The Encyclopedia of Algebraic Thinking consists of 66 entries that articulate students\u2019 misconceptions and ways of thinking about algebra. The research on students\u2019 algebraic thinking typically includes a discussion of the mathematics involved in a misconception or way of thinking, assessment problems, statistics on students\u2019 responses to problems, transcripts of interviews with students describing their thinking, and suggested instructional strategies. Construction of entries in the Encyclopedia of Algebraic Thinking were guided by five questions:<\/p>\n
- \n
- What CCSS does this research address?<\/li>\n
- What is the symbolic representation of thinking with the idea? (What does it look like? How does students\u2019 written work indicate how they are thinking about the idea?)<\/li>\n
- How do students think about the algebraic idea? (What does it sound like? What do students say when they discuss the idea?)<\/li>\n
- What are the underlying mathematical issues involved? (What prerequisite knowledge is involved, what concepts are essential, and where does the concept fit in mathematics?)<\/li>\n
- What research-based strategies\/tools could a teacher use to help students understand?<\/li>\n<\/ol>\n
The Encyclopedia is searchable by CCSS, keyword, and cognitive domain for teachers in non-CCSS states. Based on the research, typical algebra textbooks, and the CCSS, the project defined five cognitive domains around which ATP staff organized the articles reviewed: Variables & Expressions, Algebraic Relations, Analysis of Change, Patterns & Functions, and Modeling & Word Problems. Our mathematics teacher educators used the Encyclopedia for four purposes.<\/p>\n
- \n
- Make preservice teachers aware of the range of ways students may think about algebra and the nature of misconceptions they can develop.<\/li>\n
- Help preservice teachers acquire a disposition towards wanting to learn about students\u2019 thinking through exploration of the entries.<\/li>\n
- Provide preservice teachers with a just-in-time resource to help them anticipate how students might interact with an algebraic concept in their lesson, in an effort to compensate for their relative lack of experience with students\u2019 thinking.<\/li>\n
- Reflect on and interpret their experience of interacting with students\u2019 thinking during a lesson to facilitate their next instructional decisions.<\/li>\n<\/ol>\n
In order to help mathemathics teacher educators engage preservice teachers in considering students\u2019 thinking, 20 different problems were identified in the research that were representative of typical concepts that students struggle with across the five cognitive domains identified by the project. Two different sets of 10 problems were created from the original 20. The two different forms were alternated as they were given to 17 middle school students to get a breadth of response to the 20 problems. Each student worked on the problems for between 10-30 minutes. Each student was then asked by ATP staff to explain their thinking with each problem as they were videotaped. The result was approximately 170 video clips, one for each problem for each student. Of those video clips, 68 had useful explanations (other than \u201cI don\u2019t know\u201d, etc.) and were combined into a Video Database. The Video Database is freely available at the project website (http:\/\/www.algebraicthinking.org\/) when users establish a free account.<\/p>\n
Video has proven to be a useful tool for teachers to consider students\u2019 thinking (Franke, Carpenter, Fennema, Ansell, & Behrend, 1998; Gearhart & Saxe, 2004) and can provide a more dynamic environment in which to consider students\u2019 thinking than can text. The video clips were used as a significant part of 15 project-designed instructional modules for mathematics methods courses from which our math educators could choose to implement. The intent of the video is to allow preservice teachers to hear why students struggle or make choices as they engage in a problem.<\/p>\n
The National Education Technology Plan (2010) recommended that educators \u201cdiagnose strengths and weaknesses in the course of learning when there is still time to improve student performance\u201d (p. 9). However, one of the stressors for preservice teachers is their lack of confidence in monitoring the status of their students\u2019 understanding of the mathematics. One key feature of the research literature is assessment problems designed to elicit students\u2019 range of algebraic thinking. The ATP catalogued these empirically tested problems into a Formative Assessment Database for preservice teachers to assess their students\u2019 algebraic thinking.<\/p>\n
The web database is searchable by keyword, CCSS, and cognitive domain. Problems range from \u201cWhich is larger, 2n<\/em> or n<\/em> + 2?\u201d (Kuchemann, 2005) to more complex problems, such as the Smith Family Holiday Problem shown in Figure 1.<\/p>\n
![\"figure](\"https:\/\/citejournal.org\/wp-content\/uploads\/2016\/04\/v15i2math1Fig1.jpg\")
<\/p>\nThe Smith family\u2019s whole holiday is shown on the graph at left. The vertical axis shows the distance in kilometers away from home. The horizontal axis shows the time in days since the start of their trip.<\/p>\n
a) During which days did the Smith family travel fastest?<\/p>\n
b) They stayed with friends for a few days. Which days were these?<\/p>\n
c) On average, how fast did the Smith family travel to get to their destination?<\/p><\/blockquote>\n
Figure 1<\/strong>. Smith family holiday problem. Graph recreated based on problem published in “An \u2018Emergent Model\u2019 for Rate of Change” by S. Herbert & R.. Pierce, 2008, <\/em>International Journal of Computers for Mathematical Learning, 13, p. 240. Copyright 2008<\/em> Springer Science+Business Media B.V.<\/em><\/p>\n
<\/p>\n
Although the first problem may appear simplistic, it addresses complex understanding of the concept of a variable as an unknown. The student must articulate how variation in one set of values depends on variation in another set and the relative power of operations on a range of numbers. The Database provides teachers with information about how students in the study responded. For instance, with the first problem, 71% of 11 to 13-year-old students believed that 2n<\/em> would be greater, 16% believed n<\/em> + 2 would be greater or that they would be the same, while only 6% responded correctly that the answer is 2n<\/em> when n<\/em> > 2 (Hart et al., 1981).<\/p>\n
The second problem examines students\u2019 understanding of rate of change. Researchers found that common incorrect answers included that days 1 and 2 the family traveled the fastest because they are the steepest. Many students saw a steep upslope but did not see a steep downslope. Other students responded that the fastest days would be 3, 4, 5, and 6 because they are the flattest. A formative assessment problem such as this one creates the opportunity for teachers to learn the status of multiple aspects of their students\u2019 understanding of a concept.<\/p>\n
While the problems are usable in paper and pencil format, the ATP staff also developed a Classroom Response System (CRS) that directly accesses the Formative Assessment Database. The literature on CRS has shown that technology enhanced formative assessment can increase student participation and reshape teacher discourse patterns (Feldman & Capobianco, 2008; Langman & Fies, 2010).<\/p>\n
The unique aspect of our CRS is that it allows teachers to use problems from the database or write their own and deliver them to students in the classroom on mobile devices. Students answered the problem(s), then the app collected the data and instantly provided teachers with easy-to-consume, graphical evidence of the range of students\u2019 thinking. The preservice teachers could immediately see how the class understood a concept on a single problem or could look at an individual student\u2019s understanding across all problems.<\/p>\n
Finally, the ATP developed 17 iOS-based virtual manipulatives that address specific algebraic concepts identified in the research as challenging for students to understand. The National Council of Teachers of Mathematics (2000) suggested that virtual manipulatives can allow students to \u201cextend physical experience and to develop an initial understanding of sophisticated ideas\u201d (p. 27). They can be powerful tools for learning, providing significant gains in achievement (Heid & Blume, 2008; Suh & Moyer-Peckenham, 2007). However, \u201cone reason that educational software has not realized its full potential to facilitate and encourage students\u2019 mathematical thinking and learning is that it has not been adequately linked with research\u201d (Sarama & Clements, 2008, p. 113).<\/p>\n
Accordingly, the ATP used research on algebraic thinking as the basis for manipulative development. Our apps typically have students manipulate and explore a dynamic between variables. For example, one app addresses research by Monk (1992) that students tend to draw graphs that imitate reality, such as a hill, regardless of the labels of the axes. Our Action Grapher app (see Figure 2) shows a biker climbing various hills while simultaneously three separate graphs of height, distance, and speed versus time appear alongside. Student draw what they think each graph will look like, then animate the bike and compare their hypotheses against the graphs that unfold.<\/p>\n
![\"figure](\"https:\/\/citejournal.org\/wp-content\/uploads\/2016\/04\/v15i2math1Fig2.jpg\")
Figure 2.<\/strong> Action grapher.<\/em><\/p>\n<\/p>\n
\u201cTechnologies often afford newer and more varied representations and greater flexibility in navigating across these representations\u201d (Mishra & Koehler, 2006, p. 1028). The Action Grapher app provides that opportunity for students to observe changes across representations and creates cognitive conflict for those students who have the misconception described by Monk (1992). The ATP virtual manipulatives take little time and focus on a specific, challenging concept or misconception. The purpose of these apps was to nurture preservice teachers\u2019 orientation toward their students\u2019 algebraic thinking. Since these apps target hard-to-understand topics in algebra, preservice teachers needed to consider the following:<\/p>\n
- \n
- The nature of the challenge to students\u2019 thinking the virtual manipulative addressed.<\/li>\n
- How the app addressed a misconception or way of thinking about the concept.<\/li>\n
- Why that might be an effective approach.<\/li>\n<\/ol>\n
Together, the four technology-based resources were designed to be an integrated strategy to develop preservice teachers\u2019 focus on students\u2019 thinking, understanding of the nature of students\u2019 thinking, ability to assess the range of their students\u2019 thinking, and ability to intervene in a research-based way upon that thinking. It is unrealistic to believe that preservice teachers could acquire all the knowledge they need regarding students\u2019 thinking in a teacher preparation program. Accordingly, the intent of the project was not simply to fill preservice teachers\u2019 brains with research but to orient them toward their students\u2019 thinking and develop a habit of mind to use technology-based resources to prepare for and interpret students\u2019 engagement with algebra.<\/p>\n
No known research exists regarding a similar comprehensive approach toward infusing technology into teacher preparation in mathematics. While our use of a web-based encyclopedia, video database, formative assessment database, and iOS apps in combination is unique in teacher preparation, each element has a research basis, as indicated in each of the previous sections on our tools.<\/p>\n
Much discussion has occurred around the concept of technological pedagogical content knowledge (Voogt, Fisser, Pareja Roblin, Tondeur, & van Braak, 2013). The technology, pedagogy, and content knowledge (TPACK) framework captures the complex dynamic between teachers, their content, and resources\u2014focusing on the knowledge needed for implementation of technology in the classroom (Mishra & Koehler, 2006). The pedgogical knowledge element of TPACK includes knowledge about how students learn and methods of assessment that is the heart of our work (Harris, Mishra, & Koehler, 2009).<\/p>\n
Heid and Blume (2008) noted that there is a \u201cconsiderable amount of research focused on the effects of technology on the development of the concept of function and on the role of technology in enhancing symbolic manipulation skills and understanding\u201d (p. 58). Significant research also has examined how teachers use technology, particularly graphing calculators (Zbiek & Hollebrands, 2008). Although research indicates that technology is changing the way individuals think about and do algebra, research is not yet clear on technology\u2019s role in developing capable preservice teachers. This project is an effort to use technology in many forms to prepare teachers for success in the algebra classroom.<\/p>\n
The Research Study<\/p>\n
Participants<\/p>\n
The research study was implemented by four public and private universities in Oregon with mathematics teacher preparation programs. Each institution was charged with restructuring their mathematics methods courses to implement the technological resources of the ATP. In the third year of the 4-year study (the year for which results will be conveyed in the paper), the courses included 45 preservice teachers (25 female) working toward teaching licenses in basic or advanced mathematics. The public institution had 28 students, with the rest spread equally across the other institutions. Approximately 30% of participants entered the teacher education programs with mathematics degrees.<\/p>\n
The mathematics methods programs at three institutions were entirely graduate, Master of Arts in Teaching (MAT) degree programs. At one institution, undergraduate students also participated in the mathematics methods courses, so that 25% of participants overall did not yet have a Bachelor of Arts degree. Field experience and mathematics methods course requirements were the same for both Bachelor of Arts and MAT programs. Case study participants were chosen based on their being placed in an Algebra I class for their student teaching experience and willingness to participate in the study.<\/p>\n
Design of the Study<\/p>\n
Multiple differences existed across the four ATP institutions in regard to coursework and timing. However, we were all under the same state regulations for teacher education, so the content of our coursework and field experiences were very similar. Whether undergraduate or MAT student, each teacher candidate experienced the same coursework within an institution. For undergraduates, the coursework was distributed over 2 years, with the field experiences distributed across the final year.<\/p>\n
Each of the MAT programs had part-time as well as full-time options. One institution was on the quarter system, and the other three were on the semester system. The mathematics methods courses are offered at different points in the programs across the final year that includes field experiences. At two institutions the methods course occurred in the fall, while at the other two the methods course was distributed across two semesters. Participants in the study were all preservice teachers enrolled in our mathematics methods courses seeking either a Middle Level or High School Level teaching license or both.<\/p>\n
Each institution was free to choose which elements of each resource were most useful in their mathematics methods programs. All methods instructors used at least one module, used video clips of students explaining their thinking during class, referred preservice teachers to the Encyclopedia for information about students\u2019 thinking or required citations from the Encyclopedia in assignments, oriented preservice teachers to the formative assessment database, and discussed ATP apps.<\/p>\n
To determine the effectiveness of the resources and approach to developing preservice teachers\u2019 knowledge of students and content as well as their habits of mind in using the resources, <\/em>the study included quantitative and qualitative data. For quantitative data the project designed a preservice teacher survey. For qualitative data the project conducted eight case studies, including video of preservice teachers\u2019 instruction that was coded with the project-designed Teacher Dispositions Towards Students\u2019 Thinking (TDST) observation tool.<\/p>\n
Preservice Teacher Disposition Survey. <\/em><\/strong>The ATP staff constructed and pilot tested the Teacher Disposition Survey, containing 24 items that solicited self-reported perceptions about the degree to which preservice teachers valued and used student preconceptions in their teaching. The survey was based on a survey used by Nathan and Koedinger (2000) and Nathan and Petrosino (2003) that assesses six areas pertaining to the utility of symbol-based (rather than word-based) mathematical forms, student preconceptions and intuitive means for solving problems, and ways teachers work with student preconceptions and intuitive problem-solving methods in class. Items were pulled from this survey and combined with original items about whether or not teachers think students have preconceptions about math, qualities of those misconceptions, and how teachers work with student preconceptions to teach effectively.<\/p>\n
Data collected with the ATP survey was annually assessed for item reliability. In 2012, 17 preservice teachers completed the survey, which included 24 original items used to assess their beliefs about the role of students\u2019 thinking in effective mathematics instruction. These 24 items were developed with the project team to support content validity targeted at the objectives of the program. Cronbach\u2019s reliability for the items was high at 0.83, with only three items indicating item-rest correlations above 0.3. In 2013, the survey\u2019s reliability went up to 0.88 (n = 19), with only one item indicating an item-rest correlation of greater than 0.3. Each preservice teacher was given a pretest survey at the beginning of the mathematics methods program and a posttest at the end of the school year. The survey data were used in combination with observations and interviews to describe teacher dispositions and classroom strategies.<\/p>\n
Case Studies.<\/em><\/strong> In order to learn about the nature of preservice teachers\u2019 orientation toward using their students\u2019 thinking to inform their instruction and understand how preservice teachers used project resources, each of the four institutional Core Team members was responsible for two case studies of their students. The case studies started with an interview at the beginning of the teacher education program. The interviewer presented an algebra problem to the candidates, then asked them to solve it and then discuss how they believed a student might approach the problem correctly or incorrectly. Then the interviewer and candidate viewed a video of a student discussing his thinking with the problem. Candidates explained how they might work with that student.<\/p>\n
Next, the interviewer asked how the candidate might approach teaching an algebra topic. This process was repeated in a postinterview at the end of the mathematics methods course of the institution. In addition, the case studies included work samples (teaching units) from each candidate and reflections in which the candidates discussed their teaching and how they made use of students\u2019 algebraic thinking and the resources of the project.<\/p>\n
A key part of the project involved observing preservice teachers in inquiry sequences, or interchanges, where they helped students explore and better understand a topic. Video of the case study preservice teachers\u2019 instruction was taken at the beginning, middle, and end of their field experience to assess changes in their instructional practice. Our four institutions\u2019 field experiences varied from 10 weeks to a full semester (18 weeks). The project used videos to analyze preservice teachers\u2019 interactions with students using the ATP\u2019s TDST. The TDST was developed and field tested by outside evaluators on the staff of the International Society of Technology Educators during the first year of the project. The evaluators trained project team members in how to use the tool to analyze video. In order to increase reliability of the coding, two evaluators each coded a video separately and then conferred to resolve differences and came to consensus on accurate final codes.<\/p>\n
The TDST is a custom-built, classroom observation instrument based upon research and methodologies used to describe classroom discourse (Thadani, Stevens, & Tao, 2009; Wells & Arauz, 2006) and student inquiry in the classroom (Fennema et al., 1996; Franke, Carpenter, Levi, & Fennema, 2000; Puntambekar, Stylianou, & Goldstein, 2007). It allowed observers to record qualities of teacher-student interactions, including question-oriented inquiry sequences, teacher lectures, student work, and other instructional activities.<\/p>\n
Content validity for the TDST was based on an extensive review of the literature on classroom discourse. Evaluators worked with the project team to refine observational indicators for each area assessed, conducted field testing of the instrument and observer calibration\/rater reliability (both project evaluators conducted observations, discussed their coding, and refined the instrument), and conducted two trainings with the project team. The technology of the tool is built upon an existing tool\u2014the ISTE Classroom Observation Tool\u2014that has been used in hundreds of classrooms since 2008.<\/p>\n
In addition to data about setting, context, and initiation, an observer using the TDST noted frequencies of interactional qualities that characterized an interaction, including the following:<\/p>\n
- \n
- Frequency and length of inquiry sequences;<\/li>\n
- Characteristics of the setting, including class size, student demographics, subject;<\/li>\n
- Initiation of each sequence, including initiator (student or teacher) and technique (question, directive), student groupings (whole class, small group, individual); and<\/li>\n
- Types of teacher and student interactions for each sequence:\n
- \n
- Confirm: Teacher or students respond typically with \u201cyes\u201d or \u201cno\u201d to a question.<\/li>\n
- Logistics: How class activities and workload are ordered.<\/li>\n
- Status check: The degree to which student(s) understood the material.<\/li>\n
- Recall\/replicate: Access, name, or recall a piece of information.<\/li>\n
- Basic operational: Use an algebraic concept, solve a problem or execute a procedure.<\/li>\n
- Advanced operational: Use an algebraic tool in an original context, going beyond basic drill and practice. Use a math concept in a complex scenario such as a word problem.<\/li>\n
- Conceptual: Abstract qualities of and relationships among algebraic concepts, how\/why the concept functions external to a concrete situation; analysis of underlying principles.<\/li>\n
- Metacognition: Students reflect upon what they learned and how they learned it.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
For instance, a student may ask the teacher, \u201cAm I doing this right?\u201d The teacher may check the student\u2019s work, and use the opportunity to successfully segue into a discussion about the relevant concept underlying the problem solving exercise. This sequence would be recorded as Individual (context); Student, question (initiation); Student \u201cStatus check\u201d; Teacher \u201cBasic operational\u201d; Teacher \u201cConceptual\u201d; Student \u201cConceptual.\u201d<\/p>\n
With the TDST, observers constructed an empirical record describing frequencies of interactions and their qualities, permitting analysis of sequential observation data (as in Bakeman, 2000; Bakeman & Gottman, 1997). ATP staff triangulated observational data from the TDST with survey and interview data to describe changes in teachers\u2019 orientations toward student thinking over time. We hypothesized that ATP teachers learned to anticipate students\u2019 thinking and, as a result, were more interested in how students thought about a topic, rather than if students could simply solve a problem, and engaged students in more conceptual discussion as the year progressed.<\/p>\n
Results<\/p>\n
The ATP was in its fourth and final year of the grant at the time of this writing, so results conveyed in this paper are based upon data collected in the third year (first year of implementation). In this section are results from the Teacher Disposition Survey and one of the case studies. The quantitative data gathered from the Teacher Disposition Survey gave the researchers a midproject glimpse into the overall impact of the technology-based resources.\u00a0 After the completion of the project, a larger sample size will be available for additional quantitative analysis.<\/p>\n
At this point in our data collection we can examine the impact of the technology-based resources on an individual preservice teacher\u2019s practice through one of our case participants, Jessica.\u00a0 This case provides an illustration of how the project resources may be used by teacher educators in other settings.<\/p>\n
In regard to the Teacher Disposition Survey, 19 of the 45 participants took both the pre- and posttest. The low response rate was due to multiple variations in students\u2019 participation in mathematics methods courses, as only 19 took the entire mathematics methods program at the four institutions for the full school year. Noticeable gains in awareness and use of ATP resources suggest that mathematics methods instructors had some success integrating them into their courses (see Table 1). ATP applications for use on mobile devices were most popular with preservice teachers. Teacher candidates also reported moderate use of the Encyclopedia.<\/p>\n
Pre-post comparisons suffer from inconsistent timing of administration, as teacher candidates took their surveys at different times and mathematics methods programs were of different durations across schools, so time lapsed between pre and post varied by institution and individual.<\/p>\n
Table 1<\/strong>
\nTeacher Disposition Survey Results<\/p>\n\n\n
\n <\/td>\n \n Pre (19)<\/strong><\/div>\n<\/td>\n\n Post (19)<\/strong><\/div>\n<\/td>\n<\/tr>\n\n \n Level of Use<\/strong><\/div>\n<\/td>\n\n n<\/em><\/strong><\/div>\n<\/td>\n\n %<\/strong><\/div>\n<\/td>\n\n n<\/em><\/strong><\/div>\n<\/td>\n\n %<\/strong><\/div>\n<\/td>\n<\/tr>\n\n Encyclopedia of Algebraic Thinking<\/td>\n<\/tr>\n \n I don’t know what this is<\/td>\n \n 15<\/div>\n<\/td>\n\n 79%<\/div>\n<\/td>\n\n 4<\/div>\n<\/td>\n\n 21%<\/div>\n<\/td>\n<\/tr>\n\n I know what this is, but never used it<\/td>\n \n 3<\/div>\n<\/td>\n\n 16%<\/div>\n<\/td>\n\n 4<\/div>\n<\/td>\n\n 21%<\/div>\n<\/td>\n<\/tr>\n\n I used this 1 or 2 times<\/td>\n \n 1<\/div>\n<\/td>\n\n 5%<\/div>\n<\/td>\n\n 7<\/div>\n<\/td>\n\n 37%<\/div>\n<\/td>\n<\/tr>\n\n I used this a few times<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 2<\/div>\n<\/td>\n\n 11%<\/div>\n<\/td>\n<\/tr>\n\n I used this monthly<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 2<\/div>\n<\/td>\n\n 11%<\/div>\n<\/td>\n<\/tr>\n\n I used this weekly<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n<\/tr>\n\n iPad\/iPod applications from ATP<\/td>\n<\/tr>\n \n I don’t know what this is<\/td>\n \n 6<\/div>\n<\/td>\n\n 32%<\/div>\n<\/td>\n\n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n<\/tr>\n\n I know what this is, but never used it<\/td>\n \n 8<\/div>\n<\/td>\n\n 42%<\/div>\n<\/td>\n\n 1<\/div>\n<\/td>\n\n 5%<\/div>\n<\/td>\n<\/tr>\n\n I used this 1 or 2 times<\/td>\n \n 3<\/div>\n<\/td>\n\n 16%<\/div>\n<\/td>\n\n 4<\/div>\n<\/td>\n\n 21%<\/div>\n<\/td>\n<\/tr>\n\n I used this a few times<\/td>\n \n 1<\/div>\n<\/td>\n\n 5%<\/div>\n<\/td>\n\n 5<\/div>\n<\/td>\n\n 26%<\/div>\n<\/td>\n<\/tr>\n\n I used this monthly<\/td>\n \n 1<\/div>\n<\/td>\n\n 5%<\/div>\n<\/td>\n\n 5<\/div>\n<\/td>\n\n 26%<\/div>\n<\/td>\n<\/tr>\n\n I used this weekly<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 4<\/div>\n<\/td>\n\n 21%<\/div>\n<\/td>\n<\/tr>\n\n Formative Assessments from ATP<\/td>\n<\/tr>\n \n I don’t know what this is<\/td>\n \n 18<\/div>\n<\/td>\n\n 95%<\/div>\n<\/td>\n\n 9<\/div>\n<\/td>\n\n 47%<\/div>\n<\/td>\n<\/tr>\n\n I know what this is, but never used it<\/td>\n \n 1<\/div>\n<\/td>\n\n 13%<\/div>\n<\/td>\n\n 6<\/div>\n<\/td>\n\n 32%<\/div>\n<\/td>\n<\/tr>\n\n I used this 1 or 2 times<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 2<\/div>\n<\/td>\n\n 11%<\/div>\n<\/td>\n<\/tr>\n\n I used this a few times<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 1<\/div>\n<\/td>\n\n 5%<\/div>\n<\/td>\n<\/tr>\n\n I used this monthly<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 1<\/div>\n<\/td>\n\n 5%<\/div>\n<\/td>\n<\/tr>\n\n I used this weekly<\/td>\n \n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n\n 0<\/div>\n<\/td>\n\n 0%<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/p>\n
Twenty-four items assessed preservice teachers\u2019 beliefs about the role of students\u2019 thinking for instruction. This portion of the survey demonstrated good reliability with a Cronbach\u2019s alpha level of 0.88, with only one item exhibiting an item-rest correlation of less than 0.3 (\u201cEliciting misconceptions from students only reinforces bad math habits\u201d). As with the items around use of ATP resources, beliefs held by preservice teachers were supportive of the ATP project, specifically, around the idea that student preconceptions about mathematics are important (see Table 2).<\/p>\n
All negatively worded items (indicated) were recoded inversely on scales of 1-4 from Never true<\/em> to Always true<\/em> and including I don\u2019t know<\/em>. Comparing all responses from both the 2011-12 and 2012-13 cohorts, the latter exhibited more positive views about understanding students\u2019 thinking\u2014average (group) responses to every item were more positive. However, when considering the respondents who completed both pre- and postsurveys, there was generally no change in attitude over time, save a significant decline for two items (p<\/em> < 0.05): \u201cEliciting misconceptions from students only reinforces bad math habits,\u201d and \u201cIn my teaching, I feel compelled to find out what my students are thinking.\u201d<\/p>\n
Table 2<\/strong>
\nPre-Post Teacher Disposition Survey Change\u00a0<\/strong><\/p>\n\n\n
\n \n Prompt<\/strong><\/div>\n<\/td>\n\n Change<\/strong><\/div>\n<\/td>\n<\/tr>\n\n For any math topic, students have preconceptions that they will apply to what they are learning.<\/td>\n \n -0.42<\/div>\n<\/td>\n<\/tr>\n\n Students come to any new math class with misconceptions about how math works.<\/td>\n \n -0.26<\/div>\n<\/td>\n<\/tr>\n\n Students enter the algebra classroom with intuitive methods for solving algebra story problems<\/td>\n \n -0.05<\/div>\n<\/td>\n<\/tr>\n