Introduction to ICTing and Mathing Across the History Curriculum. Part 6

The study of cause and effect in the academic discipline named history is substantially different than the study of cause and effect in the academic disciplines we call the sciences. Consider the following example of cause and effect. As night fall begins to occur, the sun goes “down” and it gets dark. In the morning, the sun comes “up” and it gets light. The going down and coming up of the sun are causes. The darkness and light are effects.

Hmm. Does the sun really go down and come up? Is it perhaps a better explanation to say that the earth spins (rotates) on its axis, and this makes the sun appear to go down and come up? And if we observe carefully over a long period of time, we notice that the amount of time the sun is down varies over the course of a year. Humans developed the science we call astronomy to better explain and more accurately forecast the changes they observed in the heavens.

In the sciences, we can have careful descriptions of occurrences (the effects), and an underlying explanation (theory) for the causes of the occurrence. Here is an example. For thousands of years people have known that when an object such as a stone is dropped, it falls. One of Isaac Newton’s contributions to science was his development of a theory of gravitation that describes this occurrence (Thompson & Havern, n.d., link). The effect is that a dropped object falls, and the cause of this fall is named gravity. Researchers can study this cause and its effects in a huge variety of different situations. Eventually the researchers can convince themselves and others that the theory of gravitation is accurate enough and broadly applicable enough to be widely accepted as a scientific result that others should believe and make use of.

Newton’s research served the world well for several hundred years. Eventually, Albert Einstein developed a theory of relativity that turns out to be a more accurate description of some aspects of physics.

Classroom Activities

Now, here is a question for history teachers. What do you want students to learn about Isaac Newton and Albert Einstein? Does it suffice that students memorize these names and the approximate dates when they lived and worked? Does it suffice that students associate the word gravity with Newton and relativity with Einstein? What understanding do you want your students to have of the accomplishments of these two scientists and how their work changed the world?

Newton and Einstein were physicists as well as mathematicians. Both worked on cause and effect. Both developed theories and carried out and proposed experiments. So, when you teach history, do you stress the idea of cause and effect from the point of view of both historians and scientists? Explore this issue with your students.

Here is another example to ponder. Wars are an important topic in the study of history. We can study thousands of wars that have occurred over recorded history. While there are some similarities, no two wars are identical. We can talk about the causes and effects of wars, but we cannot carry out controlled experiments as we can do in the sciences. Thus, we cannot develop a precise science of the causes and effects of these wars. If you are a history teacher, do you help your students learn to understand the significant difference between cause and effect in the areas of history you teach versus cause and effect in the sciences that your students know and/or are learning?

In summary, humans are much better at determining cause and effect in the sciences than they are in the discipline of history and the other social sciences. However, many areas of study combine the sciences and social sciences.

Here is a business example that students might explore. Suppose that 200 years ago an inventor announces the development of a device named telephone that allows voice communication over a distance. Would it prove popular and become widely adopted? If so, how long would this take, and how will its widespread adoption change the world? There may well be examples from history that will help provide insights into possible answers to these questions. The production and distribution challenges fall in the area of science. The rate and amount of adoption, and the uses, fall mainly in the area of the social sciences. Supply and demand affect prices, which in turn affect distribution and usage. In brief summary, forecasting (predicting the future) is a challenge.

People often say that if two events seem to be related to each other, they are correlated. Mathematicians use a much more precise definition of the term. This section discusses correlation.

In statistics, the phrase “correlation does not imply causation” refers to the inability to legitimately deduce a cause-and-effect relationship between two variables solely on the basis of an observed association or correlation between them (Wikipedia, 2020b, link).

Causality … by which one event, process or state, a cause, contributes to the production of another event, process or state, an effect, where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future (Wikipedia; 2020a; link). [Bold added for emphasis.]

The word correlation has two widely used definitions:

Correlation is usually defined as a measure of the linear relationship between two quantitative variables (e.g., height and weight) [and other numerical measures of quantity]. Often a [much] looser definition is used, whereby correlation simply means that there is some type of relationship between two variables (DISPLAYR Blog, n.d., link). [Bold added for emphasis.]

The first graph in Figure 1 suggests that sales of heaters increase as temperatures decrease when the weather gets colder. That is, the temperature and sales are negatively correlated. This seems like common sense. However, perhaps the more astute shoppers buy heaters in the summer so they are available when the first cold weather hits? The second graph in Figure 1 is somewhat suggestive that sales are increased by increasing advertising.

Figure 1. Two examples of visually apparent correlation (DISPLAYR Blog, n.d., link).

The eyeball or “feels like” analysis of the examples in Figure 1 may suffice in some cases. But, mathematicians are interested in more precise analysis of such data. The applicable component of mathematics named Probability and Statistics dates back more than 1,200 years (Wikipedia, 2020c, link).

Although details are lacking, each dot in the two examples of Figure 1 is a point in a two-dimensional coordinate plane. That is, the location of each dot can be specified by a pair of numbers. The two-dimensional graphic representations are much easier to visualize and interpret then a list of the numerical values of the coordinate pairs. However, in doing mathematics to analyze such data, it is much more effective to deal directly with the numbers. Computer software called a spreadsheet is very useful in such endeavors.

Spreadsheets

A spreadsheet is computer software designed primarily for the storage, manipulation, and output of quantitative (numerical) data. It also can be used to store other types of data, such as names and addresses of the employees in a company, along with rate of pay, hours worked, tax withholdings, and all of the other information needed to calculate and print payroll checks.

I find it interesting to think about what are the most important types of software students should know about and routinely use as they study history. My current response to this question includes:

1. The Web and a Web browser.
2. Word processor and related aids to including graphics in documents.
3. Spreadsheet for storing and processing quantitative data. (A spreadsheet can also contain qualitative data, such as names and addresses.)
4. Database for storing and processing a combination of qualitative and quantitative data. The Web is a humongous database, and it includes a number of pieces of free database software that individuals can use to create their own databases. Some important questions: When should students learn to use such software? Who should provide the necessary instruction? How can history teachers at all grade levels gain the knowledge and skills to integrate routine use of these tools into their teaching?

Figure 2 provides some historical information extracted from data on the average height and weight for Wisconsin children ages 1-18 (Children’s Wisconsin, 2020, link). I used a spreadsheet in extracting and organizing the data. We will use this spreadsheet of data in several examples.

Spend some time examining the data. Does it agree with your intuitive understanding for the growth of children? The height and weight data in Figure 2 appears to show a somewhat linear relationship to age. That certainly makes sense. Do you see any obvious errors in the data? (In collecting and recording experimental data, errors sometimes occur.) These two types of questions can be applied to any data you are trying to read and understand. They provide a good first effort to determine whether the data is real or fake.

Figure 2. Growth of average female and male Wisconsin children (Children’s Wisconsin, 2020, link).

What else do you see as you examine this table of data? Is there an age when many girls are taller than boys? Do you see growth spurts? Do you wonder why, on average, boys grow to be taller and heavier than girls? Asking and possibly exploring these types of questions does not require a lot of math knowledge and skills. Such questions and explorations can help the reader to get a feel for or overall understanding of the data.

Now, let’s do a little mathematical analysis of this data. Figure 3 contains the math formula for statistical correlation. Most people are overwhelmed by the math notation and complexity of this formula, so do not be bothered if you happen to be one such person. Unless you have studied and still remember quite a bit of math, you may find that the formula does not communicate much to you. In the “good old days,” many college students taking statistics courses were expected to memorize this formula and gain skill in carrying out such calculations using a desktop calculator. Computers have certainly changed this aspect of statistics courses!

where n is the total number of samples, xi (x1, x2, … ,xn)
are the x values and yi are the y values.

Figure 3: Formula for statistical correlation.

The basic idea of statistical correlation is that there is a formula for calculating the correlation between two columns or two rows of numbers. There are many free websites that carry out this calculation (Alcula, 2020, link). In addition, typical spreadsheet software includes provisions for calculating correlations.

Figure 4 provides examples of some correlations of data from Figure 3. The first set of data analyzes age versus height for girls ages 1 to 6. The second set of data analyzes age versus height for girls ages 13 to 18. Notice that the correlation decreased. This decrease is caused by girls tending to reach nearly their full height by age 15.

Finally, just for the fun of it, I calculated the correlation between average height of girls and average height of boys. It came out 0.9914. That is a high correlation. But, correlation is not causation. Surely we should not conclude that the height of girls causes the height of boys, or vice versa!

Or, have we discovered an interesting research topic? Perhaps boys and girls who differ in height by a certain amount tend to marry more often than do boys and girls who differ in height by considerably more or considerably less? This might lead to their children having a similar difference in height pattern.

Finally, suppose that one is looking at the data in Figure 4, and wonders what the numbers might look like for people age 19, 20, or so on? Can we predict height and weight for this age group? Based just on your personal knowledge about yourself and other people, you know that people do not continue to grow taller, year after year. The data suggest that girls reach a peak well before age 18, but there is insufficient data to know what happens for boys as they grow past age 18. We cannot accurately use the data to forecast the average height for boys over 18 years of age. A later section of this newsletter examines a component of the discipline of history named future studies.

Have your students study the growth table in Figure 2 above. They can compare their personal statistics with the average for students of their age. This can help them to predict whether they are likely to have a growth spurt in the near future.

They also could check the average heights and weights of students older than themselves. Here is an interesting idea. Instead of students looking in the table for students of their own age, they can look in the table for students of their own height, and see average weights for such students. Some students may find results that suggest they are significantly lighter or heavier than average for their height. Note, however, that this table of data was not designed as a vehicle to help determine whether a student is overweight or underweight. So, students should be warned to not jump to hasty conclusions.

Students might be encouraged to make use of the Web to gain more information about whether they are overweight or underweight. They can learn to use the Body Mass Index (BMI) as a useful measure (Bachmann, 2019, link). (As an aside, BMI calculations are based on historical data, and thus are a good example of scientific analysis of historical data.)

Here are questions to explore:

1. Are their significant differences in growth patterns for students of different racial backgrounds?
2. Are there significant differences from state to state, between major areas of the country, or between different countries?

As a teacher, you want your students to learn to pose and explore such questions.

Our schools teach history as part of an overall program of studies designed to help prepare students for their own possible futures. This suggests to me two challenging questions:

1. Can we predict the future accurately enough so that what we teach students will turn out to have been appropriate and helpful as they become adults?
2. Should future studies be an explicit component of many courses that students study? Or instead, as we do with many possible content areas, should it be a self-contained course taught at some specific grade level?

For thousands of years we have had fortune tellers, also known as a soothsayers, who claim to be able to predict the future. Even today, many people believe in and act on advice from fortunetellers. Do you read your daily horoscope and then act on its advice? Do your students or their parents read horoscopes?

In recent years, an academic discipline named future studies has emerged in a number of universities as a self-contained or interdisciplinary area of study.

Futures studies is the study of postulating possible, probable, and preferable futures and the worldviews and myths that underlie them. There is a debate as to whether this discipline is an art or science. In general, it can be considered as a branch of the social sciences and parallel to the field of history. In the same way that history studies the past, futures studies considers the future. Futures studies seeks to understand what is likely to continue and what could plausibly change. Part of the discipline thus seeks a systematic and pattern-based understanding of past and present, and to determine the likelihood of future events and trends. Unlike the physical sciences where a narrower, more specified system is studied, futures studies concerns a much bigger and more complex world system. The methodology and knowledge are much less proven as compared to natural science or even social science like sociology, economics, and political science (Definitions & Translations, 2020, link). [Bold added for emphasis.]

This definition suggests to me that history teachers at all levels should be involved in helping students learn about possible futures. The previous section in this newsletter about cause and effect certainly contains some relevant content for such endeavors.

The World Futures Study Federation was established in 1973.

[The] WFSF, since its inception, has encouraged and supported a pluralistic approach to futures studies. This pluralism is reflected in the diversity of the WFSF membership and the research it supports. The WFSF uses the plural term “futures” studies rather than the singular “future” studies to counter the notion of only one future, the latter having both conceptual limitations and political implications. This pluralization of futures opens up the territory for envisioning and creating alternative and preferred futures. A major focus of futures studies for us at WFSF is how we envisage and develop desirable outcomes in the times ahead (WFSF, 2020, link).

This indicates that a large number of people are involved is the scientific study of possible futures. These people come from a wide range of academic disciplines of study.

All teachers, parents, and others who help to provide informal and formal education of children should help in the future studies education of children. A recent study suggests that we are not doing well in educating students about future careers:

But since the start of this century, the career aspirations of teenagers have narrowed, not expanded, in spite of arguably equally dramatic technological and social changes. A new global study by the Organization for Economic Cooperation and Development finds teenagers’ “dream jobs” today are nearly identical to those in 2000, and could leave many students at a disadvantage in the emerging economy.

 “What is striking is that most [dream jobs] are actually 19th- or 20th-century jobs. Very few aspire to 21st-century jobs by the age of 17,” said Andreas Schleicher, the OECD’s director for education and skills and a co-author of the study, at a discussion of the study Wednesday at the 50th annual World Economic Forum in Davos, Switzerland, which was livestreamed via Twitter. “You see the world of work becoming more diverse, but what young people cite is becoming more myopic, more concentrated. … What we know about the future of work doesn’t make its way into classrooms and experiences of young people.” (Sparks, 1/2/ 2020, link).

Classroom Activities

My recent Google search on the term “predictions of the future” produced nearly a billion results. If your students are mature enough to independently search for and make use of such articles, you can have them find, analyze, write about, and share some of their findings. Another approach is for you to use some class time, perhaps once a week, to discuss predictions that you have selected. You want your students to become accustomed to the idea that the world is changing and they can learn to observe and participate in the changes.

Reading and writing are ancient history, but still are essential components of modern life. Computers and Artificial Intelligence (AI) are more recent history. It seems clear that computers and AI are still in their infancy despite their rapid development and amazing accomplishments.

Electronic digital computers first became commercially available in the early 1950s. The field of AI was formally founded in 1956, at a conference at Dartmouth College, in Hanover, New Hampshire. In my opinion, Computers and AI are world-changing developments that rival the creation of reading and writing about 5,400 years ago. There are considerable similarities between these developments. Both facilitate the storage and retrieval of information. Both facilitate communication over distances. Both aid the human mind in solving problems and accomplishing tasks.

Human brains are quite good at detecting possible causes and effects. Consider a simple example in which one of our very ancient ancestors sees some nice-looking berries on a bush, eats the berries, and becomes quite ill. Without further research, the person posits a cause (eating the berries) and an effect (illness), and decides to not try this experiment again. The human race has survived and prospered partly because of this type of ability to recognize cause and effect.

While AI has made huge progress in solving certain types of problems, the field has made very little progress in developing AI systems that are good at determining cause and effect. According to Elias Bareinboim, Director of the Causal Artificial Intelligence Lab at Columbia University, “Understanding cause and effect is a big aspect of what we call common sense, and it’s an area in which AI systems today are clueless.” [Bold added for emphasis.]

This statement reinforces the value of placing more emphasis on cause and effect in the education of our children. Currently, humans are far better at this task than are our artificially intelligent machines.

Alcula (2020). Statistics calculator: Correlation coefficient. Retrieved 3/16/2020 from http://www.alcula.com/calculators/statistics/correlation-coefficient/.

Bachmann, C. (2019). Body Mass Index (BMI). Retrieved 3/22/2020 from https://www.smartbmicalculator.com/.

Bergstein, B. (2/19/2020). What AI still can’t do. MIT Technology Review. Retrieved 3/19/2020 from https://www.technologyreview.com/s/615189/what-ai-still-cant-do/.

Children’s Wisconsin (2020). Normal growth. Retrieved 3/7/2020 from https://chw.org/medical-care/adolescent-health-and-medicine/issues-and-concerns/adoloscent-growth-and-development/normal-growth.

Definitions & Translations (2020). Future studies. Retrieved 3/19/2020 from https://www.definitions.net/definition/FUTURES+STUDIES.

DISPLAYR Blog (n.d.). What is correlation? Retrieved 3/23/2020 from https://www.displayr.com/what-is-correlation/.

Lewis, T. (12/4/2014). A brief history of artificial intelligence. LiveScience. Retrieved 3/19/2020 from https://www.livescience.com/49007-history-of-artificial-intelligence.html.

Sparks, S.D. (1/22/2020). Students’ “dream jobs” out of sync with emerging economy. Education Week. Retrieved 3/23/2020 from https://www.edweek.org/ew/articles/2020/01/22/students-dream-jobs-out-of-sync-with.html.

Thompson, H., & Havern, S. (n.d.). The history of gravity. Retrieved 3/18/2020 from https://web.stanford.edu/~buzzt/gravity.html.

WFSF (2020). World Futures Study Federation. Retrieved 3/19/2020 from https://wfsf.org/about-us/wfsf.

Wikipedia (2020a). Causality. Retrieved 3/20/2020 from https://en.wikipedia.org/wiki/Causality.

Wikipedia (2020b). Correlation does not imply causation. Retrieved 3/20/2020 from https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation.

Wikipedia (2020c). Timeline of probability and statistics. Retrieved 3/17/2020 from https://en.wikipedia.org/wiki/Timeline_of_probability_and_statistics.