A Child’s Equations
Sticks and Ticks, Space and Time…and…Speed…
There are certain physical properties which occur in threes such that, if you can measure two of them you can calculate the third.
Simple right?
This = That/The Other Thing, and so, That = The Other Thing x This
Dr. Seuss math.
The types of experiences suggested in A Child’s Physics provide a concrete, sensory basis for just such three part equations. Which begs the question, why do we wait until late elementary and even high school to introduce these relationships to students? Why do we hold back basic physics demonstrations until well after the child has the math instead of letting students do the demonstrations early on and regularly so to eventually derive the math? The Montessori Method uses this approach, preceding the math with the object it describes exemplified in the Binomial Cube and Theorem.
Three part equations are enormously useful in that they describe physical reality. Which answers the age old complaint about the study of mathematics, “But why do I have learn this?”. Because, it describes reality. We can measure and calculate all sorts of things, from recipes to money to electricity.
And all you need is the relevant sensory experience, the idea of measurement and then, when you’re ready, arithmetic. But even before you have multiplication and division, with just physical experience and tools that quantify, you can work with and discuss ideas like this:
Speed = Distance/Time
This is the child’s first three part equation. Bear in mind, the goal at the primary level is not to solve for actual values of specific cases. The goal is realize the relationships between physical properties through experience and discussion; in the way, for instance, that alternating current can be defined by puffing on a straw.
So, lets talk about how a child measures space and time and how we can facilitate that in a prepared environment. Space and time are two of the first four primary physical aspects, the other two being mass and frequency.
Sticks…. Space
We are going to compare the physical size of one thing to another. One thing, a stick, will be smaller than the other and be easy to count. The other will be one large thing. So we have a stick and a distance. The stick is the agreed upon unit of measure. Doesn’t matter what it is as long as we agree and are consistent. To make this point at the primary level, do not start with a regular marked ruler. Just use a stick. The window is ten sticks away.
Ticks…. Time
Any consistently recurring event can be used as a clock: dripping water, running sand, rhythmic clapping, a pendulum, a mechanical clockwork, a cesium atom. The point is, we are comparing two events, one of which is regular, repeating and can be counted, the other of which may or may not happen more than once. That’s what a clock is for, it counts ticks for us while we observe or count whatever else is happening. This is an important distinction in the child’s world. Counting ticks is not the same as telling time, as in knowing that snack is when the clock says 10:00. Nor are we discovering a new dimension or substance. There is nowhere to travel, no physical thing, no stuff called time. We are counting energetic changes, events, and comparing them. That’s it.
Lets sit in circle and use rhythmic clapping as our clock. One of us walks to the window, which we already measured and know is ten sticks away. We count ten claps or ticks, while the walker walks. So how fast is that walk?
Ten sticks in ten ticks.
You can stop there if the members of the circle cannot yet multiply and divide. The important point is that there is a relationship between this numerator and denominator that is palpable right in the physical situation and the language used to describe it. Also, you don’t have to reduce 10:10 to 1:1 in order to point out that the question,
“How Fast?” is a combination of Sticks and Ticks, or
Speed = Distance/Time
This approach is not about solving a formula by doing arithmetic as an empty exercise in computation using terms that are only vaguely defined. It is about grasping relationships between particular physical realities. There are a number of these three term equations appropriate to early physics education that all spring from and depend upon simple, clear demonstrations.
Ohm’s Law is one,
Current = Voltage/Resistance.
A child can build and discuss this simple circuit; an LED as a resistance, batteries as voltage and a meter to associate quantity with current. The LED alone pegs the meter. An added resistor reduces the current to within the meter’s range and dims the LED. So, amp is a combination of volt and ohm.
We can also talk about the hand gestures associated with a push model of electricity. Clearly, there is a combining of voltage and resistance to produce the idea of current. The physical metaphor gives us another opportunity to use the language.
Acceleration is literally just a gentle push away from speed. Have a child walk across in front of you. Give her a push as she passes, just enough to alter her course. That literally, physically, is the experience of being accelerated. It can be talked about. This is where Mass and the baric exercises mentioned in A Child’s Extrasensory Experience come into play, because in our primary Newtonian world mass is weight, the idea and measure of how much “stuff” is in an entity. Like current in the push model of electricity, mass is how much gets pushed, but through a space, as opposed to a circuit. Acceleration can be compared to voltage as it is how hard (and long) you push. The two together, how much and how hard, are called Force. Force is the combination of how much you move and how much you move it. So,
Force = Mass x Acceleration.
Acceleration also brings in the idea of direction, which gives us the word vector, or how we combine motion and direction. We can draw acceleration. It is an arrow which, incidentally, is a type of stick. The longer the arrow, the harder the push. Speed can also be drawn as a vector arrow but then we call it velocity. The longer the arrow the faster the speed.
Vector arrows need to point somewhere and that brings us to a new use of sticks; measuring, in fact establishing, a space in addition to just a distance.
First, we’ll make a two dimensional grid; maybe with a pegboard or an image on a cloth, a chess board or the tiles on the floor. Floor tiles work really well because it puts the child’s body into the space placing easily cut dowels for the vertical axis, and then you can specifically locate any object in the classroom using numbers.
The first pegboard should be zero to three, or sixteen holes with numbered axes and visible grid lines. The pegs are in units 1, 2, 3, the same size as the distance from one hole on an axis to another. Now we practice counting to identify places in space. All the holes except the origin need two numbers to tell where they are. The top of a peg in a hole needs three numbers. So, for example, lets line up the pegs so we can see the tops at 1,1,1 and 2,2,2 and 3,3,3. The teacher can provide cards that have coordinates on one side and a picture of the correct peg location on the other as a control for independent use. The three by three grid can also be used to learn the names of the spaces in two dimensions.
When you get to grids of 10x10x10, you have enough room to arrange vector arrows, function curves and geometric solids within the space; you get to handle the curves before you know what a function is.
One key curve, a wave actually, is two opposite curves together, a yin/yang. This is a sine wave, which for practical purposes only has to be as sine-like as the child can draw, or bend out of a pipe cleaner. It doesn’t matter as long as there is available a standard set of waves that show different combinations of wavelengths and amplitudes.
Once we identify the shape of a single wave, we can count waves as they go by. “Going by” is an event, like a stick is a distance and a tick is a time. Waves go by, we count how many while we have a clock counting ticks. Combine “how many waves?” with “how many ticks?” and its called Frequency.
Frequency = Events/Time
And since we also know that a wave has length, say for example, one stick long, and we have a number of actual waves to work with, we can physically show that as ten waves go by the first wave will go 10 sticks further. This describes waves in terms of sticks, ticks and events. So we can combine the above ideas of speed and frequency and say,
Wavelength = wave speed/wave frequency
Yet again, this is not aimed at doing arithmetic problems. This comes before that. This is presenting physical experience and the associated vocabulary so you can combine terms, quantify and talk about the ideas, just like anything else. If you need two cups of milk for one egg then for two eggs you need four cups of milk. You don’t have to be able to reduce the ratio 4:2 to 2:1. You just need milk, eggs and a cup and the ability to count. This approach works provided, of course, a grownup gives you a graduated sequence of physical experiences to continue talking about. Billiard balls, a ripple tank and a pendulum, along with wire waves, coordinate grids, and curves and arrows are good physics additions to a well developed primary math curriculum such as that expressed in the Montessori math materials. The sensors and measuring tools suggested in A Child’s Extrasensory Experience provide the experience of actually quantifying the different forms of energy.
We are learning to observe and describe Energy in and around static objects in addition to the objects themselves.
What we are doing is learning to describe the behavior of objects in space and of waves in a medium. Behavior is dynamic. So, we are learning to observe and describe Energy in and around static objects in addition to the objects themselves. This is the contribution to early childhood education offered in A Child’s Physics. To do so, we need to count distances, elapsed times, weights of objects and particular events; in other words, space, time, matter and energy, words the child does not yet have. The particle/wave dichotomy is a basic organizing idea in this process that is easy to present, play with and talk about. If space and time can be characterized as sticks and ticks, matter and energy can be thought of as stuff and go. How does “stuff” manifest in the child’s world? as combinations of the elemental substances on the periodic table. How does “go” manifest in the child’s world? as Light, Sound, Motion, Heat, Reaction, Magnetism, Electricity and Consciousness.
