PROBLEM-SOLVING 101 · 02
Schema Instruction
Help students see the mathematics beneath the story
Some students read a word problem and immediately ask, “Do I add or subtract?” Others search for a familiar word, grab the numbers, and hope for the best.
I want students to begin with a different question:
“How are the quantities related?”
Schema instruction helps students recognize the mathematical structure of a problem. The characters, setting, and numbers may change, but the relationships can remain the same.
Different Stories, Same Structure
Consider these two problems:
STORY ABOUT TOYS
Four shelves each hold six toy cats. How many toy cats are there altogether?
STORY ABOUT STUDENTS
Four tables each have six students. How many students are there altogether?
Four equal groups of six
One story is about toys and the other is about students. Both describe four equal groups of six.
Students can represent either story with a drawing, an array, or the equation:
4 × 6 = 24
Recognizing that shared structure helps students connect a new problem to something they already understand.
What Structures Should Students Notice?
These are some useful starting points. The names matter less than students’ understanding of the relationships.
STRUCTURE
WHAT STUDENTS NOTICE
EXAMPLE
Parts and whole
Parts combine to make a whole.
There are 8 black cats and 7 orange cats. How many cats are there?
Change
A starting amount increases or decreases.
There were 15 cats. Some left. Now there are 9. How many left?
Additive comparison
Two amounts differ by a certain quantity.
Mia has 12 toy cats. Leo has 8. How many more does Mia have?
Equal groups
A number of groups each contains the same amount.
Four shelves each hold 6 toy cats. How many toy cats are there?
Multiplicative comparison
One amount is a certain number of times another.
Mia has 3 times as many toy cats as Leo. Leo has 4. How many does Mia have?
The unknown can appear in different places. In an equal-groups problem, students might need to find the total, the number of groups, or the amount in each group.
How I Bring This Into a Lesson
1
Begin With the Story
Use the Three-Read Method to visualize the situation, identify what the quantities mean, and clarify the question.
Give students time to understand what is happening before asking them to calculate.
2
Make the Relationships Visible
Ask students to represent the problem with objects, a drawing, a bar model, or another useful representation.
PROMPTS MIGHT INCLUDE
“Where is each quantity in your model?”
“What does the unknown represent?”
“How does your model show the relationship?”
A model should explain the mathematics. A quick sketch of the setting helps students visualize; a mathematical model shows how the quantities connect.
3
Connect the Model to an Equation
Ask students to explain how their equation matches their representation. Keep the meaning of each number, and the unknown, part of the conversation.
4
Compare and Vary Problems
Place two problems side by side. Ask:
“What changed in the story? What stayed the same mathematically?”
Include problems with different contexts, wording, and unknowns so students have opportunities to recognize a structure beyond one familiar example.
Moving Into Multistep Problems
Consider this problem:
Four shelves each hold six toy cats. Seven toy cats are taken away. How many remain?
The equations follow the relationships:
4 × 6 = 24
24 − 7 = 17
There are 17 toy cats remaining.
Students can identify two connected structures:
EQUAL GROUPS
Ask, “What does 24 mean in this story?”
Four groups of six make 24 toy cats.
CHANGE
This keeps the intermediate answer connected to the situation.
Seven are taken from the starting total.
Keep Understanding at the Center
Schema instruction can become another memorization task if students only label problem types. Keep returning to the relationships:
“Show me how the quantities are connected. Explain how you know.”
The goal is for students to understand a problem well enough to represent it, choose a strategy, and explain why their solution makes sense.
SEE IT IN MY CLASSROOM
On the Math Schemas page, see a worked example from my classroom and download the schemas guide.
Connections to Mathematical Practice
Schema instruction offers opportunities to develop the following practices. These connections depend on what students do and explain during the lesson.
MP2
Reason abstractly and quantitatively
Students connect the story to symbols and return to the story to interpret them. In the cat problem, they explain that 4 represents shelves, 6 represents cats per shelf, and 24 represents the total before any are removed.
MP4
Model with mathematics
Students use a drawing, bar model, array, or equation to represent the relationships in the situation. They interpret the answer in context and check whether their model fits the story.
MP7
Look for and make use of structure
Students recognize equal groups in stories with different settings. They use that relationship to plan a solution, including when the unknown changes position.
MP1 / MP3
Additional opportunities with MP1 and MP3
Students practice MP1 when they make sense of the problem, plan a pathway, and revise their approach. They practice MP3 when they justify why a model fits and discuss a classmate’s reasoning.
TEACHER PROMPT
“How does your model show the relationship, and why does your equation match?”