Multistep (advanced)

From procedural fluency to modeling

In section Multistep (core) we discussed how Multistep can help students solve core mathematical problems, such as simplifying expressions or solving equations. Although this is a powerful way to develop mathematical skills, there is more to mathematics than procedural fluency.

Consider, for example, the following types of exercises:

• Calculus problems that require completing multiple tasks:
  "Find the coordinates of the maximum of the graph of $y=2x(2-x)$."
  The student needs to find the derivative function, equate to zero to find the $x$ coordinate, calculate the $y$ coordinate, and finally write the coordinates of the maximum.
• Arithmetic story problems:
  "An aquarium of $60 cm\times 40 cm\times 35 cm$ is 80 percent filled with water. How many liters of water is in the aquarium?"
  There are multiple strategies to solve this problem. For example, find the height of the water and then calculate the volume. Or calculate the volume of the acquarium first and then find $80\%$. Students must also convert from $cm$ to $dm$ or from $cm^3$ to liters.
• STEM problems involving units and scientific formulas:
  "A car is moving at a speed of $25 ms^{-1}$. Calculate the total distance traveled to a complete stop when the driver decelerates at $2.5 ms^{-2}$."
  The student needs to solve an equation to find out how long it will take to stop the car. Substitute this time into the formula for displacement to find the total distance.

Such questions require a strategy to decompose the problem into subproblems and combine the partial results. Additionally, students sometimes need to transform a real-world problem into mathematical language. We refer to this as mathematization.

The combination of strategy and mathematization is generally called modeling. We will explore some powerful features of Multistep to support such math problems.