Mathematics can be a rich, playful subject, where students generate exciting new ideas to creatively solve tricky problems. But from worksheets to typical online practice programs, much of the math content to which students are exposed is not very rigorous and requires little or no creative thinking, leading to low student engagement, memorization without understanding, and in many cases, an aversion to math. This is so common it's become acceptable to say, "I'm just not a math person."

This is evidently a major problem, especially given the needs of today's STEM workforce.

We can use this rigor relevance chart to analyze the types of math content students are asked to engage with. Vertically, we have an axis of thinking from low-level regurgitation of memorized facts, to high-level creative problem-solving. And horizontally, we have an axis of application, from routine to non-routine questions. Worksheets and most online math practice programs fall pretty squarely in this lower left quadrant. They generally require little beyond memorized facts or procedures, and at best contain routine problems where the students already know how to get the answer. While it is necessary for students to be able to solve these routine problems, this is not truly mathematics. We really need to engage students in rigorous mathematical thinking that falls into this upper right quadrant. Non-routine problems that require creative thinking to solve them.

First, let's look at how we can modify the content. We'll take this simple first grade addition from the worksheet, and change it to this more open-ended question. Arrange the numbers one through five in these boxes so that when you add the numbers horizontally you get the same sum as adding the numbers vertically.

Notice how we don't give the students a strategy of how to find the answer. One of the big ideas of rigor is that students should have to struggle, figuring out how to think about the problem by themselves. Here's one possible solution: putting 1 in the shared square and then 2 and 5, and 3 and 4 on the other squares. The sum in each direction is 8. But that's not the only answer. Maybe you placed 5 in the shared square. Now the sum in each direction is 10. And there are more. In fact, one of the hallmarks of rigorous questions is that there are likely to be multiple solutions. At this point, it's very tempting to think that we've solved society's math problem just by modifying the content. All we need to do is provide textbooks full of interesting problems like this for every grade level, and, voila! Problem solved.

However, books like this already exist.

And this brings up another very challenging issue: implementation.

Facilitating a class full of students in solving an open-ended problem and keeping them engaged in productive struggle is a very difficult art. The teacher needs a strong combination of deep math content knowledge, pedagogy, and good classroom management. Unfortunately, when students start to struggle, the overwhelming temptation for teachers, and online math programs, is to start giving hints. For example, in this case, the teacher might say: "Why don't you try thinking about number pairs?" Let's look at how this affects the rigor of the task. While the task remains non-routine, we've reduced the creativity needed by the students, and if the students continue to struggle, the next level of hints from the teacher or program will likely make the task more routine.

So even if content has the potential to be highly rigorous the implementation will very likely reduce the rigor of the task until it's not that far removed from the original worksheets. In order to guarantee that students have access to creative problem-solving, we need a scalable method to deliver rigorous content to students.

ST Math is a visual instructional program that addresses both of the challenges: content and implementation. Let's look at how ST Math does this. Here's a puzzle called Tugboat, where students must determine how to balance the number of boats on each side of a bridge. Notice there are no written instructions. Students must explore the puzzle to determine how to solve the problem. Students drag tugboats to either side of the bridge to help JiJi the penguin cross to the other side. But as the students productively struggle and try solutions they think will work, the system doesn't offer hints. The student is given visual feedback on exactly what they tried, and they see why JiJi cannot cross. As in the open ended box problem, there are multiple ways to solve these puzzles, and in this instance, they addressed the same core mathematical addition concept. And while this example obviously belongs in first grade, the same approach could be applied to all mathematical topics, no matter what the grade level. ST Math is a standards-aligned program that allows us to systematically deliver rigorous instructional tasks requiring creative problem-solving to every student.

Learn more at stmath.com.