Scenario: Mirella teaching two-digit divisions by invariantive law

Country: Italy

Grade (student age): Year 3 (age 7-8)

Contributed by: Marco Bardelli, University of Padova, Italy

Context – public primary school, national curriculum

In Italy two-digit divisions is a topic that is usually taught in year 4. The new national curriculum in Italy  (2007), named “Indicazioni per il Curriculum,“ gives just some directions to the teachers about the competences that must be reached in each subject matter at the end of third, fifth and eighth grade of schooling and about the skills the  students have to manage in order to reach those competences. At the end of the third year of schooling  knowledge of the algorithm of the division with two-digit and remainder is not required. Students have to know how  to solve divisions with one digit and remainder. Moreover the knowledge about decimal numbers is not required. It Is left to the teachers to decide the level of complexity of the operations the students have to perform (i.e. with or without remainder).

Mirella had graduated as as a primary school teacher and was in-service three years as mathematics teacher in a  primary school when this episode took place. The episode took place on the 4th may 2009, one month before the end of the school year.


Previous event

In the previous lesson Mirella assigned some questions to serve as a guideline by which the students were to invent a story problem with a division. The pupils worked in pairs. A couple of pupils invented the following story problem:

Frederick buys 7 candy bags to be shared among 3 classes of children. There are 20 children in each class and 40 candies in each bag. How many candies will each child receive?

The problem requires two multiplications and one division: , to be solved, but the matter  was that, at that moment, the teacher had taught only divisions with one-digit divisor. Some hypothesis were formulated by the children about how to solve divisions with two-digit divisor. The teacher just managed the discussion, then the pupils concluded the division could be transformed  into an equivalent as 28:6, but without having clearly understood what to do with the zeros they cut from the numbers. So the teacher told the students to solve this story problem in written form for homework.


Mirella begins the lesson solving the problem at the blackboard for the whole class. In solving the problem she explains the procedure. She writes: 7(bags)x40(candies/bag) and 3 (classes)x 20 (children/class). She says that the 7 and the 3 become 40 and 20 times bigger respectively. She explains that to it is necessary to multiply 7x4 and 3x2, eliminating the zeros and the adding the zeros to the results. Then when the teacher has to divide 280:60, the last operation of the story problem, she writes it on the blackboard and proposes to proceed in a similar way as she did with multiplications, and as the pupils proposed in the previous lesson, eliminating the zeros from the numbers and solving it as before:




At this point some students that did the problem for homework say “no it’s 40”,. She asks what the pupils did with the zeros they proposed  to eliminate.  The teacher remembers that during the previous lesson some pupils said to add the zeros to the quotient (she writes it on the blackboard),



And someone else says that it isn’t necessary to add the zero to the quotient. Then she asks if any student checked the division at home. At this point some student answer that the result is 40 and others 4. Some pupils solved the division correctly at home with quotient 4 and remainder 40, but the teacher doesn’t check their notebooks. The teacher writes the check at the blackboard with 4 as result and 4 as remainder:




and Mirella says:

Sixty times four is two-hundred-forty plus forty we are at two-hundred-eighty. Alessandro what did you write as result?

Alessandro:     Forty

Mirella writes the check with forty as result:




The teacher at this point looks at the blackboard and she realizes that there is some error in the division. She erases the blackboard and starts again explaining the division 280:60. She recalls the invariantive law of the division and using an analogy with the multiplication she says that as a number becomes 10 or 100 times bigger if it is multiplied by 10 or 100 then, if the number is divided by 10 or 100 it becomes 10 or 100 times smaller. Mirella continues the analogy saying that as in the multiplication the zeros are added, in the division the zeros should be cut. So she writes on the blackboard:



So the teacher says that now they are able to solve the division that gives the result:


with r as the remainder. Then she claims that this is the result of 280:60: four with the remainder of four candies that will be left over.


Knowledge Quartet Coding Commentary

Contributed by: Marco Bardelli, University of Padova, Italy

Knowledge Quartet Dimension: Foundation

Knowledge Quartet Code: Overt Subject Knowledge (OSK)

Scenario: Mirella teaching two-digit divisions by invariantive law


The division between two numbers  and , , is defined in the set N of natural numbers as the operation which gives as result two numbers   and . Then the relation between  and  will be:


With  dividend,  divisor,  quotient and  remainder. The linearity of the relation (1) involves that if  is divided by a number n also  and  must be divided by the same number n. In this case, dividing by 10, the relation  becomes:


So the dividend, the divisor and the remainder must be divided by the same amount for the relation could be the same. In the division operation, the invariantive law says that the quotient doesn’t change if the dividend and the divisor are divided by the same number, but says nothing about the remainder. That’s because the invariantive law, as usually written in school-texbooks, regards the division as it is in the domain of rational numbers Q, for the division without remainder is an operation that can be performed in the rational numbers domain and not in the set of natural numbers. If one uses rational numbers written as decimals the division is the same after applying the invariantive law. But if one calculates the quotient and the remainder after having applied the invariantive law, dividing the dividend and the divisor by the same number, the remainder is no longer the same as before unlike the quotient. Dividing  and b by 10 the quotient doesn’t change, but the remainder  is divided by 10 as well (2). Then  the remainder of the division  needs to be multiplied by the same number 10.

Mirella tries to solve the division with a two-digit divisor by the invariantive law without using the mathematical definition of the division in Natural numbers. She doesn’t realize that she should multiply the remainder by the same number she used to divide dividend and divisor.

She makes the analogy between multiplication and division to help the children to better understand the division by 10. This is a quick way to calculate the operation and to link procedures but here the problem is that the analogy between multiplication and division is weak. Mirella confused the division by 10 with the invariantive law where 10 devides the dividend and the divisor. In any case teaching or talking about multiplication and division by ten or one-hundred just as a matter of adding or cutting zeros doesn’t help students nor teacher, to grasp the meaning of the operation in a conceptual way.

Mirella didn’t plan to teach divisions with a two-digit divisor. She left the pupils to invent problems with divisions on their own and the division with two digits unexpectedly came up. This is a good teaching strategy to help children understand the contexts where the division can be useful, but the teacher must be prepared, with an adequate subject-matter knowledge, to face unexpected events, especially if, like Mirella, one wants to teach in a constructivistic way (Damiano, 2006).



Damiano, E. (2006) La nuova alleanza. Temi, problemi, prospettive della nuova ricerca didattica. La Scuola Ed.

OSK: Scenario 2
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