TOMT Computers and Math Education
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This Wiki page is designed to support a series of articles that David Moursund is writing for The Oregon Mathematics Teacher (TOMT). The articles focus on computers in math education. They will be published every two months beginning in September 2009.
There is one section of this Wiki page for each article in the series. Each such section begins with the title and a very brief summary of the article. Then it provides subsections for two major things:
- Moursund's comments about and possible expansions on the article. Suggestions for applications in various math education levels and courses.
- Readers comments about and additions to the article. Some of these will likely contain specific suggestions for how math teachers at various grade and course levels can integrate ideas from the article into the math they are teaching.
At the current time this document contains a number of ideas about possible content for this page and for the articles that are in the process of being written. One way to think of this document is that it represents Moursund brainstorming with himself.
Overview and Purpose
Information and Communication Technology (ICT) affects the content, instructional processes, and assessment of each academic discipline. The effect can be particularly strong in math educational because computers are a very powerful aid to helping to solve math problems.
Indeed, computers are now such an important part of math and math education that a new subdivision of math called Computational Mathematics has emerged in recent years. The three major subdivisions of math are now named Pure Mathematics, Applied Mathematics, and Computational Mathematics. A 7/1/09 Google search using the expression "Computational Math" OR "Computational Mathematics" produced about 475,000 hits.
Computational Thinking has recently emerged as a broad field of study having to do with used of computer across all curriculum areas. Here is a definition:
- Computational thinking is a way of solving problems, designing systems, and understanding human behavior that draws on concepts fundamental to computer science. Computational thinking is thinking in terms of abstractions, invariably multiple layers of abstraction at once. Computational thinking is about the automation of these abstractions. The automaton could be an algorithm, a Turing machine, a tangible device, a software system—or the human brain. (Carnegie Mellon, n.d.) [Bold added for emphasis.]
The iae-pedia Wiki page Computational Thinking provides an education-oriented introduction to computational thinking. The following free book for preservice and inservice teachers discusses the integration of computational thinking into the K-8 math curriculum.
- Moursund, David (2006). Computational thinking and math maturity: Improving math education in K-8 schools. Eugene, OR: Information Age Education. Access at http://i-a-e.org/downloads/doc_download/3-computational-thinking-and-math-maturity-improving-math-education-in-k-8-schools.html.
This book addresses the problem that our K-8 school math education system is not as successful as many people would like it to be, and it is not as successful as it could be. It is designed as supplementary material for use in a Math Methods course for preservice K-8 teachers. However, it can also be used by inservice K-8 teachers and for students enrolled in Math for Elementary and Middle School teachers’ courses.
The book draws upon and explores four Big Ideas that, taken together, have the potential to significantly improve out math education. The Big Ideas are:
- Thinking of learning math as a process of both learning math content and a process of gaining in math maturity. Our current math education system is does a poor job of building math maturity.
- Thinking of a student’s math cognitive development in terms of the roles of both nature and nurture. Research in cognitive acceleration in mathematics and other disciplines indicates we can do much better in fostering math cognitive development.
- Understanding the power of computer systems and computational thinking as an aid to representing and solving math problems and as an aid to effectively using math in all other disciplines.
- Placing increased emphasis on learning to learn math, making effective use of use computer-based aids to learning, and information retrieval.
Article # 1 September 2009: Introduction to Computers and Math Education
Comments and Additions by David Moursund
Here is an interesting question that can be easily explored using an 8-digit calculator with 4-key memory.
What is 1/2 + 1/3 + 1/4 + 1/5 + …?
Is it bigger than 2? Is it bigger than 5?
The calculator keying sequence MC 1/2 M+ 1/3 M+ 1/4 M+ … can be used to do a number of terms of the calculation. The calculator user can stop adding terms at any point and key MR to see the current sum. Then continue to add on more terms. Thus , for example, the keying sequence might look like:
MC 1/2 M+ 1/3 M+ 1/4 M+ MR 1/5 M+ 1/6 M+ 1/7 M+ 1/8 M+ 1/9 M+ 1/10M+ 1/11 M+ MR
On an 8-digit calculator, the first MR displays the result 1.2833333 while the second MR gives the result 2.019877. You know that this total for the first ten terms of the sum is not exactly correct (due to it being done using an 8-digit calculator), but you can have confidence that the sum of the infinite series is surely more than 2.
Now, how large to you think the sum of an infinite number of terms of the series is? Here is a way to gain insight into how large the answer might be. Group terms as follows:
(1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + … 1/16) + (1/17 + … 1/32) + …
Notice the first group contains just one term and its value is 0.5. The second group contains 2 terms, and each is 1/4 or larger. So, the sum of the terms in the second group is larger than 0.5. The third group contains 4 terms, and each is 1/8 or larger. So, the sum of the terms in the third group is larger than 0.5. Perhaps you are seeing the pattern. There are 8 terms in the 4th group, and each is 1/16 or larger. There are 16 terms in the 5th group, and each is 1/32 or larger. This argument should convince you that the sum of this infinite series is larger than any positive number. That is, the sum is infinite.
Okay, how about another somewhat similar series. What is:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …?
Do you think that the sum is larger than 2? Is it larger than 5? If you use an 8-digit calculator with 4-key memory to add together a number of terms of this series, you can the sum increasing, but you can also see that the sum does not appear to be getting larger than 1. Interesting! It turns out that the sum of this infinite series in exactly 1. How might one go about proving this?
Well, let me give you a similar but perhaps somewhat simpler problem. What is the value of the sum
.9 + .09 + .009 + .0009 + …?
You can see that the partial sums are .9, .99, .999, .9999, … . Thus, the partial sums keep increasing, they get closer and closer to 1.0, and they do not get larger than 1.0 The partial sums eventually get larger than any number less than 1.0. Thus, (using ideas and definitions from calculus) we conclude that the sum is exactly 1.0.
Now, go back to the original 1/2 + 1/4 + 1/8 + … . If you are familiar with the binary number system, you can see that in binary the partial sums are .1, .11, .111, .1111, …
Thus, the sum is 1. Can you think of a way to rewrite the partial sums in the original series so that you can make a convincing argument that the sum is 1?
Here is an idea. Spend some time thinking about the partial sums represented as fractions:
(1/2) = 1/2 = 1 - 1/2
(1/2 + 1/4) = 3/4 = 1 - 1/4
(1/2 + 1/4 + 1/8) = 7/8 = 1 - 1/8
(1/2 + 1/4 + 1/8 + 1/16) = 15/16 = 1 - 1/16
(1/2 + 1/4 + 1/8 + 1/16 + 1/32) = 31/32 = 1 - 1/32
Notice that we have explored two different infinite series that add to exactly 1. Are there any other such series?
Comments and Additions by (name or screen name; date)
Comments and Additions by (name or screen name; date)
Article # 2: Automated Manipulation
Comments and Additions by David Moursund
Comments and Additions by (name or screen name; date)
Comments and Additions by (name or screen name; date)
Article # 3: Calculators
Comments and Additions by David Moursund
Computer scientists often use the term register when talking about the memory locations used in carrying out operations such as addition, division, and so on. That is, a register is a special purpose memory location. In this language, when you key a number into your calculator it is stored in the display register (the display memory location).
- Here is an aside comment from David Moursund to himself. Here is an issue that I have not adequately dealt with in the discussion so far. In order to store a pair of numbers that are to be added, subtracted, multiplied, or divided, two memory locations are needed. If memory is expensive to build, a calculator 's circuitry can make use of one memory location for both the display memory and for the storage of the second of the pair of numbers involved in the arithmetic operation. I need to decide how complicated or how simple I want to make this discussion.
- Probably the simpler approach would be to think in terms of four dynamic memory locations with names such as D (display memory), A (the first of a pair of numbers being entered for an arithmetic calculator), B (the second of a pair of numbers being entered for an arithmetic calculation) and M (the name of the memory location that is acted upon by the keys MC, M+, M-, and MR.
- How complicated does this make it as we try to exaplain what happens when one keys in a sequence such as:
- C 23 + 14 - 8 x 5 =
- And, contrast this with the sequence:
- C 23 + 14 = -8 = x 5 =
- Part of the point is that this example, and others given in this section, illustrate how complicated a "simple" calculator is. We provide students and teachers with a "black box" that has a number of characteristics of the number line and arithmetic on numbers on the number line. Think of a calculator as a model or representation of the real number line and arithmetic calculation. The model is not an exact representation. So, we expect differences between the model and that which is being modeled This is a very important concept in modeling.
- Moreover, modeling (and simulation) are very important educational topics. I wonder what students are being taught about modeling and simulation in school????? I wonder whether it is a good idea to talk about a calculator in terms of modeling & simulation. I suspect that the answer is yes. This will require rewriting some of the articles I have already written and/or adding an article just on modeling and simulation.
- Hmmm. This is definitely food for thought and a "work in progress."
6-function calculators with 4-key memory have a memory location (that I call M) and four keys: MC (memory clear, sets M to zero), M+ (adds the contents of the display memory to M and stores the result in M), M- (subtracts the contents of the display memory from M and stores the results in M), and MR (makes a copy of the contents of M and stores it in the display memory.
In place of the MC key, some of these calculators have an AC (all clear) key that sets the values in all of the dynamic memory locations to zero.
Many 6-function calculators have a repeated multiplication feature. Here is an example of this features use. Try it on your 6-function calculator and see if works the same as it does on mine. Key the sequence:
C 5 x x 8 =
You will see the result 40. in the display.
Then continue with 12 = and then with 7 =. The 12 = produces the result 60. (that is, 5 x 12) and the 7 = produces the result 35. (that is, 5 x 7). What is happening is that the keyboard strokes 5 x x not only store the number 5 as a multiplier, but also make this multiplier available for use with a sequence om multiplicands.
Here is another experiment to try. On yuo0r 6-function calculator, key the sequence:
C 5 x =
My calculator give the result 25. A little experimentation convinces me that if I do not specify the multiplicand when doing a multiplication, my calculator calculates the multiplier times itself. This provides a quick way to square a number.
What if you want to enter a negative number into your calculator? Some inexpensive calculators have a +/- key that can be used to change the sign of a number in its display memory. Others do not.
Suppose that you have a calculator that does not have a +/- key, and you want to put a -6 into the display memory,Try the sequences of key strokes:
C - 6 =
C 6 -
C 6 - =
I get varying results from the various 6-function calculators that I own. The square root key on a 6-function calculator works as one might expect. In the example that follows, the symbol √ is used to designate the square root key. Thus, the sequence C 16 √ or the sequence C 16 √ = produces the result 4., which is the exact positive square root of 16.
Similarly C 2 √ calculates the positive square root of 2 and produces the result 1.4142135. You know that the positive square root of 2 is an irrational number, and that you are working with an 8-digit calculator. What do you suppose you will get when you square this answer?
The sequence C 2 √ x = produces the positive square root of 2 and then multiples the result times itself. The result is 1.999998. Oh oh! This illustrates peculiarities of the 8-digit calculator. The e-digit calculator's number line is explored in Article # 4 of this series of articles.
