Her Scores Were Not Wrong. They Were Incomplete.

For years, I thought my daughter was doing well in math. She took an accelerated math class in 8th grade that included Algebra I concepts and scored in the 90th percentile on the state test. The following year, she was placed in another accelerated course that covered Geometry and Algebra 2 content, and once again, she performed well on the state assessment. On paper, everything looked fine. More than fine, actually. Her course placements suggested she was ahead, her scores suggested she was succeeding, and by the time she entered 10th grade, she had supposedly completed the equivalent of three years of high school mathematics.

So when she transferred to an out-of-state boarding school, she was placed in Precalculus. Two weeks later, she called me to tell me she was failing. She later said that making that call felt like calling from jail to tell me she had been arrested. She did not want to make it, but she knew she needed help.

What is important, though, is that she was not ready to leave the course. She was confident that she could stay in Precalculus if I tutored her over the phone. From her perspective, she understood the mathematics and had simply made some mistakes. I asked her to send me the quizzes and tests she had missed, and almost immediately I noticed a pattern. She was making mistakes with factoring polynomials.

I told her I thought she had a gap in her understanding. She disagreed. “I know how to factor. I just made careless mistakes.” So instead of showing her another procedure, I started asking questions. What does factoring actually mean? Why are you doing that step? How do you know the expression you wrote is equivalent to the original one?

The more questions I asked, the clearer it became that she did not really understand factoring. She knew a procedure and could sometimes get the right answer, but she did not understand the mathematics underneath the procedure well enough to use it flexibly. She had passed the courses and scored well on the state assessments, but the understanding underneath those results was fragile.

That realization has stayed with me for years. I was her mother. I was also a mathematics teacher. And I did not know. Everything I had been looking at said she was doing well. None of it showed me what she actually understood.

Her scores were not wrong. They were incomplete.

When success hides a gap

My daughter’s experience illustrates something I think we underestimate in mathematics education. Students can look successful for a long time while carrying significant gaps in understanding. They can earn good grades, pass courses, score well on state assessments, and even be placed in accelerated mathematics while relying on procedures they do not really understand.

The challenge is that procedural knowledge can work for quite a while. A student may remember what steps to follow when the problem looks familiar. They may recognize a pattern on a multiple-choice test or reproduce a process closely enough to earn credit. That success can reinforce the belief that they understand the mathematics.

The gap often becomes visible only when the mathematics becomes more demanding. Now the student has to use an earlier idea in a new situation, recognize structure, explain why something works, connect one representation to another, or decide which strategy makes sense without being told exactly what to do. That is when fragile understanding can finally be exposed.

In my daughter’s case, the problem did not begin in Precalculus. Precalculus simply revealed it. By then, the gap may have been there for years.

The cost of weak understanding is often delayed

This matters because we can easily misread when the problem began. If a student starts struggling in an advanced course, we may assume the difficulty belongs to that course. We might conclude that the content is too hard, that the student is not ready, or that the teacher needs to reteach what is currently being taught.

Sometimes that is true. But sometimes the student is encountering the delayed consequences of an earlier gap that was hidden by successful performance.

That is what happened with my daughter. She genuinely believed she knew how to factor. From her perspective, she was making careless mistakes. Her previous performance had given her no reason to think otherwise.

Students do not always know what they do not understand. That is why teachers and leaders need systems that make student thinking visible before the next course exposes the gaps.

Eventually, I had to convince my daughter that the goal was not simply to survive

Precalculus. She wanted to stay in the course and believed that with enough tutoring, she could make it work. I believed she needed something different. She needed time and space to build the understanding that her previous success had suggested she already had.

Moving her to a lower course was not about lowering expectations. It was about giving her the opportunity to strengthen the foundation she would need in order to be successful later.

What does this have to do with school leadership?

Quite a lot.

Secondary principals are held accountable for mathematics outcomes, yet many of the indicators available to them focus primarily on results. They see benchmark scores, state assessment data, course pass rates, credit accumulation, pacing reports, standards coverage, and intervention data. All of those indicators matter, and I am not suggesting that leaders stop using them.

But they cannot answer every question a leader needs answered.

A score can tell us whether students got enough answers correct. It cannot always tell us what students understand, whether they can explain their reasoning, whether they understand why a procedure works, whether they recognize when that procedure is appropriate, or whether they can use what they have learned when the problem changes.

It also cannot tell us whether the understanding students are building today will be strong enough to support the mathematics they will encounter tomorrow.

That is a significant limitation when leaders are trying to determine whether instruction is actually improving.

The missing middle between instruction and achievement

In schools, we often talk as though the relationship between teaching and achievement is direct. Improve teacher practice, and scores will improve. Adopt a stronger curriculum, and scores will improve. Implement a new intervention, strengthen PLCs, or provide more professional learning, and scores will improve.

But something has to happen between what teachers do and the outcomes students eventually produce.

Students have to think. They have to make sense of ideas, connect new learning to what they already know, reason, explain, revise, make decisions, and develop increasingly sophisticated understandings of mathematics.

That student thinking is the missing middle between instruction and achievement.

If leaders only monitor what teachers are doing on one side and what students score on the other, they can miss the very process through which learning occurs.

This is why a classroom can appear successful during a walkthrough while students are doing very little mathematical thinking. The teacher may be explaining clearly, students may be compliant, everyone may be on the same page, and students may even be producing correct answers. But the leadership question cannot stop with, “Did they get it right?”

We also need to ask, “What did students have to understand and do in order to get there?”

Student thinking is not an alternative to achievement data

Paying attention to student thinking is not an argument against test scores. Principals are accountable for results, and that reality is not going away. The question is whether leaders have enough information to influence those results before the scores arrive.

Achievement data are often lagging indicators. They tell us something about what has already happened. Evidence of student thinking can function as an earlier signal.

When leaders examine student explanations, strategies, misconceptions, representations, discussions, and written work, they gain access to information that a score alone cannot provide. They can begin asking what students seem to understand, where their reasoning is breaking down, whether they are using procedures with understanding or simply reproducing steps, and whether they can use what they have learned in unfamiliar situations.

Those questions move leadership closer to the actual learning process. And that matters because by the time a conceptual gap appears clearly in achievement data, the instructional problem may already be years old.

What leaders choose to see shapes what schools improve

For a long time, the evidence I had about my daughter told me she was doing well. I had scores, course placements, and evidence of acceleration. What I did not have was access to her thinking.

I did not know how she understood the mathematics until I sat with her work and started asking questions.

That experience changed the way I think about evidence.

If we want students to become stronger mathematical thinkers, our leadership systems have to help us see more than whether answers are correct. Classroom observations need to include evidence of what students are thinking and doing. Growth conversations need to move beyond telling teachers what to fix and help them examine the learning their instruction is producing. PLCs need to spend time looking at student reasoning, not only percentages correct. Assessments need to give students opportunities to show how they are thinking.

Instructional planning also has to consider not only what content will be covered, but what mathematical work students will actually do.

None of this requires a principal to become a mathematics expert. It requires a leadership system designed around better evidence.

The question I wish we had asked sooner

Looking back, I wish we had known to ask a different question much earlier. Not simply, “Is she passing?” or “How did she score?” but “What does she actually understand?”

That is the question I now think secondary school leaders should be asking more often.

If a student in your school were earning strong grades and strong test scores while building fragile mathematical understanding, what in your current leadership system would help you see it?

Because sometimes the data are not wrong.

They are simply incomplete.

This version should read much more naturally as a blog article, especially aloud.