Saturday, April 12, 2014

How Much Advanced Math Preparation Do Middle and High School Teachers Need?

In a recent paper appearing in The Monthly Notices of the American Mathematical Society (Vol. 61, no. 3, p. 292-95) the question was asked as to how well Mathematics Teacher Education Programs Were “Aligned with Recommendations Made in MET II”?

Here “MET II” refers to the specific document:  The Mathematical Education of Teachers II. The thrust of the article then was to examine the extent to which the current preparation of Middle School math teachers (as well as High School math teachers) not only conformed with MET II recommendations but also the degree of integration between mathematics and pedagogy – and who should have a voice in making decisions about the preparation of mathematics teachers.

The article correctly notes:

“Little research has been done that examines the requirements of mathematics teacher education programs or the effects of these requirements.”

To try to answer the questions, the authors reported the “results of a national survey of secondary mathematics teacher education programs."  The main issue addressed was:

How much do current secondary mathematics teacher education program course requirements align with the recommendations described in MET II?”

In terms of the recommendations, they were categorized under those for: 1) prospective Middle School teachers and 2) prospective High School teachers,  as follows:

1)     Should be required to complete at least twenty four semester hours of mathematics that include at least fifteen semester hours on fundamental ideas of school mathematics appropriate for middle grades teachers.

2)     Should be required to complete the equivalent of an undergraduate major in mathematics that includes three courses with a primary focus on high school mathematics from an advanced viewpoint.

Whether given teaching programs conformed with the preceding or not was based on how respondents selected courses from a given list – for (1) and (2) respectively. The respondents were also to identify the total number of courses and credits for each course type. The final analysis of the paper used the data from 64 programs in respect of category (1) and 78 programs in respect of category (2).

The results obtained were interesting to say the least, including the following:

1) For Middle School Teaching Programs:

i) None of the programs reported using the MET II recommended fifteen semester hours of courses in fundamental ideas of school mathematics designed for middle grades teachers.

ii) The maximum number of required credits reported by any program was twelve semester hours (in four programs) and the average number of required credits of this type was three.

iii) The most commonly required programs were 'Geometry for Teachers' (13 programs), 'Statistics and Probability for Teachers' (4 programs) and 'Algebra for Teachers' (3 programs).

iv) 'Numbers and Operations' courses recommended by MET II were almost completely absent.

v) All 64 programs met the requirement for 9 semester hours of additional, advanced mathematics courses. Most university programs required teachers to take courses selected from statistics, calculus, number theory and discrete mathematics.  

vi) Calculus dominated, required in 63 of 64 programs, with Probability and Statistics next at 58 programs, and discrete mathematics with 45 programs requiring such courses.


For those unfamiliar with discrete mathematics, the following illustrate sample problems:

1. (a) Show that (p ® q)  «  (~q ® ~p)  is a tautology.                                       
    (b) Let x  Î { 2, 3, 4} and y Î {12, 16}.  Let the propositional function

P(x, y) be the statement “x is a factor of y”.  Write the following  propositions using conjunctions and disjunctions and determine the truth value of each.

            (i) "x $y P(x, y)              (ii) $y "x P(x, y)                                          

2.    (a)   Show that log n! = O (n log n).                                                                

(b)     Let f(x) = 2x3 + 3x –1 and g(x) =  log x. 

Find the least integer n such that:
(c)Use mathematical induction to prove that

       (1x2) + (2x3) + (3x4) + … + n(n + 1)  =  n(n +1)(n +2)/3.                     

3.  (a) Prove that if n is an integer  and n3 + 5 is odd, then n is even using:
                        (i) indirect proof         (ii) proof by contradiction.   

Moving on,  let's look at the requirements:

2) For High School Teaching Programs:

i) Of the 78 programs that prepare high school teachers, sixty three or 81 percent, required a three course calculus sequence and the mean number of calculus courses across the programs was 2.8.

ii) Sixty nine programs (or 88 percent) required at least probability and statistics course.

iii) Almost all programs (76) required students to take at least one linear algebra course.

iv) MET II required 18 additional semester hours of advanced math beyond calculus, statistics and linear algebra courses and all the programs satisfied this requirement.

v) The most popular 'more advanced' math program was Geometry (70 programs), with Abstract Algebra next (61 programs) and then Discrete Mathematics (52).

vi) Only eight programs (10 percent) reported meeting the nine semester hours of high school mathematics from an advanced perspective. A typical description of such a course is shown below:

Some Observations:

In terms of the Middle School teacher preparation what amazed me the most is how few 'History of Mathematics' course programs were required. Make no mistake that the history of math gives the broad perspective I believe is needed for teaching at the middle grades (5-8) level. It also exposes the connections between the different math disciplines, i.e. between calculus and analytic geometry.  It would seem to me that Middle School teaching programs would be better served by more history of mathematics courses, than say Discrete Math.

Calculus dominates as it should, but anyone who's ever taught it also knows that the content can vary widely. What textbooks are being used, for example, and how useful are they to those being prepared? How much in the way of application is present?

In the Advanced Courses recommended for High School teaching programs I was somewhat dismayed to see the short shrift given to real analysis and differential equations (29% and 35% of programs adopted, respectively), say compared to geometry which appeared to dominate (90% of programs). WHY does geometry dominate? Looking back at secondary teaching programs from a historical perspective it appears it always has - going back to the 1920s (I have a relative's Geometry text from way back then, designated for teacher ed.)  All the same stuff you might expect is covered, including the theorems of Pappus and Desargues. But, if you're really going to teach Geometry why not use the best textbook of all, David Hilbert's Foundations of Geometry?

I suspect the predominance of Geometry, say in preference to real analysis and differential equations is because: 1) Most high schools still include at least one year of geometry in the curricula, despite the fact in the Caribbean and Singapore, for example, this is regarded as passé, and 2) Teachers- to- be are more comfortable with a course high in visualization content, as opposed to the more abstract real analysis, for example.

Inclusion of geometry at the secondary level is often expected in the same way as Algebra II - as a means to "encourage logical or critical thinking" - by having to manipulate objects in 2 or more dimensions. But I say if one is going to go this route it's far better to just replace it with solid geometry which can be more easily integrated with calculus (especially for AP students) or at least analytic geometry.  I warrant it's more important for a kid to know the equation for the ellipse (or the circle) and how to change them to change the shape and dimensions -i.e.  in terms of semi-major and semi- minor axis,  than to know Desargues theorem.

But old habits die hard, and it's probably going to take a long time before geometry is finally ousted from its preferential pedagogical perch.

As for teachers' exposure to geometry, they'd be better off just going through Euclid's Elements at a detailed level (say analogous to that presented in Stephen Hawking's book, God Created the Integers, Book I: Basic Geometry. Then follow it up by basic exposure to Non-Euclidean Geometry, say as provided in the excellent monograph: Non-Euclidean Geometry by Stefan Kulczycki.  Less time would be consumed and so more time could be devoted to really looking at the underlying math from an advanced perspective. In this sense, if one is later going to be faced with teaching algebra and geometry this would appear to be the better route to take, along with Abstract Algebra and at least 3 semester hours of History of Mathematics.

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