From High School Success to Academic Challenges at University: What Are We Missing?

.Every year, students complete Grade 12 mathematics with strong averages and gain admission to university programs in engineering, computer science, health sciences, business, and other fields. Admission to some highly competitive programs typically requires averages in the 90s, reflecting the strong academic records of successful applicants.

Yet high school grades and university readiness do not always align. Some students who achieve strong results in Grade 12 encounter unexpected difficulties when they begin university. They may struggle with mathematical knowledge and skills introduced in earlier grades, such as operations with integers and fractions, algebraic manipulation, factoring, graphing, functions, or trigonometry.

These are foundational mathematical concepts and skills on which more advanced learning depends.

How can a student achieve a high average in Grade 12 mathematics while still having gaps in knowledge and skills developed during previous years?

The answer is more complex than assuming that students have forgotten what they learned or that their grades were undeserved. A final mark provides valuable information about achievement in a particular course, but it does not necessarily reveal every strength, weakness, or misconception within a student’s learning.

To understand this distinction, we need to look beyond the final percentage.

When a Final Mark Hides the Learning Profile

Ontario’s Achievement Chart recognizes that academic achievement involves different dimensions of learning. It identifies four categories through which students demonstrate their knowledge and skills:

Knowledge and Understanding: Subject-specific content and comprehension of its meaning.
Thinking: The use of critical and creative thinking processes, including reasoning, planning, and problem solving.
Communication: The expression and organization of ideas and information.
Application: The use of knowledge and skills in familiar and new contexts.

Achievement is described through four levels. Level 1 corresponds to 50–59%, Level 2 to 60–69%, Level 3 to 70–79%, and Level 4 to 80–100%. Level 3 represents the provincial standard, while Level 4 represents achievement beyond that standard.

These categories and levels help us understand why two students with similar final results may have different learning profiles.

In my tutoring practice, I have worked with two students who both achieved Level 3 overall in mathematics but demonstrated different learning profiles.

The first student demonstrated Level 4 achievement in Knowledge and Understanding but Level 2 in Thinking, particularly when selecting strategies and solving unfamiliar problems. The second student demonstrated relatively consistent Level 3 achievement across Knowledge and Understanding, Thinking, Communication, and Application.

Although both students achieved Level 3 overall, their strengths and learning needs differed. The first student demonstrated stronger mathematical knowledge and understanding but needed further development in reasoning and problem solving. The second student demonstrated more balanced achievement across the four categories.

The same overall achievement level can therefore reflect different patterns of strengths and weaknesses.

The final percentage summarizes a student’s performance in a course, but examining the evidence behind that mark provides a more informative picture of what the student understands and where additional support may be needed.

Learning Gaps Can Also Exist Within a Single Category

Differences in learning do not occur only between Achievement Chart categories. They can also exist within the same category.

In mathematics, a student may demonstrate a good understanding of transformations of functions and perform well on trigonometric problems while still making recurring errors involving fractions, negative integers, factoring, or algebraic manipulation. These weaknesses may affect only a limited number of questions in a particular assessment, allowing the student to achieve a strong overall result.

A similar situation can occur in chemistry. A student may perform well in atomic structure, periodic trends, chemical reactions, and several aspects of nomenclature while continuing to struggle with writing formulas for ionic compounds containing polyatomic ions.

In English, a student may communicate ideas clearly in writing but encounter difficulties with reading comprehension, interpreting implicit meaning, recognizing and analyzing literary devices, or constructing well-supported arguments.

These examples illustrate an important distinction between overall achievement and mastery of individual concepts or skills.

Another consideration is the difference between conceptual understanding and procedural fluency.

A student may know how to follow a memorized procedure when solving a familiar mathematical problem without fully understanding why the procedure works or recognizing when it should be applied in a different context. Conversely, a student may understand the underlying concept but lack fluency in the calculations or algebraic manipulations needed to apply it efficiently.

Both situations can create difficulties when students encounter unfamiliar, multistep problems.

The significance of a learning gap therefore depends not only on how much it affected a previous grade, but also on how heavily future learning depends on that particular knowledge or skill.

These underlying difficulties are explored further in our article, Beyond the Grades: The Underlying Struggles, which examines how learning challenges can remain hidden behind students’ academic results.

Yesterday’s Learning Becomes Tomorrow’s Prerequisite

Mathematical learning is cumulative. Concepts and skills introduced in elementary and secondary school become the foundations for more advanced mathematical learning.

For example, students taking Calculus I and II at university are expected to have a solid understanding of foundational mathematical concepts and skills, such as algebraic manipulation, factoring, functions, graphing, and trigonometry. Instructors build on these foundations to teach more advanced concepts, including limits, derivatives, and integrals, and their applications to mathematical problems.

A student may understand the concept of a derivative but struggle to solve a calculus problem because factoring or simplifying algebraic expressions remains difficult. Solving the problem requires the student to apply both the new calculus concept and mathematical procedures learned in previous years. When those earlier procedures are not sufficiently secure, they can interfere with the student’s ability to demonstrate understanding of the new material.

Universities recognize the importance of these foundations.

The University of Toronto provides preparation materials for students entering first-year calculus, covering nine areas of high-school mathematics, including algebra, functions, graphing, and trigonometry. These resources address skills such as working with fractions, rational expressions, order of operations, and algebraic equations.

Similarly, the University of Waterloo offers MTHEL 99, a mathematics-readiness course designed to help incoming students review high-school mathematical knowledge and identify areas requiring additional preparation.

These initiatives do not imply that all incoming university students lack foundational knowledge. Rather, they recognize that students enter university with different learning profiles and that reviewing prerequisite knowledge can support the transition.

The same principle extends beyond mathematics. In physics, students must apply mathematical reasoning to interpret physical situations and solve problems. In chemistry, earlier understanding of atomic structure and chemical bonding supports the study of more complex chemical concepts. Reading comprehension, analytical thinking, and written communication also remain important across university disciplines.

Knowledge and skills learned in previous years do not become irrelevant when a student completes a course. They often become the tools required for future learning.

High School Success and University Readiness Are Not Identical Measures

Research has examined the relationship between high-school mathematics achievement and subsequent university performance.

A Canadian study examining 15 years of student performance data, from 2001 to 2015, compared Grade 12 Pre-Calculus grades with first-year university Calculus grades at a large Canadian university. The researchers found a significant and increasing discrepancy between achievement in the two courses, with a particularly pronounced difference among students who performed poorly in university Calculus.

These findings raise questions about how fully high-school mathematics grades capture readiness for first-year calculus. However, the study examined students at one university, and its findings should not automatically be generalized to every Canadian institution or program.

The transition from secondary to university mathematics has also received attention in Ontario. In February 2026, Francis Duah of Toronto Metropolitan University presented a discussion at the Fields Institute examining the transition from high-school mathematics to university courses, including Calculus I and II and Linear Algebra. The presentation highlighted challenges associated with prerequisite mathematical knowledge, reasoning, and the expectations students encounter in university mathematics.

This broader perspective matters because the transition to university changes more than the complexity of the academic content.

Students may encounter a faster instructional pace, different teaching and assessment approaches, greater expectations for independent learning, and problems requiring them to coordinate several mathematical concepts simultaneously.

A student who previously relied on familiar problem types, frequent teacher guidance, or repeated practice of a particular procedure may find it more difficult to adapt when asked to select an appropriate strategy independently.

These challenges can involve both mathematical knowledge and learning skills, including time management, organization, self-monitoring, and knowing how to respond when a problem is unfamiliar.

Academic difficulties at university therefore cannot be attributed solely to gaps in high-school knowledge. They may emerge from the interaction between students’ prior preparation, their learning strategies, and the expectations of a new educational environment.

Meeting the Standard, Competing for Admission, and Being Ready Are Different Questions

Three concepts are important when considering a student’s transition to university: meeting the provincial achievement standard, satisfying university admission requirements, and being academically prepared for the demands of a particular program.

Although related, these concepts are not interchangeable.

University admission operates according to different criteria. Institutions establish their own prerequisites and selection requirements, which can vary considerably between programs and universities, even within the same field of study. Some programs require specific Grade 12 mathematics or science courses, while others accept a broader range of subjects. Highly competitive programs may also require substantially higher admission averages and additional assessments, such as supplementary applications.

For example, Health Sciences programs at Canadian universities differ in their mathematics and science prerequisites and in their published admission averages. Meeting the requirements for one university does not necessarily mean that a student meets those of another.

Academic readiness introduces another consideration.

A student may meet the admission requirements for a university program while still having specific weaknesses in foundational knowledge or learning skills that will become important after enrolment.

Conversely, a student who does not meet the competitive admission average for a particular program may nevertheless possess strong mathematical reasoning, problem-solving skills, and effective learning strategies.

Admission criteria determine whether a student qualifies for consideration and selection. Academic readiness concerns the knowledge, skills, and learning habits that will support the student once the program begins.

A strong admission average is an important achievement, but it does not provide a complete picture of readiness for every academic challenge ahead.

Perhaps We Need to Look Beyond the Final Number

None of this diminishes the importance of grades. Course marks serve legitimate purposes: they communicate achievement, document academic progress, and provide universities with information for admission decisions.

However, a final percentage cannot describe every aspect of a student’s learning.

A student who earns 92% in Grade 12 mathematics has demonstrated strong achievement in that course. Nevertheless, the mark alone cannot tell us whether the student occasionally struggles with fractions, whether algebraic manipulation is fluent or effortful, whether the student can select a strategy when faced with an unfamiliar problem, or whether graphing skills remain dependent on familiar examples and guidance.

This is where diagnostic assessment can provide valuable additional information.

Rather than relying exclusively on whether an answer is correct or incorrect, a diagnostic approach examines how a student understands the question, selects a strategy, applies prior knowledge, and explains the reasoning behind an answer.

A correct answer does not always indicate secure conceptual understanding, just as an incorrect answer does not necessarily indicate a complete lack of knowledge. Examining the student’s reasoning can help distinguish a calculation error from a misconception, a gap in prerequisite knowledge, or difficulty interpreting the problem.

This information can guide targeted review and support students in developing a more secure foundation before moving into increasingly complex material.

A strong Grade 12 mathematics mark is an achievement worth recognizing. However, understanding a student’s readiness for future learning requires looking beyond the final percentage.

Examining how students reason, apply prerequisite knowledge, approach unfamiliar problems, and respond to difficulties can reveal learning needs that are not immediately apparent from their grades. This information can help educators and students identify areas requiring further development, whether in mathematical understanding, procedural fluency, problem solving, or learning strategies.

The question is therefore not simply whether a student has succeeded in high school, but whether the knowledge and skills required for the next stage of learning are sufficiently secure.

Recognizing the difference between achievement and readiness can help students build stronger foundations and approach the transition to university with greater confidence and independence.