Liisa Näveri & al.: Planning and task introduction (6.9.2012)
Planning and task introduction in problem solving teachingLiisa Näveri, Anu Laine, Erkki Pehkonen and Markku S. Hannula
Helsingin yliopisto, Opettajankoulutuslaitos
The study compares the aims in the lessons and the grade 3 pupils´ solutions in two open problem solving tasks. The tasks have been done in the same groups when the pupils were in the third and fourth grades.This study is part of a larger 3-year (2010-13) research project, where the development of pupils’ and teachers’ mathematical understanding and performance are compared when dealing with open ended problems.Lessons (6 classes) were videotaped and transcribed and pupils´ solutions were collected and examined. In addition, the teachers´ lesson plans and meeting videos were used.The teacher assignment classification system is a model based on Pólya. This paper examines the planning and the significance of the task introduction, when the focus of the task is to justify the solution.They were foundto be in related to pupil´s performance. Also teaching practices remained more or less the same.
Keywords : Mathematical thinking, problem solving, primary school.
This article compares the assignment in the lessons, and its connection to pupils´ solutions in two abnormal problems. At first we review three theoretical concepts of mathematics teaching: thinking skills, problem solving and teachers´ performance. There were six teachers, who are in the project and have doneopen-ended problems dealt with in this study. The other task was carried out in January / February 2012 and compared with a task that is made in March 2011. The data is based on lesson plans, lessons on video, findings, teacher meetings and pupils´ achievements.
THEORETICAL BACKGROUND
According to the curriculum (2004) it is not enough that pupils are able to calculate mechanical, but they should also be able to make conclusions and to explain their operations both orally and in writing (The National Board of Education 2004). Already in pre-school curriculum (2010) states that in the development of mathematical thinking it is important that children learn to observe their own thinking. At the end of the second class the target is good learning and working models, and also the development of mathematical thinking and reasoning by presenting.
Thinking skills
In Finland the curriculum (2004) uses the term 'thinking skills‘.It is not enough that the pupils know how to calculate mechanical, but they should also be able to make arguments and conclusions, as well as to explain orally and in writing (NBE 2004). – what, how and why?This is recognized in the curriculum (2004) criteria for grades 1-2 mathematics teaching core mission.
According to the curriculum (2004) the task of the mathematics teaching is" to contribute opportunities for the development of mathematical thinking ". There are many conceptions and definitions, whatis the mathematical thinking. The curriculums used the concept of thinking skills. There are also elements about mathematical knowledge (conceptual, functional and strategic) (see Joutsenlahti 2005; Sternberg 1996). All these are characterized by a high-level thinking.
Problem solving in mathematicsteaching
Problem solving has generally been accepted as a mean with advancing thinking skills (e.g. Schoenfeld 1985). We will look at problem solving in mathematics teaching with a teaching model developed from the Polya model.About sixty years ago Polya (1945) introduced his 4-step model for problem solving. From this model we have modified a model for teaching problem solving. This model, which we shall call A teaching model developed from the Polya model (Laine at al, 2012):
0. Planning phase(addition to the Polya model)
1. Understanding the problem – the significance of thetask introduction
2. Devising and carrying out the plan – the significance of guidance
3. Looking back – feedbackon the pupils’ solutions
The most important factor in problem solving situation at school is a teacher and teacher´s performance to develop students' problem solving persistence (Pehkonen 1991).This can be achieved by allowing the experience of solving problems (Nunokawa 2005). According to the study understanding and creative thinking helps to learn (Kretschmer 1983). Kretschmer (1983) carried out a teaching experiment.The different groups, he used, were the thinking aloud group,the creativity group, the basic operations group, the strategy group.Only the creativity group improved its performance in the delayed measurement.Creativity takes place in the open problems.Schurter (2002) has found that the own understanding about the tracking techniques are a crucial factor.Creativity is visible(for oneself and for the others) in reasons.
The research problems
This paper examines elementary teachers' (grade 3 and 4) conceptionsabout mathematical problem solving, particularly in planning and the task provision by Polya´s model for teaching problem solving.In addition these connections were looking in the pupil´s assignment. Thusthe researchquestionscan beformulated as follows:
( 1 ) How do the teachers plan and give the introduction in their lessons?
( 2 ) How do the teachers develop in planning and giving the introduction in their lessons?
( 3 ) What kind of connections there are to the pupils´ achievements?
IMPLEMENTATION THE STUDY
This research is part of the larger Academy of Finland 3-year (2010-13) research project ( Project # 135 556 ), where the Chilean and Finnish teachers performance and pupils understanding and development are compared, since there are used once a month problem solving tasks. The present tasks are of those experimental tasks, the implementation of which is collected from a wide range of information.
Assignment and control
In order to triangulate research results (cf. Cohen & al. 2000), we collected both pupils’ performances and other data sources. We collected from every teacher about half a page with the main points that she gave beforehand to the researchers. We observedthe problem solving lesson (two researchers in the class). One of the researchers recorded the teacher’s actions on video, and the other one focused on some target pupils’ performances. Furthermore the project group consisting of the research personnel of the project and the experimental teachers discussed the tasks andthis discussions were also recorded on video. Thus we had several data sources about both pupils’ and teachers’ actions.
Two researchersin the first and one research in the second task has classified in cooperation the variables connected to the teachers and their actions (introduction of the task, guidance, looking back, introduction of the critical feature of the task). First, existing obtions were charted, and then the researchers decided the categories for the teachers. In this classification, the lesson plans, the videos recorded during the lessons, and the video from the meeting of the project group were used.
The experimental group
The study included six teachers and their pupils.They discussed the same aritmagon task in third grade and track construction in the fourth grade. Teachers used in this text pseudonyms: Ada, Bea, Carla, Dana, Elinor and Fiona.
Tasks
Pupils had to solve the three-phase aritmagon task.facilitated aritmagon task containing the natural numbers, finding solutions to the reasoning behind the strategy and to do an additional task to friends. The ordinary task was made easier so that there were two same figures as a sum.
Aritmagon is a triangle, where the numbers are in the cups and their sums on the side of the triangle (e.g. Brown & Reid 2006).
Picture 1: a) The structure of aritmagon ja b) the essential task.
The real task was: Aritmagons can be solved in many ways. How did you solve it? Did you find a method for solving any aritmagons, when the numbers on the sides are given and two of those are the same? The addition task was to invent for the classmate an easy aritmagon and one not so easy.
About one year later the same groups (the same teachers and their pupils) did a new open problem. Grade 4 pupils´ task was to build a three-dimensional object using cocktail sticks and peas (as edges and cusps). The number of the options were limited by using less than 12 sticks. For example, the cube needed to build a 12 cocktail stick and 8 peas (see picture below)
Examine the constructions and explain why there are all the possibilities.
Data collection
Time to do the task in all the groups was used one hour. Two groups had also pre-processing lessons anda summary of the task was also done. Teachers work was videotaping and transcribing. In the teacher meeting before the lesson we were talk about the implementation of the task. After the lesson in the teacher meeting we discussed about teacher ´s experiences. Also teacher meeting was videotaped. All the students´ achievements were collected to the researchers. The task was carried out in January / February 2012.
RESULTS
Pupils
The basis for classification about both tasks´ solutions are examined in this chapter.All problem solving tasks in Chile-Finland-project have a specific target. Both tasks in this article the task structure was the same: to do the essential task (1), the strategy of the processing (2) and providing firms for the faster students the additional task (3). This article focuses on the first two, and in particular, to analyze the solutions which teachers´ design and assignment gendered.
Table 1.Pupils´ performance categories in both tasks.
Kategories
I / The pupil explain the strategy when solving the essential problem.
II / The pupil solve the essential problem.
III / The pupil solve the addition problem.
Pupils' results are reported in Tables 3 and 4 in connection with the teachers´results.
Combining the teachers´ assignment and the students´ performance
Six teachers participated in this study. They were handled both of the examined tasks, aritmagon and object construction. When you look atteachers´ placement in the essential task in aritmagon and in contributing to the sticks and peas problem, you see that the assignment of different categories were made according to the following classification:
Table 2. Teachers´ assignment categories.
Category A / To find the strategy solving the essential problemat the beginning.
Category B / To find the strategy solving the essential problem at the guidance phase.
Category C / Just to do the essential problem without the strategy.
In aritmagon task category A means, that the teacher gave the introduction to find the rule to solve the essential problem, when aritmagon has been given two same sums. Category B means that the teacher did not given the introduction to find the rule to solve the essential problem at the beginning of the lesson but at the guidande phase. C means, that the teacher gave the introduction to solve the essential problem.
Here is the classification in the aritmagon task. The criterion for classification you see below the table.
Ada / Bea / Elinor / Fiona / Dana / Carla
task provision/
solutions / A / A / B / B / C / C
I / 14 / 0 / 6 / 0 / 2 / 0 / 22
II / 1 / 7 / 2 / 4 / 8 / 8 / 30
III / 2 / 1 / 7 / 9 / 6 / 0 / 25
Table 3.Teachers assignment and pupils´ performance in aritmagon.
Preliminary results on the Aritmagon task are given in Näveri & al. (2012).
In the other task, in the sticks and peas-problem, teachers contribute is classified and the categories are explained below. In contribute task category A means, that the teacher gave the introduction to find the strategy how to know, that is found all the solutions.Category B means, that the teacher gave the introduction to solve the essential problem andcategory C means that the teacher did not given the introduction to find the rule, but to build objects without rule.
Table 4. Teachers assignments and students´ performance in the sticks and peas problem.
Ada / Carla / Elionor / Dana / Fiona / Bea
solutions / A / A / B / B / C / C
I / 18 / 14 / 0 / 3 / 0 / 0
II / 2 / 0 / 16 / 4 / 0 / 0
III / 0 / 0 / 4 / 9 / 19 / 6
Ada, Elionor, Bea and Dana had a written lesson plan. Categories have been made from lessonplans, lesson videos,observations and teacher meetings.
Ada´s lesson manages as planned except the arcument´s recording. The third hour was the puddling hour. The pupils discussed and documented their objects systematically with teacher. The written report was submitted to the researcher. Reflection how to know that all are found, was not conducted. Ada´s category is A
Carla´s group made objects, which had less than 12 sticks. The aim was to make as many as possible. When 30 minutes had passed, moved to large objects. Carla said also, that objects need not mention by name, more important is to find different kind of objects. Carla gave plenty of time to look for different solutions. Carla directed the pupils systematic search for alternatives so that they were looking at all options for the different amounts of sticks. Carla´s category is A. Elionor gave instructions to do objects with less than 13 sticks. She highlighted discussion. Two pupils designed the objects together discussing. Each of them did however the own object. The compaction was to differentiate little objects (less than 13 sticks) and the bigger ones. Elionor´s category is B.
Fiona gave the task to build objects. The smallest objects were named. When the time had passed 13 minutes, the students were no longer willing to make small objects. Fiona described the situation in a meeting : “It was rapidly important to the pupils, how large object still remains intact.” The classification is C. The target in Bea´s lesson plan is to construct objects with less than 12 sticks. Bearecommended to write the names of the objects on the paper and the names were searched actively. The pupils moved quickly to make bigger objects:“It will be agreed, if you are doing two, where it is less than 12 sticks and then you can continue the other so that there are more sticks.” The classification is C.
Summary.By giving time and giving the task step by step (first objects less than 12 sticks and not until pupils are ready in this then larger ones) (Ada, Carla, Elionor) we see in the table 3, that the solutions are better. Similarly, if the second part of the test was given too early, pupils were not willing to stay and make objects with small number of sticks (Dana, Fiona).If the goal was to mention objects by name (Dana, Bea) it looks like that the number of different alternatives remains small.
Comparison in two open-ended problems
In our project there was made in March 2011 an open-ended aritmagon task (Table 3) and in January/February made open-ended construction task (Table 4). It is no justified to compare pupil´s performance levels with each other, instead that developin planning and in the task provision levels could compare, because there are the same teachers.
Table 5. Assignments in two open problems.
Ada / Bea / Carla / Dana / Elinor / Fiona
Task 1. / A / A / C / C / B / B
Task 2 / A / C / A / B / B / C
Teachers have different targets to their lessons. So they guide their pupils in different ways and emphasize different things. Four of the teachers (Ada, Dana, Fiona, Elinor ) practices seem to be stabilized in planning and the task provision. Ada´s aim is to present arguments. The only teacher (Ada) in both tasks to give an excellent introduction, focusing on finding a general solution method and giving good examples, achieved most of students who reached the highest quality of performance.On the other hand, the teacher (Elinor) who limited her instructions and did not focus on a specific objective and provide responsibility to pupils. Elinor emphasizes students responsibility in their working.
Dana´s lessons targets are elsewhere than to find various strategies.When the teacher does not emphasize the proper task and give different levels at the same time, students move quickly forward and bypass the reasoning (Fiona), repeatedly. In addition two teachers (Carla, Bea) seem in these tasks attempt to find arguments, though there is shown variation in the implementation. In all of the courses where their teachers established the task objective as doing simple calculations without mention of finding a general solution method, or teacher gave incorrect or misleading instruction, the pupils did not submit valid justifications for the results of the task,
Thereby showing that in the case of the Finnish classes, the quality of the introduction given by the teacher clearly influences the quality of the results and performance achieved by their students. Our project has also a developing element. Three years, once a month, in the teacher meeting we are talking about the previous task, and planning to the next. In these meetings we also discusshow to teach open problem solving tasks. From this perspective it seems that changes in teacher behavior are slow.
DISCUSSION
The Finnish curriculum demands that it should be taught in school (also in elementary school) besides calculation skills also problem solving and mathematical thinking (NBE 2004). But this does not seem to happen in the ordinary mathematics teaching, where the teacher too eagerly uses the textbook and its tasks. Therefore, new elements should be connected in instruction: tasks with which the teacher can practice pupils’ problem solving and thinking skills. Problem solving is important in the development of mathematical thinking (eg Mason, Burton & Stacey 1982, Schoenfeld 1985, Stanica & Kilpatrick, 1988). In the Finnish primary school curriculum, mathematical thinking is consist of the conceptual thinking and problem solving. Pehkonen et al (1991) consider the problem solving tenacity as a key aim in the problem solving teaching.The problem solving tasks will include the development of the creative component.The teachers´ targets will be guided and will be realized during the lessons.This article concerned with the problem solving exercises has evidence of the importance about targets and assignment in the problem solving.