Abstract
Authentic tasks can promote higher-order learning. In the student task discussed in this paper the student teachers developed, tried, graded, discussed and revised mathematics test questions, in connection to an amusement park visit. The task was designed in an effort to bring the teacher educators in the mathematics and educational departments and those in the schools closer to each other through participation in a joint project, which was part of the "VFU" - the time that student teachers spend in school as part of the subject studies. The task involved different aspects of authenticity, both in the mathematics questions created in and in the creation of questions Qualities in students' problems are discussed, and also student reflections on pupil and teacher reactions in relation to the authentic contexts. Traditional textbook problems were found to have a strong influence, as evidenced by many of the student-created problems and some of the supervisor reactions. Students' descriptions of teacher comments indicated that these focused more on classroom management than on mathematics content, consistent with traditional expectations on the VFU.
"the mathematics teaching shall give the pupil possibilities to practice and communicate mathematics in meaningful and relevant situations, in an active and open search for understanding, new insights and solutions to various problems."One of the goals for mathematics teacher education is to guide student teachers to develop the ability to identify relevant math and also to help them to learn to assist and assess this development in their pupils.
Previous research shows that pupils often view the mathematics as irrelevant and are not able to identify applications in everyday life. This view is often confirmed in encounters with students. It has been found that much of today's mathematics education in Swedish schools consists of pupil's individual text-book based problem- solving (Skolverket, 2003). Mathematics is often viewed as a difficult subject, creating considerable anxiety among learners (e.g. Tobias, 1993, Ashcraft and Kirk, 2001). Making mathematics visible in local applications may not be a primary goal for the university mathematics, but is an important ability for mathematics teachers in school. Children are often curious about many aspects of the world around them. By bringing mathematics student teachers in contact with children's curiosity, students can be inspired to look beyond "Why do we need to know this? - It's not in the curriculum". A collaboration between the different departments involved, i.e. mathematics, mathematics education and schools, can provide applications to new contexts, which invite observations, estimations, creativity and discussions.
Mathematics, in itself, does not need applications, but can be cultivated as a pure and beautiful art in its own right. However, by experiencing a variety of applications of a particular equation or concept, a learner is likely to develop a deeper understanding of the concept (Marton and Booth, 1997). Abstractions are powerful, but to discover their power, students need to gain familiarity with their application in a variety of situations. Literature provides many examples where students fail to apply knowledge from one area to another (e.g. Lave and Wenger 1990, Säljö, 2000).
Authentic tasks can promote higher-order learning. Berge et al (2004), in their review of the research on authentic tasks, found that important design criteria for the problems to be challenging were that they were realistic, meaningful, open and demanding. This is important both in schools and at the university.
Assessment of learner's knowledge and understanding is a complex, and mastery takes a long time to develop. It benefits from discussions with colleagues and with teachers more experienced in the area. A focus on assessment in a course which involve teachers from different academic cultures, can make more explicit the different principles and traditions in examination and lead to a discussion among teachers about how examination affects learning outcomes.
The work, which includes an iterative design of a student assignment in an authentic setting attempting "simultaneously to understand and improve educational processes" can be seen as a design study (di Sessa and Cobb, 2004). The theoretical basis is drawn from research on authentic tasks as a way to promote higher-order learning (Berge et al. 2004).
The primary research questions in this work can be expressed as:
A teacher education needs to help students develop a wide variety of knowledge, skills and competencies. A well-prepared exam can be a useful learning experience, challenging the mathematical thinking of the students. The national tests, currently available for grades 5, 9, 10 and 11, provide examples of inspiring problems, and instruction for grading, together with a few typical solutions. This provides a framework for discerning qualities in student mathematical understanding, and for providing more precise feedback to learners. The creation of national tests involves a number of researchers and practitioners and represent a larger volume of work than can be expected of individual teacher for every test occasion. Boesen (2006) has compared teacher-constructed tests to national test in upper secondary school, with respect to mathematics context and requirements on creativity of the students and found considerable differences.
The data collected includes the student-created problems and grading instructions, before and after revision, and a few grading examples. In additions, notes were taken during the final joint discussion and the students were requested to send in a reflection on the task. This form of the assignment was the result of a development over a few semesters, as described below.
The students identified tasks with among other things geometric concepts, to measure angles, length and time, and to use body measures to estimate heights and lengths. In general, the students found that it was possible to discuss different methods of measurements and accuracy with the pupils.
During this pilot project, we found that many students needed additional instructions to be able to formulate mathematics problems for the pupils. Encouragement to focus on curriculum goals for a particular school year was found to work well. This was included as part of the assignment in the development of the project during the academic year 2004-2005, as discussed below.
Students were asked to produce one short problem, where they should indicate the number of points assigned for different types of solutions. They should also develop one larger problem, which would allow for assessment of different quality aspects. The task required students to analyse the curriculum requirements for different school years, and how they could construct problems to test various abilities. Students again had to reflect on whether the problems were best suited for individual written solutions or for small-group discussions.
The problems were discussed in groups of 8-10 students during a seminar with the teacher educators, and feedback on the usability of the problems was received from a few active teachers.
In the next step, groups of 3-4 students, with VFU in the same age group, selected problems for testing in their classes. They revised the formulations and refined the grading instructions before testing the problems on a group of pupils. Students then graded the solutions, and reflected on the applicability on the grading instructions, bringing their work for sharing at a seminar.
The results in this paper are based on the problems and grading instructions created by students, and on student reports on reactions from pupils and teachers, as well as their own reflection. Section 4.1. presents a description and discussion of problems created by the students.
This work has been presented at conferences (Lindberg and Pendrill, 2005a, 2005b, 2006) and discussed with colleagues in a few different contexts. Comments and reflections from these discussions are included in the discussion in section 5.3.
Some of the problems could be described as small variations of problems given on some of the web-pages assigned in the beginning of the project or familiar textbook problems, but often they were adapted to the local school situation. An example was for the pupils to work out how many tours with a particular roller coaster would be required to travel the same distance as the bus from their school to Liseberg.
Section 4.1 presents a few examples of student-created problems relating to the Ferris wheel and wheels of fortune. Several of the students presented individual problems for these attractions, and they were also used in the group assignments. The evolution of the problems was mainly in terms of "survival of the fittest", where groups selected the more creative and elaborate examples. Some revision of wording took place, as well as revision of the grading instructions (section 4.3). Discussions of specific aspects are given in direct connection to the problems, with the discussion of more general points saved for section 5. The results also include student and supervisor responses (4.4).
If you go on the Ferris wheel, you have a good view over Gothenburg, and it can be a nice way to finish an enjoyable evening at Liseberg, when you have been on the date with someone you like. How far do you get to travel?The eye-catching Ferris wheel was a popular ride for problem creation. Some students wrote a short introduction, e.g. as the one quoted above. Others simply stated facts from the WWW pages:
The Ferris wheel has a diameter of 25 meters and rotates with a velocity of 2.4 turns per minute. It has 20 gondolas, and can take 120 persons.
One report noted that "It has a handsome geometrical shape, which makes it suitable for use in mathematics problems." A circular ride invites common problems related to circles, e.g.
Also incorrect problem formulations occurred in students original versions, e.g. "What is the speed of a gondola, expressed in m/s, if we assume that we count the velocity from the seat in one of the gondolas which we assume to be placed 2.4 m from the outer rim of the wheel." The suggested solution then reduced the radius of the circle. However, the gondola remains at the same distance below its pivot point throughout the ride and thus the distance travelled in in a complete turn is the same as the circumference (neglecting the small horizontal motion at the sides due to acceleration in the slow circular motion). Still, a pupil might easily ask a similar question, and by asking this question, the student inspired the whole group to work out the motion with more confidence.
The most popular of the student-created problems connected to the Ferris wheel, which was also the problem kept after the group discussions, was to work out how many turns the Ferris wheel would have to make in order to make it to the school and how long it would take, given the radius of the Ferris wheel and the time required for a full turn. This can be described as a surreal problem, which connects real-world properties in a surreal way, as discussed in 5.1.
The Chocolate wheel has 20 numbers to play. Each number occurs in four places on the wheel. In three of these, there are 3 green dots (2:nd prize) and 2 white dots (1st prize). The fourth time, there are 4 green dots and one red dot (star prize).Some students chose to simplify the task, stating e.g. that the possibility to win a star prize is 1/12 or that one of 20 fields give a star prize and that 2 give the second prize (both claims are inconsistent with the real situation).
Most students who used a wheel of fortune in their individual problem started with a mini-story, e.g.
When at Liseberg, it is nearly a "must" to play on all wheels of fortune. Let us calculate the probability for Peter to win.When the stage is set, typical problems follow, e.g.: "What is the chance for him to win if he plays one number? Three numbers? If Peter plays all numbers, 1-20, thirty times in a row, how many star prizes do you expect him to win? How much money would he spend?" One of the student gave a more open formulation, stating that Peter, who is only interested in the star prize, is going to spend 40 kr, and that the cost of one play is 2 kr. "He wants to play all numbers at once, whereas his friend, Sven, claims that it is better to play the same number 20 times. Who is right?"
The wheel-of-fortune story emerging as a joint problem in the group discussions was an even more open-ended problem formulated and tested by one of the students. The task involved a discussion of how many people you should expect to carry around giant chocolate box, which is the star prize for one of the wheels of fortune. The question to be answered was if there could be any ground for a suspicion that the park might send out fake winners walking around the park to attract more people to the wheels. The problem was further developed by a group of 3-4 students. It was formulated as an assignment to the Slovak detective Proba Bility. In the original individual problem, the task was given to the French "mathematics detective" Geo, working under cover as the father of the Metry family.
The "detectives" have to work out the probability to win the large chocolate box, which is a combination of the probabilities that your choice of number comes up, and that the prize is the star prize. This part of the problem is well-defined, but to answer the assignment, the group also has to make a number of estimates, e.g. of what fraction of the numbers are played every time, how often the wheels are played, how often you meet someone with a giant chocolate box, how many people there a the park and how many of them you meet during the visit.
The student who did the initial problem formulation had also tested it in class and reflected on the difficulty for pupils to think on two levels at once, both as the detective doing observations, and still sitting in the classroom, estimating values for what he could observe.
The pupils testing the problem commented "Nice to work with math in real life, instead of just in the book", "Somewhat difficult without numbers, but a fun challenge. Pity there was no correct answer.", "Tricky problem. But fun, you learned to think something through properly." This group of students then tested the problem in their different secondary school classes. They found that it worked well as a group problem and that the groups came up with interesting discussions and creative solutions. The comparison of results obtained with different methods provide additional insight.
The grading matrix for the larger problems assess three quality aspects: Understanding and choice of method, execution and analysis, and, finally, presentation and mathematical language. For the first two aspects, students generally managed to give specific criteria for the different quality levels, whereas the assessment of "the presentation and mathematical language" was found to cause most problems. E.g., one of the criteria given by one of the groups for the highest level was stated as "The presentation is well structured and easy to follow. The pupil uses a correct mathematical language", which can be applied to most problems. In grading authentic student solutions, the questions of mathematical language became more concrete: Does it mean that the students write only numbers no words? Initial disagreement led to deepened appreciation. Where students disagreed on the grade points assigned to a particular solution, different possible views became visible.
Many students noted that the authenticity of the tasks provided extra incentive for the pupils to get involved. One student expressed:
The pupils enjoyed when there were many sub-questions to one problem. At the same time, they thought that it was fun with problems relating to their experiences, in particular the new ride Kanonen. Many of them asked if the data, such as the height of the loop, were correct. When I said that they were, it felt like they were more motivated to solve the problems. I think it became more interesting when it was about things they are going to try for themselves later.Still, a number of students used clearly unrealistic data, sometimes on purpose. Two of the individual student-created problems stated that seats in a ride were 5 foot wide. The students claimed that the intention was to see if the pupils would note that this was a bit extravagant. Only one pupil objected. Torulf Palm (2002) describes in her thesis how pupils often fail to build on everyday experience and reflect on if the results are reasonable. We can conclude that students, as well as pupils and often teachers, are too used to problems where the only interest in the result is a comparison with the key in the end of the book. In real life, the ability to assess if a result is reasonable is important and may in some cases be a matter of life and death.
Many students discovered that the formulation of a problem sometimes demands more knowledge and work than solving the problem. The situation may need to be adapted and the problem given must be solvable. In some cases students backed off from problem ideas, because the formulation seemed to raise too large difficulties - and assumed that then also the solution would be too difficult for the pupils. The result was sometimes that the final problems were relatively trivial.
The most popular Ferris wheel problem, asking the time required for the wheel to roll to a neighbouring town, can be described as a surreal problem, which connects real-world properties in a surreal way (Gellert and Jablonka 2009). It is so obviously surreal that students accept the premises that it is just a "Gedanken" problem, where there is no need to worry about the technical details concerning how to remove the wheel from its stand, or the intrinsic strength of the structure of the wheel. In this way, students get to work on the problem without being tempted to mixing in irrelevant everyday complications. Another surreal problem was created by a group of students who noted that the teapot in the centre of the classical "Teacups" ride is rather small compared to the teacups where the riders sit. The observation was converted to a problem where the teapot shape is approximated by a cylinder and the cups by half-spheres. How many cupfuls in a teapot? Although pouring tea from the pot is not a realistic situation this problem uses the authentic experience of volume of the objects in the ride, and makes the visitor recall this amusingly non-realistic problem during subsequent visits in the park. These surreal problems were popular both with student teachers and pupils.
The authenticity of this student task involved not only the mathematics, but also the fact that the problems were discussed with active teachers and tried on real pupils. We find that the task satisfies many of the criteria found in the review by Berge et al (2004) for authentic tasks to contribute to higher-order learning. The work encouraged students' reflections over and investigations of their pupils' level of mathematics skills. The motivations increased to study and interpret the steering documents. The task also opened for creativity.
In connection with the Ferris wheel problem in Section 4.1, one of the initial individual problems hinted to the fact that the distance between two gondolas could be approximated by a circular arc. This could have been developed into a problem investigating the validity of this approximation and a discussion of a circle as a limiting case of a polygon.
One student attemped to create a queue modelling problem for a ride opening later than the park, when, as he stated, 120 people were waiting, and 2 persons per minute join the queue. "Three hours later, the queue is stable at around 30 people. ... The number of people joining the queue turns out to be inversely proportional to the number of people already in the queue." In the initially attempted problem formulation, the task was then to work out the time between the tours in the ride, given that 12 persons ride each time, with a bonus task to work out a formula describing the number of people in the queue. We note that this situation could be an interesting modeling application for use e.g. in a spreadsheet program. However, the problem was instead simplified to state that 2 persons per minute join the queue, independent of length. Pupils were also asked to draw a graph and describe in words what happens to the queue during the day. Finally, they were asked if graph seems credible, and invited to draw their own graph of the variation of the number of people in the queue during the afternoon. This problem did not survive the selection into the common group problems.
The problems avoided hint to various factors resisting change, found in textbook traditions, student teachers' own understanding or perceived capability of the students. Some open-ended problems became less open-ended when students were facing the requirement to provide well-defined grading instructions. One important factor for the problem selection was that the classes assigned for the students' VFU were in primary or lower secondary school, which often did not allow students to use the university mathematics in constructing problems. In authentic assignments, the real conditions must be taken into account, even if they may provide inertia to change.
In the mathematics departments, as in many other university department with large enrolment, the grades at the early undergraduate level is based nearly exclusively on a written exam at the end of the course, even if it contains other required activities. This form of examination can be characterised as a summative evaluation, to ensure that students have acquired a sufficient grasp of the required material. To provide detailed feedback on qualities in need of development is then not necessarily a task given high priority. Some teachers thus question the need for a grading matrix, with its requirement to discern different qualities in the solutions.
Most of the examination problems that students encounter in the mathematics department as part of their teacher education are essentially context-free. Teachers justify this approach by emphasizing that they want to be sure that what they examine are mathematical skills and knowledge. Although the teachers often welcome collaboration to bring in different contexts, they do not necessarily take initiatives to put the mathematics into context. In relation to this discussion, it might be worth noting results from physics education research (e.g. Redish, 2003): In diagnosing student understanding of concepts, it has been found that students can often solve traditional examination problems in physics, without understanding the concepts.
Some of the students in this project encountered teachers who never constructed test of their own, but relied on test provided in connection with the textbooks or on national tests.
For us, as teacher educators, it was a challenge to formulate a task that inspired student creativity. Our initial open tasks, without external constraints on problem area, left many students at loss. We found that the requirement that the problems should be suitable for a particular age group provided sufficient restriction to get most students actively and creatively involved. The task also meant that students had to reflect on the wording of the problems in a way understandable by the pupils, including a consideration of pupils who were not native Swedish speakers.
The realistic situation meant that data was often available only in indirect form or after a number of approximations. Sometimes, in the construction of problems, students needed computations to convert the data to a form usable for the pupils. This challenge helped them develop more familiarity e.g. with spreadsheet programs. Many students were not used to open-ended problems. Some students initially disbelieved our level of ambition for the assignment. Student expectations resisting change are shaped by previous experiences and demands, in schools and in teacher education. Through group discussions of the different problems, students had a chance to perceive different level of student teacher achievements, possibly also sensing that "ill-defined problems become well-defined with practice". The same information is perceived and understood differently, depending on the knowledge someone brings to the situation. The review by Berge et al (2004) on the design of authentic tasks concluded that for the tasks to contribute to higher-order learning, "both structure and complexity need to be carefully matched to the level students can cope with".
The organisation of the VFU in teacher education is often, in itself, challenging, since not all students are dealing with the same context areas at the same time. The task of creating problems for the pupils could easily be adapted to the class at hand, although the assignment of classes sometimes made it difficult to connect to university mathematics. The tradition of general school practice often overrides the ambition of the VFU as part of the subject studies. At the time of this project, no central university instruction clarified the contents and goals of the VFU in different subjects. The integration of VFU in subject course is often easier in smaller universities, as e.g. in projects described by Nilsson (2005,2008).
... compare, estimate and measure length, area, volume, angles, quantities and timeThese are, in fact, important aspects of any project studying physics in the park, and, indeed, physics can provide many applications of mathematics throughout the school years. Wigner has noted "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (1960). However, applications may be found not only in natural sciences and technology, but in all aspects of life. Mathematics does not need to ask for applications - the generalisability constitutes the mathematics.
Still, the agility to move between the abstract and the concrete is
important for successful applications. Richard Feynman (1988) described
his briefing for the investigation of the Challenger disaster. In the
beginning, he had to ask everything about the function and the meaning
of all symbols, but soon, he was able to make inferences based on
knowledge of other physics phenomena, described by analogous
equations.
Real-life applications are among the goals for mathematics teaching in schools. The teacher education system includes many actors: the student, the school and the different university departments. Using a mechanically coupled system as a metaphor, we note that the teacher education system can be characterized by its components, the coupling between them and by the boundary conditions. The boundary conditions for a course in teacher education include curriculum, teachers, examination system and previous courses. These also shape the knowledge and expectations of the students. The response to perturbations provides information about the system itself. The response depends on the location of a perturbation, strength of the couplings, and on the damping in the system. The design project, which constituted the perturbation of the system in this study, probed the tensions between pure and applied mathematics, between schools and the education and mathematics departments, between curiosity, openness and tradition. As common with design experiments, the results and observations lead to questions beyond the original scope (di Sessa and Paul 2004). A deeper understanding of the teacher education system would require more detailed data collection at different levels and parts of the system.
The inclusion of student-created assessment as part of the mathematics teacher education, bringing many actors together in a collaborative effort, is possible within the current teacher education (Utbildningsdepartementet, 1993), but the proposition (Utbildningsdepartementet, 2008) will make this form of collaboration more difficult.
Although inspired by the work preceding the previous revision of the teacher education (Utbildningsdepartementet, 1999) and well connected to the official mathematics curriculum (Skolverket, 1994), this project still challenged many traditions, including textbooks (Skolverket 2003) and assessment cultures in mathematics, but also school practice in itself, and school practice as part of mathematics studies in teacher education.