Guided notes: an interactive method for success in secondary and college mathematics classrooms.
Montis, Kristine K.
Abstract
This paper reports the results of an action research project that
examined the use of interactive guided notes in two sections of freshman
level college algebra. This method unifies lecture, in-class guided
practice, and cooperative learning into the students' note taking.
Student success and satisfaction were dramatically higher in the course
sections using the guided notes. The use of guided notes also made it
possible to include discussion, inquiry, and group problem solving in a
course that is otherwise taught entirely by lecture. The paper also
describes how the author used principles from concept and information
mapping to inform the development of the guided notes.
Introduction
In mathematics classrooms at the secondary and college level there
are institutional norms and policies that hinder the process of changing
to reform-based practices (McDuffie & Graeber, 2003). One of the
most entrenched norms found in these classrooms is the emphasis on
traditional lecture and student note-taking format. This paper reports
the results of an action research project on the use of interactive
guided notes as an alternative to the traditional lecture method. The
paper also reflects on how this process improved student success and
satisfaction in freshman level college mathematics courses and supported
the inclusion of discussion, inquiry, and group problem solving into
courses that are otherwise taught entirely by lecture. In conclusion,
this paper suggests possible ways to encourage the use of guided notes
in mathematics courses at the secondary and post-secondary levels and
discusses the need for ongoing inquiry into the effectiveness of
instructional methods and educational policies as the culture in which
they function continues to change rapidly.
Action research is a form of investigation designed for use by
teachers to solve problems and improve professional practices in their
own classrooms. Action research involves systematic observations and
data collection, which can then be used by the practitioner-researcher
in reflection, decision making and the development of more effective
classroom strategies (Parsons & Brown, 2002).
The problem addressed by this action research project was the high
failure rate in freshman level mathematics classes at a small state
university campus in the Midwest. One project had already created a
mathematics learning center with developmental courses and required labs
for incoming freshmen placed in the program using the placement test
developed for this purpose by the Minnesota State Colleges and
Universities (MinnSCU) Center for Teaching and Learning. While
participation in the math learning center by these students
significantly improved their subsequent success rate in college algebra,
there were still 20-30% of the students who were unable to successfully
complete college algebra on their first try.
The impetus for this action research project came from reading the
observations of others studying typical mathematics lessons in Japan and
Germany as well as personal observations of how student note taking
actually interfered with student interaction and learning in the
classroom. Trelfa (1998) notes that in all levels of Japanese schools mathematics is normally taught, not directly from the textbook, but from
"printouts" that the instructor makes for each class. The
printout, or worksheet, contains the lesson objectives and problems
related to each day's lesson. These are typically clear and well
organized in order to help students follow the lecture, study and
review. They are not typically graded by the teachers but rather kept by
the students for reference and review purposes.
Additionally, the following four classroom observations contributed
to my interest in the development of guided notes of freshman college
mathematics courses. First, many students are often unable to write
coherent notes while at the same time listening to and thinking about
what the instructor is saying. It then becomes a choice for them between
listening and taking notes. So they either come away from the lecture
with notes they do not understand or with 50 minutes or longer blocks of
lecture they have "heard" but have little in the way of
coherent notes to which they can refer.
Second, the notes student do take are often incomplete,
error-ridden, and/or illegible to the point of being useless to them
because of the speed and manner in which the material is being
"delivered". Third, students are increasingly unable or
unwilling to sit passively for 50-75 minutes of lecture. Attendance
typically drops sharply after the first two weeks of the semester as
students develop the perception that the lectures are useless to them
because they are treated as passive receptacles rather than active
participants during class time. This student preference for interactive
modes of engagement can only be expected to increase as video and
computer interactions become the students' norm for learning and
entertainment.
Fourth, even when transparencies are prepared ahead and class time
for copying them is allotted, the "scribe" process of copying
down definitions, theorems, examples, and proofs takes a
disproportionate amount of class time. This time could better be spent
discussing and developing ideas and concepts and actively applying
concepts to selected problem situations.
Therefore this action research project was designed to investigate
the development and results of using "guided notes" which were
teacher-made and tailored specifically to the objectives of the lesson.
The goals of the project were to reduce wasteful "scribe"
time, assist students in keeping accurate and well-organized notes,
engage students interactively during the class period, and improve
student performance in and satisfaction with the course.
Guided Notes and Related Instructional Methods
Guided notes and their historical development are succinctly described by Sweeney et al (2002) in their review of the literature. The
quote below is from that review and includes Sweeney's references
so that the historical development of guided notes can be traced.
Guided notes are a guide or skeleton outline that assists the
Student in identifying and recording the main points of a lecture or
reading (Lazarus, 1988, 1993, 1996; Palmatier, 1973). Guided notes
provide the student with a standard set of cues, prompts, and space to
write important concepts, facts, key points, and/or relationships that
pertain to the instruction (Courson, 1989; Heward, 1996). In addition,
guided notes are a systematic and structured system that will provide
each student with a standardized set of notes for future study. Guided
notes can be viewed as a compensatory instructional strategy that can
be easily adapted to the needs of the individual student, while
promoting active participation and responding in the classroom
(Lazarus, 1993).
The development of the guided notes for this project was informed
by research in information science including concept and information
mapping. Concept mapping is currently being researched both as a
presentation and as an assessment methodology (Baroody, & Bartels,
2000; Kinchin, 2000, 2001; Kinchin, Hay, & Adams, 2000; Romance,
& Vitale, 1999; Zwaneveld, 2000). Information mapping is a
methodology with a 30-year history of use in business and industry for
analyzing, organizing and presenting information so that writers can
communicate a wide variety of complex information in a simple and
effective way (Information Mapping Inc., 1999). Table 1 describes how
the organization of the guided notes developed for this project utilized
the principles described by these researchers.
Description of Project
In 2002 a small grant entitled "Interactive Instructional
Design for MA 127 College Algebra" was funded through Minnesota
State College and Universities (MinnSCU) Center for Teaching and
Learning. The grant supported the creation of a pilot set of guided
notes for the college algebra course that went far beyond a simple copy
of the instructors notes or course outline.
The guided notes were designed as an instructional method of
facilitate student learning by making class time itself less tedious by
providing some of the narrative notes already written out. More
importantly, the rest of the guided notes were designed to make class
time useful to the students in ways that the students could immediately
recognize as beneficial to them and in which they could readily
participate. By observing student reactions and performance when using
the guided notes and by interviewing the students themselves, the
following five strategies were identified as particularly dynamic and
effective in achieving these goals.
The first and overall strategy involves engaging students in an
interactive process throughout the class period by having them thinking
and talking about the concepts rather than just copying them down.
Figure 1 shows an application of this technique to learning to apply the
order of operations correctly. Figure 1 also employs concept and
information mapping techniques to intentionally create visual cues to
make the symbolic parts of the notes easy to decode and comprehend. For
instance, the way the example is aligned spatially by writing each
transformation on a new line and using color (or different thickness of
line in this case) to show where the transformations are taking place
emphasizes the train of logic that connects the various steps.
A second strategy that is basic to the guided notes used in this
study involves keeping students actively engaged with the material
during the lesson by building in short "lab times" where
students are actually applying concepts and skills for themselves and
receiving immediate feedback from other students and/or from the
instructor. The two problems at the bottom of Figure 1 are an example of
such "a lab time." In class discussion following completion of
the lab problems, students are encouraged to ask questions and challenge
statements (written or oral) that made no sense to them.
For instance, when going over the order of operations, students
generally believe they understand how to apply it, but inevitably there
are a number of students who will add the 8 and the 2 at the beginning
of the first problem in Figure 1. Having students do the problem in
class, on their own, and then compare their work with that of other
students automatically moves them into discussing "what does this
really mean" as they try to determine who has the correct answer.
As they determine how their work differs, the students also have the
opportunity to find their own errors and make the cognitive adjustments
necessary to prevent making the same errors again. As we check the
problems later in class, students are expected to suggest corrections
and alternative methods. The idea that there are reasons why certain
procedures are correct and others are not is emphasized by expecting the
students to know and be able to verbalize those reasons during these lab
time discussions.
[FIGURE 1 OMITTED]
The third strategy is cluing students into important points being
made during discussions by instructor-modeled marginal notes added on
the overhead transparency during class. The note about "the
opposite of '8 squared"' in Figure 1 is an example of
this. These extra marginal notes can be used to assist students in
developing skill in prompting themselves through lengthy sequential
processes. One way to do this is for the instructor to do "think
aloud" commentary while working examples in class. The content of
the "think-aloud" does not show up in the guided notes except
as notations the students are encouraged to write in beside the steps.
It is necessary to keep such additional notes short so that students can
easily include them.
A fourth strategy basic to creating effective guided notes is
emphasizing and connecting the mathematical reasons for the steps of the
procedures rather than simply stating procedural rules. Information is
presented in ways designed to encourage mathematical thinking about how
and why the rules "work" rather than simply requiring the
memorization of rules without concurrent development of understanding.
Figure 2 demonstrates how guided notes can help students understand how
the exponent rules relate to material they already know. The way the
guided notes supply part of the material already on the page simplifies
the process of recording those ideas and at the same time directs
students' attention to the pertinent portions of the symbolic
representation.
Figure 3 shows an example of guided notes using this technique to
help students differentiate among the exponent rules. Students need
these differences stated explicitly and then need the experience of
differentiating among various situations involving exponents and
connecting them with the correct exponent rule. Giving such rules short
names that are descriptive of the differences among them helps this
process. Note that the lab problems use the rules in combinations so
that the students must differentiate among the rules each step of the
way. This is critical to helping the students develop skill in
differentiating among situations and applying the rules appropriately.
Having students state the name of the rules this way helps students
focus on the features that are pertinent to deciding which rule to use.
[FIGURE 2 OMITTED]
One final strategy for creating effective guided notes is providing
"scaffolding" within the guided notes to help students make
connections and summarize the lesson or unit material. Students often
need assistance making connections among what they perceive as
completely unrelated ideas discussed during the course of the lesson. A
concept map at the end of a section that pulls the various
representations and ideas together and visually shows how they are
related ideas is one way to provide such scaffolding. Figure 4 shows a
"skeleton" concept map that students might be asked to fill in
with examples of each of the different representations in order to help
the students make the connections among the multiple representations of
a function. This section of the notes could be done in class or assigned as part of the homework.
[FIGURE 3 OMITTED]
Action Research Method
The project produced a set of guided notes totaling 132 pages. This
was a Microsoft Word Document totaling 3.68 MB. In the Fall Semester of
2002, there were nine regular sections of MA 127 College Algebra. Two
sections were taught with the guided notes and seven sections were
taught without guided notes. There was no attempt to make the situation
a formal experiment. Enrollment was by student choice, not random
assignment. Both sections that used the guided notes were taught by the
author. The other sections were taught either by another professor or by
experienced adjuncts.
[FIGURE 4 OMITTED]
All sections were required by departmental policy to cover the same
sections of the departmentally chosen textbook, College Algebra by
Stewart, Redlin, and Watson. While instructors created and graded their
own exams, the senior mathematics faculty maintained careful supervision
to ensure that all sections covered the required material and that the
level of expectation on the final exams was comparable among sections.
Assessment of the project effectiveness was done using 1) formal
student course evaluations, 2) student comments on both midterm evaluation and the formal course evaluations, 3) informal student
interviews, 4) comparison of attrition rates in sections using the
guided notes with those not using the guided notes, and 5) comparison of
pass/fail rate of sections using the guided notes with those not using
the guided notes.
Results
The guided notes were required in the sections that were using
them. A set of the guided notes cost less than $10.00. There was only
one student out of the 90 who started out in these sections who
complained at the end of the course about having to purchase the guided
notes. Of the seventy-one students who completed the course sections
using guided notes, 52% specifically mentioned the guided notes as
something they "liked best" about the course on
university's official, anonymous course/instructor evaluation.
Keeping in mind that this was not a controlled experiment,
comparisons with performance of students in other sections that semester
showed student attrition in the sections using the guided notes was less
than half the rate of attrition in the sections that did not use the
guided notes. The attrition rate was measured as the percent of the
original enrollment that stopped attending before the final exam. In
this study the attrition rate in sections that did not use the guided
notes was 47%. That is, 47% of the students who began the course had
dropped, withdrawn, or simply stopped participating at the time of the
final exam. In comparison, only 22% of the students who began the course
in sections using the guided notes were lost due to attrition.
The success rate of students in the sections that used the guided
notes was also substantially higher than in the sections that did not
use the guided notes. In sections using the guided notes, 76% of the
students who took the final exam completed the course with a grade of C
or better as compared to only 58% of the students in sections that did
not use the guided notes. Of those who took the final exam in sections
using the guided notes, 90% completed the course with a D or better
compared to 73% in sections not using guided notes. Note that the
"success" figures only include students who actually took the
final exam, and not the students already accounted for as "lost due
to attrition." Again, these statistics are from an action research
project and not a controlled experiment.
The midterm course assessment was part True/False survey of opinion
and then students could write narrative at the end. Of the students who
turned in the assessment 97% marked the statement "The guided notes
have helped me take better notes in this course" as TRUE. Of those
responding, 93% marked the statement "The guided notes have helped
me learn and understand the material covered in this course" as
TRUE.
In the narrative part students were specifically asked what they
liked best and what they liked least about the class. They were also
asked to comment specifically on what was or was not helpful about the
guided notes. Only one student had a negative comment about the guided
notes, saying they were boring. The following are representative of the
numerous positive comments:
* Guided notes--really helped me learn the material. Extra
explanation helped us understand the material more thoroughly.
* Guided notes helped with examples.
* I liked the guided notes. They were easy to follow and
understand.
* Guided notes. They are great because it is written out in a form
that is readable and they are interactive.
* I loved having the guided notes to refer to! It was a great tool
for studying!
* I liked working in groups on our Guided Notes. The guided notes
were a good thing because we worked through actual problems that made
more sense than the book.
* The Guided Notes--they were easy to read and study off of and
also well organized.
In informal interviews with student volunteers, students mentioned
specifically that it was a relief not to have to worry about getting all
the notes written down. Students enjoyed being able to listen and think
about what the instructor was saying and then being able to do some of
the problems themselves in class and check them there. They also
appreciated the atmosphere where they were encouraged to ask questions
and make sense out of mathematics being discussed in class.
Other instructors' interest in using guided notes in
mathematics classes at the college and high school level was
demonstrated at an April 25, 2003 presentation at the joint Spring
conference of the Minnesota Council of Teachers of Mathematics/Minnesota
Mathematics Association of Two Year Colleges (MCTM/MinnMATYC) entitled
"Guided Notes: An Interactive Method for Success in Freshman
Mathematics". The session was well attended. Participants followed
up and requested electronic copies of the College Algebra Guided Notes
to work at their own institutions.
As other instructors and professors see guided notes used in
classrooms and hear colleagues and students talk about how much they add
to the effectiveness and efficiency of the classes, many will be willing
to try using guided notes in some form. This is particularly true if the
guided notes could somehow be created for them, instead of all
instructors having to create guided notes for themselves. For instance
there are now three other math instructors in the math department who
are using some form of guided notes in their classes. Two of them are
modifying guided notes that I have created. The third is experimenting
with creating her own for specific lessons she feels can be improved by
using guided notes.
Discussion, Conclusion, and Implications for Future Study
As indicated at the beginning of this paper there are entrenched
institutional expectations at the secondary and college level that often
hinder the implementation of instructional methodologies other than
traditional lecture. As expected, there were several sources of
resistance to deviating from the traditional lecture mode. The main
argument against the use of guided notes came from instructors who had
not seen the guided notes used in a classroom. Their initial negative
reaction seemed to be due to the notion that guided notes are just
copies of the instructor's notes handed out to the students so
students do not have to take notes at all. Other instructors believed
current methods were sufficiently effective and that part of a college
education should be learning how to dig out the information from lecture
and standard texts. Some faculty feared that students would like the
guided notes so well that they would come to expect all faculty to
provide them and that the time necessary to create guided notes exceeded
what was reasonably expected of faculty in terms of preparation time.
It is true that the initial development process is extremely
time-intensive and requires a synthesis of knowledge about mathematics
as well as information science and learning theory. Creation of the set
of guided notes developed for this project required more than 200 hours.
An initial strategy to encourage instructors to try guided notes
may be to encourage them to try some of the strategies on individual
lessons that they want to improve, If the strategy proves useful, that
will provide motivation to create more lessons using guided notes. The
only problem with this approach is that making copies is expensive and
handing them out is time consuming. But once done, they can be made into
a course packet that can be made available for students in subsequent
semesters.
Another way to encourage other instructors to try a guided
methodology might be for a mathematics faculty to work together to
create guided notes tailored to the needs of a specific course and work
cooperatively on the process of continuous revision. Commitment of a
mathematics faculty to working cooperatively on such a process could
become a professional development activity designed to improve the
quality of student outcomes in mathematics courses at the institution.
Guided notes can be used to significantly improve student outcomes
in mathematics classes at the secondary and college level. However it is
important not to use guided notes to enable or perpetuate the emerging
student attitude that "just getting by is good enough". Ease
of learning should not be confused with insubstantial learning. There is
a need to look carefully at how guided notes may or may not enhance the
student's ability remember and apply what is learned, to become a
self-motivated and to become an independent, self-disciplined learner.
Guided notes are not a magic cure for all the problems that we
face. Even with guided notes, a lecture section with 300 students will
not be as effective as one using guided notes and also small enough that
the instructor can monitor the quality of the student work on a daily
basis and interact with students as individuals. Likewise, guided notes
designed only to provide a coherent record of the lecture will not be as
effective as guided notes designed as a way to integrate interactive
components of instruction such as instructor-modeled
"think-aloud", in-class guided practice labs applying the
material being presented, opportunities for problem solving, cooperative
learning, and concept mapping into the regular classroom routine while
maintaining a coherent record of the entire learning experience. In
conclusion, using guided notes is a way to blend these components of
instructional design into a traditional lecture style environment
thereby providing an integrated classroom experience that meets the
needs of students who are increasingly expectant of being active in
their learning environment.
Author Note
Kristine K. Montis is currently an Associate Professor in the
Department of Mathematics at Minnesota State University Moorhead.
The guided notes discussed in this article were created with
support from a grant from the Minnesota State College and Universities
(MinnSCU) Center for Teaching and Learning for a project entitled
"Interactive Instructional Design for MA 127 College Algebra.
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Kristine K. Montis
Minnesota State University Moorhead
Table 1: Information Science Principles Applied to Development of Guided
Notes
Information Science Principle Way Incorporated into Guided Notes
Diffusely organized concepts prevent Guided notes regularly make
learners from fully understanding explicit the connections among
the material (Romance and Vitale, concepts being explored by use
1999). of brief summary narratives,
color coding and use of
analogous positioning within
series of mathematical
transformations. Concept maps
are also used for this purpose.
In order to move beyond memorization Guided notes should provide the
students must learn to structure students with examples of how to
the mathematical knowledge and make note of these underlying
skills with which they are working mathematical structures and
(Zwaneveld, 2000). organizing principles and
provide them with opportunities
to recognize and record such
structures and organizing
principles on their own.
In information maps the arrangement Careful attention is paid to
of the words and illustrations on matching how information is
the page provide spatial analogues presented on the written page to
to the interconnections and the actual structure and meaning
relationships within the of ideas being developed. For
information (Information Mapping, instance use separate lines for
Inc., 1999). each step of an algebraic
transformation
