ERIC EJ942879: Questions for Practice: Reflecting on Developmental Mathematics Using 19th-Century Voices

2010·0.22 MB·English

by  ERIC

Questions for Practice: Reflecting on
Developmental Mathematics Using 19th-
Century Voices
By Marcus E. Jorgensen
ABSTRACT: In this article the author has used make a connection with history and realize that
19th-century arithmetic and algebra textbooks a basic understanding of mathematics is some-
as a way to reflect on current practices in devel- thing that, for years, has been considered to be
opmental mathematics education. Five areas important to what an educated person should
of special interest were found: motivation, rel- know. The contexts of the word problems are
evance, depth, pedagogy, and textbooks. Philo- sometimes humorous to modern readers, but
sophic and practical statements from vintage the problems are remarkably similar to those
textbook authors remind educators of a num- that students do today. For example, instead of
ber of questions and issues within each of those solving for how many hours it will take for two
areas of interest. In some respects, little has planes to meet, students in the 1800s calculated
changed over the years and many issues remain the number of days it would take for two people
unresolved or little progress has been made. walking to meet or the number of hours for a pi-
“An event or situation One hundred years from now will things be the rate ship to overtake a schooner. But, essentially,
same, or is it time for a change, a rethinking? the problems are the same.
beyond the individual’s
typical experience must Many authors of 19th-century arithmetic and Areas of Special Interest
algebra textbooks have made profound state-
occur if the reflective process In reviewing the philosophical statements from
ments that have remarkable relevance for math-
the introductions of several textbooks, I have
is to be triggered.” ematics instruction today. Listening to their
found five areas of special interest for today’s
voices has made me question much of what is
developmental mathematics educators: motiva-
currently done in the teaching of developmental
tion, relevance, depth, pedagogy, and textbooks.
mathematics. In this article I review the writings
These are important educational concepts in
of some historic authors, as found in their book
any academic discipline but they have special
introductions, as a way to reflect on current phi-
significance to developmental math instruc-
losophies and methodologies.
tion. Motivation and relevance are closely re-
Just as a mirror helps one to see their own
lated and must be strongly addressed by the
physical reflection, educational statements from
developmental instructor, especially in teaching
the distant past can be used to help instructors
students who are potentially at risk. Pedagogi-
better see their own educational practice. Con-
cal approaches and selection of textbooks are
cerning the concept of reflection in education,
typically more complicated in developmental
Rogers (2001) analyzed a number of theoreti-
math. Classrooms often have a mix of tradi-
cal approaches and determined, with regard to
tional and nontraditional students who all have
antecedents to reflection, that “most authors
had some level of math instruction earlier in
agreed that an event or situation beyond the
their educational experience. Many students
individual’s typical experience must occur if
have developed attitudes and emotions towards
the reflective process is to be triggered” (p. 42).
math that can inhibit self-confidence and learn-
I propose that the voices of 19th-century text-
ing. Also important to developmental math in-
book authors can provide that reflective trigger
struction is the depth of the students' learning.
by speaking from a different time and a different
Going beyond rote memorization of math con-
context, but with an air of familiarity.
cepts and skills is important especially because
In this article I provide a number of quotes
developmental math is preparation for at least
from the introductions of textbooks from the
one more college-level math course. In addition,
Marcus E. Jorgensen 1800s pertaining to various issues that are rel-
the importance of effective instruction is criti-
Assistant Professor of Developmental evant today. In one respect, relevance is under-
cal, especially given that developmental math
Mathematics standable because, after all, math is math and
courses are must-pass courses for students who
Utah Valley University certainly the basics of algebra have not changed
may have had previous negative experiences in
MS 272, 800 West University Parkway much over the years. In fact, at least once a se-
learning math and are seeking a higher educa-
Orem, UT 84058-5999 mester I have students work problems directly
tion degree. This is contrasted with the fact that
[email protected] from these antiquarian books. I want them to
26 JOURNAL of DEVELOPMENTAL EDUCATION
developmental math courses often top the list ning at Math (Nolting, 2008), is devoted to mo- How is curiosity raised, especially when
of “graveyard” courses at many institutions. So, tivation and math-related emotions. The Ameri- some students are in survival mode, meaning
can these voices from the dust trigger reflection can Mathematical Association of Two-Year Col- that they just want to know what to memorize
on important aspects of practice? I explore each leges’ (AMATYC) Beyond Crossroads has noted so they can get a passing grade in the course?
area of interest and discuss questions and impli- that “the beliefs and attitudes that students bring Do instructors enable that attitude by telling
cations for practice today. with them to the classroom play a major role in students what to memorize, because there is
how they learn mathematics” (Blair, 2006, p. 23). so much content to cover (Yopp & Rehberger,
Motivation Many of my unsuccessful students seem to lack 2009)? A common test-taking technique that
Questions. Almost 200 years ago Thomas the motivation necessary to attend class or to some textbooks and teachers preach is the brain
Dilworth published several editions of an arith- take their homework seriously. Some students dump. Students are told that as soon as they get
metic textbook entitled, The Schoolmaster's As- even acknowledge their lack of effort. One stu- their exam, they should write down the memo-
sistant. He included "An essay on the education dent, on the last page of her final exam, apolo- rized formulas at the top or on the back of the
of youth, humbly offered to the consideration of gized for her lack of effort in recognition that exam. What then is the expectation of what they
parents" as a preface. In the 1816 edition of the she would not pass the course. To her credit, she learn for the long term? Is rigor the amount of
text, he listed a number of suggestions to assist took responsibility, but I think her story is one temporarily memorized algebraic algorithms or
parents in the education of their children. His shared by many. should it be something different? Do instructors
very first point is: "A constant attendance at Some students, on the other hand, have take the time to raise curiosity or just lecture to
school is one main axis whereon the great wheel shown strong motivation, particularly nontra- the students as the most efficient way to cover
of education turns" (p. iv). His next point is: ditional students for whom education is now of the material?
But whatever task he [the teacher] imposes more importance. In talking with a student re- More progress is needed in this area with
on his pupils, to be done at home [emphasis cently about math test anxiety she commented, research specifically targeting developmental
added], they should be careful to have it per- math students (e.g., Howard, 2008). Given these
formed in the best manner, in order to keep students' prior history with math, research with
Some students even
them out of idleness: “For vacant hours move K-12 students or even college-level students may
more heavily, and drag rust and filth along acknowledge their lack of not be generalizable to the developmental math
with them; and it is full employment, and a population.
effort.
close application to business, that is the only Will future educators be able to cite improve-
barrier to keep out the Enemy, and save the ment, or will the same issues related to motiva-
future man.” [He notes that the quote is from tion exist 100 years from now? Bonneycastle, as
Watt’s essay but gives no further information noted in the previous quote, has said that in-
"This is why I dropped out of college 20 years
on the reference.] (p. iv) structors should "point out to them a valuable
ago." The difference was that, years later, she was
acquisition, and the means of obtaining it" (p.
Translation: Go to class and do your home- now motivated to do something about it and,
iv). How well is the purpose for math require-
work. I find it interesting that 200 years later I by the way, successfully completed the course.
ments in higher education articulated, especially
give that same advice to my students, although Howard (2008) conducted an interesting quali-
when it is a general education requirement? Ap-
not as eloquently as Dilworth. I tell my students tative study of developmental mathematics stu-
parently, previous generations have also asked
that the two most important things they can dents who eventually became very successful
the perennial question, "Why do we have to take
do to be successful in the course are to come to but who previously had not been. Her study
math?”
class and to do their homework in a meaning- supports the idea of the importance of affective
ful way. In one sense it is somewhat comforting factors in learning math.
Relevance
to know that today’s instructors are not alone in Why is it that there is continued trouble over
Questions.
dealing with basic motivational issues of keep- the years in motivating students to study math?
ing students engaged. However, it is a little so- How much control do educators really have A parent often inquires, “Why should my son
bering to realize that in over 200 years the same over motivation and other affective factors? Ad- study mathematics? I do not expect him to be
motivational challenges still exist. The following ditional research in this area is needed, particu- a surveyor, an engineer, or an astronomer.”
quote is from over 100 years ago yet it still ap- larly for developmental math students. Levine- Yet, the parent is very desirous that his son
plies today to far too many students: “Algebra Brown, Bonham, Saxon, and Boylan (2008) should be able to reason correctly, and to ex-
has not always proved to be an interesting sub- review several instruments that can be used to ercise, in all his relations in life, the energies of
ject to the younger classes in our secondary or assess affective factors. a cultivated and disciplined mind. That is, in-
lower schools; indeed, in very many instances it Nearly 200 years ago, John Bonneycastle deed, of more value that the mere attainment
has been greatly disliked by the students in such made this statement relating motivation to rel- of any branch of knowledge. (Ray, 1848, p. v)
institutions” (Milne, 1894, p. 3). Is this human evance of the content:
For those students not majoring in mathemat-
nature, or could it be that a solution that makes To raise the curiosity, and to awaken the list-
ics, science, or engineering, why do they take
learning math meaningful and motivating still less and dormant powers of younger minds,
math? What is the relevance? All math teachers
evades mathematics educators? we have only to point out to them a valuable
have to answer that question, and I suspect that
Implications. For most of my students who acquisition, and the means of obtaining it;
the answer usually has to do with using math ev-
are not successful in a developmental math class, the active principles are immediately put into
ery day and that, for those concepts that do not
the reasons may be varied, but I cannot help but motion, and the certainty of the conquest is
have direct application to someone’s everyday
think that motivation and other affective factors ensured from a determination to conquer.
life, learning math helps develop general prob-
play a major role. Much of Nolting’s book, Win- (1825, p. iv)
lem-solving and critical-thinking skills. Dudley
VOLUME 34, ISSUE 1 • Fall 2010 27
(2010), a retired university mathematician, makes thinking skills. Textbooks contain some prob- with those “who can apply a score of general
a strong point that the vast majority of Americans lem-solving guidelines but little else to develop principles to millions of particulars” (p. 4).
will never use higher math skills, like algebra, in critical thinking. Several instruments for mea- Implications. What is the underlying phi-
their lives. He states that the more important rea- suring critical-thinking and reasoning skills are losophy of math education today in terms of
son for learning subjects like algebra is to increase reviewed by Levine-Brown, Bonham, Saxon, depth of learning? Specific math skills deterio-
reasoning skills. Here are two other quotes from and Boylan (2008). There is also some recent rate with time if not used. This is reflected in
the first half of the 19th century that concur thru evidence that students can be taught critical the time limits that many institutions place on
with Dudley’s sentiment: thinking skills through a math course (Amit, course prerequisites. For example, at some insti-
But, of all the sciences which serve to call 2010; Sanz de Acedo Lizarraga, Sanz de Acedo tutions, if a student took beginning algebra, but
forth this spirit of enterprise and inquiry, Baquedano, Goicoa Mangado, & Cardelle-Ela- it was over 2 years ago, it will not count as fulfill-
there are none more eminently useful than war, 2009; Benander & Lightner, 2005). Should ing the prerequisite for the intermediate algebra
Mathematics. By an early attachment to there be more emphasis on teaching critical course. The student must retake the placement
these elegant and sublime studies, we acquire thinking? Do faculty need training in the teach- test or retake the course. This type of rule is an
a habit of reasoning, and an elevation of ing of critical thinking? What is the connection acknowledgement of the fact that if students do
thought, which fixes the mind, and prepares between learning math and becoming as Day not use it, they will lose it. This begs the question
it for every other pursuit…. it is, likewise, (1841) would call “sound reasoners?” then about what expected learning outcomes
equally estimable for its practical utility. will endure for students, especially for students
(Bonneycastle, 1825, pp. iv-v) Depth in majors where math is primarily a general edu-
Questions.  cation requirement. Is content the master over
If the design of studying the mathematics
depth? Is the content a mile long and an inch
were merely to obtain such a knowledge of An attempt has been made to render the
deep? Are students memorizing procedures just
the practical parts, as is required for busi- science [mathematics] as easily attainable
to get by? Is more focus needed on outcomes re-
ness; it might be sufficient to commit to
lated to mathematical understanding and criti-
memory some of the principle rules, and to
Some students even cal thinking, or even attitudinal outcomes?
make the operations familiar, by attending to
The quandary is content versus depth. With
the examples.… But a higher object is pro- acknowledge their lack of
more content to cover, teachers often resort to
posed, in the case of those who are acquiring
effort. lecture and the students resort to memorization.
a liberal education. The main design should
It has been suggested that content be reduced so
be to call into exercise, to discipline, and to
that selected concepts could be covered in more
invigorate the powers of the mind. It is the
depth (Jorgensen, 2005). A common response
logic of the mathematics which constitutes as possible, without prejudice to the main
was that this approach would just “water down
their principal value, as a part of a course of result; not to save the learner the trouble of
the curriculum.” However, the suggestion that
collegiate instruction. The time and atten- thinking and reasoning, but to teach him
content be reduced was to actually increase the
tion devoted to them, is for the purpose of [sic] to think and reason; not merely to sup-
rigor with depth rather than providing surface
forming sound reasoners, rather than expert ply a series of simple exercises, but to insure
coverage of broad content. I suggest that rigor
mathematicians. (Day, 1841, pp. iii-iv) a good knowledge of the subject. (Sherwin,
can be thought of as a function of both depth
1846, p. vi)
Implications. Are students better problem- and content. Imagine a graph with the verti-
solvers as a result of taking math courses? Are What should be the “main result” of teaching cal y axis as depth and the horizontal x axis as
they better critical thinkers? Some would dis- mathematics? What constitutes “a good knowl- content. Now picture a plot of a developmen-
agree (Lemire, 2002) and past research on trans- edge of the subject?” Sherwin suggests a happy tal course’s content matched up with the depth
fer of skills to other domains has been inconclu- medium between making the material “as easily at which each content element is taught. From
sive (Atherton, 2007). A search for experimental attainable as possible” but still allowing the stu- my perspective, the area under the curve would
studies conducted during the last 10 years on the dents to think and reason while learning math. represent the rigor. Given this view of rigor, less
transfer of reasoning skills with a focus on alge- The extent to which students think and reason content does not mean diminished rigor if the
bra and higher level math at the secondary and is what I refer to as depth. Bloom’s taxonomy of learning is deeper.
postsecondary levels has produced few reports; cognitive domain objectives is a way to think Maybe there is some room for discussion
this is surprising because this topic is so criti- about depth of learning with deepness increas- about appropriate amounts of content. Johnson’s
cal to why math is required of so many students. ing as one progresses in order: knowledge, com- (2007) case study of a developmental mathemat-
But, if it is assumed that taking a mathematics prehension, application, analysis, synthesis, and ics program has found that “the vast majority of
course increases students’ reasoning abilities in evaluation (Bloom, 1977). students in the Intermediate Algebra course will
general, and it is further assumed that critical Joseph Ray, M.D. (1848) believed that stu- use almost none of what they actually learn in
thinking is a desired outcome of developmen- dents should be able to do more than just mem- that course in their college level work in mathe-
tal and general education mathematics courses, orize steps to doing problems. He stated, over matics” (p. 287). Alignment of curricula is worth
then how should that inform pedagogy? I be- 150 years ago, “the pupil should not be taught looking at, and maybe there should be different
lieve that I am a good critical thinker; however, merely to perform a certain routine of exercises pathways to the different college-level math
I have never been educated in what constitutes mechanically, but to understand the why and courses; one size may not fit all. Johnson further
critical thinking, and I suspect that this is the the wherefore of every step” (p. v). Bailey (1892) states, “It is not sufficient to take the content of
case for most mathematics instructors. referenced depth of learning with a different developmental mathematics courses ‘for grant-
In many cases math is taught with the hope perspective. He said that “he who relies upon
that somehow students pick up some critical thousands of special rules” is nothing compared continued on page 30
28 JOURNAL of DEVELOPMENTAL EDUCATION
continued from page 28 ing, especially for deeper understanding of con- opmental education practice. They found that
cepts. However, there may be a fine line between “the most effective developmental teaching
ed’; it is essential to consider the actual needs of useful struggling and frustration. And, since strategies in the literature are characterized by
college mathematics students” (p. 288). Depth of some developmental math students already have dynamic student-and-student and teacher-and-
learning is as important a need as the content. a history of frustration with math, then finding student interactions as well as by efforts that
AMATYC’s Crossroads document supports where that line is located becomes critical for an aim to awaken students’ innate desire to acquire
efforts to improve depth of learning: “One of inductive approach. knowledge” and that “most of these approaches
the most widely accepted ideas within the math- As an administrator at a community college, fall within the category of active or student-cen-
ematics community is that students should un- I started receiving numerous complaints about tered learning, which has been demonstrated
derstand mathematics as opposed to thought- a math instructor. This was unusual for this in- to be effective with adult, nontraditional, and
lessly grinding out answers.” But, that statement structor so I asked him what was going on. He developmental students” (p. 7). They note, how-
is followed by: “But achieving this goal has been explained that he was taking a different approach ever, that “most strategies have not been empiri-
like searching for the Holy Grail” (Cohen, 1995, and letting the students work in groups to figure cally tested to determine their efficacy with large
Chapter 2). out problems on their own so that their learning numbers of students over long periods of time”
could be deeper than a traditional approach, in but “informal studies and substantial anecdotal
Pedagogy his opinion. The students, however, complained evidence suggest that they can help academi-
Questions. that “he is not teaching us.” They expected lec- cally unprepared students achieve” (p. 7). High-
The best mode, therefore, seems to be, to give tures on his part and memorization on their part. lighting the need for more pedagogical research,
examples so simple as to require little or no Changes to how teaching is approached may re- an extensive search of the developmental educa-
explanation, and the learner reason for him- quire that students are informed about and pre- tion literature by the U.S. Department of Educa-
self [sic], taking care to make them more dif- pared to change their approach to learning. tion’s Office of Vocational and Adult Education
ficult as he proceeds. This method, besides (2005) found only a limited number of studies
giving the learner confidence, by making on adult developmental mathematics, none of
him rely on his own powers, is much more There may be a fine line which were randomized controlled trial experi-
interesting to him, because he seems to him- ments (p. 3). The field needs additional research
between useful struggling
self to be constantly making new discoveries. to evaluate such active approaches with develop-
Indeed, an apt scholar will frequently make and frustration. mental math students.
original explanations much more simple
Textbooks
than would have been given by the author.
This mode has also the advantage of exercis- Questions. When I tell my students that I
I question the extent to which students are
ing the learner in reasoning, instead of mak- collect algebra textbooks, sometimes a student
required to memorize for traditional, closed-
ing him a listener, while the author reasons will joke by saying, “Do you want mine?” Then
book exams. Some memorization is critical,
before him. (Colburn, 1834, p. 3) I explain that my collection is from the 19th
such as the multiplication tables, laws of expo-
century. When I show them some of the books
Colburn advocates for an inductive approach nents, order of operations, or any mathematical
their first impression is always the convenient
that allows students to discover general prin- fact or concept that students should be able to
size. Most of these books are 5 inches by 8 inches
ciples. I often find it tempting to be the sage on do by rote. These are typically frequently ap-
and weigh just ounces. One student recently said
the stage and make the students listeners in- plied concepts or operations that are essential
that he might actually bring his book to class if
stead of exercising the learner in reasoning. But to having a fundamental number sense. How-
it was that size. Today’s books are enormous
when I do that am I depriving them of the joy of ever, what is the point of memorization beyond
by comparison. I looked at the size of four re-
discovery? Teachers are naturally interested in these types of skills and concepts? Although this
cent editions of developmental mathematics
their discipline of study and may want to tell it may be heretical to some, consider why students
intermediate algebra textbooks and found that
all. Maybe students need to be given the chance memorize the quadratic formula. If they ever
they averaged a staggering 882 pages with ship-
to have some of the same discovery experiences actually had to use it outside of math class they
ping weights on Amazon.com that averaged 4.2
that the teachers have had when they learned would look it up or use a calculator. Isn’t it more
pounds. Are these large, expensive textbooks
math as students. important that they know what it is and how it is
really necessary? Does the additional bulk add
Implications. The current primary peda- used? As noted, instructors promote temporary
much in effectiveness?
gogy, judged by developmental mathematics memorization by teaching test-taking skills such
I reviewed the introductions from five new
textbooks, seems to be as follows: (a) present the as the brain dump. I propose that deeper learn-
intermediate algebra textbooks from three dif-
principles, rules, and/or steps used in solving ing on appropriate concepts could be assessed
ferent publishers to identify any information
some type of problem; (b) show some examples; if students were allowed to use their books or
on pedagogical philosophy or any special fea-
(c) have the students do some problems on their notes during testing; the tests could have more
tures that they mention. They were all relatively
own in class; and (d) assign practice problems difficult questions. Instead of memorizing a lot
similar and noted their content sequence, clar-
for homework. This more deductive approach of information, students could spend their time
ity, page design, study helps, and supplemental
has some value, but some of the voices from the learning deeper concepts and use their book,
information (including accompanying software
past are reminders to think about other options. notes, and other information sources as refer-
and web site programs). Most touted their use
One of the issues with an inductive or discov- ences for solving problems requiring higher
of example exercises. One book stated that “ex-
ery approach to learning is the degree to which level critical thinking.
ercises are often the determining factor in how
students are allowed to struggle. To a certain Schwartz and Jenkins (2007) summarize key
degree, mental struggling can be useful to learn- findings from the literature on effective devel-
continued on page 32
30 JOURNAL of DEVELOPMENTAL EDUCATION
continued from page 30 life applications” (Larson, 2010, p. xii). My expe- seems to be an unstated understanding that
rience is that most application problems are not, what students need is lots of examples, lots of
valuable a textbook is to a student” (Sullivan & in fact, “real-life.” In this particular textbook, the explanations, lots of help, and plenty of home-
Struve, 2010, p. xii). following problem is given as an example of a work problems for drill and practice. However,
Exercises are mentioned as key in the 19th real-life application: no research is cited to support this assertion.
century textbooks as well. E. Bailey (1835) stated The first generation of the iPhoneTM has an Easiest may not always be best. Added features
that one of the leading principles which he ob- approximate volume of 4,968 cubic inches. can become unnecessary and distracting clut-
served in writing his book was “to show the rea- Its width is 0.46 inch and its face has the di- ter. I think students would welcome a straight-
son of every step, without perplexing the learner mensions x inches by x + 2.1 inches. Find the forward, simple presentation with real-life rel-
with abstruse demonstrations” (pp. 4-5). Later dimensions of the face in inches. (p.xii) evance. They might actually bring their books
in the century, Milne (1894), lamenting about to class.
This example is not real life. No one would
textbooks, said, Within a few years many new text books on
ever have to solve that type of problem. The use
Two causes, chiefly, have conspired to produce algebra have appeared in different parts of
of iPhone gives the appearance of being a mod-
this unfortunate condition of affairs [student the country, which is a sure index that some-
ern application to which students can relate.
dislike of algebra]—one, the unattractive and thing is desired–something expected–not
However, the underlying math is not connected
uninteresting method of presenting the sub- yet found. The happy medium between the
to real life. Here is another example, from a dif-
ject; the other, the difficulty of the examples theoretical and practical mathematics, or,
ferent developmental textbook, of a supposedly
and the complexity of the problems presented rather the happy blending of the two, which
real-life application, again featured in the intro-
to the pupils for solution. (p. 3) all seem to desire, is most difficult to at-
duction of the textbook as an example of their
tain; hence many have failed in their efforts
Clear examples at appropriate levels of diffi- “new and improved applications” that “include
to meet the wants of the public. (Robinson,
culty have been goals for years. However, are the real data and topics which are more relevant and
1856, p. iv)
students being spoon fed by providing too much
in the way of examples? Looking at Colburn One hundred and fifty years later and “not
“Explanations rather
(1834) again, he states, “The learner is expected yet found.”
to derive most of his knowledge by solving ex- embarrass than aid the
amples himself; therefore care has been taken Conclusion
learner, because he [sic] is
to make the explanations as few and as brief as
What I appreciate most about the vintage text-
is consistent with giving an idea of what is re- apt to trust too much to
books is that the authors typically express their
quired” (p. 3). His inductive approach calls for
them.” teaching philosophies. This provides a good
the learner to do a little more work at increas-
mirror in which to reflect on current practices.
ingly appropriate levels of difficulty. He even
I have noted five areas of interest in reviewing
says that giving the student too much may be
a number of introductions of 19th-century text-
harmful: “In fact, explanations rather embarrass interesting to today’s student” (Miller, O’Neill, &
books: motivation, relevance, depth, pedagogy,
than aid the learner, because he [sic] is apt to Hyde, 2010, p. xi):
and textbooks. I have identified a number of is-
trust too much to them, and neglect to employ An athlete’s average speed on her bike is 14
sues and questions in each area but few answers.
his own powers” (p. 3). Day (1841) concurs: mph faster than her average speed running.
Although there is a need for continued research
In the colleges in this country, there is gener- She can bike 31.5 mi in the same time that it
in each of these areas, it may be more important
ally put into the hands of a class, a book from takes her to run 10.5 mi. Find her speed run-
to look at the overarching system because these
which they are expected of themselves to ac- ning and her speed biking.
topics are all interrelated. In this approach the
quire the principles of the science to which
Real data I suppose, but not real life in terms unit of analysis would be the system, a system
they are attending: receiving, however, from
of the math involved. built on a strong philosophical foundation. I am
their instructor, any additional assistance
We insist on the application of every princi- encouraged by the recent efforts of the AMA-
which may be found necessary. (p. iii)
ple, for it is the application that gives life and TYC Developmental Mathematics Committee
Are students just searching for example ex- importance to theory. Who would study the to address the system in their New Life Project.
ercises that are exactly like the homework so steam engine if it were a mere philosophical Starting with a new mission statement for devel-
they can get it done and out of the way without toy – if it were not for its utility and mechani- opmental mathematics, the project’s purpose is
striving for deeper learning? The issue of how cal power? We hear much of studying mathe- to develop a “new life vision of developmental
much help is truly helpful to a student demands matics for the improvement of the mind, and mathematics” (AMATYC Developmental Math-
further research. we would not detract in the least from that ematics Committee, n.d.).
Modern books advertise the motivating and object; but it is attained in its highest sense The historical perspective given by these
relevant features of their books. These concerns only when we combine theory and practice. voices from the past gives mathematics educa-
are echoed in older books as well: “One of the (Robinson, 1856, p. vi) tors pause to wonder whether or not the cur-
designs of this book is to create in the minds of riculum and pedagogy are tied to tradition in
Implications. Textbooks are key and they
the pupils a love for the study, which in some which little has changed over the years. Review-
drive curricula and pedagogy. Publishers obvi-
way must be secured before success can be at- ing these vintage books allows instructors to
ously want to develop effective texts, but I won-
tained” (Robinson, 1856, p. iv). I am disappoint- step back and look at the big picture. One hun-
der if the profit-motive drives them to produce
ed, however, in today’s attempts to relate math dred years from now will things be the same,
books that are more attractive and easier for the
to life. One of the newest textbooks describes its
students and easier for the instructors. There
applications as including “a wide variety of real- continued on page 34
32 JOURNAL of DEVELOPMENTAL EDUCATION
continued from page 32 Day, J. (1841). An introduction to algebra, being
the part of a course in mathematics, adapted to
or is it now time for a change, a rethinking, a the method of instruction in American colleges.
resolve to make math relevant and meaningful New Haven, CT: Durrie & Peck.
to students? Dilworth, T. (1816). The schoolmaster’s assistant:
No mathematician thinks of using the Being a compendium of arithmetic, both prac-
clumsy and antiquated processes by which tical and theoretical. Edinburgh, Scotland: J.
we have been accustomed to teach our pu- Ruthven and Sons.
Michigan
pils in algebra…Why not, then, dismiss for- Dudley, U. (2010). What is mathematics for? No-
DevelopMental ever these processes, and let the pupil enter tices of the American Mathematical Society,
eDucation at once upon those elegant and productive 57(5), 608-613.
methods of thinking which he [sic] will ever Howard, L. (2008). Developmental students’ per-
consortiuM
after use? (Olney, 1873, p. iv) ceptions of unsuccessful and successful math-
Presents... ematics learning (Doctoral dissertation, Utah
References
State University). Retrieved from http://
26th Annual Conference
AMATYC Developmental Mathematics Com- digitalcommons.usu.edu/cgi/viewcontent.
What Works? mittee. (n.d.). New life project [Wiki com- cgi?article=1210&context=etd
Best Practices in ment]. Retrieved from http://dm-new-life. Johnson, P. (2007). What are we developing? A
wikispaces.com/ case study of a college mathematics program.
DeveloPmental eDucation
Aizikovitsh, E., & Amit, M. (2010). Evaluating School Science & Mathematics, 107(7), 279-292.
April 14th & 15th, 2011
an infusion approach to the teaching of criti- Jorgensen, M. E. (2005).Quantity versus quality:
Macomb Community College cal thinking skills through mathematics. Pro- Finding time for innovations in the classroom.
Clinton Township, MI cedia—Social and Behavioral Sciences, 2(2), NISOD Innovation Abstracts, XXVII, 1.
3818-3822. doi:10.1016/j.sbspro.2010.03.596 Larson, R. (2010). Intermediate algebra (5th ed.).
call for proposals Atherton, M. (2007, April). A proposed theory of Belmont, CA: Brooks/Cole, Cengage Learn-
the neurological limitations of cognitive trans- ing.
We welcome proposals addressing fer. Paper presented at the annual meeting of Lemire, D. (2002). Math problem solving and
areas of Developmental Education /
the American Educational Research Associa- mental discipline—The myth of transferabil-
Learning Assistance while promoting
tion, Chicago, IL. ity. Journal of College Reading and Learning,
student success. We especially
Bailey, E. (1835). First lessons in algebra, being an 32(2), 229-238.  
welcome proposals from developmental
educators in neighboring states. easy introduction to that science; designed for Levine-Brown, P., Bonham, B. S., Saxon, D. P., &
the use of academies and common schools. Bos- Boylan, H. R. (2008). Affective assessment for
proposal cover sheet to include:
ton, MA: Russell, Shattuck, & Co. developmental students, Part 2. Research in
name, title, institution, address,
Bailey, M. A. (1892). American mental arithmetic. Developmental Education, 22(2), 1-4.
phone number, email address,
name of presentation, 50 to 100 word New York, NY: American Book Company. Miller, J., O’Neill, M., & Hyde, N. (2010). Interme-
abstract of the proposal (2 copies) Benander, R., & Lightner, R. (2005). Promoting diate algebra (2nd ed.). Boston, MA: McGraw
transfer of learning. The Journal of General Hill.
subMit proposals to:
Education, 54, 199–208. Milne, W. J. (1894). Elements of algebra; a course
Dr. Michael Oliver
Blair, R. (Ed.). (2006). Beyond crossroads: Imple- for grammar schools and beginners in public
Schoolcraft College
menting mathematics standards in the first two and private schools. New York, NY: American
18600 Haggerty Road
Livonia, MI 48152 years of college. Memphis, TN: American Math- Book Company.
ematical Association of Two-Year Colleges. Nolting, P. (2008). Winning at math (5th ed.).
DeaDline for proposals Retrieved from http://www.beyondcrossroads. Bradenton, FL: Academic Success Press, Inc.
March 14, 2011 com/doc/PDFs/BCAll.pdf Olney E. (1873). A university algebra. New York,
Bloom, B. (Ed.). (1977). Taxonomy of educational NY: Sheldon & Company.
MDEC Board Members objectives: Cognitive domain. (Handbook 1; Ray, J. (1848). Ray’s algebra, part first: On the ana-
21st printing). New York, NY: Longman, Inc. lytic and inductive methods of instruction: with
Deborah Daiek, President
Bonneycastle, J. (1825). An introduction to algebra numerous practical exercises. Cincinnati, OH:
[email protected]
with notes and observations; designed for the use Winthrop B. Smith & Co.
Linda Spoelman, President Elect
of schools and places of public education (J. Ryan Robinson, H. N. (1856). A theoretical and practi-
[email protected]
Ed./Trans.). New York, NY: Evert Duyckinck. cal treatise on algebra: In which the excellences
Denise Crudup, Secretary Cohen, D. (Ed.). (1995). Crossroads in mathemat- of the demonstrative methods of the French are
[email protected]
ics: Standards for introductory college math- combined with the more practical operations of
Annette Magyar, Tresurer ematics before calculus. Memphis, TN: Ameri- the English; and concise solutions pointed out
[email protected]
can Mathematical Association of Two-Year and particularly inculcated (24th ed.; univer-
Karel Asbury, Membership Colleges. Retrieved from http://www.imacc. sity ed.). Cincinnati, OH: Jacob Ernst.
[email protected] org/standards/ Rogers, R. R. (2001). Reflection in higher educa-
Joe LaMontagne, Membership Colburn, W. (1834). An introduction to algebra tion: A concept analysis. Innovative Higher
[email protected] upon the inductive method of instruction. Bos- Education, 26(1), 37.  
ton, MA: Hilliard Gray & Co. Sanz de Acedo Lizarraga, M. L., Sanz de Acedo
34 JOURNAL of DEVELOPMENTAL EDUCATION
Baquedano, M. T., Goicoa Mangado, T., & For Your Information
Cardelle-Elawar, M. (2009). Enhancement
of thinking skills: Effects of two intervention
methods. Thinking Skills and Creativity, 4(1),
February 22-24, 2011–Association for the Advancement of Computing in Education’s (AACE)
30-43. doi:10.1016/j.tsc.2008.12.001
online conference, “Global Time, Global Conference on Technology, Innovation, Me-
Saxon, D. P., Levine-Brown, P., & Boylan, H. R.
dia & Education.” For more information visit www.aace.org/conf/gtime

(2008). Affective assessment for developmen-
tal students, Part 1. Research in Developmental 23-26, 2011–National Association for Developmental Education’s (NADE) 35th Annual
Education, 22(1), 1-4. Conference, “Capitalizing on Developmental Education,” at the Marriott Wardman
Schwartz, W., & Jenkins, D. (2007). Promising Park Hotel in Washington, DC. For more information see ad, back cover, or visit the
practices for community college developmental website http://nade2011conferencede.com/
education. Retrieved from the Community
March 7, 2011–Society for Information Technology & Teacher Education’s 22nd International
College Resource Center web site: http://ccrc.
Conference at the Sheraton Muxic City Hotel in Nashville, TN. For more information
tc.columbia.edu/ContentByType.asp?t=1&Co
visit www.site.aace.org

ntentItemTypeID=0&PagePos=2 
Sherwin, T. (1846). The common school algebra. 8, 2011–Call for papers, submission deadline for the Developmental Education Summit
Boston, MA: Phillips and Sampson. at Lake Land College, Mattoon, IL, October 25, 2011. For more information see ad, page
Sullivan, M., & Struve, K. R. (2010). Intermediate 18, or contact [email protected] or [email protected]
algebra (2nd ed.; annotated instructor’s ed.).
20-23, 2011–Teaching Academic Survival Skills’ (TASS) 22nd Annual Conference at
Upper Saddle River, NJ: Pearson Education,
Embassy Suites Hotel, Fort Lauderdale, FL. For more information visit the website
Inc.
http://www.tassconference.org

U.S. Department of Education, Office of Voca-
tional and Adult Education. (2005). Strength- 31 to April 2, 2011–On Course 6th Annual National Conference, “Festival for Learn-
ening mathematics skills at the postsecondary er-Centered Educators,” at the Hilton Long Beach & Executive Meeting Center, Long
level: Literature review and analysis. Retrieved Beach, California. For more information visit the website http://www.oncoursework-
from https://www.ed.gov/about/offices/list/ shop.com/conference.htm
ovae/pi/hs/factsh/litreview905.doc
April 3-5, 2011–New York College Learning Skills Association’s (NYCLSA) 34th Annual Sym-
Yopp, D., & Rehberger, R. (2009). A curriculum
posium on Developmental Education, “Brain Based Learning: Taking it to the Top,” at
focus intervention’s effects on prealgebra
the Kaatskill Mountain Club in Hunter Mountain, Hunter, NY. For more information
achievement. Journal of Developmental Edu-
visit the website http://www.nyclsa.ort/annual_conf_current.htm
cation, 33(2), 26-36.
7-8, 2011–Pennsylvania Association of Developmental Educators’ (PADE) 2011 confer-
ence, “The Many Faces of Developmental Education: Working Together for Success,” at
the Willow Valley Resort, Lancaster, PA. For more information visit the website http://
www.pade-pa.org/conference-info.html

PROMOTING EXCELLENCE
14-15, 2011–Michigan Developmental Education Consortium’s (MDEC) 25th Annual
AMONG LEARNING
Conference, “What Works?” at Macomb Community College, Warren, MI. For more
CENTER PROFESSIONALS
information see ad, page 34, or visit the website http://www.mdec.net/MDEC%20Con-
ference/Index.htm

15, 2011–Call for Proposals Deadline; College Reading & Learning Association’s
(CRLA) 44th Annual Conference “Hands across the curriculum” in San Diego, Cali-
fornia. For more information see ad, page 29, or visit www.crla.net

May 29 to June 1, 2011–NISOD’s 32nd Annual International Conference on Teaching &
Leadership Excellence, “Putting Teaching and Leading on the Map,” at the Hilton Aus-
WWW.NCLCA.ORG tin in Austin, TX. For more information visit the website http://www.nisod.org/confer-

ence/
A few benefits of membership:
June 25 to July 22, 2011–Kellogg Institute for the Training & Certification of Developmental
♦Journal Subscription (TLAR)
Educators at Appalachian State University in Boone, North Carolina. For more infor-
♦Discounted registration
mation visit www.ncde.appstate.edu/kellogg
Annual Conference (Fall) and
Institute (Summer) October 25, 2011–Developmental Education Summit at Lake Land College, Mattoon, Illinois.
♦Professional Certification For more information visit ad on page 18 or contact [email protected] or
♦Online Workshops [email protected]
♦Quarterly Electronic Newsletter November 9-12, 2011–College Reading & Learning Association (CRLA) “Hands across the cur-
♦Members Only website resources riculum” 44th Annual Conference in San Diego, California. For more information visit
www.crla.net
We are the association dedicated to
learning center professionals.
VOLUME 34, ISSUE 1 • Fall 2010 35