How to Identify and Close Math Learning Gaps Before the New School Year
By Lusine Baghdasaryan, PhD, Founder of APRISE Learning
The beginning of a new school year is a natural time to reconnect with math.
Children do not need to remember every problem, formula, or procedure from the previous year. What matters most is whether the key ideas they will need next are reasonably secure.
A short review before school starts can help parents answer three useful questions:
What has stayed strong? What needs a quick refresh? And is there one important concept that should receive a little more attention before new material builds on it?
The goal is not to repeat an entire year of mathematics. It is to identify the few skills that will make the transition into the new year smoother.
What Does a Math Learning Gap Actually Look Like?
A math learning gap is a concept or skill that has not yet become secure enough to support the mathematics that follows.
Sometimes it is obvious. A student may not remember how to divide fractions or solve a basic equation.
But many gaps are less visible.
A child might:
Remember a procedure but not understand why it works
Complete familiar examples but hesitate when the problem looks different
Know individual steps but lose track when several steps must be combined
Calculate accurately but have difficulty deciding which operation to use
Understand a new concept until an older skill appears inside the problem
For example, a student may understand the idea behind solving an equation but get stuck whenever fractions appear.
In that case, the main issue may not be algebra at all. A small amount of focused fraction review may make the algebra much easier.
That is why identifying the first point where the work becomes uncertain is often more useful than simply counting incorrect answers.
One Mistake Is Not Necessarily a Gap
Every student makes mistakes.
A single incorrect answer may come from rushing, misreading a question, copying a number incorrectly, or simply having an off moment.
Look instead for patterns.
A skill may deserve more attention when the student:
Makes the same type of error repeatedly
Cannot explain a concept that was previously taught
Needs an example every time before beginning
Can solve a problem in one format but not another
Consistently avoids a particular type of problem
Needs an unusually long time to complete a familiar skill
Has difficulty using an earlier skill inside newer mathematics
Patterns tell us much more than isolated errors.
A Simple Back-to-School Math Check at Home
Parents do not need to create a formal test.
A brief, low-pressure review can provide useful information.
Choose a small number of problems from material your child learned last year. Include several different types:
1. A Familiar Skill
Choose something your child should recognize readily.
This helps answer:
Has the basic skill remained comfortable?
2. A Problem That Combines Two Skills
For example, a problem might require both fractions and measurement, or multiplication and a word problem.
This shows whether the student can use familiar ideas together.
3. A Word Problem
Ask the child to explain what the problem is asking before calculating.
This helps separate mathematical understanding from simply recognizing a familiar procedure.
4. A Visual Problem
Use a graph, number line, diagram, table, geometric figure, or model.
Seeing the same mathematics represented differently can reveal how flexible the understanding is.
5. An Unfamiliar Problem
Choose a problem where the method is not immediately obvious.
The goal is not necessarily to get the correct answer.
Watch what the student does when they need to think.
Do they draw something? Try an example? Look for a pattern? Break the problem into pieces? Ask a useful question?
Those strategies provide valuable information about mathematical readiness.
Ask for the Thinking, Not Just the Answer
After a problem, ask:
“How did you decide what to do?”
“Why does that method work?”
“Could you solve it another way?”
“How could you check your answer?”
“Where did this problem become difficult?”
“What part felt familiar?”
A correct answer with no understanding and an incorrect answer with strong reasoning tell very different stories.
Listening to the explanation can help you determine whether the child needs:
A quick reminder
More practice
A different explanation
Review of an earlier concept
More experience applying the idea in new situations
What Should Elementary Students Review?
The exact skills depend on the child’s grade, but elementary mathematics builds heavily on number relationships.
Useful areas to check may include:
Place value
Addition and subtraction strategies
Multiplication and division
Basic fact fluency
Fractions
Decimal concepts
Measurement
Understanding mathematical symbols
Explaining simple problem-solving strategies
For younger students, use concrete examples whenever possible.
Coins, clocks, rulers, measuring cups, number lines, blocks, drawings, and everyday quantities can make mathematical relationships easier to see.
For example, instead of asking only:
“What is 1/2 + 1/4?”
you might ask:
“If half of a pizza is left and someone gives you another quarter of a pizza, how much pizza do you have?”
Then ask the child to draw it.
The goal is to see whether the child understands the quantities as well as the calculation.
What Should Upper Elementary and Middle School Students Review?
As mathematics becomes more connected, several foundations become especially important.
Consider checking:
Fraction operations
Decimals and percentages
Ratios and proportions
Positive and negative numbers
Order of operations
Multi-step word problems
Variables and expressions
Basic equations
Coordinate graphs
Area, perimeter, and volume
Fractions deserve particular attention because they appear inside so many later topics.
A student may understand ratios, equations, slope, or probability conceptually but still find the work unnecessarily difficult if fraction operations require too much effort.
That does not mean reviewing every fraction topic.
Find the specific operation or relationship that is interfering with current work and focus there.
What Should Students Preparing for Algebra or Higher Math Review?
For students entering Algebra I, Algebra II, geometry, precalculus, or other advanced courses, the most useful review is often prerequisite mathematics rather than previewing large amounts of the new course.
Depending on the upcoming class, useful areas may include:
Fractions and rational numbers
Negative numbers
Ratios and proportional reasoning
Exponents
Order of operations
Solving equations
Equivalent expressions
Coordinate graphs
Slope
Function notation
Basic geometry relationships
The Institute of Education Sciences recommends that algebra instruction help students recognize the structure of expressions and equations, compare solution strategies, and connect different representations—not simply memorize procedures.
A student who begins the year with secure prerequisites has more mental space available for those deeper ideas.
Do Not Review Everything
One of the easiest ways to make back-to-school math unpleasant is to turn it into a complete replay of the previous year.
If a child remembers a topic comfortably, move on.
A useful review is selective.
Try this sequence:
Check → Identify → Refresh → Apply
Check
Use a few varied problems to see what the student remembers.
Identify
Notice one or two skills that would benefit from attention.
Refresh
Review the idea using a concise explanation and a small amount of focused practice.
Apply
Use the skill in a different or slightly more complex problem.
This final step matters.
If a student reviews fraction multiplication, for example, use it afterward inside a measurement, ratio, geometry, or algebra problem.
That helps reconnect the skill to the larger mathematical system.
How Much Practice Is Enough?
More practice is not always better practice.
Once a student demonstrates comfortable understanding and can apply the skill independently, endless repetition may add little value.
A better question is:
Can the student use the idea accurately, explain it, and recognize when it is needed?
Short practice sessions spread across several days can be easier to sustain than one long review session.
A simple session might include:
5 minutes of familiar review
10–15 minutes on one focused skill
One application or reasoning problem
A short explanation from the student
That is often enough to reconnect with a concept without turning the end of summer into another school day.
The Institute of Education Sciences includes spaced review, worked examples, visual representations, and opportunities to revisit important content among evidence-based approaches for mathematics instruction.
What If Your Child Has Forgotten Quite a Bit?
First, keep the review focused.
After a long break, some skills may simply need reactivation.
Start with a familiar example, provide a reminder if needed, and then try a similar problem again.
If the idea returns quickly, it may have been a memory refresh rather than a significant gap.
If the student continues to have difficulty after explanation and practice, move backward one step.
For example:
If dividing fractions is difficult, check whether the student understands multiplication of fractions.
If solving equations is difficult, check operations with negative numbers and fractions.
If percentages are difficult, revisit fractions, decimals, and proportional relationships.
The objective is to locate the point where understanding becomes secure again and build forward from there.
When Is Extra Support Worth Considering?
Parents may want additional help when:
The same foundational skill continues to interfere with several topics
Review at home repeatedly leads to frustration
The student needs more explanation than a quick refresher provides
Important prerequisite material was never fully learned
The upcoming course depends heavily on a skill that is still developing
The child would benefit from a more structured review plan
It is difficult to determine exactly where the confusion begins
Outside support can be most effective when it is targeted.
Instead of simply assigning more practice, the goal should be to determine which concept needs attention and what explanation or experience will make it more secure.
Use the First Weeks of School as a Second Readiness Check
Back-to-school review does not end on the first day of class.
The first few weeks of school provide valuable new information.
Pay attention to:
How independently homework is completed
Which topics seem immediately familiar
Where the student needs repeated reminders
Whether new concepts make sense when taught
Which prerequisite skills the teacher expects students to use
Whether assignments are taking much longer than expected
Which topics generate curiosity or enthusiasm
Do not react to every difficult assignment.
New material is supposed to require effort.
Instead, watch for recurring patterns.
If one underlying skill appears repeatedly, addressing it early can make the rest of the course more manageable.
How APRISE Learning Approaches Back-to-School Readiness
At APRISE Learning, our evaluation helps us understand how a student approaches mathematics, what knowledge is already secure, and which concepts would be most useful to strengthen before building further.
Because students differ in age, coursework, experience, interests, and goals, the evaluation is adapted while following the same thoughtful framework for every learner.
For back-to-school readiness, we may look at:
Key prerequisites for the upcoming course
Fluency with frequently used skills
Conceptual understanding
Application in unfamiliar problems
Mathematical reasoning
Organization of multi-step work
Ability to explain a solution
Connections between earlier and upcoming topics
The purpose is to identify a focused starting point.
A student may need only a short review of one prerequisite skill. Another may benefit from a more structured reinforcement plan. Another may already have the foundations in place and be ready to move directly into the new material.
The recommendation should reflect what the student actually needs—not how much material can be assigned.
Begin the Year With a Clearer Picture
Preparing for a new school year does not require mastering everything in advance.
A more useful goal is to make sure the mathematical foundations most likely to matter next are ready to support new learning.
A few carefully chosen problems can reveal a great deal.
Look for patterns.
Find the earliest point of uncertainty.
Review selectively.
Connect the refreshed skill to something new.
Then move forward.
That approach turns back-to-school review into preparation rather than repetition—and gives students a stronger foundation for the mathematics ahead.
Want a clearer picture of your child’s math readiness for the new school year? Schedule a Free Evaluation to identify the skills that are already secure and the most useful next steps for the year ahead.
Sources and Further Reading
Institute of Education Sciences, U.S. Department of Education. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades
Institute of Education Sciences, U.S. Department of Education. Improving Mathematical Problem Solving in Grades 4 Through 8
Institute of Education Sciences, U.S. Department of Education. Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students
Institute of Education Sciences, U.S. Department of Education. Evidence-Based Strategies for Improving Student Literacy and Teaching Mathematics in Grades PK–9
This article provides general educational information. Individual learning needs vary, and the appropriate type and level of academic support should be based on the individual student.