Scenario: Franca solving a story problem with the students.
Grade (student age): Year 4 (age 9-10)
Contributed by: Marco Bardelli, University of Padova, Italy
Context – public primary school, national curriculum
The Italian National Curriculum emphasizes the development of students’ number sense through realistic mathematical problems where the estimate of numbers can be performed. The mental calculation has taken great importance and the students should manage the control of their reasoning. Moreover the National Curriculum underlines that students should be able to solve mathematical problems with different strategies.
Franca is teaching volume measures to her pupils. She has already explained the litre and its submultiples as a measure of capacity. In this lesson she wants to show her pupils how to use mental calculation in a story problem context to give them a chance to manage the change of unit in meaningful way.
Franca tells her students the following story problem:
In a kindergarten a teacher wants to give every child 200ml of chamomile. She has some thermos. The capacity of each thermos is 500ml. How many thermos it is necessary to fill to give the chamomile to all the 18 children?
She writes the data on the blackboard in this way:
200ml every child
and asks to all the pupils how they would start to solve the problem. Manuele says that they have to find how much chamomile it needs in all and that the operation is 200 times 18 (200 x 18). Franca writes on the blackboard 18 x 200ml and asks what is the result of this multiplication. Manuele answers that the amount of chamomile is found. Then Franca tells Manuele to calculate 200 x 18. Manuele says 18x2 is 36 therefore you obtain 3600. Franca writes to the blackboard 3600ml and asks to all the pupils how many lits are 3600ml. Manule says it is 3,6 lits. Franca asks Mattia the reason why it is so. The student says that 3600 are millilitres. So the teacher underlines that you need to divide by 1000 to convert mls to lits. Then the teacher asks how many thermos are needed if every thermos is filled with ½ lit of chamomile. A student answers that 6 thermos are needed, but Franca forces the pupils to tell their reasoning. So another pupil says that it needs to divide 3600 by 500. Franca acknowledges that it is correct but she would like the pupils found a strategy to do the operation in mental way. Then Franca asks to the pupils:
Franca: If a thermos is filled with ½ a lit how many thermos are necessary for one litre?
Franca: Then what can I do?
The students try to find the answer telling numbers randomly until a student says that you need six thermos. Franca asks for the reason behind this answer. The student answers that if a thermos contains ½ a litre then 6 thermos contain 3 lits, and that he did a multiplication by two to find this result. Franca wants to know why he did a multiplication. Another student explains that the classmate made a multiplication because if a thermos is filled with ½ lit, for one lit you need two thermos. Franca approves this answer. Then she asks to do 3,6 x 2 in their mind. She pushes the students to make (3 + 0.6) x 2 = 3 x 2 + 0.6 x 2= 6 + 1.2 = 7.2 and when they find the result, Franca writes 6, 1.2 and 7.2 on the blackboard. She points out that 7.2 are thermos and then she asks to her students how is it possible to fill 7.2 thermos. She writes on the blackboard 7 and 0.2 saying that the question is what “0.2 thermos” means. Then a student answers that 0.2 is half a thermos. Franca asks:
Franca: How much is one half?
Some students: 0.5!
Franca: So half thermos is 0.5 not 0.2.
She writes 0.5 on the blackboard. While she’s saying that 0.2 is less than one half and a student says that it is a quarter. So she asks what a quarter is in decimal numbers. The students try some numbers randomly until one says 0.25 and Franca writes it on the blackboard. Then she explains that 0.25 is ¼ of a thermos because 25 times 4 is 100. Then she encourages the students to compare 0.2 with 0.25 to find which is bigger. The students easily find that 0.2 is less than 0.25 and then Franca wants to know from them how to calculate 0.2 of 500ml. Manuele says 500 divided by 0.2. Franca writes it on the blackboard but she says that the result of the division is more than 500 and that it is impossible to have more than 500ml of chamomile in a thermos. Then she asks which is the fraction of 0.2 knowing that 0.25 is ¼. She writes on the blackboard 0.25= ¼ and 0.2 = 1/? She reminds her students that two is contained five times in ten. Then some students say 1/5. She writes it on the blackboard. Then she explains that is easy to calculate 1/5 of 500. The students find that is 100. Franca asks which is the operation and the students answer that 500:5. Then Franca writes on the blackboard 7 thermos and 100ml. Again Franca asks what it is needed to fill 8 thermos. At the end some students answer that 400ml are needed.
Knowledge Quartet Coding Commentary
Contributed by: Marco Bardelli, University of Padova, Italy
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Awareness of purposes
Scenario: Franca solving a story problem with her students
Franca is bringing the pupils step by step to the problem solution. In so doing she makes clear which are her purposes related to the way to solve the problem and to use the mental calculation.
First of all we can recognize the flexibility she asks from her students in changing the unit of measurement to perform a mental calculation (in the solution process Franca converts 3600ml in 3.6lit and 500ml in ½ lit to reduce the size of the numbers). This flexibility in making conversion of units opens the way to make easier the calculations and to link the decimal representation of rational numbers with the representation by fractions.
The number of thermos is found by a sort of change of unit of measurement. When Franca asks how many thermos are needed for one litre and then for three lits, if one thermos is half a litre, this reasoning can be understood like a change of kind of units from lits to thermos. This strategy gives the opportunity to extend the meaning of unit of measurement and to perform a calculation in a simpler way than to divide 3600 by 500. Moreover it makes clearer why the result of the operation is the number of thermos. The division between 3600 and 500 is among numbers with different kind of unit, lits and lits/thermos, so the result is the number of thermos but it must be made explicit to the students, as Franca does, because it is not so obvious for them.
Another step of the solution that emphasizes the purpose of Franca can be found in the transformation of 7.2 thermos in millilitres. The research on mathematics education underlines the troubles that student find in giving sense to the numbers in the school problems (Schoenfeld, 1988). Franca makes a realistic consideration separating 7 from 0.2. Seven is a whole number that can properly represent the number of thermos but 0.2 is a part of a thermos. How much is it this part? Again Franca tries to force her pupils to make explicit their reasoning and moreover she encourages them to find a strategy to make calculation by mind. Then 0.2 is transformed in 1/5 by progressive adaptations. First the students find the fraction corresponding to 0.5, then the fraction corresponding to 0.25 and finally the students find the fraction corresponding to 0.2. Franca wants the students have a number sense, knowing that 0.5 is a half then that 0.25 is half of a half ( a quarter) they can grasp the solution by continuous adjustments.
Franca’s awareness of the purposes in this lesson is about the kinds of mathematical learning the students should achieve. The good subject matter knowledge that the teacher shows in this episode allows to connect different parts of the contents of mathematics teaching (story problems, units of measurement, fractions) and this connection is made following some close purposes. The story problem is not a tool to perform calculation in an algorithmic way, but a tool to develop some competences as the number sense and the more useful representation of a number in a context where mental calculation are a target of mathematical instruction (Arcavi, 1994; Sowder, 1992). Franca considers different strategies to confront numbers and measures. The focus is on the strategies to adopt and not on the procedures to perform. Moreover Franca wants her students to be aware of the meaning of each number they find in the process of solving the problem, because this is one good way to find a number sense and a more economic strategy for solving the problem.
Arcavi, A. (1994) Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24-35
Schoenfeld, A.H. (1988) When good teaching leads to bad results: The disaster of “well taught” mathematics courses. Educational Psychologist 23(2), 145-166.
Sowder, J. (1992). Estimation and number sense. In D.A. Grouws (Ed) Handbook for research on mathematical teaching and learning . New York: McMillan.