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I've always been puzzled by the unequal treatments of square roots and logarithms in school mathematics. In the United States, most students know what a square root is before they enter high school (9th grade, age 14 or so), and can answer questions such as $$ \sqrt{25} \,=\, \text{?}\qquad\text{or}\qquad \sqrt{64} \,=\, \text{?} $$ Most students also learn about the algebraic properties of square roots in a first algebra course.

At the same time, it seems to be very rare for 8th grade students to be able to answer questions like $$ \log(100) \,=\, \text{?}\qquad\text{or}\qquad \log(10\text{,}000) \,=\, \text{?} $$ Indeed, students don't usually learn anything about logarithms until Algebra 2 or even Precalculus. One result of this is that calculus students always seem very comfortable with square roots, but have a very shaky knowledge of logarithms, even though the two concepts have about the same difficulty level.

So my question is, why the unequal treatment? Is the idea that square roots are somehow more common or more useful than logarithms? Or is there something that's harder about logarithms that I'm missing?

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I learned about logarithms in around 8th grade. No calculators around then... –  vonbrand Apr 20 at 23:14
    
@vonbrand I think I remember learning a bit about logarithms in 8th grade (Algebra I) even though the TI-83 Plus was a standard calculator for us –  Izkata Apr 21 at 3:05
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It's very easy to draw a square. What the heck do you draw for a logarithm? –  user2357112 Apr 21 at 6:51
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Since there's no longer any motivation for logarithms (and log tables) as an aide to calculation, I would suggest that the natural point to introduce logarithms is alongside $e$, and that's why it comes late in the typical syllabus. –  Steve Jessop Apr 21 at 15:16

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The appendix A in the Common Core State Standards for Mathematics seems to agree that traditional curricula put logarithms somewhere after Algebra I. CCSSM isn't a curriculum, and it doesn't suggest teaching logarithms later, it just offers "pathways" that are supposed to help districts using traditional curriculum find a way to meet the standards. I mention it just to show that, yes, this seems to agree that a traditional time to introduce the subject is later than Algebra I.

The Massachussetts Curriculum Frameworks put logarithms into "Algebra I" under a category like "Functions." Although this is also considered to be "Common Core Aligned."

One short answer is that "it's traditional." The order you describe is seen as early as 1898 in Bull Wentworth's New School Algebra. (Donoghue, 2003)

Curriculum design is (should be) about structuring the subject matter in a way that students can form a coherent understanding. At the least, it is an arrangement that the designers find coherent. This means that it's not necessarily true that the options are: square roots are more common, square roots are more useful, or square roots are easier. Nor is it necessarily true that there is one correct way. So, one way may be produced, become traditional, and then be perpetuated.

Reform curricula have come up against this time and again. Sometimes, the way we do things in math education are traditional and people don't want to question them, even in the face of evidence.

While I think you have a good question, perhaps a more forward-looking question might be whether we can find reasons to teach logarithms earlier.

Squares and square roots are connected to a particular 2D representation (squares) that can be used as a model for multiplication. It's popular for assisting students beyond the tenacious idea of multiplication as repeated addition. A representational leg up like that may be (just speculating here) one convenient reason that squares and square roots can be placed early in a curriculum "pathway." I'm really only guessing here; IANACD (I Am Not A Curriculum Designer). But I doubt that's a hard and fast rule that cannot be broken. Some people have successfully introduced algebraic thinking to early grades students (Smith III & Thompson, 2008). Although getting students to reason algebraically is a little bit different (in intention) from introducing a particular concept or operation at a different time.

Cited:

Donoghue, E. F. (2003). Algebra and geometry textbooks in twentieth-century America. In G. M. A. Stanic & J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 329–398). Reston, VA: National Council of Teachers of Mathematics.

Smith III, J. P., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–131). New York, NY: Taylor & Francis Group.

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Thanks for your fantastic answer! From my point of view, the main reason to teach logarithms earlier would be so that students are more comfortable with them by the time they take math and science courses in college, though certainly this doesn't apply to every student. You raise a good point about square roots being connected to squares and multiplication. Logarithms are connected more closely to exponentiation and to ideas such as orders of magnitude, scientific notation, exponential growth, and so forth, none of which are quite as compelling as multiplication. –  Jim Belk Apr 21 at 0:45
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(cont.) Also, as you mention, there are other factors in curriculum design that might be playing a role here, such as curricular coherence, tradition, and simply the fact that there isn't always time to cover everything. –  Jim Belk Apr 21 at 0:47

Historical comments.

Early on, the study of logarithms and logarithmic tables was incorporated into trigonometry. For more on this background from the perspective of the history of trigonometry education, search through the following dissertation for the word logarithm:

Van Sickle, J. (2011). A History of Trigonometry Education in the United States: 1776-1900 (Doctoral dissertation, Columbia University). Link.

Example: You can find a book from 1914 entitled "Trigonometry with the Theory and Use of Logarithms."

Here is an excerpt:

enter image description here

At the least, the author seems to realize that finding the hypotenuse using the Pythagorean Theorem is more sensible; but it is strange that this would even need saying. JPBurke's informative answer mentions Eileen Donoghue, which made me wonder about the history of logarithms with regard to the New York Regents. (Donoghue is an excellent source for historical information about curricula, textbooks, and the Regents in particular; I mentioned the same 2003 article by her in an earlier MESE post.) It seems that four years after the above-mentioned book was published, a 1918 article in The Mathematics Teacher argues that logarithms should be learned in intermediate algebra, rather than "consigning" it to trigonometry.

The citation for this article is:

Decker, F. F. (1918). The New York State Regents Syllabus in Intermediate Algebra. The Mathematics Teacher, 11(1), 1-8. Jstor.

A free copy of the article can be found here, from which I quote:

The introduction to logarithms is placed in intermediate algebra. Consigning this topic to trigonometry has several disadvantages. Many students who carry their mathematical study through the course in trigonometry seem to get the idea that the usefulness of logarithms is confined to trigonometry. Those who do not reach trigonometry lose what is at once an important tool and an elegant system. In the separation of the treatment of exponents and logarithms the theory seems to contain a discontinuity. The laws for commensurable exponents are carefully proved and then the theory of logarithms is made to appear to rest on that of exponents. But most of the logarithms used are incommensurable! The plan here proposed is to introduce incommensurable exponents with reference to the continuity of an exponential curve and to pass immediately to logarithms.

The author also cites a separate document arguing that logarithms be studied along with exponents:

[A report] of the Committee on the Teaching of Mathematics to Students of Engineering, calls for the teaching of logarithms in logical connection with the subject of exponents rather than with trigonometry.

This report is foot-noted as:

"Syllabus of Mathematics," Society for the Promotion of Engineering Education, 1912.

but an APA citation is:

American Society for Engineering Education. Committee on the Teaching of Mathematics to Students of Engineering. (1912). Syllabus of mathematics: a symposium. New Era Printing Co..

The report can be found on google books; here is the relevant excerpt:

enter image description here

Evidently, the study of logarithms within trigonometry is widespread enough by 1912 to warrant comments from a group of engineering educators. For now, I will not pursue the connections between these two subjects any further, but I would not be surprised if the disentangling of logs from trig resulted in the former being placed later in the curriculum than you might consider ideal from a more modern perspective.

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Your mention of my Donoghue citation made me go back and check it. There was info in my citation database that was incorrect, and now I've fixed it. The source is a chapter in the Stanic and Kilpatrick "A History of School Mathematics." –  JPBurke Apr 21 at 11:14
    
Re:Historical Reasons. The main use of those things were in navigation. See for example math.stackexchange.com/questions/168897/… –  belisarius Apr 21 at 12:02
    
Re: connection of logs and trig: I don't have a good source for it, but I read in one book about history of math that before the invention/discovery of logarithms, it was trigonometric functions that were used for calculations (using e.g. the formula for $\cos x +\cos y$ and cosine tables - you could change a multiplication into a bunch of things like additions, doubling and halving). –  mbork Apr 21 at 21:21

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