Issue Number 252 February 28, 2019

This free Information Age Education Newsletter is edited by Dave Moursund and produced by Ken Loge. The newsletter is one component of the Information Age Education (IAE) and Advancement of Globally Appropriate Technology and Education (AGATE) publications.

All back issues of the newsletter and subscription information are available online. In addition, seven free books based on the newsletters are available: Joy of Learning; Validity and Credibility of Information; Education for Students’ Futures; Understanding and Mastering Complexity; Consciousness and Morality: Recent Research Developments; Creating an Appropriate 21st Century Education; and Common Core State Standards for Education in America.

Dave Moursund’s newly revised and updated book, The Fourth R (Second Edition), is now available in both English and Spanish (Moursund, 2018c). The unifying theme of the book is that the 4th R of Reasoning/Computational Thinking is fundamental to empowering today’s students and their teachers throughout the K-12 curriculum. The first edition was published in December, 2016, the second edition in August, 2018, and the Spanish translation of the second edition in September, 2018. The three books have now had a combined total of more than 35,000 page-views and downloads.

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

David Moursund
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 Reading, ‘Riting, ‘Rithmetic, and Reasoning/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:

  1. It behooves our educational systems to help students learn to math across the various curriculum areas taught in schools and in their everyday lives.
  2. 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.
  3. 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:

  1. The types of problems, tasks, and activities it addresses.
  2. Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, and so on.
  3. Its history, culture, and language, including notation and special vocabulary.
  4. Its methods of teaching, learning, assessment, and thinking. What it does to preserve and sustain its work and pass it on to future generations.
  5. Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
  6. 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 Reading, ‘Riting, and ‘Rithmetic (math), and integrating use of them into our everyday lives. We have just scratched the surface on teaching the 4th R of Reasoning/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 Reasoning/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:

  1. They are fun to play.
  2. 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:

  1. 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.
  2. 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.
  3. They are interesting to and intrinsically motivating to teachers and to their students.
  4. 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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About Information Age Education, Inc.

Information Age Education is a non-profit organization dedicated to improving education for learners of all ages throughout the world. Current IAE activities and free materials include the IAE-pedia at http://iae-pedia.org, a Website containing free books and articles at http://i-a-e.org/, a Blog at http://i-a-e.org/iae-blog.html, and the free newsletter you are now reading. See all back issues of the Blog at http://iae-pedia.org/IAE_Blog and all back issues of the Newsletter at http://i-a-e.org/iae-newsletter.html.