This class is an introduction to scientific modeling for children ages 8-14 using as support fun experiments, which can be safely made in a kitchen. Adults (parents, other teachers) are welcome too!

Motivation

We are all born scientists: since childhood, we experiment with the surrounding world to infer cause effect relationships between events. What will happen if we drop a ball? If we drop a glass? If we touch our nose? If we touch the fire? Etc. Hence we all build simple models allowing us to describe the basic functioning of systems or phenomena and make predictions of the consequences of actions: what will happen if I touch or drop this new object?

In recent school curricula, an effort has been made to sensitize children to the difference between facts and opinions. For instance, “fire is hot”, “ice is cold”, “the sky is blue on sunny day”, and “the earth rotates around the sun” are thought of as facts. Conversely, “this show is hot”, “ice cream is good”, “blue skies are pretty”, and “dogs are fun” are thought of as opinions. Kids get it immediately and have fun making their own riddles. It very quickly appears that there is not always such a sharp distinction between facts and opinions.

One reason for the apparent difficulty in distinguishing between facts and opinions is that many statement are neither facts nor opinions, there are models. If we can agree on definitions of fact and opinion such as:
Fact: A piece of information that can be verified or proved.
Opinion: A personal belief or judgment that is not founded on proof or certainty.
What should we call Newton’s law: “F = ma: the net force on an object is equal to the mass of the object multiplied by its acceleration”? Is this a fact or an opinion? Scientific laws are usually developed by induction, from experiments. They are hence a blend of facts and opinion: they can be verified according to an established standard of evaluation, but they can also be eventually falsified; they are never really proved or certain. We call them models.
Model: A schematic description of a system or phenomenon accounting for its known or inferred properties that allows for investigation of its characteristics and, in some cases, prediction of future outcomes.

Now what? How does this help us in our everyday life? Or in our life of scientist if we are one or want to become one?

We are constantly flooded by information, which try to classify between fact and opinion by applying “critical thinking”. We need to all become experts at critical thinking to make good decisions: Who should I vote for at the next elections? Should I eat organic food? Should I send my kids to public school? Should I invest in real estate? Ideally, we should verify all information on which we base our decisions, but this is impractical because it is too time consuming, costly, infeasible or unethical. So we must rely on our judgment on the quality and diversity or the sources of information and detect possible i. We propose to go one step further: build models and conduct real experiments to verify them, when possible. Let’s see how much difference this can make

Causal models: from physics to engineering 

Causal models are used in medicine, epidemiology, social sciences, econometrics, and other domains to predict the consequences of actions. Regular models built on simple associations (correlations) allows us making predictions (diagnosis) when the system under study is not intervened upon.

For instance, the risk of being sick with a cold can be predicted from its causes (having been exposed to the germs of another sick person, having stood in an air current) and its consequences or symptoms (sneezing, having a fever, shivering, aching muscles). But if we want to administer an effective treatment, we better distinguish between causes and consequences. For instance, giving a medicine to stop sneezing or to remove muscle pain might remove the symptoms, but not cure the cold. Administering an antibiotic to kill the germs might. In Figure 1, we show a very simplified model illustrating these causal relationships.

Figure 1: Causal model of the common cold.

These models are very useful to scientists and engineers, but hard to establish. How do we know that germs cause disease for instance? This required a lot of observations and some experiments that are not so easy to reproduce in a kitchen laboratory. So, we will explore other systems using some simple physical mechanisms to illustrate the concept of causal modeling. The model of Figure 2 is equivalent structurally to the model of Figure 1 and illustrates the same concepts, but it is much easier to experiment with.

Figure 2: Causal model of the mass of some matter.

To predict the mass M (the variable of interest) all connected variables might be used (Quantity Q of material, Volume V, and Weight W). But only intervening on the cause Q can influence M. V is correlated to M but not causally related (It is a consequence of a common cause Q). W is a consequence of M.

We will progressively establish such causal relationships. To that end, we will use the scientific method:
- observations
- hypothesis
- validation of hypothesis by experiments.

Class plan

1. Proportions and correlation: Observation: Volume and weight seem to be interchangeable in cooking recipes, both are used. Hypothesis: Weight must therefore be correlated to volume.  Experiment: In the first lesson, we will use a simple kitchen scale and a measuring glass to weigh various volumes of liquids (water, milk, oil, beaten egg)  and powders (sugar, flower) and plot M as a function of V (we are not making yet the distinction between mass and weight, but since we are using mass units, we refer to the variable as M). We will establish the proportionality of the two quantities and explain graphically the notion of correlation. We will denote our model V <-> M with a double arrow to denote correlation. Conclusion: The hypothesis is confirmed, V and M are correlated. Discussion: Is there a causal relationship between M and V? If M is the number read on the scale, can we force the number to change without changing V? If so, we can conclude that weight (as read on the scale) does not cause volume. How about the opposite? It looks like when we change volume, we change weight. Can we conclude that volume causes weight? Within this particular experimental setup it looks like V->M. We'll revise that in the next lesson. Application: We'll have two teams making pancake dough: Team A will use a recipe with ingredient amounts indicated  in weight but will be forced to use a measuring glass. Team B will use a recipe with ingredient amounts indicated  in volume but will be forced to use a scale. We'll bake the pancakes.

2. Experimentation, interventions: Observation: With our system of the first lesson (measuring glass + kitchen scale) we could observe correlation between V and M. We hypothesized that V -> M because we can change the reading of M without changing V, but does changing V always results in a change in M. Can we modify the volume without changing the weight? Do you know examples? How about pop-corn? Hypothesis: We can intervene on volume without changing the weight. Experiment: We will first define a good standard operating procedure (SOP) for the use of the scale and the measuring glass such that, if we respect it, weight and volume are proportional.  We'll then weigh a few grams of corn kernels, microwave them and weigh them again. Conclusion: The hypothesis is confirmed, the volume changes but not the mass.  Discussion: In putting the finger on the scale to change the weight without changing the volume (lesson 1) did we respect the SOP? In microwaving the pop-corn, did we respect the SOP? To determine whether one variable causes the other, we deviated from the SOP and performed "interventions" on our system to try to change the value of one of the variables without changing the other. This is an operational test of causality. This allowed us to rule out both that weight causes volume and that volume causes weight. If neither variable causes the other, what can explain that they are correlated? Not all correlations mean a cause-effect relationships. Two variables may be correlated because they are the consequence of a common cause. There must be a common cause to both volume and weight. What could it be? New hypothesis: There is a common cause to volume and weight called Q: V <- Q -> M. If we assume, like the Greek did, that matter is composed of tiny identical elementary units (atoms or molecules), what really affects the weight and the volume might be the quantity (Q) of such units. New experiment: We'll use a model with beads to perform experiments and validate this hypothesis. New discussion: In  what respect can we say or not that this experiment validates our causal model?
Note: we are still not making the distinction between mass and weight.

3. Measuring weight: In the third lesson, we will play with a number of simple scales (hand-made  with simple common materials). Observation: "Standard weights" generate a reading on the scale corresponding the their declared weight. In Figure 3, we show different kinds of scales.  Model: For each scale, children will propose a causal model. Experiment:  We will calibrate a scale with "standard weights" and validate our models with various interventions (including those used in the previous lessons). We will learn to read scales properly to avoid systematic errors.  Conclusion:  Many different physical phenomena can be used to measure the same quantity. To make sure we all measure the same thing, we need standards. See the International Prototype of the Kilogram. Application: Take various packaged snacks and verify their weight with the scale you built and calibrated. Eat the snacks! 

(a) Spring scale (M, the weight/mass we want the measure is a hidden variable)

(b) Hydraulic scale

(c) Two pan balance scale

(d) Roman balance scale

Figure 3: Various weight scales.

4. Forces, gravity, and weight: In the fourth lesson, we will (finally) make the difference between mass and weight. Observation 1: By pressing on the scale with the finger, we will note that we can change the weight. Observation 2: When moving, heavy objects require more force to stop quickly than light objects. When resting, heavy objects need more force to get in motion quickly then light objects. Preliminary experiment: Use the spring scale,  measure force horizontally. Throw balls of various weight horizontally on a table and observe the spring displacement. Hypothesis: A force proportional to the tendency of objects to resist change in motion causes the mass reading on the scales. We will now distinguish between mass and weight, the force caused by the mass will be called weight. The hypothesized model is that of Figure  2. Experiment: Newton defines a mechanical force as F=Ma, where a is the acceleration and M is an intrinsic property of objects called inertial mass, proportional to their tendency to "resist" change in motion.  We can define as weight the force that attracts objects to the ground (accelerates them). Is the inertial mass of Newton the same as the gravitational mass that we measure with scales? We will perform Galieo's experiment. Two jingle bells of identical volume and different masses will be dropped.  We will observe that they follow exactly the same trajectory and arrive at the same time. Hence they have identical accelerations a₁=a₂. Conclusion: Denoting the gravity forces (weights) as W₁= M₁ a₁ and W₂= M₂ a₂, since a₁=a₂, we have  W₁/W₂=M₁/M_2. Therefore,  the weight we measure with the scales is proportional to the inertial mass.  We can identify the  two concepts of mass. We call g (gravitational acceleration) the coefficient of proportionality, such that W=Mg. Application: Wear a seat belt to avoid getting hurt because of inertial force. We will see how we can package an egg in a box such that when we drop the box the egg does not break. If not packaged properly, the egg will break. We will bake a French omelet with the broken eggs.

5. Density and buoyant force: Cooking: At the beginning of the lesson, we will put ice cubes in glasses of water and mark the level of the water. We will let them melt during the lesson. Meanwhile... Model: Putting together all we have observed so far, we can quantify the model of Figure 2 using "structural equations". We call V the volume, Q the number of atoms or molecules, M the mass, and W the weight. We have, for a given material:
V = v Q (v=the unit volume of matter; note that it depends on temperature and pressure, we assume those are constant in our SOP)
M = m Q (m=the atomic or molecular mass)
W = g M (g is the gravitational acceleration)
As a result, we see that for a given material the quantity D (density) defined as D=M/V=m/v is a constant for a given material in the standard conditions of temperature and pressure. Problem (Archimedes story): Is a crown really made of gold or of a cheap alloy? Solution 1: Using a balance scale, weigh the crown with pure gold. Then, put the crown in a bucket of water and note the change in height. Redo the same thing with the pure gold. If the two height are different, the jeweler cheated. Explain, using the notion of density. Experiment: Let us go back to the hydrolic scale (Figure 3 b). Push on the bucket. Feel the force. This is called the buoyant force. We will combine the spring scale and the hydraulic scale to measure the buoyant force (Figure 4). We will find that the buoyant force is a force upwards compensating the weight and proportional to the volume of water displaced. Explain how the hydraulic scale works (Figure 3b); refine its model. Propose a model for the system of Figure 4.  Archimedes' problem, solution 2: Using a balance scale, balance the crown with pure gold. Dip the scale in water. Observe whether the balance looses its equilibrium. Discussion: Why is solution 2 better than solution 1? (Tip: what are the measurement errors made). Cooking: What happened to our ice cubes. Where are they now? What is the water level? Explain with the buoyant force. Why will the ocean not raise if the ice of the North Pole melt?

6. Closed systems, mass and energy: In this class we will introduce the concept of a closed system (conserving mass and energy) and interventions made on this system via adding (or removing) energy to disrupt it (experimentation). We will note that a system that we can observe is not really closed. What kind of energy or mass gets in and out our system (for instance our spring scale, Figure 3 a).  We will wonder what we can neglect and what we cannot neglect. Observations: Water evaporates, if we do not close the bucket with a lid and let the experiment last long, we might loose matter.  Temperature might change.  Volume changes with temperature (see thermometer). Pressure might change. What happens when we take a seal packet of chips to the mountains? Do you think pressure affects our experiments?  Hypothesis: A temperature change can significantly affect a weight measurement.  Experiment: Fill a balloon with 1/3 water and 2/3 air (without inflating it) and close it tight with 2 knots. Weigh it. Put it 30 sec in the microwave or as much as it takes to boil the water and inflate the balloon. Weigh it again. Let it cool down and weigh it again. Observe the inflated balloon weighs less even though the same amount of matter is being weighted. Propose an explanation. Is there something we might have neglected in our model? Complementary experiment: Fill balloons with helium and let them flight away.  Conclusion: Air is a fluid like water. It exerts a buoyant force.  Modeling: Propose a change in the model of Figure 2 to take into account the buoyant force in the case of the balloon experiment. How does this model differ from that of Figure 4?

Figure 4: A combination of the spring scale and the hydraulic scale to illustrate Archimedes' principle.

Class organization

Each class will include:

- A new concept
- A new problem to be solved

For a 90 minutes, we anticipated that the classes would be divided as follows:

-    Welcome: Introduction to the topics of the day (the concept and the problem) – 10 minutes.
-    Practical work: Observe experiment, do something fun in groups of two illustrating the concept and solving the problem – 30 minutes.
-    Clean up – Five minutes.

-    Bathroom break – 10 minutes
-    Sharing: Share the results of the experiments.  Build a model –15 minutes
-    Story time: A story from History: how scientists addressed similar problems – 15 minutes.
-    Questions. What did we learn? What should we do next? – 5 minutes.

But for a 45 minute class, we will right away start with the practical work (30 minutes), we will then share the results of the experiments and conclude. The instructors will set up things in advance and do the cleanup. We will skip the story time.

The materials will be provided by the instructors. There are no prerequisites.

	Concept	Problem	Practical work	Model	Story
Week 1	Proportions	Use a scale to “measure volume”. Weight is proportional to volume.	Bake either cup cakes using recipes that need weight/volume conversions.	V->M?	How proportions are used in science. The example of Thales.
Week 2	Controlled experiments	Invalidation of the model V-> M. Interventions on weight and volume.	Using leavening agents in baking.	V<-Q->M correlation does not mean causality	Wrong  causal explanations based on correlations, later disproved by experiments Jacob's flock: Spontaneous generation
Week 3	Measurements	Measure weight.	Build a scale.	M -> some physical manifestation of weight	Falling objects in History and Philosophy.
Week 4	Gravity	Distinguish mass and weight.	Galileo’s experiment.	V<-Q Q->M->W	When Newton's theory does ot work, relativity
Week 5	Buoyancy	Archimedes’ problem	Merging the spring scale and the hydrolic scale.	V’=volume of displaced water V’->W’	Archimedes' story.
Week 6	Closed systems	Errors due to unintentional exchanges of mass or energy.	Microwaved water balloons and Archimedes’ buoyancy force.	V<-Q Q->M->W V=V’ V’->W’ W->Wa<-W’ Wa=apparent weight	Closed systems in physics and in philosophy.

About the instructors

Address questions about science to: Isabelle Guyon
causality @ clopinet . com

Address questions of phylosophy to: Teed Rockwell
mcmf @ california . com

Copyright © Isabelle Guyon and Teed Rockwell, 2008

Sponsors

This project is supported by the National Science Foundation under Grants N0. ECCS-0725746. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.