We learned this month that in a year when the state raised standards and increased the difficulty of the tests, almost threequarters of New York City’s fourth-graders and almost 40% of our eighth-graders passed their math exams, up 19 and 9 percentage points, respectively, from four years ago when the mayor took control of our schools. Our students’ progress in reading has been equally dramatic.

We cannot afford to be too pleased with these results when hundreds of thousands of students have still not grasped the fundamentals of math and reading and when we see such a fall off between elementary and middle school performance.

We can, however, be optimistic.

We’re moving ahead because of the Mayor’s commitment to education and the hard work that principals and teachers across our city do every day.

We launched our reform strategy by bringing standards, stability, and coherence to a school system that had been hampered for decades by patronage and a jumble of instructional practices.

Our critics said our initial reforms — training new school leaders, installing a new curriculum, ending the “social promotion” of failing students, creating weekend and afterschool sessions for struggling students, and providing schools with additional resources so they can use data to identify trends and address students’ individual weaknesses — would doom our students to failure.

In fact, these reforms accomplished just the opposite. Our students are making real progress and their gains are outpacing their peers’ throughout the state.

Since 2002, the number of New York City fourth-graders achieving math proficiency has jumped by almost 19 percentage points. The number of students in the rest of the state achieving proficiency has increased by only about 5 points. The number of New York City eighth-graders achieving math proficiency has grown by about 9 points. The number of students in the rest of the state attaining proficiency has increased by only about 5 percentage points.

We’ve seen the most remarkable gains for students in the earliest grades, the children who grew up under our reforms. Our thirdgraders have gained 28 points in math since 2002 and are closer to performing at the overall state level than any cohort since testing began. Our fourth graders have gained almost 19 points on state tests.

Last year, when the State’s Department of Education raised the bar, making the exams more difficult, students across New York struggled, but New York City’s students held their ground. New York City’s fourth-graders lost 6.5 points in math, while students in the rest of the state lost 7.2 points, for example. In English, our eighth-graders gained 3.8 points while students in the rest of the state lost 0.3 points.

The overall picture is looking increasingly bright: We are narrowing the “achievement gap” that separates our black and Hispanic students from our white and Asian students. The graduation rate has climbed by 7.4 percentage points in the past four years, and in the first cohort of our new small high schools, the graduation rate is almost double that of the large, comprehensive high schools they replaced.

Our charter schools, which serve student populations that are almost 90% African American and Latino, have also made enormous progress. Almost 66% of the students at our charter schools are proficient in mathematics, according to the new results.And charter students scored an average of 12.4 percentage points above the city school districts in which they’re located.

We have a long, long way to go, but the first phase of our reforms has already put our schools on a new trajectory to success. Because of our work, thousands more elementary school students have learned enough to be promoted to the next grade at the end of the school year and thousands more children have entered high school with the skills they’ll need to graduate.

The mayor and I are now confident that we are ready to take the next step: infusing our schools with leadership, empowerment, and accountability—creating a new environment in which the leaders in the system accept greater authority and are truly willing to be held accountable for results.

When we take responsibility for students’ academic results, kids win. This is nowhere more visible than in our 300-plus Empowerment Schools, where, starting this year, principals have embraced accountability for performance in return for the power to innovate and make decisions about their budgets and their programs.

As we face the challenge of taking the next bold steps—advancing the results we’ve achieved to a new, higher level — we do not have time for complacency. It’s time for a breakthrough. We must harness our strengthened schools, our investments in principals, and a new willingness to be held accountable for student results.

The experience of the past four years has taught us that this kind of progress is not only necessary but possible.

Mr. Klein is the Chancellor of the New York City Department of Education.