Knowledge, Understanding and Application

The work over the last two weeks in our exploratory class has been really focused on three pieces that I am working to weave together for our students. In the common core standards for mathematics there is an emphasis on three domains of learning: procedural knowledge (I will refer to as knowing) application and understanding. In a sense, the work that we are trying to do in both mathematics and science is to build not just the knowledge of things like definitions or how to solve problems, but putting them into real-world context so that we can use this knowledge to solve problems, and to apply that information to novel situations.

So how does this play out over the last two weeks? In our mathematics work, we have been looking at solving algebraic expressions which is a foundational skill that will allow the mathematical models that we are developing on our science side to be more fully understood and accessible so we can solve for a wide range of problems. As a school, we are working with Dr. Milfreid Olson from the University of Hawaii Curriculum research development group (CRDG) to help us reflect, strengthen and deepen our approaches in mathematical work throughout campus.
In our class, that has meant doing some traditional math practice work with the equations and eventually inequalities as well as moving on to more complex functions like quadratic equations which will help us better express and understand motion that is both linear and constant as well as changing and accelerated.

As an example, we just did a investigation on Wednesday that had students rolling a golf ball down an inclined plane to try and better understand its motion through the mathematical model generated by looking at its position as a function of time. I’ve included an example of graphs and mathematical models that the students generated as a part of that study below.

investigating motion of golf ball

investigating motion of golf ball

golf ball rolling in action

golf ball rolling in action

analyzing video capture of motion

analyzing video capture of motion

Cami explains the math model in numeric, graphical and motion map forms

Cami explains the math model in numeric, graphical and motion map forms

What is notable here to me is in order to get to the kind of representation you see, it requires students to do all three phases: knowledge, application and understanding. We can deepen this more in the post investigation that will occur on Friday as we look at different inclines, try to use these models as predictive representations of the real world and try to come to a more complete understanding of what the unit analysis and the mathematical function tell us about the motion.

The ultimate goal is to make this real world applicable and we are in the midst of learning about the mechanics of bicycles as a part of our year-long investigation in transportation. This week we spent our double block on Tuesday taking apart cleaning up and reassembling the different bearing systems that allow a bicycle to translate force into motion by reducing friction.

using torque to take apart the crank shaft

using torque to take apart the crack shaft

taking apart the bottom bracket to expose the bearings

taking apart the bottom bracket to expose the bearings

galen explains the roll of bearings

galen explains the roll of bearings


All of these activities lend themselves to rich meaningful conversations around terms like friction, torque, force, procession, rotational mechanics, mechanical advantage, simple machines and much more. We will be using part of our time on Friday to unpack some of the terminology and physical understanding that was easily evidenced by their work on the bicycles.

There isn’t always a perfect fit between the work we are doing with our math, science and technology (STEM) but as much as we can, we are trying to draw meaning between these investigations so that we are weaving together these strands into a more tightly integrated and understood unit about our investigations of the physical world.

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