# Commentary

Samar’s inclusion of Arabic/Islamic mathematics in this lesson is a fine example of how history can serve as a vehicle for motivating students to learn mathematics while highlighting important contributions from a non-European culture. O’Connor and Robertson (2003) write:

Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.

*(1)*

Samar’s inclusion of ethnomathematics in the lesson reveals to students that their ancestors were important contributors to mathematics and that they too, as students, can engage as mathematicians to discover an algorithm—a word invented by al-Khwarizmi. What is truly a gem in this unit is that the source for developing the algorithm arises out of situations connected not only to the history presented but also to a real-world problem still relevant to Muslims today.

Observations of this lesson show students readily leaving their seats to compare answers with others (SMP3), busily working on partitioning their “land” to discover patterns, developing the algorithm for the multiplication of fractions (SMP4, 7), and, finally, applying it to real life (SMP1, 2, 6). Samar incorporates NCTM’s PtA in that her goal is clearly focused on having students reinvent an important algorithm through tasks that require problem-solving situations from her students’ culture. She supports productive struggle through questions as students work in small groups to discuss the problems. Students’ work demonstrates the use of correct procedures for the concepts, and she elicits and uses evidence of her students’ thinking to guide the current lesson.

The unit reflects the *Principles and Standards for Numbers and Operations* and adheres closely to research on helping students to build on informal knowledge for understanding the multiplication of fractions. In her research, Mack (2001) cites work that supports Samar’s pedagogy Mack lists research that:

proposes that knowledge of partitioning lends itself to understanding the concept of fractions and the various interpretations associated with the concept. One such interpretation is “operator” where a fraction such as “V/t” represents a multiplicative size transformation in which a quantity is reduced to three-fourths of its original size by both partitioning and duplicating various portions of the quantity

*(269)*

For example, once Samar’s students interpret the problem, “^{2}/з of ^{3}Л” as two thirds of three-quarters of the land inherited, they then physically reduced the three-quarters strip of land to two-thirds of its original size through paper folding. In her own research with fifth-graders, Mack (2001) reports that this method allows students to build on their informal knowledge to solve problems involving multiplication problems in ways meaningful to them—similar to what occurred with Samar’s students.

Samar’s persistence in having students find their own rule or algorithm is to be commended. She easily could have just given the rule and have had students complete practice problems. Instead, she had them solve challenging verbal problems that are approachable to all students through paper folding. She then deliberately used the board to organize the results from students’ paper folding so that a search for patterns would be fruitful. Interestingly, while many students quickly detected patterns, one of the students, Yasser, called out what he thought was a pattern by saying, “You switch the second upside down.” He then said, “Oh. No. That’s division. Here you just multiply” and he proceeded to show how he agreed with the other students. Did Yasser really detect a pattern from the work shown? Apparently not. In her articles on the effect of rules on children’s understanding, Behrend (2001) writes, “Teachers often try a quick fix, such as giving a rule to follow The rule may appear to solve the immediate problem but could actually interfere with students’ development of mathematical understanding” (36). It appears that Yasser may have learned operations with fractions from a quick fix approach that was now making it difficult for him to focus on Samar’s conceptual approach: He preferred to try to remember those rules rather than to reinvent one quickly and accurately from the class’s work.

Samar’s approach to helping student develop the concepts is from a problem-solving approach in that she presents students with inheritance problems for which students have no immediate way of reaching the solutions. Her sequence of tasks to help students develop the algorithm for the problems aligns with recommendations from research by Simon et al. (2016). This research builds on the work of Tzur and Simon (2004) on two stages of development in learning a mathematical concept, which they labeled participatory and anticipatory Simon et al. (2016) modified the stages and introduced a structured approach to trace the development of a concept from initial activity, through the participatory stage, and to the anticipatory stage:

We use the participatory-anticipatory distinction in assessing every abstraction that we foster. The goal in each segment of our teaching experiments is to foster an anticipatory stage of the concept. When a learner demonstrates an abstraction as the result of her engagement with a task sequence, we only have evidence for a participatory stage.

In Samar’s lesson, the *initial activity* occurs during the discussion she has with the students where she uncovered their understanding of fractions and inheritance laws. The *participatory stage* occurred when her student used paper folding to find a fraction of a fraction and when they discovered the algorithm. The *anticipatory stage* occurred when her students solved an inheritance problem *without *paper folding. The researchers also identified steps to a sequence of tasks for promoting conceptual development (67). Samar’s application of the steps is in parentheses:

- 1. Assess the relevant understanding of the learner. (Initial discussion questions)
- 2. Specify the learning goal-intended abstraction. (Algorithm for multiplying fractions)
- 3. Identify an activity or activity sequence that the learner already has available that could be the basis for the new abstraction. (Paper folding)
- 4. Design a sequence of tasks that is likely to bring forth the learners’ use of this activity and lead to the intended abstraction. (Computing fraction of a fraction through paper folding)

It is clear that Samar demonstrated respect for students’ ways of thinking. She often asked, “Does anyone have a different way for doing it?” Her welcoming of students to the board to show a different approach is only one such example of her view of a classroom as a space to share and enjoy the challenges of thinking. Teachers at all levels sometimes lament that students are not prepared to do the work at hand. Samar demonstrates how students can be kept moving forward in the curriculum while teachers help supply the missing concepts. For example, guided by a student’s question, Samar decided to jump to the next day’s lesson by introducing an inheritance as money instead of land. In their attempt to solve two-thirds of $18,000, students’ uncertainties prompted Samar first to simplify the problem to an inheritance of only $18 and then to teach the property of dividing a number by one. Once students understood how to represent 18 as a fraction, they then solved the problem. Samar’s approach to the teaching the multiplication of fractions shows that, given appropriate tools and teacher facilitation, students can discover important algorithms.

Both authors enjoyed interacting with the students who were so quick to smile and ask questions or help from the authors.