Issue Number 252 February
28, 2019

This free I

All back issues of the newsletter and subscription information are available online. In addition, seven free books based on the newsletters are available:

Dave Moursund’s newly revised and updated book,

The current newsletter is part of a series on improving education. The first newsletter in the series is titled ICT Tools and the Future of Education, Part 1: A Brief History of Tools in Education. It and subsequent newsletters in this series are available at https://i-a-e.org/iae-newsletter.html.

ICT Tools and
the Future of Education

Part 5: Mathing Across the Curriculum

Part 5: Mathing Across the Curriculum

David
Moursund

Professor Emeritus, College of Education

University of Oregon

Professor Emeritus, College of Education

University of Oregon

“The heart of learning mathematics is
mastering a particular way of thinking—what I (and some others) call
“mathematical thinking,” sometimes also described as “thinking like a
mathematician.” (Keith Devlin; British mathematician, math educator, and
author; 1947-.)

“It isn't enough just to learn—one must
learn how to learn, how to learn without classrooms, without teachers,
without textbooks. Learn, in short, how to think and analyze and decide
and discover and create.” (Michael Bassis; American educator and author;
1946-.)

Background

Each of us experiences the world through a filter that includes all
of our informal and formal education, and our life experiences. My
first career was as a mathematician. By the time I was 35 years old, I
had probably spent more than 20,000 hours developing my knowledge and
skills in that area. I learned to view the world through *mathematics
colored glasses.*

Here is a personal math example from a few weeks ago. I am interested in the quality of life of people throughout the world, and the hugely uneven distribution of wealth between the very wealthy and the very poor. I had recently read a short article stating that the 26 wealthiest people on earth have as much wealth as the least wealthy half of the people on earth (Lubby, 1/21/2019, link). My brain immediately told me that there are about 7.6 billion people on earth, and that half of this is 3.8 billion. I also remembered reading that the wealthiest person on earth has about $150 billion. So, suppose the 26 wealthiest people have an average wealth of $50 billion. My brain make a rough estimate that (50 x 26) /3.8 is somewhat over $300 per person.

That is consistent with a vague memory I had of reading in another article that a few of the richest people in the U.S. together have more wealth than the poorer half of the people in the country, with the poorer half having an average wealth of $400.

So, in just a few seconds, I was able to “fact check” the assertion and decide that it was reasonable. Notice how I used some memorized data and some mental estimations to come up with my conclusion. As I was writing this section, I took out my calculator and used it to do the calculations. My estimate was a little low, which was consistent with what I had tried to do. I then used the Web to look up the article about wealth in the U.S. (Kirsch, 11/9/2017, link). That article actually named the three wealthiest people in the U.S. and drew on other data that it referenced in order to conclude that these three had as much wealth as did the entire bottom half of the total U.S. population.

What does this example have to do with* mathing across the
curriculum?* Each of us is routinely faced by problems and tasks
that involve numerical measurements of quantity. We make decisions
based on the math knowledge and skills we bring to dealing with these
problems. We also bring to the task our memorized knowledge about
those aspects of the world that seem to be related to the problem. We
can use the Web and other information sources to learn more.

Schools provide students with the opportunity to gain knowledge and
skills across many disciplines. Because math is useful in so many
different disciplines, we want students to learn to use math across
each of the various discipline areas they are studying. We want them
to develop the ability of *mathing across the curriculum* just
as routinely as they are *reading across the curriculum.*

The Fourth R

I have written extensively about the **4 Rs** of **R**eading,
**‘R**iting, ‘**R**ithmetic, and **R**easoning/Computational
Thinking (Moursund, 2018b, link).
The **4th R** uses the combined powers of human brains
and artificially intelligent computer “brains” to help solve problems
and accomplish tasks. Each of the **4 Rs** is an
important discipline of study in its own right. However, for most
people, their importance lies in the applicability of these
disciplines as an aid to representing and solving problems across many
different disciplines of study.

Several earlier* IAE Newsletters *in this series have
explored the use of computers as an aid to solving math problems. So,
computers are an important aid to* mathing across the curriculum.*

Of course, computers do much more than just help people solve math problems. The Web provides people with access to a collection of materials that is many times larger than that stored in the world’s largest “hard copy” libraries. Every discipline of study is aided by people being able to access a huge collection of the accumulated knowledge in that discipline. In addition, the artificial intelligence of computers is such that computers can solve or greatly help in solving a very wide range of problems in the various disciplines.

My conclusion is three-fold:

- It behooves our educational systems to help students learn to math across the various curriculum areas taught in schools and in their everyday lives.
- It behooves all teachers in these various curriculum areas to teach their students to make effective use of computers for doing the math relevant to their disciplines, and also to make effective use of other capabilities of computers within their disciplines.
- Out educational systems need to add to their curriculum instruction that helps students to understand the ongoing progress in artificial intelligence and other areas of technology that are changing the world they will face in their adult lives.

In summary, teachers are now faced by the task of helping students
learn to **4th R across the disciplines** they teach and
to prepare for an adult life that routinely makes use of the **4th
R**.

**What Is a Discipline of Study**

Here is the definition I use when talking about various disciplines of study:

Each** academic discipline** or area of study can be
defined by a combination of:

- The types of problems, tasks, and activities it addresses.
- Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, and so on.
- Its history, culture, and language, including notation and special vocabulary.
- Its methods of teaching, learning, assessment, and thinking. What it does to preserve and sustain its work and pass it on to future generations.
- Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
- The knowledge and skills that separate and distinguish among knowledge and skills in the discipline from a rank beginner to a world class expert. The school curriculum in each discipline area works to move a student up this expertise scale.

The sixth item is about increasing expertise. Students see and understand their own increasing expertise as they grow in cognitive and physical capability, and as they undertake learning tasks. In some curriculum areas it is difficult for a student to recognize the progress he or she is making. Perhaps the basic issue is, “In terms of things that are of personal interest to me, what can I do better now than I could do a week or so ago?” In many courses, it is hard to see that one is making progress. Contrast this with computer games that typically are designed so that a player moves up levels. Moving up a level in the game is a clear indication of increasing performance. Depending on the game, it may take only a few minutes to move up a level, or it may take a week or more.

Earlier in this* IAE Newsletter, *I indicated that I had
spent about 20,000 hours developing a reasonably high level of
expertise in mathematics. Just for the fun of it, a few years ago I
looked up data on the world’s leading chess players. Most of them had
spent 30,000 hours or more achieving their world class level of
expertise. Wow! These top chess plays devote full time to maintaining
and increasing their chess knowledge and skills.

Hmm. In 1997, a computer program named Deep Blue beat the world’s best chess player in a six game chess match (Wikipedia, 2019a, link). In more recent years, computers have achieved a level of expertise above the world’s best in a number of competitive games and in other areas. So, when we talk about students achieving an increased level of expertise in a particular discipline, we need to be thinking about three approaches to problem solving: human alone, computer alone, and human plus computer working together. Sometimes a human alone is best, sometimes a computer alone is best, and sometimes the combination of human plus computer working together is best.

Students deserve an education in which they learn what they can do better than computers can do, what computers can do better than they can do, and how best to work with computers to solve the problems and accomplish the tasks in each discipline they study.

Remember that we have had thousands of years of experience in
teaching the **3 Rs **of **R**eading, ‘**R**iting,
and ‘**R**ithmetic (math), and integrating use of them
into our everyday lives. We have just scratched the surface on
teaching the** 4th R **of **R**easoning/Computational
Thinking.

Reading Across the
Curriculum

We can gain some insights into *mathing across the curriculum *by
examining how our schools have handled the issue of* reading across
the curriculum.* My 1/31/2018 Google search of the expression *reading
across the curriculum *produced about 133 million results.

**Reading across the curriculum**,
also called content literacy or active reading, is defined as "the
ability to use reading and writing for the acquisition of new content
in a given discipline. Such ability includes three principle cognitive
components: general literacy skills, content specific literacy skills
and prior knowledge of content.” (McGivern, n.d., link).

Interestingly, this definition includes writing as part of*
reading across the curriculum.* Reading and writing are
thoroughly intertwined. The basic idea is that not only do we want
students to learn to read and write, we want them to gain sufficient
skill to be about to use their reading and writing skills to learn the
materials being presented in the various discipline areas they are
studying at the time.

Schools use a variety of approaches to accomplish the goal of having students learn to read across the curriculum. One approach is merely to get students to read for pleasure. An approach that has been widely used in this endeavor is called Sustained Silent Reading (Wikipedia, 2019b, link):

Sustained Silent Reading (SSR) is a form
of school-based recreational reading, or free voluntary reading, where
students read silently in a designated time period every day in
school. An underlying assumption of SSR is that students learn to read
by reading constantly. Successful models of SSR typically allow
students to select their own books **and require neither
testing for comprehension nor book reports.** [Bold added for
emphasis.]

SSR is a valuable approach to developing students’ reading skills and habits of reading. But, reading across the curriculum is more than just reading for pleasure. We want students to learn to use their reading skills as an aid to learning the content materials in each curriculum area that they are studying. This is certainly a standard approach in today’s schools. Teachers expect students to make use of reading and writing in each course they take. Even as students begin to take specific courses in either reading or writing, they are expected to do both in such courses.

The** 4th R** of **R**easoning/Computational
Thinking is now a commonly used component of reading and writing. It
is helpful that so much of the materials people want to read and study
are readily available on the Web. The Web also makes available access
to both audio and video materials. So, reading today now includes
providing students with computers that enable them to access a
combination of online print, audio, and visual materials—which may
well be in an interactive format.

Writing in the “good old days” was done with a quill pen on paper, or
chalk on a slate. The fountain pen and typewriter were great
inventions. And now our students’ routine use of word processors
certainly has changed the process of writing.

*Here’s an aside.* The improvements
in aids to writing reminds me of a story from my distant past. My
doctoral students working in the field of computers in education asked
me if they could use a word processor when writing their comprehensive
exams. I agreed this was all right, and I got permission from the
“powers that be.” When the timed exam ended, my students were allowed
to print out what they had written.

This was acceptable until some of the faculty found out that the word processor being used included a spelling checker. They considered this to be cheating and said my students should not be allowed to use a word processor. I eventually won this argument!

Now, of course, it is routine for students to write using a word processor that includes both spelling and grammar checkers. While writing, they also can easily check the definition of any word they write, and they can easily access the Web for information, references, and so on.

However, relatively few teachers give open book tests, and still fewer teachers give open computer tests. Contrast this situation with the writing that people do outside of school tests, such as writing reports and other papers in school and on the job. Hmm.

A broad definition of reading today includes listening and viewing. A
broad definition of writing includes developing audio and visual
materials. Just think of the challenge to teachers who were only
trained to teach traditional reading and writing of print text, and
who now have students who want to be learning this more broadly
defined discipline of reading and writing! Students teaching
themselves to make use of emoticons and voice texting when writing are
only one very small step in that direction. Computer graphics has
become a huge and vibrant discipline.

Mathing Across the
Curriculum

The term* mathing across the curriculum* can be considered as
a takeoff from* reading across the curriculum. *Sustained
Silent Reading is based on the idea that if students learn to read for
pleasure, there will be transfer of this learning to their more
effective use of reading across the curriculum. What about trying to
get schools to implement Sustained Silent Mathing (SSM)?

I often do SSM for pleasure. I do this in three different ways. One way is to read math articles, papers about math education, and so on. A second way is to play math-like games. Bridge, dominos, Monopoly, poker, solitaire card games, and Sudoku all fall into this category, as do many computer games. A third way is that I look for the use of math in the various (non-math) articles and books I read. I find it to be both fun and somewhat frightening to find errors in an author’s math-related knowledge and thinking skills.

A huge number of people do math for fun. My 2/1/2019 Google search of
the expression* math puzzles *produced about 284 million
results, and my Google search of* math games* produced about
584 million results. So, suppose that we include playing math games
and solving math puzzles in the activity we call SSM. (Let’s not
quibble about whether playing a math game with one or more other
people is a silent activity. What we are interested in is having
students engaged in math-related activities that combine pleasure and
learning.)

My professional colleague and friend Bob Albrecht has spent much of his long professional career in developing math games that have these dual characteristics:

- They are fun to play.
- Learning and playing the games helps students gain math knowledge and skills that are important in their own right and transfer to other math learning endeavors.

Many of Albrecht’s free materials are available in the Information
Age Education publications (Albrecht, 2018, link);
(Moursund & Albrecht, 2011, link);
Moursund, 2016, link).
The last reference is to my book,* Learning Problem-solving
Strategies through the Use of Games: A Guide for Teachers and
Parents. *The book presents a number of ideas that are
applicable in problem solving across the curriculum.

Improving student skills in mathing across the curriculum takes a
combination of what math teachers can do and what non-math teachers
can do. Math teachers can and should place an increased emphasis on
developing *math sense* and *math modeling,* and also
on developing the ability to solve challenging word problems that
require careful thinking. They will have more time to do this as they
spend less time teaching students “by-hand” methods that compete with
things that computers excel at doing.

*Here’s an aside, another story from my
past.* In teaching secondary school math teachers about uses of
computers in math, I would ask the question, “Why do we teach students
the quadratic formula for solving quadratic equations? Can you give me
an example from outside of math classes in which a student might want
to solve a quadratic equation?”

The answer to the first question is, “It is in the required curriculum.” Few could give any answers to the second question.

Quadratic and other functions are applicable in many different subject areas. In addition, moving up from linear equations to quadratic equations to higher order polynomial equations exposes students to some very nice/interesting mathematics. Scientists and engineers routinely make use of such math.

This type of example suggests to me that math education at the precollege level should spend more time helping students to understand some of the uses of the math they are learning.

One way to improve math education is to increase the use of math in the non-math disciplines. This can be done by a combination of staff development and locating and/or developing math materials that have the following characteristics:

- They are relevant to the content in the curriculum and discipline the non-math teachers are teaching. It is easy enough to say that math is an important aid to representing and solving problems across the curriculum. But, what we need are quite specific examples that are relevant to the curriculum content currently being taught, and that also add a new math-related dimension. Many of these are problems that computers can solve or greatly help in solving. A teacher or student does not need to be a math whiz to make effective use of computers to solve some of the important math-related problems in the various non-math disciplines.
- They are understandable by both the non-math teacher and their students. Thus, the math knowledge and skills needed to understand the topic must not be beyond the math preparation of the teacher and the students.
- They are interesting to and intrinsically motivating to teachers and to their students.
- They are supported by lesson plans, possible reading assignments
including online resources, in-class activities, formative and
summative assessment aids, computer software, and so on. The online
resources should illustrate and make use of the ability to
**4th R across the curriculum.**

There are many computer simulations that satisfy requirements 1-3.
Such* math modeling* is an important aid to both teaching and
using math across all curriculum areas.

An individual teacher may choose to make use of these materials. But, the goal should be a school-wide approach to integrating appropriate use of such materials across the curriculum.

Final Remarks

For many years, the average math achievement of students in the U.S.
has remained fairly steady. During this time, people who need to solve
math-related problems and accomplish math-related tasks in their
vocational areas have learned to use computers and to make routine use
of them. That is, they have learned both *mathing *and *computing
*across their professional areas.

Our math educational system is lagging behind the ready availability of computer aids to help students solve math-related problems (that is, applied math problems) that they will encounter in their adult lives.

*Mathing across the curriculum *is an important idea for
improving math education. But, using computers across the curriculum
(including math) to *4th R across the curriculum* is a still
more important idea about improving education.

References and Resources

Albrecht, R. (2018). Robert Albrecht. *IAE-pedia*. Retrieved
2/1/2019 from http://iae-pedia.org/Robert_Albrecht.

Kirsch, N. (11/9/2017). The 3 richest Americans hold more wealth than
bottom 50% of the country, study finds. *Forbes.com. *Retrieved
2/2/20109 from https://www.forbes.com/sites/noahkirsch/2017/11/09/the-3-richest-americans-hold-more-wealth-than-bottom-50-of-country-study-finds/#48e9ce7a3cf8.

Lubby, T. (1/21/2019). The top 26 billionaires own $1.4 trillion—as
much as 3.8 billion other people.* CNN Business.* Retrieved
2/2/2019 from https://www.cnn.com/2019/01/20/business/oxfam-billionaires-davos/index.html.

McGivern, K. (n.d.). Reading across the curriculum.* Sidney
Silverman Library. *Retrieved 1/31/2019 from http://bergen.libguides.com/RaC.

Moursund, D. (1/15/2019). ICT tools and the future of education, Part
3: ICT and math education. *IAE Newsletter.* Retrieved
1/18/2019 from https://i-a-e.org/newsletters/IAE-Newsletter-2019-249.html.
All newsletters in this series are available at https://i-a-e.org/iae-newsletter.html.

Moursund, D. (2018a). Communicating in the language of mathematics. *IAE-pedia.
*Retrieved 1/27/2019 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.

Moursund, D. (2018b). *The fourth R* (Second Edition).
Eugene, OR: Information Age Education. Retrieved 1/25/2019 from http://iae-pedia.org/The_Fourth_R_(Second_Edition).
Download the Microsoft Word file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/307-the-fourth-r-second-edition.html.
Download the PDF file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/308-the-fourth-r-second-edition-1.html.
Download the Spanish edition from http://iae-pedia.org/La_Cuarta_R_(Segunda_Edici%C3%B3n).

Moursund, (2016). *Learning problem-solving strategies through
the use of games: A guide for teachers and parents.* Eugene, OR:
Information Age Education. Download the Microsoft Word file from https://i-a-e.org/downloads/free-ebooks-by-dave-moursund/278-learning-problem-solving-strategies-through-the-use-of-games-a-guide-for-teachers-and-parents/file.html.
Download the PDF file from https://i-a-e.org/downloads/free-ebooks-by-dave-moursund/279-learning-problem-solving-strategies-through-the-use-of-games-a-guide-for-teachers-and-parents-1/file.html.

Moursund, D., & Albrecht, R. (2011). Math games and word
problems. *IAE-pedia. *Retrieved 2/1/2019 from http://iae-pedia.org/Moursund_and_Albrecht:_Math_games_and_word_problems.

Picha, G. (10/17/2018). Effective technology use in math class. *Edutopia*.
Retrieved 1/27/2019 from https://www.edutopia.org/article/effective-technology-use-math-class.

Wikipedia (2019a). Deep Blue versus Garry Kasparov. Retrieved 2/2/2019 from https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov.

Wikipedia (2019b). Sustained silent reading. Retrieved 1/31/2019 from
https://en.wikipedia.org/wiki/Sustained_silent_reading.

Author

**David Moursund** is an Emeritus
Professor of Education at the University of Oregon, and editor of
the *IAE Newsletter*. His professional career
includes founding the International Society for Technology in
Education (ISTE) in 1979, serving as ISTE’s executive officer for 19
years, and establishing ISTE’s flagship publication, *Learning
and Leading with Technology* (now published by
ISTE as *Empowered Learner*).He was the major
professor or co-major professor for 82 doctoral students. He has
presented hundreds of professional talks and workshops. He has
authored or coauthored more than 60 academic books and hundreds of
articles. Many of these books are available free online. See http://iaepedia.org/David_Moursund_Books .

In
2007, Moursund founded Information Age Education (IAE). IAE provides
free online educational materials via its *IAE-pedia*, *IAE
Newsletter*, *IAE Blog*, and IAE books.
See http://iaepedia.org/Main_Page#IAE_in_a_Nutshell .
Information Age Education is now fully integrated into the 501(c)(3)
non-profit corporation, Advancement of Globally Appropriate
Technology and Education (AGATE) that was established in 2016. David
Moursund is the Chief Executive Officer of AGATE.

Email: moursund@uoregon.edu.

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