Many people and organizations (Sir Ken Robinson, New Classrooms, 2Revolutions, etc.) have spoken eloquently about the fact that progressing students in age-based cohorts through courses and grades does not make a lot of sense when the reality is that students of the same age are at wildly different places in their mastery of standards. The proposed solution by many, including my team at Gates, is to develop new models that “personalize” learning by allowing each student to progress through content and attain credit once they demonstrate mastery. Most people would agree in principle that makes sense but nonetheless struggle when trying to implement it within existing classroom and school structures.

I recently had an experience that reinforced that challenge. On a recent visit to a middle school that was piloting Khan Academy in some of its math classes, I saw firsthand what happens when a teacher (“Mr. T”) wrestles with this very issue in a 6th grade class. It went like this:

- The day before I arrived, Mr. T had delivered a whole-class lecture on a grade-level standard (common practice)
- When I visited the class, Mr. T asked all students to use Khan Academy to practice that standard. He previously assigned paper-based problem sets but Khan Academy has largely replaced them this year. (Note: for those of you who are not familiar with Khan Academy, it has a built-in online assessment engine in which students solve problems known as “exercises”)
- By observing students as well as looking at the real-time data flowing in through Mr. T’s dashboard, I noticed three general categories of students as they were working with the assigned exercise:
**The Strugglers**: these students were clearly having trouble with the problems. Despite using the hint feature, asking the teacher and/or firends for assistance, and in some cases watching videos, it was clear they were not quite ready for this standard. They needed remediation on prerequisite standards.**The On Trackers**: these students initially had fits and starts with the exercise but ultimately persevered successfully.**The High Flyers:**these students clearly had already mastered the standard and finished quickly.

- When students had successfully completed the assigned exercise, Mr. T told them that they could “play” on Khan Academy. This meant that they could do anything on the program they wished.
- One student, a representative member of the “high flyer” camp, finished the assigned exercise in about 30 seconds. When he finished, he immediately skipped about three years of math exercises and clicked on Trig exercises. He proceeded to convert degrees to radians and vice versa successfully. Impressed but bewildered, I asked him a few questions:
- (Q) How did you learn how to do these problems? (A) I clicked the hint and learned the formula.
- (Q) Why did you decide to do Trigonometry? (A) It seemed hard.
- (Q) Do you know what Trigonometry is? (A) Not really.
- (Q) What are you going to do next year? (A) I’m skipping 7th grade math.

- At the end of the class, Mr. T announced that he would be introducing a new concept, the next standard in the 6th grade scope and sequence, the next day. Presumably they would then use Khan Academy to practice that standard.

This experience was not unique. I have visited many classrooms around the country, some using Khan Academy and others using ST Math, DreamBox and similar programs that allow students to progress through content at their own pace, and my takeaways from each are similar.

**Disrupting Problem Sets**: Digital math content is replacing (or “disrupting”) traditional problem sets. Teachers are embracing these tools because they automate common practice, and they are simply better on multiple levels than traditional approaches to assigning math problems. For example, they provide students with real-time feedback and support, and they provide teachers with rich data sets on student progress and eliminate the need to grade problems. Given these advantages, I am confident that digital content with embedded assessments will be the rule rather than the exception in math classrooms within a few years.**Wreaking Havoc on Pacing Guides**: As the above example illustrates, implementing digital tools that allow students to self-pace through content makes it that much more obvious that traditional methods of instruction are dated. It really does not make sense for a teacher to deliver a lecture to 25 students because they are “supposed to teach factoring on October 2.” Another school that wrestled with the same challenges made the comment, “We only want our teachers to instruct kids only when they need it,” which I thought was a compelling and pithy instructional philosophy.**Courses Are Outdated:**The school’s solution to the “high flyer” above was to allow him to skip a full-year course. While perhaps this was the best solution within the current course structure, it still feels imperfect. By comparison, Khan Academy does not present math as a series of courses. Rather, it shows a learning map that is simply a series of interconnected nodes (“standards”) that follow a trajectory from the top of the screen to the bottom, starting with Addition and ending (for now) in L’hopital’s Rule. Supporting this student’s logical progression through that “learning map” makes as lot more sense than allowing him to skip entire 180-day groupings of standards called “courses.”**Students Are Naturally Curious**: I love the student who skipped ahead to Trig because it “seemed hard.” I believe that all students have that natural curiosity and am always excited to see instructional approaches that elicit it from them. Conversely, imperfect learning models that put “the strugglers” in positions in which they feel inadequate can easily destroy that curiosity. The challenge, therefore, is to design instructional models that address the unique needs and interests of all students, every day.**Context and Application Matters**: I was struck that the “high flyer” had no concept of what Trig was while he was getting the computational aspects of it correct. As Sal Khan and many others have said before, it’s important that a holistic approach to math ensures that students understand the context behind the discrete problem sets and that they are able to apply their computational skills in real-world settings.

In closing, I am optimistic that this school, and schools across the country like it, will adapt. Mr. T faces are not easy problems to solve, but the exciting part is that these technologies are forcing the issue. Ultimately, what starts as a better alternative to problem sets will ultimately catalyze fundamental change to the entire instructional model in many of these schools.

**Math as a Learning Map**