European Mathematical Society - 97D40
https://euro-math-soc.eu/msc-full/97d40
enCalculus Reordered
https://euro-math-soc.eu/review/calculus-reordered
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
For reasons of sequentiality, an elementary calculus course is usually organising the topics that are discussed in the order where first limit is defined, which is needed to define continuity, then the derivative as the limit of a ratio, and only then the integral as the limit of a Riemann sum. The same order can be repeated for functions of a discrete variable (sequences) where the variable is usually denoted as an index. The limit can only be considered when the index goes to infinity (the only accumulation point), continuity and derivative don't make much sense (unless one wants to discuss finite differences) and (improper) integrals correspond to (infinite) sums, and the convergence of series. At the beginning some functions are assumed known: algebraic functions are a minimum. Sometimes also goniometric functions, and possibly the exponential and logarithm, but in principle all of these and the more "advanced" ones can be defined later. Functions are "defined" when no name exists for a converging series or a primitive function of an integral. The logarithm for example is the integral of 1/x, and the exponential function is its inverse. At least this is how I organized my lecture notes, but this is not always the order used, and it is also not in this order that all these concepts were developed historically.</p>
<p>
What Bressoud does in this book is looking at the contents of a calculus course from an historical perspective. In what order were all these concepts developed, and where do all the well known theorems come from? In fact analysis came relatively late. The Greek were mainly doing geometry using rational numbers, and yet they computed the volume of a sphere and the area of a circle. Then the algebra came to Western Europe through the Arabs, and only then analysis took off with Newton and Leibniz who were already manipulating series or at least truncated series as interpolating approximations. It is only when analytic geometry bridged the gap between algebra and geometry, that analysis took over as the dominant tool for solving the practical analytic problems. The limit and its geometric interpretation came only very late.</p>
<p>
The first chapter of this book covers the period up to and including Newton and Leibniz. The second is about the further evolution of calculus, analytic geometry, the logarithm, differential equations, waves and field theory, culminating in the Maxwell equations. The emergence of Taylor and Fourier series are covered in chapter three, but only in chapter four, we find convergence criteria for series. This convergence, just like the proper definition of a derivative, and other concepts that are defined as the result of a limit, can only come to a conclusion by including the limiting value by upper and lower bounds approaching each other. Only using these bounding inequalities, will eventually lead to the proper concept of a limit. This new concept allowed a previously unseen expansion of analysis, requiring to rethink the concept of a function, since it was realized how exotic some functions can be, like everywhere continuous and nowhere differentiable. Bressoud's historical survey illustrates that the order of our calculus course is inverting history. First came accumulation (integral), then ratios of change (derivative), then sequences of partial sums (series) and only in the end the algebra of inequalities (limit).</p>
<p>
In an appendix, Bressoud adds his thoughts about how calculus should be taught. It is often the case that because of time restrictions that the integral is introduced as an anti-derivative, leading to cookbook recipes to compute integrals. Given the current technology, this is indeed a waste of time when there is no insight of the integral as a summation brought to a limit. Similarly teaching formulas to compute derivatives without seeing the derivative as a limiting process of ratios of change doesn't make much sense either, and series should be introduced as the limit of the sequence of its partial sums. And finally the limit is the result of a sandwich principle where upper and lower bounds approach each other. All of these insights are essential when looking at numerical analysis where exactly these insights are the elements that compute integrals, derivatives, and evaluate transcendental functions.</p>
<p>
As I have been teaching elementary courses in algebra, analysis, and numerical analysis, I can fully appreciate Bressoud's conclusions that I described in the previous paragraph and I fully realize the importance of algebra as an essential element in the development of calculus, certainly when one moves to functions of several variables, (linear) algebra, matrices and vector spaces become essential. However, it is somewhat unclear if Bressoud is promoting to keep also the historical order as he describes it in this book: integration, differentiation, series, limit. In my opinion the limit is the missing link that lets the whole building of calculus fall into a logical sequence and hence it should come first. It would be a waste of time to re-live the trial and errors of history in a calculus course. Apart from the pedagogical conclusions, the book gives a nice survey of how the main achievements and theorems that any student meets in a calculus course came about. Moreover, it is shown that the historical approach is sometimes quite different from what is written in the lecture notes, because the mathematical tools and the objectives of the person developing them were in those days quite different from what is available to a student in the 21st century.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The book is a survey of how the main ideas that underpin a modern calculus course were developed in their historical context. Based on this, Bressoud draws some conclusions about how we should teach a calculus course. An approach following the historical origin will be much closer to intuition and have didactic advantages.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-m-bressoud" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David M. Bressoud</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2019</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691181318 (hbk), 9780691189161 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£ 24.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">242</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://press.princeton.edu/titles/13397.html" title="Link to web page">https://press.princeton.edu/titles/13397.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D40</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97i99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97I99</a></li></ul></span>Mon, 05 Aug 2019 09:54:15 +0000Adhemar Bultheel49603 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/calculus-reordered#commentsHow to Solve It: A New Aspect of Mathematical Method
https://euro-math-soc.eu/review/how-solve-it-new-aspect-mathematical-method
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Since its original edition in 1945, this classic has not been out of print. With this reprint, Princeton University Press keeps the marvelous ideas of Polya alive.</p>
<p>
How come people do like solving crossword puzzles, brain teasers, riddles, or nowadays why gaming in so popular. All these challenge the intellect ant require creative solutions. And yet, as Polya writes in his preface to the second edition: <em> ``...mathematics has the dubious honor of being the least popular subject in the curriculum ... Future teachers pass through the elementary schools learning to detest mathematics ... They return to the elementary school to teach a new generation to detest it.''</em></p>
<p>
So his focus is on the interaction between student and teacher. It is not only how to learn to solve the problem, it is even more so about how to teach someone how to solve a problem. The first section makes this clear: <em>``The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help or with insufficient help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.''</em> Note that this is a general rule that applies not only to mathematics, although most of the examples in the book are mathematical, and often geometrical. Another aspect that shows in this quote is that it is not a problem of the student or of the teacher, but that the focus should be placed on the interaction between both. This explains why most of the book is written in some kind of dialog, whereby the teacher is most often just asking an appropriate question, followed by a possible reaction of the student. This reflects on a meta-level as the author teaching the reader-student how to approach the solution of a problem and the reader-teacher how to help a student.</p>
<p>
In the first part, Polya explains that the solving process goes in four phases, and that the proper questions have to be asked in each phase.</p>
<ol>
<li>
<em>We have to understand the problem.</em> What is the unknown? What is given? What are the conditions? (introduce a figure if possible, introduce proper notation, check if a solution is possible)</li>
<li>
<em>We have to make a plan.</em> Is there a related, familiar, or simpler problem that we can solve? Are all the data used? Do we satisfy all the conditions?</li>
<li>
<em>We have to carry out the plan.</em> Can we check each step and prove that it is correct?</li>
<li>
<em>We have to reflect on the solution.</em> Can we check the result? Is there another derivation possible? Is there a shorter derivation? Have we used all the data? can we use the method in another problem?</li>
</ol>
<p>
These issues are illustrated with several examples.</p>
<p>
</p>
<p>
The second part is very short (only 4 pages) and reflects a dialogue between a teacher and an ideal student in very general terms.</p>
<p>
The bulk of the book is part 3, that is written in the form of an alphabetically ordered dictionary. It has entries that are names of mathematicians (Bolzano, Descartes, Leibnitz, Pappus), but most entries somehow give further reflections on the the questions asked above and their answers with more concrete examples.</p>
<p>
Another short fourth part formulates some problems, then some hints and finally solutions. The first one has become famous among riddle lovers: <em> A bear, starting from the point P, walked one mile due south. Then he changed direction and walked one mile due east. Then he turned again to the left and walked one mile due north, and arrived exactly at the point P he started from. What was the color of the bear?</em> One might be tempted to think that the color is white because <em>P</em> is the North Pole, but there are other solutions!</p>
<p>
The focus is clearly on mathematics and almost all examples are indeed mathematical, but the same principles can be applied in any other problem solving situation. It takes some time to go through all the examples but the time is not waisted. It pays to think about them and see the spark of the generality of the idea. Some might find it frustrating that either the examples are too specific, not touching on their own problem or they find that the advise given is too general, not helping to solve the particular problem they have at hand. In an age that all solutions should be provided with the least possible effort, this book brings a very important message: mathematics and problem solving in general needs a lot of practice and experience obtained by challenging creative thinking, and certainly not by copying predefined recipes provided by others. Let's hope this classic will remain a source of inspiration for several generations to come.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a reprint of a classic, originally published in 1945. Polya explains his views on how to learn to solve (mathematical) problems. The books addresses students as well as teachers, because the students have to practice to collect experience, but they should be guided in a proper way by the teacher. The crux is to ask the proper questions at all stages of the process: understand the problem, design and execute a strategy and verify the solution.<br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/george-polya" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">George Polya</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-pres" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university pres</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9780691164076 (pbk) </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£13.95 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">288</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://press.princeton.edu/titles/669.html" title="Link to web page">http://press.princeton.edu/titles/669.html</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A10</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97b50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97b50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97c70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97C70</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/97d40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97D40</a></li></ul></span>Tue, 25 Nov 2014 12:42:32 +0000Adhemar Bultheel45849 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-solve-it-new-aspect-mathematical-method#comments