Final Paper

How Do Community College Mathematics Instructors Engage Their Students in the Learning and Mastery of

Mathematics? A Qualitative S tudy

Matthew R. LeachBoston University, School of EducationRS 652 - Qualitative Research Methods

Dr. Bruce Fraser and Steve LantosDecember, 16, 2011

Abstract: Two community college mathematics instructors were interviewed and observed to

determine what methods, if any, they use in their mathematics classrooms to engage their

students in the learning of mathematics. One teacher was a retired mathematics teacher at a

medium-sized suburban high school, in her second year teaching at the community college level.

The other teacher was a mathematics instructor for thirty years at the community college level,

prior to which she served for a little over ten years as an elementary teacher. Each teacher

participated in an initial interview followed by two observations. There was a brief final

conference upon completion of the second observation. In general, it was found that both

instructors used a number of different techniques to engage their students. Through the use of

real world examples, humor, connections to prior knowledge, individual attention towards

students, student board work, and interaction between student, teacher, and subject matter, it was

evident that these instructors' strategies were effective in engaging their pupils.

Preface: I am currently between my second and third year as a doctoral student in Boston

University's Ed.D. program in Curriculum and Teaching with an emphasis on Mathematics

Education. I am also a mathematics teacher myself at a small private school. In addition to

learning more about how other teachers in the field engage their students in learning

mathematics, it is my hope that I may apply my knowledge on engagement to future studies I

plan on undertaking. This is my first experience with qualitative research in the field of

education. I would like to acknowledge Dr. Bruce Fraser, Dr. Yolanda Rolle, and Steve Lantos

for their support, feedback, and encouragement, as I develop my abilities in the field of

qualitative research.

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Introduction: Student engagement in learning mathematics is viewed as an important

component for creating high quality mathematics instruction. States Corey in his summary of

the Intellectual Engagement Principle derived from qualitative observations of Japanese

mathematics lessons: "High-quality mathematics instruction intellectually engages students with

mathematics (appears to be most important feature in a high-quality mathematics lesson)" (p.

450). Much call for more student engagement has resulted from a combination of results

reported by the Trends in Mathematics and Science Studies (TIMSS) as well as a national push

for process strands (e.g. Communication, Connections, Problem Solving, Reasoning and Proof,

Representation) in addition to content strands (Number Sense, Algebra, Geometry,

Measurement, Data Analysis, Statistics and Probability) (e.g. NCTM, 2000).

Kidwell (2010) views the domain of engagement as "an emerging field of research and the

initial results are both startling and profound." (p. 29). Some recent studies on engagement have

focused on engagement vs. school size (e.g. Weiss et al, 2010), engagement with respect to

dropout rates (e.g. Archambault et al, 2008), and the possibility that some students will

deliberately choose not to engage (e.g. Sullivan et al, 2006). One question that has resonated

with my own was that of Yonezawa et al (2009) with their question "What keeps students

interested and engaged in school?" (p. 29). While Yonezawa et al (2009) is asking the question

from the students' viewpoint, I am considering a similar, but complementary question from the

teacher's viewpoint.

What are some ways in which mathematics educators create a classroom that engages the students?

By looking at teachers' engagement techniques, other teachers of mathematics in the field

may gain a better understanding of the importance of engagement and in turn, think of creative

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ways to create engagement in their own practice. Looking at examples of engagement may also

better inform professional develop and in-service programs within the American educational

system.

When considering some of the initial findings on engagement, many of the initial findings

seem to be more preventive in nature (e.g. reducing or eliminating dropout rates) but perhaps the

most important purpose of engagement should be for students to develop a love for learning.

Granted that textbooks have some prescribed ideas to help aim to engage students for teachers to

follow, it is perhaps more likely that the teacher's own creativity may better engage the students

rather than strictly following a textbook activity.

One major debate in the field of mathematics is the debate between what is more important:

procedural fluency or conceptual understanding (need to reread chapter in NCTM Handbook).

While procedural fluency is important, it seems that engagement will be more likely to occur

through the teacher aiming to develop students' conceptual understanding.

Being a teacher myself, I am always seeking new ways to engage my own students. This is

probably the main reason why I have chosen to undertake this research. I have some other ideas

about applying music concepts within the mathematics curriculum which may increase student

engagement. Further research is needed on my part, however, before I pursue that avenue. As of

now, my main purpose if to look at general forms of engagement, with the hope of better framing

future inquiries I have.

Review of Relevant Literature: As stated earlier, there has been some initial findings on

engagement as cited by Kidd (2010). Engagement has been considered with respect to the size

of the school, student choices with respect to engagement, dropout rates and teacher choices

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with respect to engagement.

Engagement and school size. Weiss et al (2010) conducted a quantitative study to test the

relationship among high school size, school engagement and achievement using the public-use

data file obtained from the Educational Longitudinal Study of 2002. Weiss et al (2010) found

significant differences related to student engagement between schools of different sizes. It was

further noted that school size was not significantly related to mathematics achievement.

Compared with students attending schools of the smallest size (the omitted category), those in

schools with 1,000–1,599 students or with more than 1,600 students have lower levels of

engagement (p. 170).

Engagement and dropout rates. Rodriguez (2010) contributed a short but rather descriptive

narrative in regards to the actions schools can take in reaction to the dropout crisis. A student,

"Ramon" was described as a student who was "Not learning anything". He was further

described as one who "thirsted for intellectual engagement." While this narrative took place in a

school where minorities were the majority, it seems that regardless of the student population at

hand, there is a demand for engagement in learning. Archambault et al (2009) found in a

longitudinal study of students that over time, students who are successful and engaged will

graduate from high school while others, alienated and disengaged, will eventually dropout.

Archambault et al (2009) mentions that adolescents at risk of dropping out report behavioral,

affective, and cognitive differences that affect their high school experience. One limitation in

this study was that there was an exclusive focus on low-income schools. While there is indeed a

need for intervention in these schools, it seems that any school, regardless of income level, can

benefit from knowledge of the importance of student engagement.

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To engage or not engage? That is the question. Sullivan et al (2006) in a study titled,

"Perhaps the decision of some students not to engage in learning mathematics in school is

deliberate", utilized qualitative research to determine whether a performance/mastery distinction

(Dweck, 2000) would be useful in describing student responses to schooling and mathematics.

They administered questionnaires and conducted roughly 50 interviews in the form of a teaching

conversation with students, with students experiencing difficulties in learning mathematics were

well-represented (p. 85). While it seemed that the performance/mastery distinction might be

meaningful, in this case it was not able to be measured by the degree of perseverance, since all

students persevered (pp. 96, 97). Sullivan et al (2006) suggest developing alternative

instruments instead of using the Dweck-based questions to see if different results might be

found. It was also mentioned that this study focused on students individually rather than those as

a collective group where peer pressure might lead to students choosing not to engage in learning.

States Sullivan et al (2006): "Teachers may well appreciate research into possible interventions

for classes in which an overall culture censuring apparent effort predominates, especially

interventions focusing on assisting students in overcoming such negative influences." (p. 98).

Taking into consideration some of the prior research, there seems to be a view of engagement

being important to reduce dropout rates. There is a valid concern of student engagement but now

we shall turn our attention to how teachers might engage their students.

Methodology and Data Collection: My original idea was to collect data from one mathematics

teacher from a suburban high school and one mathematics instructor from a local community

college. I submitted two letters for consent, one to the high school and another to the local

community college. For approximately two weeks, I played telephone tag with the "gatekeepers"

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at both institutions and even made appearances once at each location in an effort to express my

desire to conduct my research. Having spoke with the high school's principal, it was evident that due to a number of student teachers teaching in mathematics classrooms, an extra observer was

not in the cards. At that point, I opted for two mathematics instructors at the suburban

community college. The headmaster at the school at which I teach, was acquainted with the

Dean at the community college and was gracious enough to assist me with finding an "in" with

regard to interviewing two mathematics instructors. Having introduced myself to the Dean, I was referred to two mathematics instructors. I then

scheduled two preliminary interviews with the instructors and arranged for observation times

for both of them. Due to time constraints for both myself and the mathematics instructors

(several of days for holidays and registration), I observed two classes for each mathematics

instructor over the course of two weeks before the Thanksgiving Holiday. I observed two

50-minute lessons for one instructor (she taught Mondays, Wednesdays and Fridays) and two

75-minute lessons for the other instructor) (she taught Tuesdays and Thursdays). Preliminary

interviews were conducted in a small faculty lounge, lasted between 12 and 20 minutes, and

were recorded on a small digital voice recorder. Interviews were then transcribed on my

computer. Observations were not taped although raw field notes were typed on a small laptop

computer in an unobtrusive location so as not to distract students in the classes. Both instructors

treated me cordially and introduced me to their students ahead of time so as to ease any potential

anxiety for noticing a newcomer using a computer in class. Unfortunately, I was unable to

access my second observation notes with "Sarah" as my file was corrupt (according to Microsoft

Word). In an effort to balance my data analysis, I summarized briefly the second observations

6for both Christine and Sarah but provided detailed descriptions and analyses for each instructor's

first observation.

Data Analysis: In the next section, I provide some excerpts from the interviews as

well as narratives of the classes observed. Through these, some categories evolved which I then, split into more specific codes. Some recurring themes are summarized at the end of

each instructor’s narrative as well as a comparison and contrast at the end of this section. We

shall now turn our attention to our first mathematics instructor, "Christine", who was teaching a

unit on graphing linear equations using standard form and slope-intercept form.

Mathematics Instructor # 1, "Christine": Christine is in her second year teaching at the

community college level, teaching one course of Introductory Algebra and one course of College

Algebra. Her teaching experience totaled 39 years, 37 of which was at a small high school in

Southeastern Massachusetts. Her students are a diverse group. One student is home-schooled,

dual enrolled in the College Algebra. States Christine, "The majority of this semester's students

are fresh out of high school. She has had and has students who have returned to school after

raising a family looking to change careers. There are also students who are single or without

families in their 30s who have done something else in their past but have chosen to return to

school. One student's job was described as "being outsourced...not a lot of future..." and he was

returning to school to retool his skills. Some excerpts of our preliminary interview are

provided below:

Matt: What has been your experience as a classroom teacher and how have you reached students?

7Christine: I think it has changed...I entered teaching before computers...even before

calculators...now more active learning within confines of the schedule...more technology...I've had to learn how to solve equations by a graphing approach...when I went to school, we didn't do that...time involved. I suppose the lecture infused with the technology...

Matt: So to summarize, if I understand correctly before it was teacher-centered but now teacher, student, and curriculum-centered?

Christine: A combination, at least I hope that. It might not be like that every day...it's more difficult here than in high school because our schedule is so tight...don't have time to spend a day on a project on something so outcome-based as math. Here, I can't infuse as much as I want to and basically it's a time restriction...

Matt: In your own words, how would you define student engagement?

Christine: Following lesson, eye contact...students coming into class w/question on homework...you want them to go out the door and say that they enjoyed the class...or time in class went fast. Students making connections between concepts...you can tell...their eyes, their face...a student whose face goes "oh, yeah, I got it." That's engagement to me.

Matt: What are some ways that you, as a teacher, engage your students?

Christine: Go to board for volunteer points, I will try and get them to sing...a little group assignment, a little project on review of order of operations...at high school, used stations around the room, pass the pen...put up first line, someone else puts next line, etc...look at the thinking. We'll do an investigation when we get to cubics...volume of a box..

Matt: To what extent do you integrate real world situations in your teaching of mathematics?

Christine: Intro to Algebra class...little unit on mean, median, mode...graphing...that's an easy place to infuse it. We were doing inequalities...brought in pictures of road signs...speed limits. Composition of functions...we did Macy's ad...30% off, if you bring in 20% coupon...those kinds of things. Measuring radius of oil spill to determine how much over time...make composition out of that...trying not to avoid the word problems...using examples on percentages from the newspaper...brought in filler that comes with my charge card bill to look at calculation of average daily balance...we're going to do slope...looking for grade signs (e.g. 8% grade, 10% downhill)...ties in with percents.

8Matt: Many teachers have different views of student engagement. Some teachers work hard to

engage their students, others some days, and even others just work to get through the class. What factors might contribute to some of those differences?

Christine: Temperament of the students, schedule (e.g. get through material for finals, break, etc.), mindset of teacher: entrenched (does the same thing all the time, some students like), in transit (have done things this way but experiment with doing something else), and enlightened (no matter when they walk into the room, they seem to be able to do anything). Also depends on the topic...and some of the courses...

Taking into consideration some of the excerpts of the above interview, there seemed to be a

number of themes that evolved when engaging students. The use of real world models,

connections within the mathematics discipline, infusion of technology, offering of incentives

(e.g. participation credit), and the teacher's constant reading of students' verbal and nonverbal

cues. A recurrent theme which was considered as an issue was that of time constraints. Given

the expected material needed to complete the course, there is a sense of wishing being able to do

more to engage the pupils (e.g. period long projects) but given the time constraints, especially

during the month of November, there was no opportunity for students to engage in projects.

The classroom in which all mathematics classes are held was a bright room with long

windows and two entry points from the hall to the front and back of the classroom. On the side

closest to the windows, there were four long tables with four chairs in each row. On the side of

the classroom nearest the two doors, there were four shorter tables with two seats per row. The

tables all faced a whiteboard in the front of the classroom. At the front of the classroom, there

was a table and chair for the instructor. Off to the left, there was a console which contained

hookup for a teacher's laptop and access to the overhead projector.

I observed two Introductory Algebra courses taught by Christine. My first observation

9occurred on a Friday morning from 9:00AM - 9:50AM and my second observation occurred on a

Wednesday morning from 9:00AM - 9:50AM. Due to time constraints experienced in my

research study, I found it necessary to schedule the observations nearly back to back with one

another. The first observation is summarized below in greater detail whereas the second

observation is summarized more concisely as I felt that the second observation yielded very

similar findings.

Observation # 1, Christine: I entered the classroom described above to find the students taking

their seats and the instructor getting ready for the class's day. There were 8 males and 4 females

enrolled in this class. The range of ages seemed mostly between late teens and early 30s to one

person possibly in her late 40s or early 50s. Each student had three colored cups in front of

them: green, yellow, and red. The purpose of these cups was for the mathematics instructor to be

able to better assess the level understanding of her students where green meant that things were

great, yellow meant to slow the pace a little bit, and red meant to either slow down dramatically

or stop and retrace some lost steps. Students were asked to complete four review problems on

the board where they transformed linear equations into slope-intercept form (although students

were not yet explicitly taught the name of the form). While the students were working, the

instructor took attendance and made rounds through the room to check students' homework and

answer questions. During the homework check, comments such as "excellent," "nice job," and

"we'll talk about that later to clear up the vocabulary" were heard. One equation containing

fractions was discussed between a student and teacher: "you know why I chose that...review

clearing fractions out of equations." The instructor proceeded to discuss the board problems, pointing out connections within the

10discipline of mathematics (e.g. equivalent forms of equations, use of distributive property,

equations in the form y = mx + b). These connections are illustrated and discussed using

students’ written work on the board (e.g. y = (3 – 4x)/2 vs. y = -2x + 3/2, y = -2x vs. y = 0 – 2x). Christine proceeds to discuss the main differences between standard form of linear equations

(Ax + By = C) and slope-intercept form (y = mx + b). She proceeds to discuss how to calculate

x- and y-intercepts after checking that all the cups were green (checking student nonverbal cues).

As she proceeded to the next topic, she encouraged her students by praising their work (e.g. “I

saw some nice straight lines, nice t-tables”). She walked the class through calculating the x- and

y-intercepts from the new slope-intercept form and commented that many slope-intercept form

equations and standard equations are seen frequently in business and economics (connections to

real world) where intercepts are used to determine break-even points. She then distributed a

worksheet with some practice problems, placing a marker at the front, stating that volunteer

points could be earned. Students worked through the problems and different students

approached the board to complete the problems for participation points. She also cautions that

students find a third point to ensure “no dents” in the line. Dave completes a problem on the

board while Christine seeks agreement from other students that Dave is correct. Cameron nods

in agreement.

Christine proceeds to direct students’ attention (connections) to certain characteristics

(e.g. constant term is a multiple of both A and B in standard form). At this time, she hints that

students will graph from slope-intercept form without using the t-table (connections). Key points

about linear equations are also reviewed (connections within mathematics) (e.g. no exponents,

horizontal, vertical, and slanted lines) and difference between standard form and slope-intercept

11form. She proceeds to define slope as a measure of slant or steepness (a ratio of rise to run –

previous connection). There are some real world examples given (e.g. the rise and run of stairs

calculated by a carpenter, long and low steps on a local university campus). Students proceeded

to complete two more examples on their own. Given the lack of time to cover the slope formula,

the instructor modified the homework assignment and the students packed and departed at class

end.

Observation # 2, Christine: I entered the same room as before. Today there were eight males

and two females. As this class was the last class before the Thanksgiving Holiday break, there

were two females absent from the time before. Like the last time, the color-coded cups were on

the desks (recognizing non verbal cues for student understanding or lack thereof). In this class,

there was more evidence of the instructor's use of real world situations (e.g. percentages of

people's favorite Thanksgiving dishes based on a survey), use of participation points to entice

students to go to the board, making connections between prior and current knowledge, and

efforts to assist and guide students when there were questions on how to obtain the equation from

the graph or how to set up calculations in the slope formula.

Christine was not able to complete an intermediate interview due to the pressing need to

advise students for next semester’s courses and registration. We talked briefly at the end of both

observations in regards to whether the methods work (e.g. coming up to the board, real world

applications) and we both were in agreement that the students were indeed engaged.

Furthermore, during the last class, there was a comment made by a fellow English teacher who

walked by, popping her head in the room, and commenting “Oh, you’re all working hard.”

Christine did comment that some topics lend themselves better than others when considering the

12extent to which her techniques of student engagement work in the classroom.

Let us now turn toward a detailed analysis of the second mathematics instructor, "Sarah".

Mathematics Instructor # 2, Sarah: Sarah, like Christine, also teaches Introductory Algebra

and Intermediate Algebra. She has taught for 42 years, 12 years of which were at the elementary

school level and 30 of which were at the community college level. She described her students

over the years as being mostly post high school graduates. On occasion, she has had a dual

enrolled high school student. From her experience, over half her students have gone on to

four year colleges and beyond. Due to time constraints on her end, our interview lasted about

12 minutes and occurred immediately before her first observation. Below are some excerpts

of our preliminary interview:

Matt: Tell me about your experience as a classroom teacher (e.g. how do you reach your students)?

Sarah: I feel I have to establish some kind of a personal relationship with the students. Besides, we have to show them that math has relevancy (e.g. a real world connection that relates to them personally), of course not the most popular subject that there is… the personal relationship between myself and my students is important because they have to see me not only as someone who is going to impart knowledge on them but also going to give them some kind of a confidence in this subject that is not popular with them…needs to be relevant and doable.

Matt: How would you in your own words, define student engagement?

Sarah: Student engagement…OK…let’s see (thinking)…they would have kind of a personal involvement in the class. They’re not just big sponges that are trying to soak up what I’m saying and then spit it back out on the test. They are trying to develop a personal understanding of the topics and concepts that I’m putting across. And, in order to do that, they can’t just sit there with their hands folded and just expect it to go in just by osmosis. There has to be that personal activity…not to use the word engagement again.

13 Matt: And based on your own experience, what are some ways that you as a teacher engage

your students? And how would you measure their engagement?

Sarah: I try to…first of all I think humor is important certainly because math is not seen as funny. We laugh it and sometimes I just make mistakes and I show them that that’s OK. They laugh, and it makes sense, doesn’t it. I think that kind of a thing…that humor, it helps. And other things…I do think that I have to (slight hesitation) make sure that they have the feeling that they can make a mistake and many times when you’re getting answers in math, you’re not going to get the answer the first time…really important .....sometimes so afraid…not even try. So they see you don’t always have to get the right answer always the first time and it’s OK to make a mistake…try engaging them in maybe putting things on the board whenever possible so that they’re not afraid to be in front of their peers and make that mistake and so on. Things like that I find helpful.

Matt: To what extent do you integrate real world situations into your teaching of math? Like maybe some examples of real world situations?

Sarah: Well, I’m going to teach graphing, for instance…today (discusses comparing test scores between two classes using a graph). (Speaking about using and applying common denominators) let’s go over to Lambert’s and we’re going to buy 3 oranges and 2 apples so am I going to say I buy 5 apple/oranges and they say “oh no.” You’re going to buy 5 pieces of fruit so that’s the common denominator…(gives another example of common denominators by counting quarters, dimes and nickels)…lot of times I like to try and do puzzles. Some of the students who think they’re not good at math, some of them answer the puzzles very quickly. Then I’ll say “Ooh, look how smart you are.” Cause that empowerment of…and some kids are not that good at doing the step by step but I can get them into that by saying…”look how good you are at solving the puzzle.”.. So that encouraging thing I think is important to all of us.

Matt: Many teachers have different views of engagement. Some I know work hard to engage their students every day while others may engage some days and others will work just to get through the class. What factors might you think might contribute to those differences?

Sarah: Well, here and at other community colleges…I shouldn’t just say community colleges …I would say that there is a lot of pressure on teachers to get through the syllabus you know…say to my students all the time “I don’t want to go on a solo flight through this course. I know this stuff so if I don’t bring you along with me…” so…but…then I also have to think in my head “Oh my goodness, this person is going on to college algebra and I haven’t got to the quadratic formula.” Sometimes you really have to go to the middle of

14 the group and say "OK, I've got to go the speed of the majority here...and that puts pres-

...and some people I think also especially on the community college level think of themselves as professors, not teachers. But I feel strongly that at the lower level, it's about the teaching… I teach a course for...in math...for teachers... Called Math for Elementary teachers (gives example of a subtraction of signed numbers where students have to not only give the answer but explain why their process worked)..was explaining to them in a teaching situation that you can't just give a bunch of rules...I say why do you that...try to encourage them to understand them.

Matt: So it's the conceptual understanding versus the procedural fluency.

Sarah: Exactly, and then I say "But, if you're explaining it to somebody else, you can't just give them the steps, you have to tell them why." So they go, "Well, oh yeah, I never really thought about it, I just did it." And I say "Yes". And that's why so many people leave the math class…may have known how to do things for test but they don't have understanding of concept…not carrying the how-to away…you as a future teacher have to look at these things…not just teaching a bunch of steps…have to be able to explain why you do it.. I think a lot of people who are "good at" math, they don't think about as much as the person who struggles with it and they go "Well why does that subtraction become a plus and that negative number becomes a positive number." They're asking why because they want to really have an understanding of it rather than someone who says "Yeah that makes sense."

Like the preliminary interview with Christine, based on some of the excerpts of the above

interview, there seemed to be a number of themes that evolved when engaging students. Like

Christine, there was a use of real world models. Ideas such as relating personally to the students

and building their confidence were more prevalent explicitly in this interview than the interview

with Christine. Time was cited once again, as being a challenging factor to engage the pupils.

Sarah did not speak of technology like Christine did. Perhaps her experience in elementary

education did not require as much knowledge of technology whereas Christine, who was a high

school mathematics educator for many years, required technology to advance in her discipline.

I observed two Introductory Algebra classes taught by Sarah. These classes occurred

Tuesdays and Thursdays from 11:00AM – 12:15PM in the same classroom where Christine held

15her classes. My first observation occurred on a Thursday from 11:00AM – 12:15PM and my

second observation occurred the following Tuesday from 11:00 AM – 12:15PM. As mentioned

before, due to my own time constraints, I found it necessary to schedule the observations back to

back. Sarah’s classes were slightly longer than those of Christine but that was because her

teaching schedule was spread over two days per week whereas Christine’s teaching schedule was

spread over three days per week.

Observation # 1, Sarah: Thursday morning, I entered the classroom described earlier to find

Sarah handing back tests from a recent class. This class consisted of 4 females and 5 males. The

males sat on the side of the room closest to the window whereas the females sat on the side of

the room closest to the doors. Each row contained 1 or 2 students in this particular class. As

Sarah handed back tests, there was evidence of her desire to instill a sense of confidence as well

as some humor in her students, which she spoke of during our preliminary interview. There was

a comment made to a female student: “You should do better, you scored 100%, Gloria, this is a

person who had all this worried look on her face…you did all of the problems, you set them up

and made little sign mistakes, but you did get the 100.” It is clear that mistakes here were not

punished but rather acknowledged as a way to learn. After all the tests were handed back, Sarah

began the next lesson, stating that the class was going to start graphing lines. She proceeds with a

hypothetical set of test grades from two classes, each from a different campus of the same

community college. As she plots the points on the board, she reviews how a point’s position is

determined on the graph. She proceeded to use different colors to represent the comparison and

contrast between the two hypothetical classes. Upon completion of the graph, Sarah discusses

another area in real life where graphs may be used: buying a car. She poses the question that

16other than price, what might be some other factors in choosing a car. One student suggests

“safety,” and another student suggests “mileage.” Here, it is evident that students are

encouraged to participate and feel free to contribute to the lesson at hand. The teacher then

distributes a handout, which students quietly complete. Some excerpts are given below, from

this segment of the class.

Sarah: So you look at these statistics here. You get to do some research yourself. Here are some statistics on a car. The graph will come from two pieces of knowledge. One is going to depend on the other so to speak.

(Students begin working. One student makes an observation.)

Male student: It has gone down 3 miles per gallon, adding luxuries to cars.

While students are working, Sarah gives direction and guidance on using different colors

for both sets of data. Another student offers use of his multicolor pens to help others out. One

student who finished early receives some extra assistance on some questions on her recent test

with Sarah to better improve her conceptual understanding of linear systems. Upon completion of the car worksheet, Sarah announces that the class will be working on graphing in the coordinate plane (connections within mathematics). Sarah refers to the late

mathematician Rene Descartes in his construction of the coordinate plane. She identifies prior

connections to the horizontal number line and the graphing exercise the students just completed.

Terminology, such as ordered pairs, the origin, quadrants, and x- and y-axes are discusses. She

offers an easy way to remember the points in an ordered pair: X comes before Y in the alphabet

so the x-coordinate is listed before the y-coordinate in an ordered pair. There is some student-

instructor interaction outlined below as Sarah discusses the signs of numbers listed in ordered

17pairs.

Sarah: Now let’s think about what we did with the Prius and Ferrari. Didn’t we go right and then up? Let’s think about any point I’d put in Q1. Give me an ordered pair to get me in Q1. What kind of numbers?

Student: both positive

Sarah: Excellent, Diane

Sarah proceeds to summarize the remaining three quadrants accepting and asking for student

input (e.g. quadrant II, x is negative, y is positive, quadrant III, x is negative, y is negative, and

quadrant IV, x is positive, y is negative). Having assessed understanding and made connections

to prior material, Sarah directs the class to the next objective of the lesson: to be able to identify

and read plotted points. The students are administered another worksheet to practice naming

points. Having checked understanding of directions, the students went back to work. Sarah

circled the room, assisting students where needed. Students feel confident in asking questions

(e.g. “Does it matter which side the letters are on?”), which shows signs of engagement. Sarah

maintains their engagement by not only answering them but also providing them with words of

encouragement and/or additional suggestions (e.g. “OK, you are all right, show how…perfect!”,

“The side of the letters does not matter but you should use capital letters.”, “You’re on a roll

here, kid.”).

Sarah then goes over the worksheet, addressing some key ideas on plotting and naming points

that lie on the x- and y-axes. After briefly summarizing the key points, she states, “We’re going

to be using this system big time.” Writing the equation y = 2x – 1 on the board, she asks, “See

anything like that before? Instead of solving the equation, we’re going to graph the equation.”

18Students were asked to think about two contrasting equations and one commented that "one had

variables on the same side while the other had y alone on one side." Clearly, we can see the

instructor eliciting connections to prior knowledge for her students based on student comments.

Sarah proceeds to use multiple representations such as equations, tables, graphs and real

situations. Below, we can see evidence of more connections between representations as well as

the use of humor.

Sarah: I am x years old...husband is y years old... sum of our ages 50...I’m lying but we’ll use this equation anyway... husband 10 years older so can’t I call him y = x + 10? This equation explains the relationship between these ages. (y = x + 10). No matter how old I get, I can use this equation to find my husband’s age (or vice versa).

Sarah elicits student responses to questions regarding calculating one age, given the other age.

She states that ages may be calculated through the equation, a t-table, or a graph (connections

among representations within mathematics). After the age problem is discussed, Sarah models a

more abstract problem using the multiple representations of equations, tables and graphs. While

working through her table she elicits students’ conceptual understanding of past concepts (e.g.

signed numbers) through questioning (e.g. “What do we do with –(-2)?). Students answer when

asked (e.g. “–(-2) means -1(-2), -1(-2) = +2 and +2 + 4 = 6.”). Students are instructed to plot the points, being mindful of making sure they obtain a

straight line. Like during the time spent on the previous worksheet, Sarah circles the room once

again, offering encouragement and suggestions to the working students. We also see more

evidence of students’ engagement as they raise certain questions (e.g. “I got a fraction, am I

wrong?”) and more responses (e.g. “Not everything is a whole number, we now have to

estimate.”)

19And we see signs of more humor as well.

"Don’t let him get accepted into the culinary program. He hates fractions."

"What would we do without fractions? We’d have to eat the whole pie."

"They must not like fractions in Europe either?"

There is a recap and summary of key ideas, Sarah reviews directions for homework exercises,

and the class is over. Despite being over, two female students linger behind, speaking with

Sarah about some test questions they made mistakes on. There is more evidence that Sarah is

willing to reach out and relate to the students, as her students take time beyond class to speak

with her. Furthermore, we can see that Sarah continues to show that mistakes are part of the

learning process and students’ learning is evidenced by them (e.g. “I got a negative but the

answer can’t be negative…we can’t have -72 people.”).

Observation # 2, Sarah: A second observation was conducted the following Tuesday. As

with the other mathematics instructor, Christine, due to time constraints, observations were

scheduled in a short time span to ensure a thorough data collection. During the second

observation, there was still evidence of the mathematics instructor's methods of engagement

including the use of humor (e.g. joking with the student who hates fractions), aiming for

conceptual understanding of the material (e.g. reviewing the age problem from the previous

class, stating how the points on the line also satisfy the equation), and making prior connections

to past and current knowledge.

Let us now turn our attention to the results section. Here, we shall look at each teacher

20individually and then we shall compare and contrast their approaches toward student

engagement.

Results and Discussion: Through the interviewing process and observation process, a

number of themes evolved. These themes were in turn developed into the Principles of

Engagement, which are discussed below.

Principle # 1: The Personal Relationship Principle: Teachers should develop personal

relationships with their students. This means showing a genuine concern for their success in

school, infusing them with the confidence they need to master the material, and identifying the

strengths each student brings to the classroom. This personal relationship strongly resonates

with Jones et al, (2010), who found in their research that “Simply stated, the more that students,

felt personalization at their schools, the better students did academically.” Personalization refers

to the “tightening of the connections between the students and their learning environments (e.g.

teachers, other adults, students, peers, curriculum, overall school culture). The idea of

personalization requires a blending and working together between teacher, student, curriculum

and environment. Schwab (1973) in “The Practical 3” discusses the idea that no commonplace

dominates the others, where the commonplaces include student, teacher, curriculum, and milieu.

Sarah was one who emphasized the importance of relating to her students personally. Based

on our interview and observations, this seemed to be her one of her major techniques in

maintaining and increasing student engagement in mathematics learning and mastery. Sarah can

"give them some kind of a confidence in this subject that is not popular with them..." which will

in turn make the mathematics "relevant and doable." Sarah also cited humor as another way she

21engages her pupils, which was frequently observed through her in-class comments toward one of

her pupils who dislikes fractions. Humor was also used when discussing an application of linear

equations, namely modeling the ages of her and her husband. Comments such as "I'm lying, but

let's say that I am 30 years old, and my husband is 10 years older..." provided additional evidence

of humor, which provided her students with the opportunity to laugh. Earlier, she stated, with

respect to humor:

"We laugh it and sometimes I just make mistakes... They laugh, and it makes sense, doesn’t it. I think that kind of a thing…that humor, it helps."

The use of humor, particularly when being able to laugh at mistakes, is also helpful in

infusing confidence. We shall discuss confidence in the next Principle of Engagement.

Principle # 2: The Confidence Principle: Teachers need to instill confidence in their pupils,

making them feel that making mistakes can be a more valuable learning experience than getting

the answer the first time.

During my interview with Sarah, she also emphasized the importance of infusing a sense of

confidence in the students. The ability to instill confidence is dependent upon personal

relationships with the students, which was discussed earlier. One area where Sarah works to

build that confidence is making the students feel that it is OK to make mistakes. With respect to

making mistakes, Sarah states:

"...sometimes I just make mistakes and I show them that that’s OK...I do think that I have to (slight hesitation) make sure that they have the feeling that they can make a mistake and many times when you’re getting answers in math, you’re not going to get the answer the first time. And that’s really important because sometimes they get so afraid of the subject that they will not even try..."

22 Sarah was a teacher who would lead students to playing to their strengths. When working

with students who do not feel confident at mathematics, she spoke of encouraging them,

instilling confidence through praising their prowess at solving puzzles.

. Some of the students who think they’re not good at math, some of them answer the puzzles very quickly. Then I’ll say “Ooh, look how smart you are.” Cause that empowerment of…and some kids are not that good at doing the step by step but I can get them into that by saying…”look how good you are at solving the puzzle.”

At the conclusion of the second observation, it was evident that Sarah's efforts to instill

confidence in her pupils was effective. Having handed back some recent test, one female

student, ecstatic at her test grade of 100% stated "I am going to put this on my refrigerator for

my grandfather to see!" Seeing the excitement radiate from this young lady indicated that she

was determined to continue in her learning and mastery of mathematics.

Observing the various interactions between the students and mathematics instructors not only

showed evidence of confidence, or building of confidence, on the students' part but also led to

the development of the third Principle of Engagement, namely the Interactive Classroom

Principle.

Principle # 3: The Interactive Classroom Principle: Teachers need to create an interactive

learning environment to engage their pupils. This includes allowing students to use the board to

work out problems and perhaps teach one another. This also includes the allowance of

opportunities to discuss mathematics problems with one another or with the teacher. Interactive

classrooms should be full of individualized attention, particularly when the class size is small.

Interactive classrooms, depending on time available, should provide project-based learning

23where appropriate.

When interviewing Christine, she briefly spoke of her experience when she taught at the high

school. With respect to active learning, she stated:

"I mean when I first started teaching I did grades by hand because we didn't even...so I think the growth (of technology and pedagogical changes) has been engaging the students I think with...with more active learning as much as we can do within the confines of the schedule. Ah..I think that's really where...what I've seen the progression of, when I think back. I can't really say that because my consumer math class was...we did a lot of project work."

Sarah, like Christine, believed in interactive classrooms, through her discussion of students

being involved in class. In her interview, she stated:

"…they would have kind of a personal involvement in the class. They’re not just big sponges that are trying to soak up what I’m saying and then spit it back out on the test. They are trying to develop a personal understanding of the topics and concepts that I’m putting across. And, in order to do that, they can’t just sit there with their hands folded and just expect it to go in just by osmosis. There has to be that personal activity.

While project-based learning may be a challenge, particular with the amount of time allotted

and the need to complete the course syllabus, mathematics instructors clearly need to find other

ways to create that interactive classroom. In Christine's class that I observed, students frequently

approached the board to earn participation points. Furthermore, all students in the class could

see one another's thought processes and compare their own work with that posted on the

whiteboard.

"I have them come to the board for volunteer points believe it or not as an enticement. You know, one little point added toward their cumulative point total. It just sometimes, especially for a student, it gets them going... I do a pass the pen even at the high school. Even at the college put up the first line of the problem and then someone else contributes the next line...and then, gets to share so that we see...even if it's not the way they'd have tackled the problem, they got to now pick up on what the person ahead of them did."

24 By seeing fellow student work posted on the whiteboard, there would occasionally be

questions raised to further develop one's own conceptual understanding of the topic at hand.

This leads to the fourth Principle of Engagement, namely, the Conceptual Understanding

Principle.

Principle # 4: The Conceptual Understanding Principle: Teachers need to focus on the

"why" in mathematics in addition to the "how" in mathematics. If even one student asks "why,"

there is an instant teachable moment to sustain that student's engagement for learning.

Having taken time to interview and observe these two mathematics instructors, I gained a great

deal of insight into their practice when engaging their students. Much of what I observed was

consistent with what I was told during the interviews.

Sarah was particularly vocal on the importance of conceptual understanding. During her

interview, she spoke about a class she was teaching that prepared elementary school teachers in

mathematics teaching:

"I was explaining to them in a teaching situation that you can't just give a bunch of rules...I say why do you that (change the negative number to a positive when evaluating -5 - (-7))...try to encourage them to understand that you may know how to do something... but, if you're explaining it to somebody else, you can't just give them the steps, you have to tell them why."

A brief excerpt from Christine's second observation reveals her efforts to provide assistance

to a student, who "didn't get how they were getting the equation." In her one on one interaction

with him, she worked to provide him with the required conceptual understanding for determining

a linear equation from its graph.

25 (Referring to the slope-intercept form of a linear equation) "We have y = mx + b, where m is the slope and b is the y-intercept. When we substitute 0 in for x, we found the graph crossing the y-axis at -4 (writes -4 under the b). Now, let's pick two readable points from the graph.

We wish to measure the rise and run between these two points and we find it is 1/2. Using the slope formula, we see it is actually -(1/2). See how the line falls from left to right? Let's pick two more readable points...is the slope -1/2 again? We need to make the slope negative, so we write -1/2 under m and we can now see our equation is y = -(1/2)x - 4."

References to the "rise over the run" as well as how the y-intercept is determined reveals the

deep conceptual thought in addition to the procedural understanding of the problem. Through

this interaction, we can see the student's engagement sustained through answering why the y-

intercept is at b and why the slope is at m.

Reflecting more on the above interaction between Christine and her student, we not only see

an effort to provide a conceptual understanding, but we see Christine trying to lead the student to

make connections to prior knowledge (e.g. rise over run, x = 0 provides the y-intercept,

verifying results using a t-table). This leads to the development of the fifth Principle of

Engagement, the Connections Principle.

Principle # 5: The Connections Principle: Teachers should show the students that

mathematics is connected within the discipline as well as to many real life situations. This

means leading students to connect new knowledge to prior knowledge or connecting new

knowledge to real world situations where students may encounter mathematics.

During the classes taught by Christine and Sarah, students were led to make connections to

the real world. In Sarah's class, the class worked through creating a line graph that compared

two brands of automobile. When the students were presented with the assignment, she tells

them, "Here are some statistics on a car. The graph will come from two pieces of knowledge."

26As the class proceeds to graphing on the coordinate plane, Sarah tells the class to "think about

what we did with the Prius and Ferrari," as they review how they plotted the coordinates from

two sets of data. It is at this point that we can see a real world connection made to the discipline

of mathematics, namely plotting points on the coordinate plane. Like Sarah, Christine made real

world connections in her classes as well. When introducing slope as a measure of rise over run,

she drew students' attention to stairs and their rises and runs. She spoke of some stairs with

longer runs than rises and vice versa (e.g. long and low steps on a local university campus).

In addition to illustrating connections to the real world, both Sarah and Christine led students

to connect within mathematics to prior knowledge. Writing the equation y = 2x + 1 on the

board, Sarah asks the class if they have "seen anything like that before." The students

recognized the equation's form from a prior unit on solving systems of linear equations.

Like Sarah, Christine made connections within mathematics where appropriate. At the

beginning of her first observed class, the students were manipulating some linear equations.

When one student was confused about one of the equations containing fractions, Christine,

assisted the student. Anticipating other student difficulties with the equation with fractions, she

told the class: "You know why I chose that: to review clearing fractions out of equations." There were more instances of connections both with mathematics and outside mathematics.

Some more of these instances are summarized graphically after our discussion of the Principles

of Engagement.

We shall now turn our attention to the final Principle of Engagement. While I admit that

there was very little observed in the classrooms, this idea was discussed during the interview

with Christine. I decided to include it because I feel there are some important ideas that might

fuel future research.27

Principle # 6: The Technology Principle: Teachers should, whenever possible or applicable,

use technology to further engage students. Technology may be as simple as a scientific

calculator or more advanced such as a graphing calculator, computer, or relevant internet web

sites. As technology is hands on, this approach is necessary for teachers to further increase

student engagement.

When interviewing Christine, she commented on the role of technology although did not

use much in her observations other than a scientific calculator. Earlier, Christine spoke of her

experience, which tends to shape many teaching related decisions.

"I mean when I first started teaching I did grades by hand because we didn't even...so I think the growth has been engaging the students with technology... I've had to learn how to use technology, how to solve equations by a graphing approach which when I went to school, we really didn't do because we would have had to have hand graphed everything so time involved made that a method of solving that we didn't use, you know where now, I mean you begin that in algebra 1 so I think that um...pretty much I suppose the standard...you know, the lecture infused with the changes in pedagogy that have come along the way."

Sarah did not discuss technology. I felt it unwise to prod the topic from her as I did not wish

to taint the data. It is possible that she believes in technology but the topic was not raised in our

interview. Christine, like Sarah, did not encourage much technology use from students, with the

exception of using scientific calculators, but she did emphasize during our interview that time

constraints, particularly in the Fall semester, made it very difficult to explore certain ideas in

greater detail and to incorporate additional technology.

Limitations: Both instructors agreed that time constraints did influence some of the ways in

which they engage their pupils. Christine, in particular, states the following with regards to time:

28 "I think it's more difficult even here than high school because our schedule is so tight...um, this semester (e.g. meeting 39 times due to so many holidays in the month including a day for registration)... and there's a change but they don't change the content that needs to be covered

so some days you feel like you're shoveling...you know...you can't take that...you don't have that time to spend a day doing a project...not in something as outcome based as math because these guys need this course to be able to get to the next course to be able to get to the next course."

With respect to the time constraints both instructors were facing, it is certainly possible that I

did not necessarily get a true picture of each instructor's teaching styles, particularly with

technology. Perhaps if my study was conducted in the Spring Semester, there might be less time

constraints as there are generally fewer holidays but that might be reserved for future research.

Figure # 1: Some methods used to engage students.

29Figure # 2: A progressive flow through the Principles of Engagement.

Figure # 3: The relationship development between student, teacher, curriculum and environment.

Overall Center of Classroom: Based on J. Schwab's "The Practical 3: Translation Into Curriculum."

Teacher-centeredStudent-centeredCurriculum-centeredCombination of Above

30Conclusion: Based on the results of the study, there are a number of ways that mathematics

instructors engage their students. A number of main ideas typically flow as well, as illustrated in

Relating to the

students

Mathematics is doableMathematics is relevant

Confidence

It is OK to make mistakesRecall success with other areas (e.g. puzzles)

Interactive Classroom

Students working out problems at the board.Following one another's thinking.Mathematical discourse.

Conceptual Understandi

ng

Focus on the "why" in mathematics, not just the "how" or "what.".Seize teachable moments tofurther develop understanding.

Connections

Real world connections.Prior mathematical knowledge connections.

Technology

Use of calculators, computers, etc. to supplement paper and pencil work.Engage students through multiple representations of the same problem.

Figure # 2, p. 31. Teachers need to find ways to relate to their students to gain their trust. It is

through this relationship that teachers can start to build or increase confidence in a subject area

that is not always popular among students. By continuous support and encouragement,

particularly when the students make mistakes, engagement may be sustained or increased by

using these mistakes as a learning process rather than something that is wrong. Through

developing a conceptual understanding and making connections with and outside of

mathematics, students in turn, might be able to self-assess their own mistakes but through their

understanding of the concept, remain engaged to do better and further develop their skills and

knowledge in mathematics.

The use of technology may be another area in which teachers may sustain and increase

student engagement in mathematics. Perhaps if teachers are willing to put time in to their

professional development, they can further add tools to their teaching repertoire. More research

on the effects of technology on student engagement might reveal some valuable insight for both

teachers and researchers.

If there is one thing where teachers may lack control, it is through the pressures of time.

There is so much time allotted in the semester or year to complete a given curriculum. Time can

have an adverse effect on certain engagement techniques but even with a little effort, with proper

planning, perhaps time-restricted educators might be able to truly discover the benefits of a

mathematical project in class or using technology to teach a required topic in the syllabus, or

some combination thereof.

31References:

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Corey, D.L., Peterson, B.E., Lewis, B.M., and Bukarau, J. (2010). Are There Any Places That Students Use Their Heads? Principles of High-Quality Japanese Mathematics Instruction. Journal for Research in Mathematics Education, Vol. 41, No. 5, pp. 438 – 478.

Kidwell, C. F. L. (2010). The impact of student engagement on learning.. Leadership v. 39 no. 4 (March/April 2010) , p. 28-31.

McClure, L., Yonezawa, S., & Jones, M. (2010). Can school structures improve teacher-student relationships? The relationship between advisory programs, personalization and students' academic achievement. Education Policy Analysis Archives, 18(17), 1-18.

Principles and Standards for School Mathematics (2000). Reston, VA: National Council of Teachers of Mathematics.

Rodriguez, L. F. What schools can do about the dropout crisis. Leadership v. 40 no. 1 (September/October 2010) p. 18-22.

Schwab, J.J. (1973). “The Practical 3: Translation into curriculum.” School Review, 81, 501-522.

Sullivan, P. Tobias, S. & McDonough, A. (2006). Perhaps the decision of some students not to engage in learning mathematics in school is deliberate . Educational Studies in Mathematics v. 62, no. 1, p. 81-99

Weiss, C. C., Carolan, B.V., & Baker-Smith, E.C. (2010). Big school, small school: (Re)Testing assumptions about high school size, school engagement, and mathematics achievement. Journal of Youth and Adolescence v. 39 no. 2, p. 163-76

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