the words but miss the structure. "Mia had 12 stickers and gave some away. Now she has 5" and "Mia has 12 stickers and 5 fewer than Leo" use similar words but describe different situations. Schema-based instruction teaches students to recognize the situation first, then reach for a matching diagram and equation.
This page is the teaching how-to: the order of steps, what each lesson looks like, and one fictional student's weekly data. For goal wording and the full list of problem types, see math problem solving IEP goal examples. For the wider math area, see the math IEP goals pillar.
Why teach structure instead of keywords
Keyword lists ("altogether means add," "left means subtract") work on simple worksheets and fail on real problems. "Jon has 8 more cards than Sam. Jon has 20. How many does Sam have?" contains "more" but needs subtraction. Students taught keywords add, get 28 and never notice.
The federal RtI math practice guide recommends instruction on solving word problems based on their common underlying structures.[1] The newer elementary intervention guide makes a similar recommendation and adds explicit instruction and visual representations.[2] For grades 4 to 8, the problem solving guide recommends teaching students to use visual representations and to monitor and reflect on their work.[3] Schema-based instruction puts those three ideas into one routine.
What you need before you start
- A short pretest: 10 mixed word problems, scored for correct setup and correct answer.
- One diagram for each schema you will teach, printed on a card.
- A solving routine card with 5 or 6 steps.
- Sets of stories for each schema: some with every number known, some with one unknown.
- An accuracy data sheet with a column for setup and a column for answer.
Step-by-step: teaching schema-based instruction
Step 1: Choose the schemas and the order
Start with the three additive schemas: total (parts make a whole), difference (two amounts compared) and change (an amount goes up or down). Teach multiplicative schemas (equal groups, comparison by times, ratio) only after the additive ones are solid. Most students in grades 2 to 4 need the additive schemas first; older students may start there too if pretest errors show they confuse them.
Step 2: Teach one schema with complete stories
Begin with stories that have no unknown: "There are 9 red apples and 6 green apples. There are 15 apples in all." Ask the student to find the parts and the whole, and to explain why this is a total story. Do 4 to 6 stories in one lesson. Because nothing is missing, the student can focus only on the structure.
Step 3: Add the diagram
Map each complete story onto the schema's diagram. For total: two part boxes joined under one whole box. For change: start, change, end. For difference: a bigger bar, a smaller bar and the gap. Then cover one number and ask which box it came from. Now the student is solving for an unknown with the structure already visible.
Step 4: Teach a short solving routine
Use the same 6 steps on every problem:
- Read the problem and say what it is about.
- Name the schema: total, difference or change.
- Draw or fill in the diagram.
- Write the equation, with a letter or blank for the unknown.
- Solve.
- Check: does the answer make sense in the story?
Model it aloud on 2 problems, solve 2 together, then let the student solve 2 alone with feedback.
Step 5: Teach the next schema, then mix
Teach the second schema the same way, from complete stories to the diagram. Then give mixed sets of the two. The key skill is now telling them apart: "Is this a change story or a difference story? How do you know?" Add the third schema and mix all three.
Step 6: Teach transfer features
Real problems are messier. Add one feature at a time: extra numbers that are not needed, the unknown in a new position (start unknown in a change story), unfamiliar words, a chart instead of sentences, or two steps. Name the feature when you introduce it. "This one has extra information. Cross out what the diagram does not need."
Step 7: Monitor with mixed weekly probes
Give 10 mixed problems each week with new numbers and contexts. Score setup (correct schema and diagram) and answer separately. Repeated short probes graphed over time show whether the instruction is working.[4]
Worked example: one student from start to finish
Jayden (fictional) is in grade 4 with a math problem solving goal. On a pretest of 10 mixed additive problems, he sets up 3 correctly and gets 2 right answers. He adds the two numbers in 7 of 10 problems.
Goal. Given 10 mixed one-step total, difference and change problems, Jayden will name the schema, draw the diagram and solve 8 of 10 correctly on 3 consecutive weekly probes.
| Week | Instruction focus | Setup correct out of 10 | Answer correct out of 10 |
|---|---|---|---|
| Baseline | None | 3 | 2 |
| 1 | Total schema, complete stories | 3 | 3 |
| 2 | Total diagram and routine | 4 | 4 |
| 3 | Change schema | 5 | 4 |
| 4 | Total and change mixed | 6 | 5 |
| 5 | Difference schema | 6 | 6 |
| 6 | All three mixed | 7 | 6 |
| 7 | Extra information added | 7 | 7 |
| 8 | Unknown in new positions | 8 | 8 |
| 9 | Mixed | 9 | 8 |
| 10 | Mixed | 8 | 8 |
Setup improved before answers did, which is the expected order. In week 6 his answers lagged setup by 1, all from computation slips. The team allowed a hundreds chart on probes so the goal stayed about structure. By week 10 Jayden met the goal on 3 consecutive probes.
Common mistakes and better choices
- Weak: Teaching all schemas in one week. Better: One schema at a time, then mix.
- Weak: Starting with problems that have an unknown. Better: Complete stories first, so the structure is the only task.
- Weak: Keyword charts on the wall. Better: A schema chart with a diagram and one example for each type.
- Weak: Scoring only the final answer. Better: Score setup and answer separately to see which part is improving.
- Weak: Practicing only clean textbook problems. Better: Add extra information, new wording and unknowns in new positions.
- Weak: Dropping the diagram once the student is accurate. Better: Fade it slowly, and bring it back for any new problem type.
Tools and templates
- Problem solving goals for templates such as naming the schema and drawing a bar model.
- Math problem solving IEP goal examples for goal wording and the schema table.
- Concrete-representational-abstract math for moving from objects to diagrams.
- Percent accuracy measurement and the accuracy data sheet.
- IEP goals for math calculation for computation goals when slips hold back answers.
- The goal checker to test a draft goal.
Frequently asked questions
What is schema-based instruction in math?
A way of teaching word problems by their structure. Students learn a small set of problem types, such as total, difference and change, with a diagram for each. They sort a new problem into a type, fill in the diagram, then write and solve the equation.
Which grades is schema-based instruction for?
Additive schemas fit grades 1 to 4 and older students who still confuse problem types. Multiplicative schemas, such as equal groups and ratios, fit grades 3 to 8. The routine is the same; only the types and numbers change.
How long does it take to teach one schema?
Many teachers spend about a week per schema in short daily lessons, then a week or more on mixed practice. Weekly probes tell you when to move on: when setup is correct on most problems of that type, add the next one.
Use it in IEP Atlas
- Problem solving goals (math)
- Schema based instruction in the IEP glossary
- Accuracy data sheet
- IEP goal checker
Related guides
- IEP goals for math: calculation, problem solving and functional math
- Math problem solving IEP goal examples
- Concrete-representational-abstract (CRA) for math goals
- IEP goals for math calculation
- Functional math IEP goal examples for high school
Sources
- Assisting Students Struggling with Mathematics: RtI for Elementary and Middle Schools, What Works Clearinghouse, Institute of Education Sciences
- Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades, What Works Clearinghouse, Institute of Education Sciences
- Improving Mathematical Problem Solving in Grades 4 Through 8, What Works Clearinghouse, Institute of Education Sciences
- Progress Monitoring: Data & Goal Setting, National Center on Intensive Intervention
Written and checked by the IEP Atlas editorial team under our editorial policy. This guide describes general practice and federal rules. It is not legal advice; your state may add requirements, and the IEP team makes the decisions for each student.