# 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

- 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. - 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?

Brian, we just started with vector addition/subtraction and forces in our algebra-based course. We’re not using the “standard algorithm,” but we are spending a lot of time on developing techniques for dealing with vectors. Some students are already familiar with vectors and are struggling to “forget” what they know about components etc. and just “add the vectors graphically.” We’ve not yet got around to actually deal with forces on objects (we’ll get to that somewhat tomorrow), but so far, we’ve been working a lot on algorithms for vectors.

What your posts made me think about is: Maybe instead of trying to strategically putting an algorithm in place (whether it be algebraic, graphical, etc.) before we actually start talking about forces, maybe it’d be worth playing with and experiencing forces first to create a need for and motivate some sort of formal representation. Along the way, we could (re-)introduce vectors as a neat tool for this purpose and build on what students know about them and about representing things/phenomena.

What you are saying makes sense about not trying to get the one algorithm right; and I, for the most part, feel like I would know how to do that well in a slower pace course that I was solely responsible for. In a course that is supposed to be taught the same way by everyone (across 10 courses with common pacing and exams), I have a harder time navigating that. That said, students have had good experiences with single forces acting on objects… the leap we made too quickly was to multiple forces. We essentially made the leap to multiple forces while also the leap to strange algebraic algorithm. That’s where we lost them. I think I am suggesting that we could have better leveraged their understanding of what a “single” force does better, by guiding them toward strategies that are “net force” thinking focused. Unfortunately, our textbook teaches them the standard algorithm, so doing will be at odds with that.

I’m in a similar boat (regarding the course being taught the same way by everybody at the same pace across many sections), and I’m having an equally hard time navigating that, especially knowing that many of my students are struggling hard with what we’re asking them to do (mostly without telling them why we’re asking them to do it)…

It is (and has been) an interesting challenge, in the sense of its a puzzle. It how to I do best within the constraints I am working within. And finding out what are things you thought were constraints and are not. Or ways you constrain yourself unknowingly. It makes me more empathetic to the challenges of teaching in public schools.