ss ee oo ww dd oo HH yy pp ll pp aa tt nn ee mm ee rr uu ss aa ee mm ? ss oo nn pp ii gg ee oo tt T his article tells the story of one for numeracy that these shifts pose and student who chooses her the need for tasks to reflect the actual personal interest in pigeons and practice of real-world problems in order their lifestyle to demonstrate her under- for them to be realistic and provide standing of measurement concepts. The evidence of students’ abilities to use student’s work is used to consider the mathematics in varying and complicated possibilities for assessment afforded by situations. Making the boundaries tasks designed within reform-based between school and the world less distinct curricula. by bringing authentic contexts into the Anita (pseudonym) was a Grade 6 classroom still needs to be considered in Jane Skalicky tells student who was part of a research terms of the effect these contexts might project investigating how classroom have on student learning (Anderson et al., the story of a girl’s teachers are positioning numeracy in an 1996). This poses considerable challenges emerging values-based curriculum for the devising of assessment tasks in a mathematical setting and how this positioning influ- trans-disciplinary framework and the ences student experiences of numeracy need to assess multiple objectives within inquiry and the and their development of mathematical contextually rich tasks. understanding. The motivation for the Of interest in Anita’s case study is: assessment larger project is the reform-based (cid:127) exploration of the mathematical curriculum in Tasmania, centred around understanding demonstrated by her in opportunities it five Essential Learnings: Thinking, a context chosen by her Communicating, Personal Futures, (cid:127) examination of the assessment oppor- presented. Social Responsibility, and World Futures tunities that an open-ended (Department of Education, Tasmania mathematical inquiry provides [DoE], 2002), and encouraging trans- disciplinary activities. Other Australian states and territories are also re-concep- Learning context tualising curricula in terms of similar over-riding big ideas. Anita’s teacher was implementing a unit Alongside curriculum reform move- of work entitled ‘How do you measure ments, conceptualisation of assessment up?’ with the teaching, learning and has moved to a focus on assessment that assessment plan guided by goals in two of is educative, formative, an integrated the Essential Learnings key element part of teaching and learning, and aimed outcomes: ‘Being numerate’ at improving student performance. (Communicating) and ‘Inquiry’ Wiggins (2003) discusses the challenges (Thinking). APMC 12 (3) 2007 23 How does measurement apply to pigeons? Culminating Performance Being numerate. The DoE embraces a comprehensive view of numeracy. Being numerate involves having those The task Anita completed is presented in Figure 1. concepts and skills of mathematics that are The overriding aim of the task was to provide required to meet the demands of everyday students with an opportunity to demonstrate an life. It includes having the capacity to select understanding of their learning in a context of their and use them appropriately in real settings. own choosing by posing a ‘big’ or ‘rich’ question to Being truly numerate requires the knowl- investigate. The students were provided not only edge and disposition to think and act with the task but also with the goals being addressed mathematically and the confidence and intu- by the task. ition to apply particular mathematical Culminating Performance principles to everyday problems… (DoE, 2002a, p. 21) Understanding Goals: Being numerate – measurement: Inquiry. The central Thinking Essential 1. What can be measured and how do we describe and Learning brings the processes of inquiry and communicate those measurements? reflection to all questions and investigations. In 2. How can knowledge and understanding of measurement be particular, the aim of the ‘Inquiry’ key element used to answer questions about our world? is to develop the capacity of learners to pose Inquiry Goals: problems, gather relevant information, consider 3. Understand how to pose and define a problem, clarify possibilities, make decisions, and justify the issues involved and select and monitor the most conclusions. effective process to use. 4. Be able to collect and record information, with an under- Effective learners need the capacity to ask standing of accuracy and reliable results. good questions, persevere in a line of inquiry, 5. Understand that reflective thinking is a deliberate be systematic, set goals, and plan and follow a course of action… The ability to communi- process and that it is used to develop and refine ideas cate what has been learnt and thought about, and beliefs and to explore different and new perceptions. and to do so in a consistent, coherent, relevant and persuasive way, is essential in enabling Open Investigation: learners to participate fully in schools, Thinking about all you have learned during our measure- communities and workplaces. ment unit, what are some of the BIG questions related to (DoE, 2002a, p. 14) measurement that you may want to investigate? You will need to consider: It was within the broader goals of ‘Inquiry’ (cid:127) What makes a good question? and ‘Being numerate’ that the twelve week unit (cid:127) What are some big issues in our world that relate of work was planned. During this time students to measurement? undertook many activities and open-ended (cid:127) What would you like to find out more about? investigations building their mathematical skills and understandings particularly, but not Pose the problem, ask the question. solely, in relation to the measurement strand of Conduct your investigation. ‘Being numerate’. The goals for the unit of work As you discover some possible or partial solutions to your were brought together in a culminating question make a note of new questions you need to ask as performance. they arise. [Presentation format negotiated with the teacher] Figure 1: Culminating performance task 24 APMC 12 (3) 2007 How does measurement apply to pigeons? Table 1: Rubric for Culminating Performance Key areas for assessment Essential Learning Understanding Goal Assessment Criteria (addressed across 4 levels of performance) Key Element Outcome (from Task) Posing of relevant question and identifying Questioning Inquiry 3 problem Formulation of plan, accessing of sources, collec- Investigating tion and judgment of relevant data, representation Inquiry 3 & 4 of data for purpose and audience Can identify other Identifying new problems and issues arising from Inquiry 5 issues arising investigation, and how these are addressed Measurement: understanding of relationships and standard units, how measurement is used to Content Being Numerate 1 & 2 answer questions about our world, communication and justification of measurement understandings Neatness and organisation of work, appropriate- Presentation Being Literate Beyond task goals ness of presentation for purpose and audience Effort & Time Quality and quantity of work in relation to time and Reflective Thinking Beyond task goals Management effort. The assessment rubric for the task was collaboratively 1. Introduction to choosing the big question constructed by the teacher and her students. It included Anita originally chose as her question: ‘Why do we criteria relevant to the ‘Being numerate’ and ‘Inquiry’ have seven days in a week and why are there twelve goals declared for the task. Additional criteria for pres- months?’ She describes in this section where she took entation, effort and time management were included this investigation and that “despite my best efforts I and valued by both teacher and students. These criteria couldn’t come up with a solution… So I had to come up also contributed to the assessment of curriculum with a new question.” Anita’s question became ‘How outcomes that were beyond the scope of the goals of the does measurement apply to pigeons and their lifestyle?’ task. The rubric consisted of four levels of performance for each of the specified criteria (Appendix A). The aim 2. My life around pigeons of the rubric was to provide a clear set of criteria for Anita describes how she was introduced to pigeons by students to refer to that would guide their progress and her grandfather and tells a story about her grandfather support the learning experience. Upon completion of the raising his own pigeons. She details her personal story of task students self-assessed their work prior to teacher breeding and training pigeons. assessment. Table 1 presents a modified version of the final rubric, describing how it was constructed in terms I got my first four pigeons from my grandad when I of its connection with both the Essential Learnings key was seven, the same age he got his. I started to breed elements and the understanding goals, which were a and train them and soon after that my loft was too focus of the culminating performance. small to keep them all inside with enough room to move around. Dad brought me a new aviary, which is the one I have now … In a few years I went from four Anita’s work pigeons to seventy-eight. It was hard but we had to get rid of some. I refused to let Dad dispose of them so Anita presented her completed work in a coloured spiral I gave some away to my uncle… I got back down to bound book. The book had seven sections and these are twenty pigeons and then I had to stop breeding them so described below with particular excerpts drawn from the it wouldn’t all happen again… They never really went work to demonstrate the issues being considered in this far away from home but then their numbers started article: the situating of the measurement understand- creeping up again and I was only allowed to keep fifteen ings in the context chosen and also the areas of of them when we moved so I packed them into their opportunity for assessment. racing boxes and my dad put them on the back of the work truck that was going up to Devonport. We did APMC 12 (3) 2007 25 How does measurement apply to pigeons? that twice and ended up with only three of the ones we loft or aviary and then let them look on top of it. Push took. So altogether I had seventeen but I kept the extra them through the bolten wires then let them go… two anyway. Once a pigeon is trained you can start taking them to far away places and see if they come back. They must be able to fly at this stage though. Start off taking 3. My babi girl them a little way down the road so that it will prob- Anita gives a very personal and emotive account in this ably take them less than a minute to get back just so section of her hand-reared pigeon ‘Babi’ and how Babi that they get used to the idea. Each time take them a came to a tragic end. little bit further … This is the story of my bird, Babi …Then one year two of my pigeons laid two eggs. I waited patiently for them to hatch and shortly after they did one of the hatchlings died. I felt sorry for the now single chick … I watched her grow up and I put a small blue ring to tell who she was when she was flying … One day I went down to the loft to see Babi and I found that she wasn’t there … That was when I realised that she hadn’t been one of the escapees ... I got to the fence it was her. She was in the pond (which was empty) and her head was covered in blood. The flesh on her neck was gone … I held out my arms and she ran into them. A hot burning sensation erupted on my cheeks. I was crying. I knew she would die… She was born in 2003 and only lived until 2004… One year later I have one minute silence on the 17th minute of the 17th hour of the 17th day in the 11th month (in other words 17 past 5 on the 17th of November)… 4. Training In this section Anita writes about both the moral reasons for pigeon training and also the details of training itself. Anita describes her own method of pigeon training. She includes her own drawing of a pigeon loft and the design of the bolten wires (Figure 2) that enable the pigeons to fly back into their loft but not fly out. Pigeon training is very important. Some people just have pigeons cooped up in a cage all day every day but that isn’t right … Training starts for young pigeons when they have all their feathers and are able to glide to the ground from a reasonable height. If they cannot then the long drop into the loft could result in a broken leg or wing. You hold the pigeon the correct way and walk around your yard in a large circle... Do Figure 2: Anita’s pigeon loft as many circles as you like and then walk them to the 26 APMC 12 (3) 2007 How does measurement apply to pigeons? 5. Racing 6. Life as a pigeon The details of racing pigeons is the focus of this section, Anita describes the life cycle of the pigeon in this it introduces pigeon clubs, and how clocks are used in section. racing. Anita also describes two kinds of racing pigeons: the sprinter and the glider. Pigeons only lay two eggs at a time. Pigeon eggs take about 18 weeks to hatch… It takes it about two - four After pigeons are trained you can join a pigeon club. days to peck away the whole egg. The babies are born Pigeon clubs take pigeons to far away places to release with their eyes shut and they stay like that for about 5 them, this is known as a race. The first pigeon to days, sometimes a week. … At about the age of two return home and into its loft wins. …They use clocks. weeks I ring the babies. This is where I put a registered These are no normal clocks. They are clocks that when ring on their leg… Over the weeks the pigeon grows the rubber racing ring a pigeon has on its leg is put feathers and starts to become more independent. They into the clock it stops the timer… The night before still rely on their parents for food but occasionally try the race all contestants go to the pigeon club and picking at it themselves… Soon the pigeon learns to synchronise their clocks… There are two kinds of fly and is trained to return home. This is when the racing pigeons. Sprinters and gliders. Sprinters have fight for survival can begin… These pigeons miss the lighter bones, longer feathers and a small build. This first molting season since their feathers are still new enables them to fly super fast over a short distance and unharmed. But when the next breeding season where as gliders have heavier bones, shorter, fatter comes around they are old enough to lay their own feathers and are stockier than sprinters… When eggs and the whole life cycle happens again. Fact: entering a race you must take the pigeons to your breeding season starts in September and finishes in pigeon club on the scheduled night… The trucks used March. That is when the molting season begins. Fact: are divided into four parts. Two levels, higher and pigeons live for around 20 years. lower so as to fit more birds in… The driver then drives the pigeons to the releasing location and lets them all go at the same time. This is when the race has 7. Measurement with pigeons begun. This final section includes a list of facts Anita draws together from the specific measurement aspects of a pigeon’s lifestyle from her overall investigation. A sample of these is included in Figure 3. How does Anita’s work illustrate the objectives of studying her case? Measurement understandings in context There is much measurement embedded within Anita’s investigation. It begins more incidentally within the story she is telling about her life with pigeons and becomes more specific as she describes the training regime of the birds including construction of the aviary, the competitive race information, and the pigeon’s life cycle. Anita completes her work with a list of measure- ment-related facts to tie together some of the areas already included and fill some gaps that she identified in her previous sections. She demonstrates that although Figure 3: Measurement with pigeons sometimes deeply involved in the story-telling she does APMC 12 (3) 2007 27 How does measurement apply to pigeons? not forget the purpose of the task being to demonstrate performance enabled the teacher to tie together the her understanding of measurement as it applies to her assessment rubric with other forms of assessment into question. Throughout the seven sections of Anita’s her final judgment concerning the placement of pigeon investigation she demonstrates her under- students against the pre-defined standards of the standing of the main ‘big ideas’ of measurement as Essential Learnings curriculum. informed by the national curriculum profile in mathe- The teacher did not anticipate the richness that the matics for Australian schools – choosing units, culminating performance offered her in terms of assess- measuring, estimating, time, and using relationships ment opportunities. Beyond the criteria in the rubric, (Australian Education Council, 1994). These aspects of Anita’s piece of work presented clear opportunities also measurement were a focus of the twelve week unit of to assess outcomes in the key element ‘Being literate.’ work leading up to the culminating performance task With respect to ‘Being numerate’, Anita’s work could and Anita shows that she could transfer her understand- have been used to assess outcomes for other mathemat- ings from the broader unit to a context of her own ical strands, additional to measurement, if required. choosing. The incidental evidences of other mathematical When considering the mathematics involved in the topics–number, space, and chance and data–supports the many aspects of a pigeon’s lifestyle that Anita discussed, value of the task in encouraging the connectedness of evaluation of other learning outcomes of the wider topics within mathematics itself and to contexts socially Essential Learnings curriculum may also be possible. and culturally grounded in Anita’s life. This is particularly relevant for World Futures, with Anita’s work being related to ‘Investigating the natural Assessment opportunities and constructed world’ and ‘Understanding systems’, The assessment rubric designed for this task was two key elements within this Essential Learning. predominantly a tool to support student learning and to enable students to self-assess their work. For the teacher, it contributed to her overall assessment of her students Concluding remarks for the unit of work ‘How do we measure up?’ but was in no means a stand alone assessment tool. Spending Anita’s work provides an example of the possibilities for twelve weeks observing the classroom enabled the numeracy within reform-based curricula. She demon- formation of a total picture of the means by which strates her ability to evidence her mathematical students such as Anita were assessed. The teacher has understandings alongside her thinking in an environ- realigned her pedagogical practices to suit the changing ment where the process of student thinking about a curriculum environment in which she works. She problem is valued, not just the solution to it. This was assessed in the three key ways espoused by Earl (2003) made possible in this task by clearly stated goals as making assessment ‘an integral part of learning’ assessed against pre-defined learning outcomes in addi- (p. 21). Assessment for learning, having a formative tion to a teacher laterally looking at other evidences of emphasis, was utilised throughout the unit of work. She understanding not initially intended. It became a piece constantly kept observational records of the students of work that was a rich and valuable source of assess- using sticky notes, and used this to guide her teaching. ment. She never sat at her desk; she walked the classroom Anita’s case study has also shown that when our view interacting and questioning her students to stimulate of assessment is broadened to include consideration of their thinking, and assisting them to scaffold their the context of learning and how mathematics is applied learning. Assessment as learning is demonstrated by the within that context, then multiple opportunities to use of the collaboratively designed rubric. It assisted assess student learning are exposed. It also demonstrates students to identify their learning goals and enable the value of students using personal interests as mean- them to self-monitor their work. Assessment of ingful opportunities for them to demonstrate their learning, occurred throughout the unit of work with the understanding of many aspects of the curriculum. assessment of student artifacts. The culminating 28 APMC 12 (3) 2007 How does measurement apply to pigeons? References Acknowledgement. Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated The author wishes to thank Anita and her inspira- learning and education. Educational Researcher, 25(4), 5-11. tional teacher for sharing their work with me. The Australian Education Council. (1994). Mathematics – a curriculum author also thanks Jane Watson and Noleine profile for Australian schools.Carlton, Vic: Curriculum Corporation. Fitzallen for reading this paper. Department of Education Tasmania. (2002). Essential Learnings Framework 1.Hobart: Author. Earl, L. (2003). Assessment as Learning: Using Classroom Assessment to Maximise Student Learning. Thousand Oaks, CA: Corwin Press. Wiggins, G. (2003). "Get real!" Assessing for quantitative literacy. In B. L. Madison & L. A. Steen (Eds), Quantitative Literacy: Why Numeracy Matters for Schools and Colleges.New Jersey: The National Council on Education and the Disciplines. Jane Skalicky University of Tasmania <[email protected]> Appendix A: Assessment rubric for culminating performance Criteria Yes Yes But... No But... No Has posed an Open-ended question with Has posed a relevant ques- Has attempted to pose a Has not identified a ques- relevance to topic. tion but too big or too question but tion, problem or issue to be Questioning Has clearly identified the problem narrow. not relevant to topic. investigated. Unclear in purpose or intent. Formulated clear plan in order to proceed Has a plan to proceed with Has a plan but has not No plan formulated. with investigation. investigation. followed it. Relied on one source. Has accessed variety of sources. Has accessed a couple of Relied on one source of Data not relevant or evalu- Has collected relevant data and had made sources. information. ated for quality. judgments about the quality of informa- Has collected data but has Not enough data collected to No evidence of consideration Investigating tion. not made judgments about find a solution. given to audience or Has selected best way to represent data quality or relevance. Has not considered purpose purpose. for purpose and audience Has not considered purpose and audience. and audience in representa- tion of data. Has clearly identified new problems and Has identified new problems Has identified new problems No consideration given to Can identify issues arising as the investigation and issues arising but has not and issues arising but has not new problems or issues as other issues proceeds. adequately addressed those attempted to address them. they arise. arising Has addressed those problems and issues. problems and issues. Has demonstrated understanding of Has demonstrated some Limited understanding of No understanding evident of measurement relationships and standard understanding of measure- measurement relationships measurement relationships units of measurement. ment relationships & nits of and units and has not or standard units of measure- Has demonstrated understanding of how measurement. connected measurement ment. measurement can be used to answer ques- Has demonstrated some concepts to real world situa- Has not addressed original tions about our world. understanding of how meas- tions. question or problem. Has justified and communicated under- urement can be used to Partial solution provided. Content standings. answer questions about our Has provided a solution to original ques- world. tion or problem. Has not clearly communi- cated understanding hand has not managed a clear solu- tion. Neat, well organised used heading/sub- Neat and organised with Untidy in places with little Untidy, difficult to read. headings some use of headings. organisation. Lacks organisation. Presentation Appropriate presentation for purpose and Has not considered purpose Has not considered purpose audience and audience in presentation and audience in presenta- tion. Quality & quantity of work indicates a lot Has done what was required Quality & quantity indicates Poor quality and too brief. Effort of time and effort. but nothing over. little time & effort. All areas All areas lack care & detail. & Time lack care and detail. Management APMC 12 (3) 2007 29