Do not worry about your difficulties in mathematics; I assure you that mine are still greater

Einstein

Though open to a great many interpretations, for me this quote draws a direct parallel to what happens in my classroom.

My students’ difficulty: calculations.

Result: random rearrangement, misplaced multipliers and unruly units.

My own difficulty: working out why.

Result: confusion, frustration, and a lot of edu-reading.

At least, that was the case five or so years ago. Students, though proficient at equation recall and identifying the physical quantities involved, often fell at the hurdle of processing calculations. As such, I set myself a challenge: work out an effective way to approach calculations that works for me and my students.

**So, what exactly ***has*** the USSR ever done for me?**

Well, besides an eye-opening trip to Moscow in 2015 (which included an excursion to Russian Mission Control Centre!), USSR is the acronym that I use with students to support them in self-directing through calculation problems in physics.

- U – units
- S – substitute
- S – simplify
- R – rearrange

Through careful introduction followed by explicit modelling in lessons, this has developed into a tool that my students now possess, one that they can employ when tacking mathematical questions. In this article, I’ll take you through the USSR structure and include examples along the way. Ideally, I would have shared some actual student work to show here, but lockdown has meant that you’re stuck with my own workings.

**U – Units**

The ‘units’ step is where students will identify quantities and their units in a question, noting above them what the quantity is. They then check whether or not conversions are necessary and complete said conversion if needs be.

In the example above, mass is the unknown quantity, but kinetic energy and speed are given. Kinetic energy, however, requires a unit conversion.

**S – Substitute**

Having identified the units and quantities in a question, students then select the appropriate equation. They write it down and substitute the values into their respective places. It is important to stress that no rearranging has happened yet. This step is simply to put the numbers in the correct location in the equation, as below.

**S – Simplify**

This, for me, is the magic step. Get students to look at the equation and simplify the expression as far as they can, but still without doing any rearranging. In this example, the expression can be simplified in two ways. Firstly, square the speed. Secondly, perform the halving calculation on the right hand side of the equals sign.

The inclusion of this step came directly from experience; I found that students were prone to rearranging incorrectly, especially when there were more than three parts to a mathematical expression. The simplifying stage, therefore, was included as it should always take the expression to having just three parts.

**R – Rearrange**

Now that we have a more simple expression, the rearranging has a higher likelihood of being successful. Rearranging only as the final step is important; by this time, students are working with the most simplified version of the problem.

**USSR: a summary**

Taking all of this into account, the whole answer from the student might look something like the below.

Firstly, the unit conversions are clear. This is great for teachers as it’s easy to spot any mishaps with this whilst circulating the classroom. Secondly, the relevant equation is stated and values are substituted. Similar to step one, it’s also extremely clear to me as a teacher whether or not values have been correctly identified from the question. Simplification comes next, which reduces the expression to having only three parts and allows the final step, rearrangement, to happen when the expression is in its most simplified form.

Below I’ve included two further examples: one more simple, one more complex. Both, however, are approached using the same routine. In the mass & weight example, the ‘simplify’ step is redundant, because the expression is already in its more simple form once the values are substituted. In the equation of motion example, the rearrange step is more complex than seen previously as it involves a square root calculation.

There are, of course, drawbacks to this routine for approaching calculations. Thinking purely mathematically, it relies on a number of pre-requisites. These include:

- Knowledge of the standard units for physical quantities
- Knowledge of the meanings of mathematical prefixes
- Understanding of how to employ the rules of order of operations (e.g. BIDMAS)
- Recall of the prescribed equations

Time and energy must be spent ensuring the above are embedded if the USSR method is to work in a classroom, a worthwhile consideration when curriculum planning.

As much as there are drawbacks, there have been a number of positive outcomes since embedding the USSR method:

- Getting down to the brass tacks of it all, the success rate in terms of gaining marks for mathematical questions has increased.
- Student voice suggests that the method has built their confidence. This is because their workings are clear and mishaps more readily identifiable. Moreover, the structure provided means they can approach any equation-based problem in the same manner thus reducing panic levels when tackling the questions.
- As a teacher, the structure means that student responses are laid out in such a way that errors can be identified quickly and with ease, allowing for ‘live’ feedback on the process.
- It has proved an ideal structure for other teachers in the department, particularly non-specialists, to follow in their teaching.
- Linked to the above, the simplicity of the method means that we are beginning to achieve consistency across teachers and year groups, and so students are more well versed in what a successful approach looks like.

**Final thoughts**

Developing mathematical proficiency is, to me, a fundamental aspect of moulding our students into well-rounded physicists. I hope that this article has given you some food for thought in approaches to teaching calculations, particularly at Key Stages 3 and 4. I’ve shared one approach that works for me and my students, but I’m aware that there are a great many more successful ideas out there that are worth sharing; please use the comments section to discuss other approaches, or to reflect on what I’ve shared.

I prefer students include the units when they substitute numbers and do manipulations, to help catch algebraic errors.