Alternatives to the Standard Algebra Algorithm for Forces

In my previous post, I was writing about my dissatisfaction with teaching the standard algorithm for solving Newton’s 2nd Law problems. The standard algorithm is to sum the forces algebraically and set them equal to mass times acceleration.

A viable alternative to this approach is Force Vector Addition Diagrams. This alternative has lots of upsides, but one in particular that I like is the emphasis on Net Force. My argument here is going to be that the standard algorithm mostly avoids ever explicitly thinking about Net Force.

For example, consider a problem where a Tension force of 30N is accelerating a 7kg block at 3 m/s/s. The problem asks you what is the friction force acting on the block?

The standard algorithm would look like

 ∑ F = ma

T- f = ma

f = T -ma

f = 30 N – (7kg)(3m/s/s)

f = 9 N

A conceptual / numerical approach that focuses on net force might look like this

  1. OK, so how much force would a 7 kg object need to experience in order to accelerate at 3 m/s/s?

    a = Fnet / m
    (3 m/s/s) =  Fnet / (7 kg)
    21 N of force would be needed.

  2. OK, well how do we get 21N of force from these two forces?  Well, we have 30 N pushing forward.  That must mean we have 9N of force opposing.

These approaches are logically equivalent, but conceptually miles apart in terms of the thinking that a person does. The first one basically just uses Newton’s 2nd law as instructions for how to write your algebraic sum of forces statement. And then it’s mathematics. It basically never explicitly says, “This object is experiencing a net force of 21 N”.

The 2nd approach treats thinking about Newton’s 2nd law separately from (but connected to) thinking about the sum of forces. It first asks the question, “How much force would get the job done?” and then, “How did the individual forces conspire to make that happen?”

So in my previous post, I suggested that the standard algorithm may not be right for students who are weak in algorithm. I want to make a stronger claim here. Whether one takes a more graphical approach (like in the link above) or any another approach (like the approach laid out here), I’ll venture to propose the following: Any algorithm that skirts explicit thinking about Net Force is likely to be a mistake (especially for students just learning Newton’s laws  and/or those with weaker math skills).

Note 1: Part of this has me thinking about the idea of “standard algorithms” in mathematics, and how the issue here is very similar. While this paper is about prompting force diagrams, it’s basically related in the sense that forcing students’ to use standard algorithms has unintended negative consequences. In the paper, there are examples of more intuitive approaches, where students successfully solve problems by calculating in bits and pieces rather than the standard algorithm.

Note 2: A second questions relates to if/ how / when to move students toward something more like the standard algorithm. What contexts help motivate it? What scaffolding helps bridge it? What populations of students should this even be a goal for?

When Formal Approaches Disenfranchise Students’ Natural Problem Solving Skills

One of the reasons why I dislike teaching “algebra-based” physics in a fast pace manner is that for many students algebra is barely in their grasp. This becomes most apparent when we first starting learning about how to solve problems related to Newton’s 2nd Law.

For example, today students were solving a problem, where a 4 kg is being pushed with a 30N. The coefficients of static and kinetic friction are 0.5 and 0.3 respectively. The question is, does the block move, and if so, with what acceleration?

Students all correctly identify that the maximum static friction force is 20N (based on normal force being equal to weight), and so 30N is enough to be in the kinetic regime. They can then calculate the kinetic friction force as 12N. Many of these students can readily say that 18N is the net force acting on the object. Then most can at this point say that a = Fnet / m  =  4.5 m/s/s.

Students’ productive and natural problem-solving approach does not look like what we teach them do:

F_net = ma

F – f_k = ma

F – u F_n = ma

F – u mg = ma

a = 1/m (F-umg)

Students of course just simply do this in “chunks”, calculating pieces along the way, comparing values, doing arithmetic calculations. The formal approaches codify these chunks into the algebra and string them together. Logically that’s true, btu students do not experience it that way.

I think that teaching the formal approaches has the promise of being generally powerful (that’s the allure to teach it this way), but in reality it mostly disenfranchises students from their natural problem-solving skills, their ability to apply insight into problems, and to make sense of what they are doing.

What compounds the negative effects of teaching abstract algebra approaches is that it has quite varied effects.

  • I do have some strong algebra students and sense-makers who can apply the formal approaches and make sense of them as they do.
  • I also have some good “algorithm” followers who can apply formal approaches, but are not making sense of them. They follow the script.
  • I also have strong sense-makers who will approach problem from their own intuitive approach, basically ignoring the approaches you are trying to teach them.
  • I also have students who cannot follow the algorithms, and feeling disenfranchised from anything resembling sense-making, they cobble together strange algebra.

Having one or two of these different types seems manageable to me, but having all four I find hard to navigate.

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