effect of teaching mathematics concepts within a science...

MNS Physics

July 2009

Effect of Teaching Mathematics Concepts Within a Science Context Adrian Boyarsky, Russell Bray, and Mark Henrion Arizona State University Action Research required for the Master of Natural Science degree with concentration in physics.

Table of Contents Abstract ..........................................................................................................................3 Introduction and Rationale..............................................................................................4 Literature Review ............................................................................................................5 Method ...........................................................................................................................6 Data and Analysis ............................................................................................................9 Investigator 1 ............................................................................................................ 15 Data....................................................................................................................... 19 Student Interviews ................................................................................................ 25 Analysis ................................................................................................................. 27 Conclusion............................................................................................................. 28

Investigator 2 ............................................................................................................ 28 Method ................................................................................................................. 28 Data....................................................................................................................... 29 Post-‐instruction Interview Problems...................................................................... 34 Analysis ................................................................................................................. 41 Conclusions ........................................................................................................... 45

Investigator 3 ............................................................................................................ 47 Method ................................................................................................................. 47 Data....................................................................................................................... 56 Student Interviews ................................................................................................ 65 Student Surveys..................................................................................................... 69 Analysis ................................................................................................................. 70 Conclusion and Implications .................................................................................. 72

Overall Conclusion ........................................................................................................ 72 Implications for Instruction ........................................................................................... 73 Implications for Future Research ................................................................................... 73 Bibliography .................................................................................................................. 75 Appendices ................................................................................................................... 76 Math Concepts Inventory 2003 ....................................... Error! Bookmark not defined.

Abstract Traditional mathematics courses have consistently been taught based upon procedure, rules, and memorizing information. The investigators, however, have noticed that students have difficulty in proportional thinking and graphing quantities. High school math students were treated with a course design that introduced proportional reasoning through a modeling approach in physics concepts. Results indicate that students who were treated demonstrated higher gains in understanding of common algebraic concepts of proportional reasoning and graphing than students who received a traditional approach to mathematics instruction. Effect of Teaching Mathematics Concepts Within a Science Context Principal Investigator: Colleen Megowan Co-‐Investigators: Adrian Boyarsky, Russell Bray, and Mark Henrion

Introduction and Rationale Proportional reasoning and interpreting graphs are both complicated mental processes that are prevalent in all areas of mathematics and science. (Lesh, Landau, 1983). A vast majority of the students we encounter on a daily basis can state that slope is equal to rise over run, and, since they can give this basic definition of the algebraic term, no one ever asks them to give it a second thought. Few students make the connection that for a linear relationship between two variables, as the independent variable increases or decreases by a certain increment, the dependent variable increases or decreases by the slope multiplied by that increment. In some high school algebra classes, students can make it through an entire school year without ever seeing that the slope of a line is not just a number, but a rate of change that not only has units, but implications that affect our world. The vertical intercept of a graph is also a key element of graphical understanding whose importance can be overlooked. What is the physical interpretation of a non-‐zero vertical intercept and why do some graphs intercept the vertical axis at zero while others do not? Classroom experience by the researchers shows that, as the relationship between variables gets more complicated, even fewer students can reason proportionally about how changing the independent variable changes the dependent variable. Not all quantities change in even increments. Sometimes graphing one quantity versus another can produce curves, peaks, and bends that can baffle high school students. What do the numbers in front of our independent variable mean, and does our vertical intercept make any sense at all? The graph is now much more interesting than a straight line, but being able to use that graph to answer questions and make predictions is an enormous jump in the process of understanding graphs. The goal of this study is to investigate student conceptual and procedural understanding when proportional reasoning and interpretation of graphs are taught in a scientific modeling context as opposed to a traditional mathematics classroom. Students were tested on the Mathematics Concepts Inventory (MCI) before and after working through various labs and data collection lessons that emphasize proportional reasoning and interpretation and understanding of graphs. Having students derive proportional equations from lab demonstrations that use both linear and non-‐linear relationships, we introduce important relationships that the students will face in many courses and disciplines. The students graphed these situations, found slopes of the relationships, and interpreted the data conceptually. By introducing proportional reasoning and interpretation of graphs in this way, we examined how students’ understanding shifts toward a more coherent conceptual model while arming them with the tools to relate these concepts to other novel problems.

Literature Review Walk down the street and ask the average person what he/she thinks of mathematics and, chances are, a majority of responses will range from dreadful stories of sadistic high school teachers to blatant proclamations of ignorance of a subject that is often viewed as a silly set of predetermined rules and facts for dealing with numbers that, if one does not care to memorize, cannot be understood or used in any useful manner (Paulos, 2001). In schools across America, students are taught mathematics in ways that often overlook the useful contexts with which people use mathematics on a daily basis in everything from grocery shopping to calculating fuel mileage (Borasi, 1992). Learning mathematics as a set of rules and algorithms for solving for a letter of the alphabet that has, for some mysterious reason, snuck into a math problem, results in the failure of average citizens to see mathematics as the most creative and imaginative of disciplines (O’Shea, 2007). People associate mathematical concepts with how they are originally introduced. Several different people may be able to recite the quadratic formula by heart, but the formula itself may carry an extremely different meaning for each of these people based on how quadratic equations were taught in school (Steffe, Nesher, Cobb, Goldin, & Greer, 1996). In order to be truly competent in mathematics, people need to have a solid understanding of both conceptual and procedural knowledge. Students must be able to construct relationships between previously established and newly minted mathematical concepts and be highly skilled in using the rules, algorithms, and procedures used to solve mathematical tasks. Falling short in procedural knowledge can leave a student with a solid conceptual understanding but no ability to quantitatively solve problems, while a lacking in conceptual knowledge can force a student to use rote memorization of equations to solve a problem without ever seeing any meaning or richness in the task at hand (Hiebert, 1986). Establishing the complicated relationship between conceptual and procedural knowledge is no easy task, but some think that the link to learning mathematics and using it in other subjects, areas, and contexts is modeling instruction (Blum, Niss, & Huntley, 1989). Even though the teaching styles found in most college courses across America do not corroborate this idea, many teachers and experts contend that knowledge is acquired by construction, not by transmission alone. Learning something is far different than merely hearing something, and when we learn, the new piece of knowledge is fitted into our now reorganized existing body of knowledge. Humans always associate knowledge with other pieces of knowledge and with contexts that help reconcile or make sense of the new idea. This is only part of the reason why teaching mathematics in relation to a variety of real-‐world problems in domains other than mathematics not only enhances understanding but makes the problems themselves and the subject as a whole more significant and meaningful to students (Steffe et al., 1996). In 1980, the National Council of Teachers of Mathematics made a focus on problem solving the primary goal of

mathematics education for the 1980s, stating that acquiring facts is futile if they cannot be used to solve novel problems (Borasi, 1992). According to constructivist theory, students need to solve problems that are considered their own before any sort of meaning is applied to knowledge that is learned in a classroom (Laborde, Glorian, & Sierpinska, 2005). By teaching proportional reasoning and graphical understanding through modeling with scientific contexts, students get countless opportunities to form relationships between these mathematical ideas, experiences from their daily lives, and the role of proportional reasoning and graphing in other subject areas (Harel & Confrey, 1994). Principles of effective teaching of math applications through a variety of contexts as suggested by Steffe et al. (1996), reads like a checklist of techniques that a student in a Modeling Workshop at Arizona State University would encounter. In a high school classroom where science concepts are being used to teach proportional reasoning and graphing, interesting problems are posed in familiar contexts, students use prior knowledge, peer interactions enhance motivation and guide construction of knowledge, and students have opportunities to reflect on problems and state what knowledge has been developed. The goal of teaching proportional reasoning and graphing through scientific contexts to high school students will be to develop meaningful, useful, conceptual and procedural knowledge through problems that carry significance for the learners. Students will connect pieces of information to a grander overview of these mathematical topics and acquire visualization and conceptualization skills without missing out on the algorithmic procedures needed to perform well on more traditional assessments that emphasize rote math manipulation skills. In a study of mathematical modeling instruction for proportional reasoning in pre-‐adolescents, students’ conceptual understanding exceeded procedural understanding (Harel & Confrey, 1994. All students need a firm, connected foundation of conceptual and procedural knowledge, and teaching proportional reasoning and interpretation of graphs through scientific contexts may be the basis for this foundation.

Method Investigator 1: Investigator 1 teaches at Red Mountain High School in Mesa, Arizona. Mesa is a suburb of metropolitan Phoenix, and the high school has an enrollment of approximately 2800 students in grades 10-‐12. The population of the school is approximately 82% Caucasian, 11% Hispanic, 4% Asian, 2% African American, and 2% Native American. 18% of students are economically disadvantaged. Investigator 1 worked with 36 regular level math students who had earned a D in at least one semester of either their algebra 1 or geometry courses. Most of these students were juniors. Investigator 2: Investigator 2 teaches at Desert Vista High School in Tempe, Arizona, a suburb of Phoenix. Desert Vista has an enrollment of approximately 3000 students in

grades 9-‐12. The population of the school is approximately 78% Caucasian, 8% Hispanic, 8% Asian, 6% African American and 1% Native American. 8% of the students are economically disadvantaged. Investigator 2 worked with 21 regular physics students. The math prerequisite for physics is to have completed Algebra 1-‐2. Investigator 3: Investigator 3 teaches at Buckeye Union High School in Buckeye, Arizona, a suburb of Phoenix until 2010. Buckeye Union has an enrollment of approximately 1300 students in grades 9-‐12. The population of the school is 18% Caucasian, 10% African American, 70% Hispanic, 1% Asian and 1% Native American. 90% are economically disadvantaged. He worked with 50 Algebra 2 students. Algebra 2 is primarily populated by juniors and is part of the required progression for most students. Control Group 1: This control group was comprised of 33 regular algebra 2 students from the same school as Investigator 1. These students had earned a C or higher in every semester of their algebra 1 and geometry courses. Most students were juniors. Control Group 2: This control group was comprised of approximately 22 regular algebra 2 students from the same school as Investigator 2. Most of these students were juniors. Control Group 3: This control group was comprised of 23 regular algebra 2 students from the same school as Investigator 3. Most of these students were juniors. Pre-‐assessment of student abilities: The assessment used in this study is the Mathematics Concepts Inventory (MCI), version 7. The pre-‐test was given during the first week of school to determine initial mathematical reasoning abilities of the students before treatment, specifically in proportional reasoning and interpreting graphs. Treatment: Students in the treatment group were introduced to proportional reasoning and graphical interpretation through modeling based labs and activities. The treatment was implemented in multiple lessons throughout the 1st semester and third quarter of the 2008-‐2009 school year.

1. Students began by doing an investigative lab in which they collected data and observed relationships between variables first-‐hand. More detail about specific labs done by the individual investigators will be described later in this paper. 2. Students then examined the data and created graphs that they felt were appropriate. The students created these graphs by hand and with the use of technology available in the classroom. 3. Students then examined the elements of their graphs, including the meaning of slope, y-‐intercept, shape of curve, and units. Students generally whiteboarded their findings and articulated this meaning as a group.

4. Students then generalized the relationship they found between the variables being investigated in the lab. They expressed their findings in terms of proportionality and, through a whiteboard session, had an opportunity to articulate their findings. The instructor focused the conversation on the proportionality between the variables. 5. Students finally were given various application word problems in which they applied similar rationale as they did in the labs. These were designed to allow students to apply their knowledge of both concepts in different contexts using a similar method. These were also discussed as a class in a whiteboard session after they are completed.

Assessment: Student thinking was assessed in a variety of ways. First, the students took the MCI at the beginning of the year. Students were asked to solve application problems using the math reasoning skills developed during in-‐class activities. Each investigator interviewed several students utilizing a think-‐aloud protocol. In these interviews, the investigators elicited students to explain their reasoning out loud as they worked problems without any leading questions. Various student whiteboards were photographed and some student worksheets were kept so that the progress of students could be monitored. At the end of the third quarter, students took the MCI again. Mathematics Concepts Inventory (MCI), vs. 7: The MCI was developed by the Physics Underpinnings Action Research team at Arizona State University (ASU) in June 2000 and revised six times in the next three years. It has 23 questions and is intended for 8th and 9th grade students. The first eight questions are paired, on scientific thinking skills (conservation of mass and volume, proportional reasoning, control of variables). They were recommended by Professor Anton "Tony" Lawson, ASU School of Life Sciences, from his Classroom Test of Scientific Reasoning, a widely-‐used research-‐informed instrument. Other questions are released TIMSS, AIMS, and other standardized test questions: they are on graphing interpretation skills (#10-‐12,19-‐23); relating linear equations to other representations (#9,15,18); estimating area (#13) and volume (#14); measurement (#16) and mean value (#17). In an ASU study of 8th and 9th grade students in science and math classes in four suburban Phoenix public school districts, the baseline MCI posttest mean score in spring 2006 was 52%. In 2006-‐2007, after their teachers took a three-‐week summer physical science with math Modeling Workshop, their mean class MCI score was 50% pretest (August 2006) and 58% posttest (spring 2007) (173 students; matched students, teachers, and courses). The MCI reliability estimate (Cronbach’s alpha coefficient) for the 2007 posttest is 0.83. When drawing inferences from group level data (like much educational research), a reliability estimate over 0.80 is often considered sufficient. The MCI is at http://modeling.asu.edu/MNS/MNS.html.

Data and Analysis The data compiled for the experimental and control groups were examined in several different ways in order to draw conclusions about the method implemented in this study. Below is a summary of some significant findings from an examination of the data regarding pre and post-‐test scores on the Mathematics Concepts Inventory. Raw data comprise the appendix. F-‐tests showing that the experimental and control groups in this study can be assumed to be from the same population are available from the authors. Experimental Group – 107 Students Comparing pre and post-‐test data is an effective way to measure the success of a given teaching style. For the experimental group, pre-‐test scores showed a mean = 12.4 (~54%) with a standard deviation = 3.65. Post-‐test scores showed a mean = 14.1 (~61%) with a standard deviation = 3.99. A two-‐sample for means z test shows this to be a significant increase in scores from pre to post-‐test. z-‐Test: Two Sample for Means

Pre-‐Test Post-‐Test Mean 12.4 14.1 Standard Deviation 3.65 3.99 Known Variance 13.3 16.0 Observations 107 107 Hypothesized Mean Difference 0 z -‐3.09 P(Z<=z) one-‐tail 0.000993 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.00199 z Critical two-‐tail 1.96

A comparison of histograms for pre and post-‐test scores shows this positive shift in mean scores.

Mean = 12.4 S.D = 3.65 n = 107

Experimental Group Pre-‐Test Control Group Pre-‐Test Mean 11.7 13.1 Variance 16 11 Observations 50 23 df 49 22 F 1.46 P(F<=f) one-‐tail 0.17 F Critical one-‐tail 1.91

Mean = 14.1 S.D. = 3.99 n = 107

Control Group – 75 students The control group used in this study did not show quite as significant gains between pre and post-‐test scores on the mathematics concepts inventory. Pre-‐test scores showed a mean = 14.5 (~63%) with a standard deviation = 3.46. Post-‐test scores showed a mean = 14.6 (~63%) with a standard deviation = 3.62. A two-‐sample for means z test shows that this increase is not significant. z-‐Test: Two Sample for Means

Pre-‐Test Post-‐Test Mean 14.5 14.6 Standard Deviation 3.46 3.62 Known Variance 12.0 13.1 Observations 75 75 Hypothesized Mean Difference 0 z -‐0.161 P(Z<=z) one-‐tail 0.435 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.872 z Critical two-‐tail 1.96

Histograms of the pre and post-‐test scores show the similar distribution for each test.

Mean = 14.5 S.D. = 3.46 n = 75

Mean = 14.6 S.D. = 3.62 n = 75

Experimental versus Control Directly comparing the experimental and control groups in this study revealed some things that support the researchers’ belief that teaching mathematical concepts through scientific experiments is advantageous for high school students. It has already been stated that the experimental group showed a significant increase in mean scores on the MCI pre and post-‐test, while the control group increased by a relatively small amount.

The above chart shows that the control group did in fact have a higher post-‐test mean than the control group, but a two-‐sample for means z test shows that this difference in means is not significant.

z-‐Test: Two Sample for Means

Experimental Post-‐Test Control Post-‐Test Mean 14.1 14.6 Known Variance 13.1 16.0 Observations 107 75 Hypothesized Mean Difference 0 z -‐0.870 P(Z<=z) one-‐tail 0.192 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.384 z Critical two-‐tail 1.96 One method of comparing gains between different populations is by analyzing the normalized gain for each group (Hake, 1998). This statistic measures a student’s gain from pre to post-‐test by taking the difference between pre and post-‐test scores and dividing it by the total possible gain; thus:

score)test -pre(score) (maximumscore)test -(pre score)test -(postg

!!

>=<

Hake’s study of high school and college physics instruction led him to consider a value of g < 0.3 as a low gain, 0.3 < g < 0.7 as a medium gain, and g > 0.7 as a high gain. Average normalized gains for the experimental and control groups are shown in the chart below.

We saw previously that the control group had a very small increase in means from pre to post-‐test. Here we see that the average normalized gain for the control group is near zero. A two-‐sample for means z-‐test shows that the difference in mean normalized gains is significant between these two groups.


Experimental <g> Control <g> Mean 0.140 -‐0.040 Known Variance 0.096 0.23 Observations 107 75 Hypothesized Mean Difference 0 z 2.91 P(Z<=z) one-‐tail 0.00179 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.00359 z Critical two-‐tail 1.96 The above data give some indication that teaching mathematics concepts through scientific laboratory activities may significantly improve student scores on the MCI. It is evident that students in the experimental group made greater gains from pre to post-‐test than students taught in a traditional lecture style format. Because both the experimental and control groups consist of students from three different schools, it is important to examine data collected at each individual school.

Investigator 1 Russell Bray, Mathematics Teacher, Red Mountain High School

Method The following are the basic blueprints of the labs used for the experimental group at Red Mountain High School. Helicopter lab Focus: This lab was introduced as the first in a series to have the students focus on how two quantities change relative to each other. The directions are vague by design so there can be classroom discourse on what the quantities are that the students choose to compare. This opened dialogue for the students and introduced the nature of the course structure.

Equipment: The following picture and whiteboards.

Activity: The students were asked to graph the position versus time of the helicopter’s path based on the picture. In small groups, they presented what their graphs looked like and discussed the similarities and differences, all the while focusing on specifically what “position” they were measuring. Some students chose the crow’s distance from the original spot of the helicopter while others chose the odometer reading of the helicopter as their position. Others chose the helicopter’s altitude above the Canyon as their position. Through class discourse and variations of possible graphs presented by the teacher, the students were able to focus on a wide sample of different position versus time graphs. Though little numbers were involved, the idea of co-‐variance of quantities was introduced, leading to equation building in future labs. Bowling ball lab Focus: This lab focused on developing the students’ ability to reason proportionally, interpret linear position versus time graphs, explain the physical meaning of the slope of a graph, and find the equation of a line of a data set. Equipment: Bowling ball.

Activity: A demonstration of a bowling ball’s apparent constant motion along the ground is demonstrated to the class. In lieu of collecting data, a variety of different tables of values are given to the students. Each table of values can easily be interpreted as data collected from an actual bowling ball rolling on level ground. Some samples are described below:

Time (seconds)

0 1 2 3 4 5 6

Position (feet)

0 4 8 12 16 20 24

Time

(seconds) 0 1 2 3 4 5 6

Position (feet)

20 22 24 26 28 30 32

Time

(seconds) 0 1 2 3 4 5 6

Position (feet)

15 11.5 8 4.5 1 -‐2.5 -‐6

The students were asked to first describe the physical motion of the bowling ball. From there, they were asked to determine characteristics of the motion, paying special attention to the original position and the change in position per unit of time. Using that information, the students developed the equation of motion of each of the data situations. Through discourse and class discussions, they related each of the graphs and equations to each of the others to develop the concepts of slope and y-‐intercepts. Spaghetti lab Focus: This lab focused on developing the students’ ability to reason proportionally, explain the physical meaning of the slope of a graph, and find the equation of a line of best fit for a data set. Equipment: Dry spaghetti, paper clips, and coffee mugs. Activity: The students were asked to place set amounts of dry spaghetti across two coffee mugs (or any two heavy objects) and then count the number of paper clips needed to hang on the spaghetti before the dry spaghetti broke. The students recorded the data in individual groups and then all groups combined their results into one data set. The students then graphed the data, made a line of best fit, and calculated the equation of the line.

The students then discussed the results and interpreted the meaning of the line’s slope and y-‐intercept. This lab also introduced the effect of actual data collection and graphical interpretation of a non-‐linear model. The data showed a slight “up” curve, and students were led into a discussion that not all graphs are linear in nature. Circle lab Focus: This lab focused on developing the students’ ability to reason proportionally, interpret linear circumference versus diameter data, explain the physical meaning of the slope of a graph, and develop the equation of the circumference of a circle. Equipment: Various circle objects and measuring devices. Activity: Students were asked to measure the diameter and circumference of various circular objects. They then pooled their data into one class set and graphed the relationship between circumference and diameter of a circle. The students discussed the value of the slope (pi) and developed the equation of the circumference of a circle given its diameter (C = πd). Golf ball lab Focus: This lab focused on developing the students’ ability to reason proportionally, interpret quadratic position versus time graphs, and verbally discuss the differences between linear and quadratic situations. Equipment: Video of golf ball in freefall and Logger Pro. Activity: Students were shown a video of a ball in freefall after being bounced off the ground. The ball travels to a maximum height and then returns to the ground. The students loaded the video into Logger Pro and determined points using the video capture feature. The data was plotted as position above the ground versus time.

The students were led into a discussion of how the quantities of position and time are varying and the meaning of the “slope” and vertical intercept. The students discussed why the graph is curved and how that compares to lines seen in the past. These data also introduced the need for another mathematical representation other than the linear f(x) = mx + b equations used in the previous labs. By doing a curve fit of the data, the students are led to the quadratic equation f(x) = Ax2 + Bx + C

Data The following data analysis is based on data that can be referenced in the appendix. Experimental Group – thirty-‐six students Pre-‐test scores on the twenty-‐three question MCI showed a mean = 12.1 and standard deviation = 3.16. Post-‐test scores showed a mean = 14.2 and standard deviation = 3.49.

A paired two sample for means t test shows this increase between pre and post-‐test scores to be statistically significant with α = .008. z-‐Test: Two Sample for Means

Experimental Pre Experimental post Mean 12.1 14.2 Standard Deviation 3.16 3.49 Known Variance 9.96 12.0 Observations 36 36 Hypothesized Mean Difference 0 z -‐2.67 P(Z<=z) one-‐tail 0.00384 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.00768 z Critical two-‐tail 1.96

A look at the distribution of pre and post-‐test scores for the experimental group shows this shift in mean and standard deviation.

Mean = 12.1 S.D. = 3.16 n = 36

Mean = 14.2 S.D. = 3.49 n = 36

Control Group – thirty students Pre-‐test scores on the MCI revealed the following: mean = 15.7 and standard deviation = 3.14. Post-‐test scores showed a mean = 16.2 and standard deviation = 2.88.

A paired two sample for means t test shows that the difference between average scores on the pre and post-‐test is not statistically significant. z-‐Test: Two Sample for Means

Control Pre Control Post Mean 15.7 16.2 Standard Deviation 3.14 2.88 Known Variance 9.87 8.3 Observations 30 30 Hypothesized Mean Difference 0 z -‐0.642 P(Z<=z) one-‐tail 0.260 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.520 z Critical two-‐tail 1.96

The distribution of pre and post-‐test scores for the control group shows little change.

Experimental versus Control Group When comparing experimental and control group pre and post-‐test scores on the MCI for students at Red Mountain HS, a few things of statistical significance are found. The average normalized gain for the experimental group in this study is <g> = 0.20. This normalized value falls into what is considered the “low gain” category by Hake. The low

Mean = 15.7 S.D. = 3.14 n = 30

Mean = 16.2 S.D. = 2.88 n = 30

gain category includes values of <g> such that 0 ≤ g ≤ 0.3. However, when compared to the average normalized gain of the control group, <g> = 0.06, it can be seen that the difference in these means is statistically significant with α = 0.000.


Experimental <g> Control <g> Mean 0.204 0.0599 Known Variance 0.024 0.026 Observations 36 30 Hypothesized Mean Difference 0 z 3.69 P(Z<=z) one-‐tail 0.00011 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.000227 z Critical two-‐tail 1.96

Student Interviews Interview with Students in Regard to Problem #5 (MCI) Student 102 Researcher: Why did you choose answer B for problem number 9? Student 102: If it was 6 on the wide cylinder, then it will be 9 on the skinny cylinder. Researcher: I realize that is what answer B says, but why did you choose it? Tell me

your thinking on how you got that answer. Student 102: If it goes 6 up on the skinny for every 4 on the fat, then when you put in

50% more, then it will go 50% more on the skinny cylinder. Student 129 Researcher: Why did you choose answer B for problem number 9? Student 129: I put down the fraction 4 over 6 and that reduces to 2 over 3. So when

this is 6, then I have to change this to 9. Researcher: But why did you “change it to 9” rather than change it to a different

number? Student 129: Because 6 is 3 times bigger than 2 so I had to multiply 3 [points to

denominator of reduced fraction] by 3 to get 9. Student 131 Researcher: Why did you choose answer B for problem number 9? Student 131: Because if the water goes up 2, then you will go up 2 on the other

cylinder. Comment: The first two students [102 and 129] demonstrate strong proportional reasoning by verbalizing the idea that as one quantity changes, the other changes as a multiple of the first. Student 102 uses percentages while student 129 literally uses multipliers, yet both convey valid logic to determine the correct answer. Student 131 does not show this same knowledge of proportional reasoning. The student uses “addition steps” rather than multipliers from one cylinder to the other inaccurately. Interview with Students in Regard to Problem #19 (MCI) Student 133 Researcher: Why did you choose answer D for problem number 9? Student 133: I knew it wasn’t moving at first so it had to be either B or D. And then I

was going to pick B, but then I changed my mind. Researcher: Why did you choose your mind? Student 133: Because the position was getting closer to the origin so it had to be D. Researcher: What do you mean “closer to the origin”? Student 133: The thing is going back to where to the start or wherever zero is. It is just

going the other way. Back towards the origin.

Student 112 Researcher: Why did you choose answer B for problem number 9? Student 112: The thing isn’t moving, then it rolls down the hill. Researcher: How do you know it is not moving at first? Student 112: Because it is not moving its direction for the first bit. Researcher: Its direction? Student 112: I mean its position. Researcher: So then, how can you tell it rolls downhill? Student 112: The thing rolls down the slope and stops at the bottom [points to graph]. Student 102 Researcher: Why did you choose answer B for problem number 9? Student 102: The object is not moving at first, but then it drops to zero then stops. It

was on top of a hill like 10 feet up, then rolls down to like 0. Researcher: What is the position a measure of in this graph? Student 102: The height above the ground. Researcher: The object is above the ground? Student 102: No. No, the object is above the bottom of the hill. Like, its sea level thing.

Altitude. Comment: Student 133 demonstrated logical proportional reasoning in describing the situation of the position versus time graph. The student eliminated two answers based on the constant change in position then realized that the origin was not the point (0,0), but rather the zero of the object’s reference point to position. This student was one of six who scored this question correctly on the post-‐test. Student 112 showed a shape thinking logic to the problem; that the object follows the path of the graph. The student recognizes that the object’s motion is zero at the beginning, but then points to the “downhill” portion of the graph. However, student 102 also chose the same answer as student 112, but gave a valid reasoning to his answer. He defined his position as distance above the ground (vertical distance) and not the horizontal distance assumed for the correct answer. This does raise an interesting point in the validity of question #19 on the MCI. If two correct answers can be determined by defining the position differently, then revision on the MCI should be considered.

Analysis Examining the class means for pre and post-‐test scores on the MCI shows a significant increase in scores from pre to post-‐test for the experimental group but essentially the same pre and post-‐test mean for the control group. Both the experimental and control groups achieved a positive normalized gain from pre to post-‐test, but the control group’s gain is statistically significant. These two tests indicate that working through various labs, collecting data, graphing, analyzing, and whiteboarding ideas may help students develop mathematics concepts. The experimental group at Red Mountain High School was chosen in large part due to the nature of the students and the course design. Each student has been, in some way, lower than average in mathematics as indicated by the D or lower they received as a grade in at least one semester of the prerequisite courses of algebra 1 or geometry. Because they lacked the skills to continue to algebra 2, a course was designed to bridge that gap. This course was very open-‐ended in its nature, but had a common focus on real world-‐based data. The control group was chosen to reflect a course that was very set in its traditional approach, and the most reasonable choice was an algebra 2 course. However, during the second half of the school year, the control group was introduced to mathematical concepts in a non-‐traditional way under the guidance of Arizona State University’s National Science Foundation Math-‐Science Partnerships grant, Project Pathways, curriculum. As part of this process, the control group also participated in the golf ball lab. Though both groups had treatments, the experimental group was engaged in this treatment from the beginning of the school year while the control group had a much smaller dose confined to the 2nd semester. The experimental group had the benefit of ample time to investigate and develop mathematical concepts introduced in classroom activities. There were no outside influences pushing the class such as district exams or the need to pass the Arizona Instrument for Measuring Standards (AIMS). The class progressed at a leisurely and fluid pace throughout the year, something that is often not possible in most mathematics classrooms. This extra time led to very enriching and lively discussions on content and application of mathematics outside the confines of the classroom. Another luxury of the course was that, even though the students had not been top performers in mathematics, the course was optional. The students had the minimum math credits to graduate, but were sufficiently motivated to continue to either prepare for the algebra 2 course or their college courses for the following year. A secondary goal of conducting the course in a non-‐traditional way was to allow communication skills to flourish in the context of math and science. As the school year

progressed, students grew accustomed to clearly expressing ideas to their classmates, while respecting, encouraging, challenging, and understanding the ideas of others.

Conclusion

Having the two experimental and control groups take the same course was not feasible at Red Mountain High. It is interesting to note, however, that the control group showed positive normalized gains while being treated in a smaller dose of the experimental treatment. The discourse of the experimental group is what really set it apart from a traditional mathematics course. The time that could be used to develop ideas, whiteboard results, and come to conclusions within the class was generous and well received by the students. The students articulated their thoughts and demonstrated it to their peers. This may be the largest factor of the significant gains on MCI pre and post-‐test scores.

Investigator 2 Adrian Boyarsky, Biology and Physics Teacher, Desert Vista High School

Method This study included a control group and an experimental group. The control group was made up of 22 juniors and seniors in a regular Algebra 2 class. These students took the Mathematics Concepts Inventory (MCI) within the first two weeks of school in fall 2008. The class ran as normal for 27 weeks, and students took the MCI again at the end of the 3rd quarter in March 2009. The experimental group included 21 juniors and seniors. Twenty of these students were concurrently enrolled in regular Algebra 2, and one student was concurrently enrolled in a regular pre-‐calculus class. These students were dispersed among five different math teachers, but all of these students were enrolled in regular physics 1, a Modeling Instruction mechanics class. These students also took the MCI during the first week of class and again at the end of the 3rd quarter. Students in the regular physics class worked through a student-‐centered, model-‐based mechanics curriculum and were measured for gains in understanding of mathematics concepts. Each unit began with an inquiry based lab activity in which students observed some phenomenon and found relationships that might exist between variables. Students learned how to express relationships in several ways including diagrammatically, graphically, mathematically, and verbally. Students presented their findings in each lab using all of these methods. Students shared findings and the class would reach consensus on any relationships that existed between experimental

variables. The second step of each unit focused on using these relationships to solve physics problems. This included some supplementary labs, worksheets, and teacher generated problems. Each unit concluded with either a lab practical or culminating project in which students had to use their understanding of generated models to solve larger problems. No special mathematical instruction was given to the experimental group in this study other than the mathematics needed to learn physics through a modeling approach.

Data QUANTITATIVE DATA Experimental Group: Pre-‐test scores on the twenty-‐three question MCI reveal a mean = 14.42 (~63%). Post-‐test scores show a mean = 17.1 (~74%). A paired two sample for means t test shows the increase between pre and post-‐test scores to be statistically significant with α = .0003. t-‐Test: Paired Two Sample for Means Experimental Pre Experimental Post Mean 14.4 17.1 Standard Deviation 3.82 3.72 Observations 21 21 Hypothesized Mean Difference 0 Df 20 t Stat -‐4.37 P(T<=t) one-‐tail 0.00015 t Critical one-‐tail 1.72 P(T<=t) two-‐tail 0.00030 t Critical two-‐tail 2.09

The following graphs show the distribution of scores for both the pretest and posttest for the experimental group. While the sample is small, a definite shift in the data can be seen. With the experiemental group, the shift is to the right.

Control Group: Pre-‐test scores on the MCI reveal a mean = 14.22 (~62%). Post-‐test scores show a mean = 15.28 (~66%). A paired two sample for means t test shows this increase between pre and post-‐test scores to be statistically significant with α = .014.

Mean = 14.4 S.D. = 3.82 n = 21

Mean = 17.1 S.D. = 3.72 n = 21

t-‐Test: Paired Two Sample for Means Control Pre Control Post Mean 14.2 15.3 Standard Deviation 3.55 2.51 Observations 22 22 Hypothesized Mean Difference 0 Df 21 t Stat -‐2.67 P(T<=t) one-‐tail 0.00720 t Critical one-‐tail 1.72 P(T<=t) two-‐tail 0.0144 t Critical two-‐tail 2.08 The following graphs show the shift between pre-‐test and post-‐test scores for the control group. Unlike the shift to the right of the experimental group, the control group shows more of a shift to the middle.

Mean = 14.2 S.D. = 3.55 n = 22

Experimental versus Control Group While both the control group and the experimental group showed statistically significant increases, the amount of increase is different. The graph below shows the pre-‐test and post-‐test means of both groups.

When comparing experimental and control group pre and post-‐test scores on the MCI for students at Desert Vista, a few things of statistical significance are found. The average normalized gain for the experimental group in this study is <g> = 0.33. This is considered by Hake to be a medium gain. The average normalized gain of the control group, <g> = 0.06, is considered to be a low gain.

Mean = 15.3 S.D. = 2.51 n = 22

t-‐Test: Two-‐Sample Assuming Equal Variances

Experimental <g> Control <g> Mean 0.328 0.0713 Variance 0.0623 0.0521 Observations 21 22 Pooled Variance 0.0571 Hypothesized Mean Difference 0 df 41 t Stat 3.52 P(T<=t) one-‐tail 0.000532 t Critical one-‐tail 1.68 P(T<=t) two-‐tail 0.00107 t Critical two-‐tail 2.02

Qualitative Data Classroom Notebooks Two qualitative instruments were used with the experimental group only. The first was class notebooks which housed all the work each student did for the entire course, including labs, worksheets, and projects. As the course progressed, these were periodically collected. For the purposes of this project, the notebooks would show any progression that had occurred in the student’s understanding of mathematical concepts and how they might be applied in different situations. Students were scored on how well they showed relationships mathematically using graphing and algebraic methods. It was clear that while students were able to create mathematical models for data collected throughout various labs and graphically represent them, they failed more times than not to be able to create meaning for these relationships. Students got better

and better at manipulating variables in a lab setting, finding co-‐variation within the variables and expressing the relationships. The most difficult part for these students, even at the end of the course, was to express the meaning of these relationships in English without using numbers. While there were a few exceptions, this was a general trend that was seen most of the time. For example, questions where numbers were given and students had to solve for one or more variables were answered correctly the majority of the time, whereas questions involving conceptual reasoning without numbers gave students considerable trouble.

Post-‐instruction Interview Problems The second qualitative data set collected was a set of interviews. During the interview, each student was given questions similar to those found on the MCI. They were then asked to solve the problem and describe their methods. A camera recorded their written work. The first question posed was a variation of a question on the MCI: To the right are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4th mark (see A). This water rises to the 6th mark when poured into the narrow cylinder (see B). Water is now poured into the narrow cylinder (described in Item 5 above) up to the 11th mark. How high would this water rise if it were poured into the empty wide cylinder? a. to about 7 1/3 b. to about 7 1/2 c. to about 8 d. to about 9 e. none of these answers is correct

Student 1 response:

A) B)

A) Student recognizes the correct ratio. B) Student lists many correct relationships using the ratio, but has

trouble since the answer is not a whole number. Chooses 7 ½ because he does not calculate the ratio correctly.

Student 2 Response:

A) B)

A) Student correctly recognizes the ratio. B) Student picks the correct answer by using the ratio 22:33.

Student divided 11 by 33 on the calculator and then multiplied the answer by 22. While the work is not shown on the paper, it does lead the student to the right answer.

Student 3 Response:

A)

B)

A) Student correctly realizes the ratio and the relationship between the 2 situations. He sets up an equation.

B) Student uses basic algebra to isolate the variable and solve the question correctly.

The last question posed to the students was as follows: Mr. Smith leaves city 1 and travels towards city 2 at a constant speed of 40mph. At the same time, Mr. Jones leaves city 2 and travels towards city 1 at a constant speed of 55mph. The cities are 500 miles apart. How long will it take for Mr. Smith to pass Mr. Jones on the road? How far from city 1 will they pass each other?

Student 1 response:

A) B)

A) Student 1 began by creating a diagram that represented the situation.

B) Student 1 then recognized the relationship between variables. He uses the given data to determine the time it takes for each man to cross the entire 500 miles instead of how long it takes for them to pass each other.

C) When the student realizes that the times are not alike, he decides to take the average of the two values to get his time. The student stops after this step.

Student 2 Response:

A) B)

C)

A) Student begins with a diagram of the situation, labeling directions, speeds and distances.

B) Student realizes that since there is a constant speed he can calculate how far the two men will travel. Rather than setting up equations, the student makes distance calculations for both men using the same time until they reach the same position.

C) Once student finds the position, he records how long it took the men to get there and how far each has traveled.

Student 3 response:

A) B)

C) D)

A) Student begins by writing an appropriate formula for one of the men. Student substitutes in one of the men’s data and gets a time. The student calculated the total time for Mr. Smith to travel the entire 500m.

B) Student substitutes his time in to solve for distance. He gets 500miles as an answer but does not seem bothered by this.

C) The student does assume that the two men are traveling for the same time and uses this to solve for Mr. Jones. The student has found how far Mr. Jones would travel while Mr. Smith traveled the entire trip.

D) The student takes several minutes to try to understand the relationship between his two distances. He subtracts one from the other to get his final answer. No explanation for this is given.

Analysis By examining the results of the pre and post MCI tests, significant increases in both the control group and experimental group can be found. However, the increases in the experimental group are significantly higher than those in the control group. All statistical data supports the fact that students in the physics class outperformed students that were only in the algebra 2 class. This suggests that a Modeling Instruction physics class might increase student understanding of mathematics concepts. The physics class provided opportunity for students to continually use mathematics to find relationships. By creating mathematical models and using these models to explain phenomena, students seem to gain a better understanding of specific math concepts. It is interesting to examine which questions on the MCI showed the most significant improvement in the experimental group. Most of the increase can be found in six specific questions. While the control group showed increased scores overall, results of the same six questions showed significantly less gain. The following are proportional reasoning problems:

5. Below are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the 4th mark (see A). This water rises to the 6th mark when poured into the narrow cylinder (see B).

Both cylinders are emptied (not shown) and water is poured into the wide cylinder up to the 6th mark. How high would this water rise if it were poured into the empty narrow cylinder? a. to about the 8th mark

b. to about the 9th mark

c. to about the 10th mark

d. to about the 12th mark

e. none of these answers is correct

6. Question number 5 is true because

a. the answer can not be determined with the information given. b. it went up 2 more before, so it will go up 2 more again. c. it goes up 3 in the narrow for every 2 in the wide. d. the second cylinder is narrower. e. you must actually pour the water and observe to find out.

In the pre-‐test, 23% of control group students got question 5 correct, compared to 19% of students in the experimental group. However, in the posttest, while the control group increased to 27% correct, the experimental group increased to 47%. While this percentage is still low, it is a drastic increase. These problems can help measure a student’s ability to do proportional reasoning. The evidence suggests that by being exposed to physics Modeling Instruction, students increase their abilities to reason proportionally. The proportional reasoning abilities can also be seen qualitatively in the student interviews. While one student struggled with the math, all the interviewed students were able to recognize the correct proportion and use their understanding of the proportions to explain the reasoning behind their answers.

15. Suppose you are moving to the right at 5 m/s. A timer starts the watch when you are 2 m from the starting line. Which equation best describes your position as a function of time?

a. p = 2t + 5 b. p = 5t + 2 c. p = 5t – 2 d. p = 5t

This problem is testing the student’s ability to represent a given situation with a mathematical equation. Scores for the control group on this question increased from 23% to 36% between pretest and posttest, but the experimental group increased from 24% to 62%. The significantly higher increase in the experimental group seems to suggest that physics Modeling Instruction causes a better understanding of how to represent relationships mathematically. In regular math classes, students are very rarely asked to do this. More often, they are given the relationship and asked to solve for variables in a given situation. By having to constantly come up with their own mathematical formulas throughout the physics course, it should make sense that these students would become better at this skill. The results seen here definitely support the idea that delivering mathematics within a context where students must investigate and express relationships for themselves leads to higher math reasoning abilities than they might attain in a conventional math class.

The following question asks students to interpret the physical meaning of a graph:

19. Here is a graph of an object’s motion. Which sentence is a correct interpretation?

a. The object rolls along a flat surface. Then it rolls forward down a hill, and then finally stops.

b. The object doesn’t move at first. Then it rolls forward down a hill and finally stops.

c. The object is moving at a constant velocity. Then it slows down and stops.

d. The object doesn’t move at first. Then it moves backwards and then finally stops.

The data show that the experimental group scored better than the control group on this question on the posttest. While the proficiency of the control group did double from 18% to 36%, the experimental group’s proficiency tripled from 14% to 42%. While both groups showed significant increases in their ability to explain the meaning of the graph, the experimental group scored higher and had a larger gain. This result should make sense to those who teach using the modeling approach. In every lab, students are asked to create graphical representations of their data and explain the meaning of the graph. This practice requires them to see a graph and determine its significance. Because this question is asked in a context with which the physics class has experience, it may be more informative to test a graphical representation that physics students are not as familiar with, to see if they still achieve better results than students not in the physics class.

p o s i t i o n

time 0

The following question tests simple graphical analysis:

In an experiment, a lab group graphed the mass and volume of samples of two substances,

A and B to determine the density of each.

!

density =massvolume

20. Which of the statements about the substances is correct?

a. The density of substance B is about 6 g/cm3. b. A 6 cm3 sample of A is has greater mass than a 6 cm3 sample of B. c. The density of substance A is greater than the density of substance B. d. A 20 g sample of A has a larger volume than a 20 g sample of B.

The results to this question were quite shocking. The control group showed no gain in answering this question between pre-‐test and post-‐test, with a total of only 23% of students answering correctly both times. The experimental group had 33% of students answer this question correctly on the pre-‐test, but a large gain to 76% by the post-‐test. This stunning result could be the result of the continuous exposure to graphs in the physics class. Students worked with graphs like these almost every day. The practice seems to have improved their graphical analysis skills, an extremely important Algebra 2 topic. The evidence suggests that by going through the modeling-‐based physics class, the students in the experimental group gained more in the area of graphical analysis than students enrolled only in a math class. The final question in which the experimental group significantly outgained the control group is shown below. It asks about measurement technique. The results should be noted, but the concept of measurement is one that is taught in the science class, not necessarily the math class. The results do not add much to the idea that a modeling science curriculum increases student achievement in math.

16. You are measuring the length of objects in the lab with the ruler below. Which of the following best expresses the width of the business card?

a. 5 cm

b. 5.0 cm

c. 5.05 cm

d. 5.50 cm While the quantitative test data indicate that students enrolled in a Modeling Instruction physics class achieve higher in specific mathematical concepts, the level of that achievement is better seen when analyzing the qualitative data. When asked in interviews to solve a complex algebraic problem, the physics students struggled. Not one of the interviewed students used successful graphing or algebra skills to solve the problem, even though it was asked in a context with which they were familiar. Also, when asked a slightly more difficult version of the proportional reasoning problem, students struggled with the math needed to solve the problem. While all students recognized the proportional relationship, solving for an unknown value proved to be challenging for two of the three students described in this paper. In addition, this trend was also seen in the collected lab notebooks throughout the year. Students were unable to express more complex relationships or solve problems with multiple unknowns with any type of consistency. The basic level of understanding, however, showed an increase as the year progressed and students became more proficient at proportional reasoning and graphical analysis.

Conclusions Before proclaiming bold conclusions about the data collected at Desert Vista, it is important to understand the circumstances that were different at that location compared to Buckeye or Red Mountain. First, the sample size of both the control group and the experimental group were comparatively small. In order to be more confident as to the meaning of the information gained, it would be nice to have worked with larger numbers of students. Second, unlike the other schools in this study, the experimental group at Desert Vista was not a non-‐traditional math class. All the students in the

experimental group were enrolled in math, most in Algebra 2. They all had different math teachers with different teaching styles that could have had a severe impact on the results obtained here. The control group, on the other hand, was all from the same teacher’s classroom and therefore exposed to the same teaching style in math the entire year. The key factor that separates the experimental group from the control group was the enrollment of the experimental group in the Modeling Instruction physics class. Because of these factors, the conclusions made, based on results found by both quantitative and qualitative observations should not be mistaken for absolute fact. With that being said, there definitely is evidence to support some general conclusions. The data obtained from pre-‐tests and post-‐tests of the MCI suggest that being exposed to the modeling method of instruction in a physics class helps improve student achievement in mathematics concepts. This result should not be shocking to those who have experienced the ASU Modeling Instruction courses. These results indicate that if similar practices were used in a mathematics classroom, students would have higher achievement than in a traditional math class. Results also suggest that schools would be wise to develop ways to integrate math and science classes. At Desert Vista, higher level classes have already taken this route as calculus and advanced physics teachers work together. This approach might work even better for students at lower levels where development of key math concepts is slowest. It was embarrassing to see that in pre-‐test data, both the control group and the experimental group had less than 30% of students be able to take information correctly from a graph. By integrating math concepts within the context of science inquiry at middle school, freshman and sophomore levels, development of key mathematics concepts might be enhanced drastically. The data also indicate which specific mathematical concepts a modeling approach develops. Proportional reasoning and graphical analysis abilities seemed to be the most drastically increased basic math concepts due to the physics instruction. Students also found it easier to express phenomena in terms of numbers and equations; these skills are practiced more in the science class than in the math class but have a huge impact on student understanding of math. By looking at the qualitative data, the level of gain in these areas can be more accurately determined. Even though basic math concepts improved, the level of improvement was limited by the quality of instruction from a first year physics teacher new to Modeling Instruction. The instructor was also juggling five biology classes that consumed most of his prep time. Since Modeling Instruction is a complex skill that improves with experience, the results shown at Desert Vista could show even greater increase as the instruction improves. The depth at which students were able to show their understanding of relationships was expected to be less than what would be seen in an experienced modeler’s classroom. With experience, the results seen here could have been far more significant.

Investigator 3 Mark Henrion, Mathematics and Physics Teacher, Buckeye Union High School

Method In an effort to develop mathematical concepts in Algebra 2, labs were implemented at every opportunity throughout the first semester of the 2008-‐2009 school year. Labs were often used to introduce new topics that the class would be studying for the next several weeks. Each lab was followed by a whiteboard meeting where students shared their own ideas and interpretations of what had occurred throughout the experiment. The classes in which these labs were implemented will be referred to as the experimental group, while the classes consisting of traditional lecture, notes, and tests, will be referred to as the control group. All students took the twenty-‐three question Mathematics Concepts Inventory (MCI), a test developed by the Physics Underpinnings Action Research Team at Arizona State University in June of 2000, during the first week of school and again at the end of the third quarter in March 2009. The following is a brief overview of the labs implemented throughout the school year. Most of the students in both the experimental and control groups were in the eleventh grade. Buggy labs Focus The first two labs of this school year focused on developing the students’ ability to reason proportionally, interpret linear position versus time graphs, explain the physical meaning of the slope of a graph, and find the equation of a line of best fit for a data set. Equipment Buggy cars, batteries, stopwatches, meter sticks, and tape. Activity For the first part of this activity, each group of students was given a single buggy car and asked to develop a way to find the speed of the car using their equipment. Instructions given to the students were kept somewhat vague to allow for their own creativity, but it was requested that each group have some kind of position versus time graph at the end of their experiment. Most groups began measuring the amount of time it took for their buggy to travel various distances, thus producing a scatter plot with time on the horizontal axis and position on the vertical axis. The students put these data into their graphing calculators to produce graphs like the one shown below.

The next step was finding and interpreting a line of best fit for this data set. Our TI-‐84+ graphing calculators label the horizontal components of graphs with an “x” and the vertical components with a “y.” The students had to modify these labels to fit the needs of our experiment. An example of the calculator’s display and a student’s interpretation is shown below.

“The letter ‘y’ represents position, the letter ‘a’ is the slope, and the letter ‘b’ is the y-‐intercept. The slope has units of centimeters per second since centimeters are the units on our vertical axis and seconds are the units on our horizontal axis. The y-‐intercept has units of centimeters. This equation means that our car starts at 0.67 centimeters behind the ruler and travels at a constant speed of 8.65 centimeters per second after that.” After finding and interpreting the equation of a single buggy car, each group was given a second, faster buggy car and asked to run a couple of different experiments. First, students were asked to start the cars at the same position and graphically represent their motion. Graphs like the one below were common.

Based on this graph and the previous experiment, students began to understand that a faster car meant a steeper slope on their position versus time graph. The fact that both graphs were linear means that both cars were moving at a constant speed, and equal vertical intercepts mean that the cars started at the same position. Students were then asked to start their faster car behind their slower car and graphically represent their motion. The class was now getting graphs like the one below.

The car with the head start is represented by the line with a greater vertical intercept, and the faster car is represented by the line with the steeper slope. The intersection point of the two lines is where the faster car has caught up to and is passing the slower car. Finally, students were asked to graphically represent the motion of their buggy cars if they were set to run in opposite directions. Graphs like the following were common.

Here, the sign on the slope simply represents the direction of motion of the car. Some students had trouble getting past the idea that one car was traveling uphill while the other was traveling downhill, but eventually became convinced of the level nature of the floor in the classroom. For each experiment, students were asked to find and interpret the slope, vertical intercept, and intersection points of the lines. Students were also asked to write a clear English sentence explaining what exactly is happening in the experiment. Gravity lab Focus This lab was designed to further develop proportional reasoning, physical meaning of slope, and skills needed to find and interpret a line of best fit. Equipment Spring scales and hanging mass sets. Activity This activity began with a group discussion about how Earth’s gravity affects objects of different sizes. Some students were aware of the fact that a ten kilogram mass and a one kilogram mass should fall to the earth at the same speed, but none, through no fault of their own, had gone beyond that and concluded that the earth would need to exert a greater force on the larger mass if it were going to accelerate it at the same rate as the smaller mass. Eventually, one student compared it to moving a toy Hot Wheels car and a real car. If one were going to give each of these cars the same speed simply by pushing, one would need to push on the real, heavier car with a much greater force. From here, each group of students was given a spring scale and a set of hanging masses and asked to produce a graph and line of best fit relating force and mass. Graphs and lines of best fit like the ones below were produced.

One student interpreted her group’s graph and line of best fit by saying, “As the mass of an object goes up, so does the force the earth puts on that object. The earth has a 9.95 Newton force for every kilogram of mass.” Although the vocabulary here was a little rough, the student’s interpretation of slope was spot on. Pendulum lab Focus The purpose of the pendulum lab was to develop proportional reasoning using equipment and collecting data that would not produce a linear graph. This application of square root graphs in our everyday lives was an effort to stretch the students’ minds. Equipment Ring stands, string, and stopwatches. Activity For this activity, students were asked to produce graphs that related the period and length of a pendulum. A majority of the groups caught on pretty quickly that they could easily change the length of the pendulum and measure the period for different string lengths. One student worked on his own, put his data into Logger Pro, and came up with the following graph and equation.

Compared to the generally accepted equation for the period of a pendulum, T = 2π√(L/g), this is remarkably close. After finding graphs relating period and string length, students were asked to interpret their results. Rather than getting a straight line here, most groups came up with points that gave them a side-‐opening parabola and an equation that had a variable raised to a decimal exponent. The groups who came up with an exponent close to 0.5 were able to approximate that with a square root function. From there we could discuss what one would need to do to double, triple, or quadruple the period of their pendulum. In order to double the period of the pendulum, the length would have to be increased by a factor of four. Tripling the period requires a string length nine times that of the original string length, and quadrupling means an increase in string length by a factor of twenty-‐five. This proportional reasoning was what we were going after, at the start of this lab. Bowling ball lab Focus This lab was an introduction to quadratic models and another kind of proportional reasoning. Students observed an experiment that produced a quadratic position versus time graph and began to explore how corresponding velocity versus time and acceleration versus time graphs might look. Equipment Bowling ball, sidewalk chalk, stopwatches, long, smooth, gradual incline, and meter stick or large tape measure.

Activity This activity was different than the preceding labs in that the class performed the experiment together and shared the same data set. Starting at the top of the incline, students took a long tape measure and used sidewalk chalk to mark off one-‐meter increments. Each student was then given a stopwatch and assigned a position on the incline. When a student released the bowling ball at the top of the incline, each student started his or her stopwatch. As the bowling ball rolled past each student, he or she stopped his or her stopwatch. After doing a few runs the students began to notice that consecutive students toward the bottom of the incline had shorter gaps in time than consecutive students at the top of the incline. One student summarized this succinctly by commenting that “The ball is going faster toward the bottom so it takes less time to cover each meter.” This type of reasoning was precisely the goal of the lesson. After doing three runs and averaging the time for each position, the class headed back inside to examine the data in Logger Pro. The following graph was produced.

For the first five seconds, the graph appears to be linear, but it clearly takes a different path from five to six seconds. From here, the class did a curve fit on the data and came up with a quadratic equation that fit the graph. One student noted, “That shape makes sense because position is increasing faster than time.” This was a perfect lead-‐in to discussion of what a corresponding velocity versus time graph might look like. Since the ball was speeding up as it moved down the incline, the class agreed that the velocity

versus time graph would be a straight line with a positive slope. These graphs also helped some students get past the notion that a graph could be increasing in the positive direction even though the object in question was moving downhill. Tennis ball lab Focus The intent of this lab was for students to have more practice with applications and manipulation of quadratic equations. Equipment Tennis balls, stopwatches, meter sticks. Activity For this lab, students threw tennis balls as straight up in the air as possible and recorded the time it took to fall back to the ground. For each toss, meter sticks were used to measure the height at which the tennis ball was released. Although it is nearly impossible to throw an object straight up in the air, students ignored any horizontal motion of the object, focusing purely on the vertical. The class was not yet at a level of ability to handle launch angles and basic trigonometric functions. After obtaining an initial vertical height and a time of flight, students used the vertical motion equation yf = yo + vy0t -‐4.905t

2 to find their initial vertical velocity. From there, the equation vyf2 = vyo

2 – 19.62(yf-‐yo) was used to find the maximum height of the object. Using these two equations to find the desired information required a bit of reasoning on the part of the students. It took some a long time to realize that the time they had on their stopwatch was the time at which yf = 0. An even bigger jump came when it was determined that, when finding the maximum height of the ball, vyf = 0. Once students had these two pieces of information, the equations could be solved for the desired variable with some basic algebraic skills. Coffee filter decay lab Focus Students used this lab to get a feel for an application of decay graphs and horizontal asymptotes. Equipment Coffee filters or Styrofoam bowls and motion detector connected to a computer with Logger Pro. Activity For this lab, students placed a motion detector on the floor of the classroom and collected data by dropping different numbers of nested coffee filters straight down onto the detector. By doing a linear fit on the straight line part of the velocity versus time graph produced by Logger Pro, students were able to find the acceleration of the nested pile of coffee filters. As the number of coffee filters increased, so did the acceleration of

the falling object. When the number of filters reaches about ten, the acceleration begins to level off pretty close to the accepted value of gravity. By running this experiment for one to fifteen coffee filters, students were able to produce the following graph relating acceleration versus the number of coffee filters.

Here, the horizontal asymptote would be 9.81 m/s2, the accepted value of gravity near the surface of the earth. Kickball decay lab Focus This lab was run during a unit on geometric sequences. The goal was to let students explore the relationship between exponential decay graphs and geometric sequences with a common ratio between -‐1 and 1. Equipment Kickballs or basketballs and stopwatches with lap/split capabilities. Activity For this lab, students were asked to come up with a graph and equation relating the time between bounces of their ball and the bounce number. One student would drop the kickball from as high as possible while another would start the stopwatch. When the ball hit the floor, the student with the stopwatch would press the lap/split button, recording the time it took for the ball to reach the floor and bounce once. After the ball bounced up and back to the floor again, the student with the stopwatch would press the lap/split button again, recording the time between the first and second bounces. Continuing in this manner until the ball is completely at rest on the floor gives the students a data set of decreasing time intervals based on the bounce number. Plotting these points on a graphing calculator and running an exponential regression gives the students an exponential decay function which is easily related to a geometric sequence with common ratio between -‐1 and 1. The class also did a trial using the video

analysis capabilities of Logger Pro. This tool allowed for more accurate time measurements by using the statistics function in Logger Pro.

Data The data compiled for the experimental and control group were examined in several different ways in order to draw conclusions about the method implemented in this study. Below is a summary of some significant findings from an examination of the data. Experimental Group – fifty students Pre-‐test scores on the MCI reveal a mean = 11.72 (~51%) and standard deviation = 4.00. Post-‐test scores show a mean = 13.06 (57%) and standard deviation = 3.72.

A paired two sample for means z test shows this increase between pre and post-‐test scores to be statistically significant with α = 0.028 for a one-‐tailed test and α = 0.057 for a two-‐tailed test. z-‐Test: Two Sample for Means Experimental Pre Experimental Post Mean 11.7 13.1 Standard Deviation 4.00 3.72 Observations 50 50 Hypothesized Mean Difference 0 z -‐1.90 P(Z<=z) one-‐tail 0.0285 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.0570 z Critical two-‐tail 1.96 A look at the distribution of pre and post-‐test scores for the experimental group shows this shift in mean and standard deviation.

Mean = 11.7 S.D. = 4.00 n = 50

Mean = 13.1 S.D. = 3.72 n = 50

Control Group – twenty-‐three students Pre-‐test scores on the MCI reveal a mean = 13.09 (~57%) and standard deviation = 3.32. Post-‐test scores show a mean = 11.87 (~52%) and standard deviation = 3.86.

A paired two sample for means t test shows that the difference between average scores on the pre and post-‐test is not statistically significant, but the mean on the post-‐test was actually lower than the mean on the pre-‐test. t-‐Test: Paired Two Sample for Means

Control Pre Control Post Mean 13.1 11.9 Standard Deviation 3.32 3.86 Observations 23 23 Hypothesized Mean Difference 0 df 22 t Stat 1.52 P(T<=t) one-‐tail 0.0718 t Critical one-‐tail 1.72 P(T<=t) two-‐tail 0.144 t Critical two-‐tail 2.07

A look at the distribution of pre and post-‐test scores for the control group shows the more scattered nature of the post-‐test scores.

Selected Questions The MCI includes eight questions that one might expect to be on a typical Algebra II chapter test at Buckeye Union High School. These questions, numbers 9, 10, 11, 13, 15, 17, 18, and 20, cover topics including finding the equation of a line from a table of values (#9), interpreting graphs (#10 & 20), extrapolating graphs (#11), estimating area (#13), finding the equation of a line given a starting position and constant speed (#15), estimating means (#17), and finding a line of best fit (#18). A look at the percentage of students answering these questions correctly in the experimental group shows improvement on every question except for number 20 from pre to post-‐test.

Mean = 13.1 S.D. = 3.32 n = 23

Mean = 11.9 S.D. = 3.86 n = 23

The average score for the selected questions on the pre-‐test was 4.38 with a standard deviation of 1.59, while the average post-‐test score was 4.94 with a standard deviation of 1.30. A paired z test shows this increase in means to be significant with α = 0.027 for a one-‐tailed test and α = 0.054 for a two-‐tailed test. z-‐Test: Two Sample for Means

Experimental Pre Experimental

Post

Mean 4.38 4.94

Standard Deviation 1.59 1.30 Known Variance 2.52 1.69 Observations 50 50 Hypothesized Mean Difference 0 z -‐1.93 P(Z<=z) one-‐tail 0.0268 z Critical one-‐tail 1.64 P(Z<=z) two-‐tail 0.0536 z Critical two-‐tail 1.96

For the control group, the percentage of students answering the selected questions correctly stayed the same for questions 17, 18, and 20, but declined for all of the other questions.

Pre-‐test scores showed a mean of 5 with a standard deviation of 1.35, while post-‐test scores showed a mean of 4.39 with a standard deviation of 1.47. Clearly, the control group did not show a significant increase in scores on the selected questions from pre to post-‐test. Proportional reasoning (questions #5 and 6 on the Math Concepts Inventory): In the pre-‐test, 13% of control group students got questions 5 and 6 correct, compared to 20% of students in the experimental group. In the posttest, while the control group stayed at 13% correct, the experimental group increased slightly but insignificantly to 24%. These percentages are very low. The lack of improvement in the control group is similar to that of the control group of Investigator 2 (page 42). The lack of improvement in the experimental group, contrasted with large improvement in the experimental group of Investigator 2, may indicate that one-‐period lab days bookended by several days of traditional lecturing, note-‐taking, and homework are insufficient for meaningful learning of proportional reasoning.

Gender Data An in-‐depth examination of pre and post-‐test scores for girls and boys in the experimental and control groups is summarized in the table below.

Group (number of students)

MCI Pre-‐Test Mean (out of

23)

MCI Post-‐Test Mean (out of

23)

MCI Selected Questions Pre-‐Test

Mean (out of 8)

MCI Selected Questions Post-‐Test

Mean (out of 8)

Girls Experimental

(31)

11.7 12.7 4.39 4.68

Boys Experimental

(19)

11.8 13.6 4.37 5.37

Girls Control (13)

12.1 10.7 4.77 4.23

Boys Control (10)

14.4 13.4 5.30 4.60

Both the girls and boys in the experimental group showed improvement on the MCI and the MCI selected questions, whereas both the girls and boys in the control group showed a decline on the MCI and the MCI selected questions. Boys in the experimental group made a significant improvement in mean scores on the selected MCI questions. Pre-‐test MCI scores on selected questions for boys in the experimental group showed a mean = 4.37 with a standard deviation = 1.46. Post-‐test scores from this group showed a mean = 5.37 with a standard deviation = 1.01. A t-‐test comparing pre and post-‐test means is shown below. t-‐Test: Paired Two Sample for Means

Experimental Boys MCI Selected Pre

Experimental Boys MCI

Selected Post Mean 4.37 5.39 Standard Deviation 1.46 1.01 Observations 19 19 Hypothesized Mean Difference 0 df 18 t Stat -‐2.24 P(T<=t) one-‐tail 0.0189 t Critical one-‐tail 1.73 P(T<=t) two-‐tail 0.0377 t Critical two-‐tail 2.10

The chart below shows the percentage of students answering selected questions correctly from the above-‐mentioned group. Improvement is seen on every question except for #20, where the percentage of students answering correctly stayed the same.

Buckeye Union High School is in what could be considered a rural suburb of Phoenix, Arizona. Many of the male students in the Algebra II experimental group spend after-‐school hours, weekends, and summers working at blue-‐collar jobs that require them to work with their hands and solve mechanical problems. Seventy-‐eight percent of the males in the Algebra 2 experimental group were also enrolled in some type of vocational education class at Buckeye Union High School. Classes such as woodworking, welding, automotive technology, and agriculture provide opportunities for students to apply mathematics to novel projects. Perhaps the significant improvements on the selected questions of the MCI for males in the experimental group is also partially a result of applying mathematics in multiple situations in their everyday lives. The labs performed in class only gave them more chances to apply mathematics to real-‐world situations.

Student Interviews Student interviews were used to get a feel for how students were thinking about the types of problems we looked at during these labs. A whiteboard was presented to the student and questions were asked by the interviewer. Interview with Student #1 – Check for proportional reasoning from the pendulum lab

Interviewer: Can you describe the experiment that gave us this period versus length

graph? Student #1: We used this experiment about using the pendulum, like how many

seconds it took to swing back and forth, and how long the string was. Interviewer: What variable did we change while we were doing the experiment? (long

pause) What did we control? How did we get these different data points?

Student #1: Isn’t it the length of the string? Interviewer: So, this is our equation that Logger Pro gave us after we did a power

regression. What does that mean for us? Student #1: That means. (long pause) I will write it in another way. (student #1

rewrites equation as T = √L. Two times the square root of L. Interviewer: If we wanted to double the period of the pendulum, what would we

have to do to the length? Student #1: To double it? You would have to raise it to four. (student #1 writes out T

= 2√(4L)) The square root of four is two, so it’s going to be T equals two times two times square root of L.

Interviewer: What if we wanted to triple the period? Student #1: Then you would raise L by nine. That’s going to equal two times nine L,

and square root. And that is two times three times square root of L.

Comment: Student #1 is an English language learner, but he shows a good functional knowledge of how one side of the equation obtained in this lab affects the other side. Student #1 was also able to make the connection that raising a variable to an exponent of one half is equivalent to taking the square root of the variable. Interview with Student #2 – Check for understanding of position versus time graphs from the buggy lab

Interviewer: So we just completed the lab using the battery –powered buggies, and

we have a couple of graphs here. What can you tell us about these graphs? This is for car A and car B.

Student #2: They started at different places. Interviewer: How do you know that? Student #2: Because car A started at zero and car B started at seven. Interviewer: What’s different about the graphs of the two cars? Student #2: One graph is steeper than the other one. Interviewer: Which graph is steeper? Student #2: B Interviewer: What does that mean, that the line is steeper for car B? Student #2: Well, car B is negative and car A is positive. So, they’re going in opposite

directions. Interviewer: Okay. What does that mean where the graphs cross? What does that

point represent? Student #2: That’s where the cars cross. Interviewer: What does the fact that line B is steeper tell us about the motion of the

cars? Student #2: That car A is slower than car B.

Interviewer: How about the fact that both lines are straight? Does that tell us anything?

Student #2: That there’s no turning or anything. They just went straight on. Comment: Student #2 seems to have a grasp on steepness of lines and vertical intercepts, but still could not verbalize that a straight line on a position versus time graph indicates a constant velocity. Scores for the experimental group on the MCI pre and post-‐test were sub-‐par on the last three questions, all of which dealt with the interpretation of position versus time graphs. Interview #3 – Check for understanding of position versus time graphs from the buggy lab

Interviewer: So we just ran through the lab using the battery-‐powered buggy cars.

What can you tell us about the graph you have here on your white board?

Student #3: Well, the y-‐axis represents the position, and the x-‐axis represents the time. And with the graph you can tell that car A was going much faster.

Interviewer: How do you know that car A was going faster? Student #3: Because of its slope, it’s much steeper. And you can tell they start at

different positions because A starts here (pointing to vertical intercept of A) and B starts here (pointing to vertical intercept of B).

Interviewer: What feature of the graph is that? What is that called? Student #3: (long pause) The y-‐intercept. Interviewer: What else can you tell us from the graph? Student #3: That they cross points here, they’re at the same position (points to

intersection point of lines). Interviewer: What does that mean in our lab? What was happening with the cars?

Student #3: That’s where the cars met, and they were in the same spot. Interviewer: These are the two equations we have written down here. What does the

slope of these equations represent? Student #3: Their speed. Interviewer: All right, how about the y-‐intercept? Student #3: Initial position. Interviewer: So if I asked you at what position and time did car A pass car B, could you

work that out for me? Student #3: Yeah. (Student begins sets the two equations equal to each other and

solves for t) Interviewer: All right, so what was that you just found? Student #3: That represents (long pause) the time. Interviewer: The time for what? Student #3: That they crossed points. Interviewer: All right, how about the position? Student #3: Then I’m going to plug time back into the equation. Interviewer: Which equation are you going to plug it back into? Student #3: Each one. (Student substitutes time back into each equation and gets the

same position for each) Interviewer: And that represents…? Student #3: They were at the same position. Comment: Student #3 has a decent understanding of the graphical representation of the buggy lab that used one fast and one slow car. She was also able to find the position and time at which one car passed the other when given position as a function of time equations for each car.

Student #4 – Check for understanding of position, velocity, and acceleration versus time graphs from the bowling ball lab

Interviewer: Can you tell us what kind of lab we did to get this position versus time

graph? Student #4: You rolled a bowling ball down a slant. Everyone lined up, and everyone

had a stopwatch, and they stopped their stopwatch whenever the ball went past them.

Interviewer: Okay. What kind of shape did we get there? Student #4: A parabola. Interviewer: Why did we get that shape for the position-‐time graph? Student #4: Because the speed went up. Interviewer: So what would the velocity-‐time graph look like that would correspond to

that position-‐time graph? Student #4: (draws a straight line with a positive slope on the velocity-‐time graph) A

straight line. Interviewer: How come? Student #4: Because the speed increases. Interviewer: How about the acceleration-‐time graph. Student #4: (draws a horizontal line on the acceleration-‐time graph) Constant. Interviewer: Why is that constant? Student #4: Because gravity is constant. Comment: Student #4 shows a solid conceptual understanding of how position, velocity, and acceleration versus time graphs relate to one another.

Student Surveys At the end of the first semester of the 2008-‐2009 school year, students were given a brief survey of how they felt about the labs and experiments implemented in their

algebra II class. Students were simply asked if they enjoyed lab days and if they felt like the labs helped them learn the new chapter topic. Results are shown in the table here.

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78.9:8;23456803<012=>;8

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The chart shows that an overwhelming majority of the students enjoyed lab days, while a slightly smaller percentage felt as though the labs helped them with understanding a mathematical topic. These percentages were evident in the way students requested more lab days throughout the school year and the overall attitude and participation of the class on lab days.

Analysis The following table is a summary of data collected at Buckeye Union High School.

Group MCI Pre-‐Test Mean (out of

23)

MCI Post-‐Test Mean (out of

23)

MCI Selected Questions Pre-‐Test Mean (out

of 8)

MCI Selected Questions Post-‐Test Mean (out

of 8) Experimental 11.7 13.1 4.38 4.94

Control 13.1 11.9 5.00 4.39 Girls

Experimental (31)

11.7 12.7 4.39 4.68

Boys Experimental

(19)

11.8 13.6 4.37 5.37

Girls Control (13)

12.1 10.7 4.77 4.23

Boys Control (10)

14.4 13.4 5.30 4.60

Examining the class means for pre and post-‐test scores on the MCI shows a significant increase in scores from pre to post-‐test for the experimental group but essentially the same pre and post-‐test mean for the control group. Students in the experimental group also showed a significant positive mean increase on the selected questions of the MCI, whereas the control group mean went down on these questions. Comparing the experimental and control groups directly is difficult because of the large difference in sample size. A control group twice the size of the control group used at this site would allow for more accurate inferences about the data from this research site. One obstacle to implementing the above-‐mentioned labs is the nature of the Algebra II classes at Buckeye Union High School. All chapter tests and quarterly exams are district-‐wide assessments. After giving such assessments, each teacher must submit data indicating the number of students who earned scores that would be labeled as Exceeds (85%), Meets (70%), Approaches (60%), and Falls Far Below (<60%) according to the Arizona Instrument for Measuring Standards (AIMS). It is required that all tests be given at roughly the same time throughout the school year. District chapter tests and quarterly exams are geared towards preparing students for the multiple choice nature of the AIMS math test. Out of sixteen tests given to the Algebra 2 classes during the 2008-‐2009 school year, students were only asked to solve fifteen word or application type problems. This is a mere 3% of the total problems seen by the students, which means that, in order to give the experimental group classes a fair shake on the common district assessments, the entire chapter could not be based around the introductory lab. At some point, the students needed to develop the “drill and kill” instincts that would allow them to crunch numbers and achieve decent grades on the chapter tests. Often times, these labs, including setup, data collection, analysis, and whiteboarding, were limited to one fifty-‐five minute class period. Several students commented that they felt rushed and that the material did not always have sufficient time to “sink in.” Student surveys indicated that 96% of the population enjoyed lab days, and 83% of the population felt that it helped them learn the topic at hand. If the class went a week or two in a traditional lecture, notes, homework, test format, students were requesting to do a lab or experiment. Doing labs helped break up the monotony that can creep into a mathematics classroom that is trying to get students prepared for the AIMS and AIMS-‐based chapter tests. Unfortunately, lab days were often bookended by several days of traditional lecturing, note-‐taking, and homework. Student interviews showed students examining problems that, had they not been introduced to the labs done in the experimental Algebra 2 class, they may never have encountered throughout their high school education. Although it was not evident that the interviewed students had an absolutely complete and thorough understanding of the problem at hand, they were able to make observations, solve problems, and draw conclusions about problems that students in the control group may have struggled with mightily.

Since using calculators on the AIMS Math test is strictly forbidden, teachers at Buckeye Union High School are discouraged from letting students use technology of any kind on district common assessments. Throughout these labs, students used a variety of software to gain understanding and develop equations to explain observations made during the labs. Students used TI-‐84+ Graphing Calculators, Logger Pro, Geometer’s Sketchpad, and Microsoft Excel. During the implementation of these labs, not one student found numbers that came out to nice, evenly divisible integers that could be easily manipulated with pencil and paper. One secondary goal of using these experiments to enhance mathematical understanding was to show students that experiments rarely work out to be nice, even whole numbers in real-‐world applications. Life is full of fractions and ugly decimals. Fortunately, powerful yet user-‐friendly software is available to handle just such situations, and the students in the experimental groups were proficient at using this software by the end of the first semester.

Conclusion and Implications The students in the experimental group were an absolute pleasure to work with and be around on a daily basis. They worked hard, stretched their brains, and jumped in head first when it came to using the technology necessary to work through these labs. Whiteboard meetings were enthusiastic and full of ideas and insights into various mathematical concepts and problems. Quantitative data from the selected questions show that running these labs in a typical Algebra 2 classroom does not hurt a student’s ability to solve basic problems that may be found on a typical chapter assessment at Buckeye Union High School. Qualitatively, it is certain that the students in the experimental group enjoyed the labs and took something, albeit the ability to use technology in a lab setting, mathematical concepts, the ability and confidence to share ideas with their peers, or otherwise, from the labs.

Overall Conclusion The above research shows that students can learn mathematical reasoning skills by engaging the material differently than in a traditional mathematics course. By teaching students mathematics concepts using science applications, the students’ skills increase at a faster rate than in a traditional math class. The most significant gains seemed to be in the concepts of proportional reasoning and graphical analysis. The increases in these areas may be caused by the prevalence of these concepts in the design of the activities and labs that were used throughout the

treatment. In each classroom lab, students were required to express relationships in proportional and graphical means more so than a traditional classroom. It seems that continuous exposure to activities where proportional and graphical skills are used increases the conceptual reasoning abilities of the treated students. The traditional mathematics courses seem to lack the context necessary to build the conceptual understanding that the treated courses were built upon. In every activity, treated students had a concrete example to refer back to, throughout their development of the topic at hand. For example, the linear buggy and bowling ball labs gave treated students a physical situation that reinforced the constant rate of change model. The data suggest a significant increase in conceptual understanding when a modeling approach is used.

Implications for Instruction Students can learn mathematics concepts using a scientific modeling approach that differs from the traditional method taught in most schools. The research shows that the effectiveness of the scientific approach can produce skills greater than those of traditionally taught students. Having instruction that allows for a more cohesive blend of sciences and mathematics courses may prove to be a more efficient and effective way to teach mathematics. By using this approach, the application and relevancy of the material may keep students more engaged in learning. Implementing the Modeling Method of instruction described in this paper requires highly trained teachers who cannot be one-‐dimensional, pencil and paper, number crunchers. Getting students to learn in a mathematics classroom where science-‐based investigations are used to introduce new topics is extremely challenging. Knowing how to solve a series of mathematics problems is simply not good enough. Teachers need training, experience, creativity, and the support of their administrators and department chairs in order to effectively use these methods in a classroom setting. Investing money for the training of educators in this teaching approach could prove to be a worthwhile endeavor for schools nationwide. Part of the challenge in doing activities in class is getting over the fear of giving some of the control of the class over to the students. With adequate training, this fear is marginalized.

Implications for Future Research The data examined in this study indicate that math and science hybrid (integrated) courses may be one possible way of improving high school students’ conceptual understanding of mathematical topics. For future research, it might be valuable to test the competencies of students involved in hybrid mathematics and science courses compared to students taking traditional separate mathematics and science courses.

This study primarily focused on developing the mathematical concepts of proportional reasoning and graphical interpretation. Future studies could focus on developing other mathematical concepts such as geometric transformations, trigonometric functions, and even basic arithmetic. Students in this study were in the eleventh grade and had already taken the Arizona Instrument for Measuring Standards in High School (AIMS HS) in the tenth grade as required by Arizona state law. An examination of how the method used to instruct the treatment group would change AIMS scores for students in the ninth and tenth grades could have pedagogical implications for students who have yet to take the state of Arizona standardized test at the high school, middle school, or even elementary levels. We can see that a broader study of this research topic is needed before wide brush claims can be made. We did not look into the long term effects of the treated students as they went to their higher level courses, nor did we compile other accepted mathematics tests such as ACT or SAT assessments. These areas should be considered in future research, as it would be a more complete picture of how the students improved over time and fared during college entrance exams.

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Appendix Data

Investigator 1 Investigator 2 Investigator 3 Experimental Student # Pre Post

Experimental Student # Pre Post

Experimental Student # Pre Post

100 8 12 136 16 17 157 12 13 101 13 16 137 15 10 158 8 10 102 19 21 138 10 18 159 12 13 103 10 9 139 16 16 160 14 18 104 14 15 140 16 22 161 12 14 105 17 18 141 14 17 162 13 13 106 8 9 142 20 21 163 13 14 107 10 9 143 13 13 164 8 9 108 11 14 144 14 19 165 18 20 109 10 14 145 17 21 166 9 9 110 15 16 146 9 13 167 20 18 111 15 17 147 10 13 168 10 15 112 13 14 148 8 10 169 14 9 113 8 14 149 15 16 170 12 8 114 14 15 150 20 21 171 6 8 115 10 15 151 23 23 172 13 16 116 9 15 152 17 21 173 8 16 117 12 15 153 13 17 174 6 4 118 15 17 154 13 17 175 5 11 119 15 18 155 10 15 176 14 14 120 14 15 156 14 20 177 16 15 121 11 12 178 10 11 122 8 9 179 9 15 123 14 15 180 17 14 124 14 19 181 9 10 125 7 6 182 14 9 126 12 13 183 8 12 127 12 15 184 16 16 128 11 15 185 10 14 129 17 19 186 15 8 130 17 18 187 6 6 131 6 7 188 8 12 132 11 12 189 14 14 133 14 17 190 14 15 134 12 13 191 11 12 135 9 12 192 13 12

193 11 11 194 5 9 195 12 15 196 16 14 197 9 9 198 16 21 199 13 15 200 13 16 201 10 18 202 8 8 203 14 13 204 17 14 205 13 6 206 19 18

Investigator 1 Investigator 2 Investigator 3

Control Student # Pre Post



300 13 14 330 12 16 352 12 10 301 15 15 331 15 15 353 16 15 302 17 18 332 17 17 354 17 12 303 18 18 333 15 16 355 19 16 304 12 13 334 6 11 356 10 12 305 18 20 335 14 14 357 6 5 306 17 16 336 14 14 358 11 9 307 17 17 337 12 13 359 12 14 308 19 20 338 8 12 360 14 16 309 16 16 339 13 15 361 12 10 310 21 21 340 17 17 362 15 11 311 15 14 341 9 10 363 14 17 312 20 20 342 17 17 364 14 9 313 18 16 343 14 17 365 11 15 314 20 20 344 18 19 366 20 10 315 11 14 345 21 21 367 11 8 316 20 20 346 12 13 368 15 14 317 14 15 347 19 16 369 8 11 318 11 10 348 15 16 370 14 11 319 20 21 349 13 16 371 12 15 320 15 15 350 16 16 372 14 5 321 11 12 351 16 15 373 15 19 322 10 13 374 9 6 323 13 14 324 15 15 325 18 19 326 15 15 327 13 15 328 16 15 329 13 15

effect of teaching mathematics concepts within a science...

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