Archive for the ‘Math Education’ Category

One of my students gave me a problem last fall that was very interesting.

The problem is posed with the following diagram:


What is the measure of \angle CDA?  That is, what is the value of x?

I’ve given a similar problem in my trigonometry class for the past few years, except that version of the problem has a side length included and the triangle is not isosceles.

A pdf of this problem is linked below:


Working from my experience with the other version of this problem, I began to write in values for the various unlabeled angles in the diagram – if we label the intersection of \overline{AD} and \overline{BC} as K, then \angle CKD and \angle AKB are both 70^{\circ}, \angle CKA and \angle DKB are both 110^{\circ}, which makes \angle KCA 50^{\circ} and \angle ADB is 40^{\circ}.

I added in new variables and created a system of four equations with four unknowns, but it was a dependent system.

The solution for this problem that was devised by the student who gave it to me is after the jump…


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About fifteen years ago, when the WorldWideWeb was still text-based, I came across some of the writings of Professor Richard Mitchell (1929-2002).  Mitchell was a Professor of English and Classics at Glassboro State College in New Jersey, now known as Rowan University.  Although he was a specialist in grammar, literature and the humanities in general, he had a tremendous appreciation for mathematics and a deep and penetrating understanding of what constitutes mathematics and what it’s good for.  These short excerpts below come from his two essays, “The Uses of Audacity” and “Wise Choices in Peoria.”


From “The Uses of Audacity”

Algebra is a world of principle, and a dramatic revelation of the power of principle. In fact, algebra, and even algebra alone, could provide a true and sufficient education out of which to understand the worth of living by principle…

…[Y]ou will have it in your mind that you can know something–truly know it, and not just believe it, or be informed of it–and maybe, since that is so, you can truly know something else. It’s interesting to wonder what such a something else might be.

…You will find that algebra shows you some truths. The first great truth is that there can be something real, and complete, and harmonious, and even, in some strange way, absolutely perfect right in your own mind, and made by you alone. You will see that you have a wonderful freedom not mentioned in the Bill of Rights, the freedom to decide what your mind will contain and how it will work.  You don’t have to copy the rest of the world.

Algebra tells sad truths too. Where there is no balance, there is no truth. What is equal is equal, and between the equal and the unequal there is no conference table, no convenient compromise. In this terrible law there is a hinting question for all of life…

Algebra will show you the inexorable, the endless and permanent chain of consequence, the dark thread of necessity that brought you to a wrong answer because of a tiny little mistake back in the second line. I know how unfair that seems, and how scary that what seems unfair is nevertheless justice. Is life like that too, as all of nature seems to be? How then shall we live? What are the laws of the algebra of our living, and where do they exist, where created? Who can show us how to learn them?

It takes some serious living to see the truth hidden in algebra…


From “Wise Choices in Peoria”

[Some people]…assume that things like geometry and the multiplication table are taught in schools only out of tradition, and they are easily seduced into believing that such arts are useless to those who aren’t going to make some money from them.

But in fact the mathematical arts are the best studies in which to learn certain truths that are essential to the making of wise choices. It is in mathematics that we most readily see that the permanent relationship between principle and necessity is not subject to appeal, that every particular is a local manifestation of some universal, that there is a demonstrable difference between what we believe and what we know, and that experience can never do the work of logic. It is in mathematical studies that a child … can have his first inkling of Justice and Truth…

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Education Reform

The Education Reform movement has been ubiquitous nationwide since at least 2010.  The Common Core standards and the testing that goes along with them is a part of this movement, as are charter schools, vouchers, Teach for America, VAM methods of evaluating teachers, MOOCs and a variety of other disruptive innovations in education.  I just read an article by Jill Lepore in the New Yorker from June 2014.  It was linked from the website of Diane Ravitch who has been blogging about Education Reform for quite some time now.

The paragraphs that are quoted on Diane Ravitch’s web page and which I think really get at the issue are as follows:

…Innovation and disruption are ideas that originated in the arena of business but which have since been applied to arenas whose values and goals are remote from the values and goals of business. People aren’t disk drives. Public schools, colleges and universities, churches, museums, and many hospitals, all of which have been subjected to disruptive innovation, have revenues and expenses and infrastructures, but they aren’t industries in the same way that manufacturers of hard-disk drives or truck engines or drygoods are industries. Journalism isn’t an industry in that sense, either.

Doctors have obligations to their patients, teachers to their students, pastors to their congregations, curators to the public, and journalists to their readers—obligations that lie outside the realm of earnings, and are fundamentally different from the obligations that a business executive has to employees, partners, and investors…

At the University of Amsterdam, staff and faculty have joined the students in protests against recent actions taken by the institution.  Jerome Roos of the European University Institute connects the events in Amsterdam to a world wide crisis in higher education:

Structurally underfunded, severely over-financialized and profoundly undemocratic, universities everywhere are increasingly abandoning their most crucial social functions of yore — to produce high-quality research and educate the next generation of skilled, conscious citizens — and devolving ever more into quasi-private companies run by an utterly detached managerial elite.

To make matters worse, these managers…are actually being paid six-sum figures to push around insane amounts of pointless paperwork, forcing destructive workloads and unrealistic expectations onto increasingly precarious staff, treating students like simple-minded consumers and impersonal statistics, and putting immense pressure on highly talented researchers to spew out mind-numbing amounts of nonsensical garbage just to meet rigid quantitative publication quotas….

The protesters at UvA thus find themselves at the front-line of what is essentially a global fightback against the commodification of higher education and the steady reduction of knowledge and learning to an increasingly unaffordable consumer good. In many countries, this neoliberal logic has resulted in dramatic tuition hikes and budget cuts, combined with the metastization of a culture of top-down managerialism, creeping bureaucratization and the systematic precarization of academic labor — with all the attendant consequences of rising student indebtedness, the proliferation of work floor bullying, and deepening anxiety, depression and burnout among university staff.

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Yesterday in Pre-Calculus class we were discussing combinatorics and came across a series of questions on how many ways can a best 2 out of 3 sets tennis match be won and how many ways can a best 3 out of 5 tennis match be won.

For a 2 out of 3 scenario there are six possibilities (AA, ABA, BAA, BB, BAB, ABB) and for the 3 out of 5 scenario there are 20 possibilities (AAA, BAAA, ABAA, AABA, BBAAA, BABAA, BAABA, ABBAA, ABABA, AABBA and ten more in which B wins).

The question then came up of how to generalize these results.  The way I approached this was to consider how many ways one of the players could win and then double the answer to include the scenarios in which the other won.

One of the issues in this type of problem is that once the winner has won the required number of games the match ends, so that a best 2 out of 3 match could never end AAB, for instance.  Another issue is that each of the possibilities of winning in a different number of sets should be considered separately.  That is, winning in the minimum number of sets, then winning after having lost one set and so forth.

Taking all of these issues into account for a competition in which the winner must win \frac{n+1}{2} out of n leads to a general formula of 2*\displaystyle \sum_{k=0}^{\frac{n-1}{2}}{_\frac{n+2k-1}{2}}C_{k}.

Or – if the winner must win g out of 2g-1 then the formula would be 2*\displaystyle \sum_{k=0}^{g-1}{_{g+k-1}}C_{k}.

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I read a very interesting article today from the Notices of the American Mathematical Society (AMS) about the intelligent use of technology in mathematics.  This article, titled “The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?” describes the experiences of these three researchers in using the mathematical software packages Mathematica and Maple.  The details of their research are interesting but the important point for students of mathematics is to be aware of the limitations of the technology you use.  A key quote from the article is:

…even more dramatically, his algorithm yielded different outputs given the same inputs.

A more detailed explanation of what was going wrong:

…given the same matrix, the determinant function can give different values!

The authors do give credit to technology as a groundbreaking aid in modern mathematical research, but as is true in other research disciplines, they recommend using multiple sources.  In this case, checking the results of one mathematical software package against another software package to compare the results:

Having made this criticism, let us stress that software systems have proved very useful to research mathematicians.  Some well-known instances are the proof of the four-color problem by Kenneth Appel and Wolfgang Haken and the Kepler conjecture by Thomas Hales….Software bugs should not prevent us from continuing this mutually beneficial relationship in the future.  However, for the time being, when dealing with a problem whose answer cannot be easily verified without a computer, it is highly advisable to perform the computations with at least two computer algebra systems.

And, for students of mathematics, I would add,

– When dealing with a problem whose answer CAN be easily verified without a computer, do so!

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I came across an interesting tidbit of math today in solving a problem from the ever-fruitful, continuously engaging book Mathematics Review Exercises by David P. Smith Jr. and Leslie T. Fagan.

Apparently \sqrt{2-2\cos 108^\circ}=\phi.

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In Defense of Algebra

The American Mathematical Association of Two-Year Colleges (AMATYC) is in the process of putting together a position statement on the Appropriate Use of Intermediate Algebra as a Prerequisite Course that will annul the requirement that students pursuing a four-year degree in disciplines other than the hard sciences take and pass a course in Intermediate Algebra.  While this won’t directly impact students who initially enroll at four-year schools, it will have an enormous impact on math education at the nation’s community and technical colleges.  Some schools may amend their Intermediate Algebra courses without destroying the core topics, but I fear that others will end up going down the same road that many high schools did during the rush to get rid of algebra at the K-12 level during the 1990s.

The “reform math” movement of the 1990s led to a plethora of poorly written NSF-backed “math” textbooks such as Core-Plus and IMP at the high school level and MathLand and Everyday Math at the elementary level.  The repercussions of this misguided effort to reform math education at the K-12 level are still felt today, and in some places the fight continues.  The Seattle School Board was taken to court in 2009-10 over its mathematics textbook selection process and a King County Superior Court judge ruled their selection of the Key Curriculum “Discovery” series to be arbitrary and capricious.  An appeals court in 2011 reversed that decision and the textbook selection was allowed to stand.

The upshot of the use of reform materials at the K-12 level is that developmental math enrollment at the two and four-year colleges and universities has exploded.  The solution – stop teaching algebra there as well.

I respectfully disagree.  Here is the text of an email I’ve sent to AMATYC regarding their position statement:

I am responding to the AMATYC position statement on the Appropriate Use of Intermediate Algebra as a Prerequisite Course.  I have also copied Jerry Kissick, the President of ORMATYC, on this email as I believe that ORMATYC should adopt a position statement on the importance of Beginning and Intermediate Algebra to a Liberal Arts education.

I have been a Mathematics Instructor at Clatsop Community College in Astoria, Oregon for 10 years and taught high school mathematics for 6 years prior to entering the MA program in Mathematics at the University of Maine in the fall of 2000.  It is my professional opinion that Beginning and Intermediate Algebra are appropriate prerequisites for college level mathematics no matter the area of specialization of the student.

Any student expecting to receive a four year degree should have demonstrated an understanding of Beginning and Intermediate Algebra at some point in their academic career.  Students who are enrolled in vocational degree or certificate programs at two-year colleges should have different requirements based on their program of study.

Most four-year colleges and universities require that their students demonstrate an understanding of Beginning and Intermediate Algebra prior to matriculation at the college or university.  This is normally done through the submission of SAT or ACT scores after the completion of high school algebra courses.  At two-year colleges, developmental mathematics courses offer an opportunity for students to demonstrate understanding of Beginning and Intermediate Algebra, after which they may then transfer to a four-year school.

Algebra is one of the foundations of a Liberal Arts education and, together with arithmetic, form the conceptual foundation for all of mathematics.  Algebra is a prerequisite to Statistics.  Statistics is not a separate discipline wholly apart from the rest of mathematics – a true understanding of statistics requires an understanding of Beginning and Intermediate Algebra.

Civil rights activist Bob Moses has spent the past 25 years running the Algebra Project, trying to help minority students achieve a better understanding of algebra because he believes that access to quality algebra instruction is a civil rights issue.  Here is a link to an NPR story on Bob Moses.

The importance of Algebra to a student’s overall education demands that we not abdicate our responsibility to teach the subject, but rather work to ensure that we do a better job teaching the subject to all students.

The reform of California’s K-12 mathematics standards in the 1990s led an explosion of developmental enrollment in the California CC and Cal State systems.  This was not related to the topics in the Beginning and Intermediate Algebra curriculum, but rather to the preparation (or lack thereof) the students received at the K-12 level.  The Complete College America organization has recently been an active proponent of “alternative math pathways.”  I find it telling that the organization is called “Complete College America,” rather than “Educate American Students.”  Their focus appears to be getting students to a diploma without the academic and intellectual underpinnings that a college diploma normally represents.

I do often see students struggling to make sense of the work in their mathematics courses.  However, I also see them mature both emotionally and academically as a result of those struggles.  This is the purpose of education – to help our students to mature intellectually as a result of their engagement with the course material.

Algebra as a discipline has a history that goes several millennia deep in human culture.  The works of Diophantus, Brahmagupta and Al-Khwarizmi are beautiful, ineluctable parts of human heritage, and this is not to mention the more “recent” work of Fibonacci, Cardano, Fermat and Des Cartes.  While their techniques and notation may be archaic, it is the power of their reasoning that shines through across the centuries.

A liberal arts education is not and should not be a form of job training.  Many of my students will not directly use much mathematics in their chosen profession and they know this.  However, they are still interested in being an educated person, which means having the background in a broad range of human knowledge so as to be able to think independently about the world and society in which they live.

Thank you for your consideration of my opinion regarding the AMATYC position statement on the Appropriate Use of Intermediate Algebra as a Prerequisite Course.  I have also attached a series of essays written by individuals who believe as I do that Beginning and Intermediate Algebra are part of the foundation of a Liberal Arts education.


Richard W. Beveridge

Mathematics Instructor

Clatsop Community College

The links for the attachments I mention in the letter are below:

Daniel Willingham at the Washington Post’s Answer Sheet blog

Jessica Lahey at the New York Times’ Motherlode blog

Nicholas Warner at the Huffington Post

Evelyn Lamb at Scientific American’s Observations blog

Jennifer Ouellette at Scientific American’s Cocktail Party Physics blog

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