President, Ontario Association of Physics Teachers
Hybrid Teacher-Coach for Science, Toronto District School Board
Once classes have settled down and our students are trained (see the Quick Guide for Introduction Lessons
), we are now ready to focus on teaching some physics! The first unit often taught is motion, both in the grade 11 and 12 courses. Most teachers feel pretty confident with this unit, but I hope to share some tips that might help you out in a few tricky spots. Read on!The Motion Unit
What makes our universe interesting is change and motion is one example of change that students can easily study. Its concepts underpin many others throughout physics, so a careful treatment really pays off. One challenge is the motion unit can be very dry if it is taught like a math class. Instead, teach students to think like scientists right from the start!(1) Science is about constructing and testing models
We handicap our students when we don’t teach students about models. This happens because we (us teachers) forget that we use them every time we discuss physics! Whenever we abstract something and create a mental representation of it, we are creating a model: a car on the highway becomes a point particle moving with a constant velocity. All models are vast simplifications of the real world complete with a host of assumptions and important limitations. When we forget to mention this process, difficulties arise because students’ mental models can be very different from our expert ones. Make the model clear with the class, agree on it, and then use or test it like a scientist would. We can’t do physics without making models, so let’s make that work explicit. I start the motion unit by introducing the swimmer Penny Oleksiak in this lesson
. Learn more about models in this article
. (2) Events vs. intervals — powerful clarifying ideas
The task of translating a physical situation into a mathematical model (see tip 1) can be very subtle. Students often do this incorrectly, producing nonsensical results. For example, throwing a ball is often modeled as something that happens instantly, which can lead to problems: do you use the zero velocity of the ball before it is thrown or the velocity when it is released from the hand? As a result, students randomly plunk these numbers into their equations. To properly model situations like this, students need to understand events and intervals. An event is something that happens at one moment in time, while an interval is a process of change between two events. Throwing a ball is an interval (a process), a collision between a bat and ball is an interval (a process). The moment the ball begins to move is an event, the moment the bat and ball have the same velocity is an event. Students won’t be able to create proper models without this understanding. Practicing identifying events and intervals will greatly help your students and pay real dividends through their two years of high school physics. We review this in grade 12 using these slides
.(3) Physical ideas should come before the math
People like to argue that physics is just math or debate which is more “fundamental”. If we examine how physicists use math, we see that it is one of many tools they use for “doing physics”. And in fact, a lot of physics can be done without using much formal math at all. According to the science of learning, the best way to approach most new physics topics is this: focus on the patterns, develop the concepts, and then introduce the formalism and math. Note that this also reflects the process of science itself. This approach helps students in a few ways. First, it reduces the cognitive burden of too much new material all at once (new definitions, equations, and concepts). Second, it helps students focus on figuring out what is physically happening rather than just manipulating an equation. Third, when an equation is later developed, it has a reason for existing: to quantify a known physical pattern or relationship. This helps the math become meaningful for students; the math “makes sense”. And this reveals the true role of math in science: to make more precise a physical relationship; but we need to start with the relationship! My favourite example of this is when we introduce momentum
.(4) Always involve sense-making with math work
Using math in science requires not only mathematical skills, but sophisticated translation skills
. Every time scientists use math, they actually do this: create a model (see tip 1) to represent the physical scenario, translate the key features of the model into equations, manipulate the equations, translate the results back into the scenario, and decide what physically happens. Every time. When we lament our students’ weak math skills, it is sometimes because they can’t manipulate the equations. But it is often because they erred in the translation process. Help students practice making sense of every mathematical result they produce, otherwise they will never notice their errors in translation. Also, the manipulation of mathematical symbols largely happens in one part of the brain and while sense-making happens in another. As a result, students can often do the algebra but have no understanding of what they have found (ask your “good” students about the meaning of their results). Another interesting facet of this sense-making is the correct epistemological
interpretation of a mathematical result in science: as a prediction based on a model that uses assumptions and is reliable only within a range of uncertainty. It is not a statement of fact about reality, as my students used to believe. These ideas can be seen in parts C (I estimate …), D (I predict … ) and E (the size is reasonable …) of our regular solution process
.(5) Train students to think before doing math
The plug’n’chug is a real enemy of good learning and thinking. When experts deal with problems, lots of thinking happens, almost invisibly, before an equation is selected and a numerical answer is cranked out. From the start of grade 11 onwards we use a five-part solution process
that enforces good physical thinking before much mathematical work takes place. The five parts are: the pictorial representation, the physics representation, the word representation, the mathematical representation, and the evaluation of the result. This process models expert-like thinking and skills, and provides students with the greatest opportunity to correctly activate their physical understanding before they do the math. This is what experts do! Rigorously train students in a process like this and watch the amazing results!(6) Math work should be highly structured.
The mathematics work of most students is appalling, mostly because teachers reward them for appalling work. Students need highly structured exemplars
for writing mathematical work. Clear communication of technical work is an important skill and high structure gives students more opportunities to make sense of the math they are doing. To pull this off, teachers need to insist that the structure be followed. This is both easy and hard. It is easy because your standards are clear: student work needs have these clearly laid-out features with no meaningful errors or omissions. It is hard because you will need persistence and patience as a teacher: work that does not meet the standard should not be evaluated, it should be sent back to the student for improvement. Evaluating work is easy: full marks (100% or 1/1) for meeting the high standard and zero for not. Tracking student work is difficult: you need to collect work regularly to enforce this and hassle those who still owe you improvements. Your persistence will pay off with very high-quality work from a large majority of your students. (7) Motion: a linguistic minefield
Watch your tongue when describing motion! Physicists are blessed and cursed by a large overlap of words used in every day English and technical physics terminology. Because we personally know what we mean by these words, it is easy to become careless and lose a lot of clarity in our descriptions. Here’s my list of tricky words:
(8) Misinterpreting vectors as scalars
- Motion: Never use this as a synonym for “velocity” because in your next sentence you will use it to mean “every characteristic or feature of the object’s movement”, as in “describe the motion of this object”. Call it “constant velocity” and not “uniform motion”.
- Uniform: This is something people wear when bowling. The word “constant” is absolutely clear. Say “constant velocity” or “constant acceleration”.
- Deceleration: No other vector quantity gets a different name when its direction changes. Students will equate “deceleration” with “negative acceleration”, which is incorrect. Say “slowing down” instead, which is 100% clear.
- “Speeding up east”: The phrase “speeding up” describes what is happening to the magnitude of the velocity. Magnitudes don’t have a direction, so the “east” just doesn’t make sense: is it describing the velocity or the acceleration? Instead say “traveling east and speeding up”.
- At rest: Ask a person on the street to picture something at rest. Are they picturing a ball at the highest point in its trip? Nope! Don’t use “at rest” to describe something with a velocity of zero for an instant of time — it’s counterintuitive. Use “at rest” for velocity of zero over an interval of time.
We all start by learning natural numbers and then have our minds blown when negative integers are introduced. And then physics drops vectors on students: an equally boggling but powerful mathematical innovation. Us physics teachers want to treat vectors quickly and get on with the physics, an instinct that can lead to confusion compounded by our use of language and notation. Treat vectors very carefully!
(9) Create a unified set of visual representations for motion
- “The velocity is increasing”: Words that work well to describe scalars can be meaningless for vectors, which are a different breed of mathematical object. “The speed is increasing” is unambiguous, as is “the magnitude of the velocity is increasing”. Students have the concept of the integer number line (hopefully) deeply ingrained, where movement to the right along the line means larger or greater than. However, if a one-dimensional vector is represented on an integer number line, movement to the right does not have a clear interpretation (speeding up or slowing down are both possible).
- “The velocity +3 m/s is bigger than -7 m/s”: this misinterprets a vector as an integer number (scalar). Also, don’t say “the velocity 5 m/s is bigger than -2 m/s”. Instead, be specific by saying the speed or the magnitude of the velocity is bigger. We must talk about vectors in a different way than scalars.
- Never say two vectors are equal unless their magnitudes and directions are equal. “Equal” has a strict mathematical definition. Otherwise, clearly state which parts of the vector are equal (same direction or same size).
- We only use full vector notation when labelling vector diagrams or working with two dimensional quantities (6.7 m/s [N30°E]); otherwise we are always using scalar notation (Δx or v1y) for a one-dimensional quantity or component. I find a religious use of the arrow notation for one dimensional quantities cumbersome and not reflecting of expert practice. There is a danger to this, however: symbols Δx or v1 could represent a vector or scalar. Students need to practice deciding which based on the context. Luckily, good questions usually include lots of context.
- Provide helpful rubrics for grade 12 vector work.
Humans are hard-wired to think visually — a huge part of our brain is devoted to this. Take advantage of this by using a consistent set of visual representations for motion (part B of this example
). When we solve problems, students need to draw: motion graphs (position, velocity, acceleration), a motion diagram (a dot pattern along a coordinate axis), and velocity vectors. Doing this routinely helps students make sense of what is happening. Also, continue using them (or a select set of them) routinely in later units of physics. Learn more from this excellent resource
.(10) Train students how to interpret graphs
Students commonly misinterpret motion graphs. Teach them to use this mental routine
every time they see a graph: identify the type of graph, the graphical feature of interest, the mathematical value, and then the physical interpretation. For example: “this is a position graph, the slope is steep, it has a large value, the object is moving fast.” Motion graphs can be introduced and analyzed before the motion equations.(11) Make the study of motion real
Students commonly believe that physics isn’t real (Redish, Chapter 3, “Connecting to the real world
”). Some students think that physical laws are ideas that only work in a laboratory and not in the rest of the world. Others think that physical laws are just interesting facts that need to be repeated back to the teacher and really have no bearing on the world they live in. (This is not crazy or rare. For example, Newton’s laws are not intuitive or obvious. And if we present counterintuitive ideas with little justification or constructivist development, students don’t buy it because they are not blank slates. They trust their long-held, experientially reinforced beliefs about the world (views that are largely Aristotelian) much more than what some teacher happens to ramble on about in class one day. Check out Redish, Chapter 2 “Corollary 3.4
”for more.) Use realistic situations as examples and ask questions that practicing professionals (not teachers) would ask. Also, base their problem solving around physical things that they measure, predict, and verify. Here is a favourite activity
of mine.(12) Make sense of units and conversions.
Train students to make sense of a quantity with its unit. For example: “v1
= 3 m/s means that if the speed did not change, it would travel 3 metres in every second of time” or “a = 9.8 m/s2
means the velocity will change by +9.8 m/s for every second the object freefalls.” Help students understand the conceptual significance of changing units and some of the finer points of using units in physics. Check out my article
on this topic.(13) Many common freefall problems require an understanding of changing forces.
Freefall is a common example used to illustration motion with constant acceleration. For an object that is strictly freefalling, grade 11 students can use their 1-D kinematic skills to analyze this motion. However, many scenarios that students are presented with are not strictly freefalling. A ball that lands on the ground suddenly comes to rest. A ball being thrown upwards changes from speeding up to slowing down – all during its ascent. In these cases the interactions the ball experiences change. Without an understanding of forces and interactions, students will regularly set the final velocity of the ball that lands equal to zero. To correctly model (see tip 1) many freefall questions requires identifying when interactions change and which interval of time (see tip 2) to focus on: this requires understanding beyond kinematics. When I introduce freefall
, we explore the scenario of a ball being thrown up and then caught. The first goal is to identify the interval when we believe freefall is actually happening. Only then do we try to settle on the characteristics of freefall. I train students to use phrases like “the ball leaves contact with the hand” to define important events, rather than the confusing “the ball is thrown”. We study freefall in our unit on forces. One final freefall tip: don’t call 9.8 m/s2
the “acceleration due to gravity (ag
)”. Instead, call it the “freefall acceleration af
”. Why? Students will confuse ag
with the local gravitational field strength g = 9.8 N/kg and start labelling objects that are clearly at rest with the value 9.8 m/s2
(nonsensical). The freefall acceleration is valid only in the condition of freefall (and all the usual assumptions that go along with), while the local gravitational field strength value is valid for any state of motion near that position on Earth.(14) Two-dimensional motion should be saved for grade 12
There is plenty to do in grade 11: introduce the foundational concepts of motion and build graph-interpretation skills, problem solving skills, and modeling skills. Doing this with one-dimensional linear motion is already challenging; students don’t need another dimension to worry about. So don’t introduce any trigonometry, any projectile motion, or any 2-D vector analysis; stick to 1-D. It is an indication of a flawed curriculum that these challenging skills (how much difficulty do your grade 12s have with these?) are briefly included in the grade 11 motion unit and then abandoned for the rest of the course. Keep the skill sets clear and focused: 1-D for grade 11, 2-D for grade 12. (Note: in grade 11 we do explore forces in two dimensions, but only at right angles — no trig is required, so it’s all 1-D work.) Also, relative velocity is an interesting but niche topic that appears and then disappears; its skills are not used again in high school physics. Leave out relative velocity in both grades so you can focus on the core ideas of one and two-dimensional motion.(15) Introduce a consistent method of notating variables
We always use deltas for quantities that describe intervals (Δt
) and reserve Δd
for two-dimensional displacements only. Events are notated using numerical subscripts: v1
= the velocity at event 1, Δt23
the time interval between events 2 and 3. The subscripts “i” and “f” are only used for generalized quantities and are never used in students’ work. For example, vf
is the general equation that is introduced, but students always write v2
with the appropriate event numbers. Letter subscripts denote the object: ΔyA
= the vertical displacement of object A in a multiple-object situation. We never use a prime notation — numbers indicate the events before and after some process (mAvA1
). We introduce some of these ideas on our lesson on defining velocity
.(16) Insignificant Digits
Help students understand the significance (the conceptual significance, that is) of significant digits. “Significant” is a label we give to a digit that is physically meaningful, based on some measurement process. Most textbook questions give values with no indication of their origin; determining the number of significant digits in these numbers is impossible. So, use very simple rules for significant digits and avoid the bogus ones that are found in most high school textbooks. Scientists and engineers do not use these rules, so students shouldn’t sweat about them. Check out my article
on this topic. Here is our lesson on measurement
in grade 12 where we summarize these ideas and put them into their proper context of determining scientific agreement. (17) Math, math, math, egg, and math
Many of my tips focus on math. There is a good chance you and I like math. Many of our students don’t, for reasons that are beyond our control. What we can control is how we use math in our physics class. Our use of math can make the difference between a course that is widely enjoyed by students and one that is feared and avoided. If used carefully, with an understanding of how our brains learn, we can both de-emphasize the role of math and increase the standards of our student’s mathematical work (not a contradiction — here is a median grade 12 motion test
). Here are my tips:
- Math is different between a physics class and a math class: Make these differences clear instead of expecting student to pick up on them on their own (most won’t). Examples: use of units, functional notation, “x” as the only thing you ever solve for, and many more.
- Use exemplars: Even for the simplest math work, have clear exemplars with detailed criteria.
- Sense-making is key: Develop a work procedure that helps students to make sense of their work before, during and after the mathematics.
- Give them the equations on tests: Experts (including you and I) routinely look up equations, facts and data — this is normal. High school tests should have a strong focus on the physics rather than memorization or mathematical acrobatics. Fluency and automaticity come from regular use and reinforcement over time.
- Explore the concept before the equation: The physics should always come first.
I have overheard my students remark while working on a freefall problem, “yeah this is like math class, except it makes sense.” I have written extensively about using math in the physics classroom here
.(18) Phew! Final thoughts on the motion unit
The motion unit is our students first real taste of physics and that taste ought to be a good one. We can change much about our teaching to help students without sacrificing rigorous standards. We should emphasize in this order: good physical thinking and modeling, a deep understanding of the concepts, making sense of mathematical work, and a disciplined approach to problem solving. With this emphasis, students will enjoy their work because it will feel more meaningful and that will encourage them to persist in developing good work habits and a confident approach to problem solving.