I’m been doing more work attending to students’ spontaneous problem-solving strategies for forces:

This comes from several students I’ve observed recently, both in class and in office hours. For example, consider a problem where one force has components in both directions: A ball with weight 4N is held by two strings. One string is held horizontal, and another string is angled at 60 degrees (from horizontal).

**After drawing FBD, the standard algorithm to solve for each tension might look like:**

ΣFy = 0

(T2)y – w =0

T2 sin(θ2) – w =0

T2 = w/sin(θ2) = 5.8 N

ΣFx = 0

(T1)x – (T2)x = 0

T1 – T2 cos(θ2) = 0

T1 = T2 cos(θ2)

T1 = w cot(θ2) = 5 cot(60) = 2.9 N

**Students I have found more spontaneously think about it this way:**

- Reason that the upward component of tension (opposite side of the tension triangle) = 5N (to hold the weight). Draw 5N on the triangle.
- Knowng 5N is opposite, and 60 degree is angle, solve for “hypotenuse” of Tension Triangle using sin(60) = 5/T2 ==> T2 = 5.8N
- Reason that the two horizontal components must be equal
- Solve for “adjacent” side of Tension Triangle (using pythagorean or any trig functions), T1 = 2.9N

They aren’t always explicit about all of these steps as I am being, but it’s the general flow.

**Basically, what I’m observing is that students are not prone to write out generic component relationships**

(T2)y = T2 sin(θ2)

(T2)x = T2cos(θ2)

This is especially true when they don’t have a value for T2. If they had a value for T2, they would. Basically, students problem-solving strategies involve calculating in steps things that you can, rather than writing out algebraic relationships. They use cosine and sine to solve for numerical unknowns sides of the triangle. They were able to reason about what the “opposite” value of their needs to be (5N), and then knowing the opposite and the angle, used sine to solve for hypotenuse.

Using the standard algorithm, we also have a tendency to go “straight” for the Tension, without calculating either component explicitly. We leverage the algebraic relationships.

**What difficulties do I observe when students try to implement their strategy:**

Sometimes, they forget to find the hypotenuse after finding the opposite side. This is even the case when students are super explicit that 5N is just the vertical component (or opposite side of their tension triangle). They seem prone to forget that this 5N is not the total tension (hypotenuse), and will proceed to step 4 as if 5N was the hypotenuse of the triangle. And so will do things like 5N*cos(60) to solve for the horizontal component.

**Thoughts about the Instructional Tension:**

I’m thinking hard about how I can do a better job of “building” on students’ problem-solving approaches, which I’ll describe as “step-wise numerical”. For me, there are some “instructional tensions” here: The textbook teaches the more standard algorithm. Even tutoring my students receive (from free from library or paid outside tutors) tend to teach the standard algorithm. While we have lots of students who benefit from approaches that are not strictly algebraic, we do have some physics majors in our algebra-based physics course–often (but not always) these students are more adept at actually using and understanding the standard algorithm. In addition, future courses will also presume facility and familiarity with approaches that are algebraic (not step-wise numerical).

How do we find a balance between “building on and refining students’ step-wise numerical approaches” vs. “providing opportunities to encounter and become proficient at algebraic approaches”?

There are certainly some problems that the algebraic approach is advantageous:

- Problems that involve systems of equations that need to be solved simultaneously
- Problems where “mass” or other factors cancel out

But it’s not just about having a better approach, the outcome of the approach is different as well…

**Different Approaches Afford Different Kinds of Sense-Making (After the Fact):**

Consider that the result of algebraic approaches also gives opportunities for sense-making that you can’t readily do with numerical approaches. Mostly, sense-making involves thinking about how variables influence an outcome. For example, coming up we have some problems to work with circular motion (What’s the maximum speed a car can take a certain turn in rain)? It’s one thing for a student to get an answer of 25 m/s, and have to ask is that reasonable (that’s about 55mph, and even though that seems too slow for normal conditions, it’s raining.)? This is the kind of sense-making I want my non-physics majors to have opportunities with. Physics majors, taking an algebra approach would get, v= sqrt(us* g * R). Their sense-making would be around making-sense of the fact that more “coefficient of friction” or “wider turn” would mean you could go faster, and their sense-making might be around why mass doesn’t effect the maximum speed.

Both are valuable kinds of sense-making, but what’s an appropriate goal for whom weighs on me, and especially how to navigate these goals in a single class.