Using the counting on mathematics strategies: an action research case study.

Using the counting on mathematics strategies: an action research case study. KATIE MEAD & TOM W. MAXWELL describe an action research projectdesigned to improve the place value, multiplication, and division skillsof a group of students Katie (1), was required to engage in an action research (AR)project as part of a 10 week internship in the eighth semester of herBEd(Primary) course. Action research is defined by Macintyre (2000, p.1) to be: 'an investigation, where, as a result of rigorousself-appraisal of current practice, the researcher focuses on a'problem' (or a topic or an issue which needs to beexplained), and on the basis of information ... plans, implements, thenevaluates an action then draws conclusions on the basis of thefindings'. AR is a "form of practical action which teachersundertake as part of, not separate from, their professional work"(Grundy, 1995, p. 7). The following is an account of an AR project whichdemonstrates how AR research can be undertaken as part of everydayteaching practice. Identifying the question Also known as the "reconnaissance phase", this is thefirst step is identifying the AR question. Katie undertook areconnaissance to "ground" her AR in the realities of herworkplace, to reflect on her professional practice in context, and toconsider the benefits of professional opinion and relevant literature. Nine of Katie's students were below the NSW average inmathematics. As she began taking on the teaching load for themathematics key learning area, she began to notice these nine studentshad significant difficulties compared to the other students whencompleting mathematical calculations mentally. This was a concernbecause these skills are fundamental to being able to develop a deeperunderstanding of mathematical concepts and perform more complexcalculations (NSW Department of Education & Training, 2004, pp.5-6). The two areas with which the students had difficulty weremultiplication and division, including the use of mental strategies.Testing revealed that the nine students had little concept of placevalue and used inefficient mathematical methods, such as counting byones, to group numbers. These nine students became target students forKatie's AR project. Katie knew that she would need to engage these students inactivities which they believed they were capable of completing and thatthey found interesting. Following discussion with colleagues, shedecided that the Counting On (NSW DET, 2004) mathematical games couldachieve both of these objectives because they are designed to beaccessible and entertaining. However, Katie had never been involved inactively implementing Counting On and realised that she would need todevelop a deep knowledge about the games if she was going to base herlessons and research around its strategies. Her supervising teacherassisted by providing the DVDs and texts that comprised the Counting Onteaching and learning package. The Counting On program introduces a learning framework with fiveinterrelated steps which are intended to move students from "naivestrategies, to increasingly sophisticated strategies in order to solvenumber problems" (NSW DET, 2008. p. 5). This is illustrated inFigure 1. The aim is to assist students to progress from the lowersteps, through to the highest step on the scale. At the upper point thestudent will have successfully acquired the skills to complete complexnumber problems. [FIGURE 1. OMITTED] Further evidence of the likely effectiveness of the Counting Onactivities was provided by a report of an intervention using similarstrategies that was conducted in Illinois (Fatta, Garcia & Gorman,2009). Other research showed that establishing goal-setting andimplementing a mentoring program to "reinforce mathematicalconcepts and skills", improved student motivation and achievement(Adami-Bunyard, Gummow, MilazzoLicklider, 1998, p. 4). Katie took intoaccount the strategies Adami-Bunyard et al. used for goal setting, bygiving students individual goals for the project, as well as the groupgoal that "every student will improve their skills in place valueand multiplication and division this term". In addition to consulting the literature, Katie acted upon theadvice of her supervising teacher, using two Counting On games per weekand starting from the third step in the Counting On Learning Framework,as this is the level at which the majority of students were working (NSWDET, 2008, p. 6). Following this intensive work Katie was able toidentify her AR question as follows: Can the third step in the Counting On mathematics program, "Building multiplication and division through equal grouping and counting", improve a group of targeted ... students' abilities to use more sophisticated numeracy strategies for multiplication and division, and allow them to progress from poor mathematical strategies to sophisticated strategies? Action research cycle The AR question prompted entry into the AR cycle (Kemmis &McTaggart, 1988). The cycle served as the analytical framework although,in reality, the four parts--plan, act, observe and reflect--were not asdiscrete as presented here. Plan After the reconnaissance phase and the development of the ARquestion, two plans were developed: for "action" and for"observation", that is, data gathering. During the remaining 8weeks of the internship was as follows; the nine target students spent15 minutes per day, separate from the rest of the class, engaging inmathematical activities based around place value, multiplication anddivision from the Counting On program. Two activities were completedeach week, with the first activity being modelled and attempted onMonday, and repeated again on Tuesday. The second activity was modelledand attempted on Wednesday and repeated again on Thursday. On Fridayeach student was given the opportunity to choose an activity toparticipate in from any of the activities learned that week. There were four data gathering strategies. Firstly, a studentsurvey was conducted in Weeks 3, 6 and 10. This survey required thestudents to assess themselves in relation to their understanding ofplace value, multiplication and division by putting either B(beginning), P (Practicing) or M (mastery) into a box against aparticular skill. Self assessments are particularly useful because theyprovide students with insight into their abilities as well ascontributing to confidence building when improvements are noted. Twodifferent pre- and post-tests were administered, one class-based test,and one interview-style test (NSW DET, 2004, pp. 2-3) in which studentsused concrete objects to demonstrate their grouping strategies. Duringthe interview style test the students were also asked a range ofquestions to gauge their mental computation abilities and methods. Thetests were age appropriate and also consistent with curriculumrequirements and Counting On. In addition, students maintained a dailyjournal noting any improvements, comments or misunderstandings. Action and observations In Week 3 of the term implementation of the Counting On programbegan. The nine students attended school on a regular basis and enjoyedparticipating in the Counting On activities, often asking if thesessions could run overtime as they were enjoying them so much! The average score for the pre-test was 6.9 out of 12 correctanswers, whereas the average score for the post-test was 10.1 out of 12correct answers. Figure 2 displays these data in terms of the number ofwrong answers recorded before and after the eight week program. Althoughit is possible that the nine students may have become test-wise, the 8week period between administrations makes this unlikely. [FIGURE 2 OMITTED] There was a decrease in the number of incorrect answers for almostall questions. This was pleasing but closer examination of individualquestions yielded further important insights. For example, for Question4, shown in Figure 2, it was apparent that every student who chose awrong answer selected the same wrong answer. The students were asked toexplain the reasoning behind their responses. Of the five students whoanswered this question incorrectly, three students admitted that theyhad not taken enough time to examine the question closely. The other twostudents seemed to have trouble reading the numbers correctly. Katiethen pointed to the correct sequence and explained. Katie believed thatthese students had not quite grasped the concept of place value andfurther work in this area would be needed. Question 11, shown in Figure2, was another question in relation to which little improvement wasshown. The question involved a division algorithm. Several students haddifficulty with understanding the concept of division, while proceduralerrors appeared to account for the remaining incorrect answers. Apartfrom these questions, there was an improvement in every student'sscores. The results from the Counting On interview style test (Figure 3)correspond to the five interrelated steps in the Counting On framework(Figure 1). The level of each student derives from their method ofanswering the questions. If the student uses a more sophisticated methodto determine their answers, they are ranked higher on the framework.During this test, students are asked, "So how did you come to getthat answer then?" and "Did you count on by ones or did yougroup the numbers?" Following pre-testing in Week 2, the majorityof students were at the second or third level of the framework (seeFigure 3, light). In Week 10 when they were re-tested, every student hadprogressed along the scale, and the majority were now on the fourthlevel (Figure 3, dark). [FIGURE 3 OMITTED] The analysis of the students' journals also provided evidenceof improvement over time. Katie noted various occasions when studentschose to use more effective methods of counting than used previously.For example, on one occasion, she noted that Student 3 was using a veryineffective method to complete a simple multiplication question:"Josh had 20 almonds for recess every day for 2 weeks (14 days).How many almonds did he eat all together?" The student used thestrategy of counting by ones--a Level 1 response on the Counting Onframework. However, in Week 7 when a similar multiplication question wasattempted and Katie noted: Student 3 ... has moved from concrete materials to mental computation for simple multiplication problems. Today I asked Student 3, "In a hotel there are eight rooms which each accommodate two people. If the hotel has no vacancies, how many people are staying there?" Student 3 was not only able to calculate the answer promptly, she was also able to explain her reasoning for her answer clearly. Student 3 had, thus, progressed from Level 1 to Level 3 of theCounting On framework. Katie monitored each of the nine students in thismanner to identify the progress of each over time. The student self assessment surveys conducted in Weeks 3, 6 and 10for times tables (Figure 4) indicated that the students believed thatthey had made improvements in multiplication. There were many fewerBeginner (B) ratings (darkest columns in Figure 4) and increasingnumbers of Mastery ratings (lightest columns in Figure 4) by the end ofthe AR. These data are consistent with results in the other two areas ofthe self assessment; place value and division. It thus appeared that thetargeted students had appeared to have gained confidence in theirmathematical abilities in relation to place value, multiplication anddivision. Reflection The implementation of Counting On strategies assisted the ninestudents to progress from relatively inefficient strategies to moresophisticated ones and at the same time to develop their competence. Byrevisiting the fundamental understandings of place value, multiplicationand division, the students developed a sound understanding of theseskills. The students appeared also to benefit from being in anunthreatening environment working with other students at a similaracademic level to their own. The students took more risks with theirlearning than they may have in other contexts and they found theactivities enjoyable. Through implementing this AR project, Katie improved her teachingpractice as well as developing her knowledge of the process and benefitof AR. Katie intends to use the processes learned during this AR projectthroughout her teaching career. Furthermore, she has developed a broadunderstanding of the Counting On mathematics program, and the importanceof ensuring that students have a firm grasp of the fundamental skillsbefore moving on to more difficult mathematical ideas. Although the ARproject has ended, the supervising teacher intends to continue theCounting On strategies into his everyday teaching, aiming to bring allof the students up to the highest step in the framework. [FIGURE 4 OMITTED] Conclusions For Katie, the AR process was an interesting and rewardingexperience. She learned that critical reflection on data gathered overtime about teaching and learning can lead to improved teaching practiceand student outcomes. Being involved in an in-depth study of studentperformance gave Katie a sense of accomplishment and satisfaction inknowing that she was able to help the students improve on some vitalmathematical skills. She is now equipped with the strategies to assessher practice through action research and will take this experience withher throughout her teaching career. References Adami-Bunyard, E., Gummow, M. & Milazzo-Licklider, N. (1998).Improving primary student motivation and achievement in mathematics.Chicago, Illinois: Saint Xavier University. Fatta, J. D., Garcia, S. & Gorman, S. (2009). Increasingstudent learning in mathematics with the use of collaborative teachingstrategies. Chicago, Illinois: Saint Xavier University. Grundy, S. (1995). Action research as professional development.Occasional paper No. 1, Innovative Links Project. Canberra: AGPS. Kemmis, S. & McTaggart, R. (1988). The action research planner.Geelong, Vic.: Deakin University Press. Macintyre, C. (2000). The art of action research in the classroom.London: David Fulton. NSW Department of Education and Training. (2004). Counting On:Re-connecting conceptual development. Sydney: Author. NSW Department of Education and Training. (2008). Count me in too:Learning framework in number. Sydney: Author. KATIE MEAD & TOM W. MAXWELL Katie Mead & Tom W. MaxwellUniversity of New England <tmaxwell@une.edu.au> APMC (1) Katie was a final year BEd(Primary) student and Tom was heruniversity supervisor.