Interpreting Standards as Sense-Making Opportunities

Interpreting Standards as Sense-Making Opportunities

Sue Pawula

Summary

Algorithms created by students can be one tool to use when teaching them to make sense of math. In this article, this approach is initiated through in lessons targeted at teaching students: 

“Understand subtraction of rational numbers as adding the additive inverse, p-q=p+(-q). Show the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. (7.NS.c)

Instead of teaching students the “keep-change-change” method, it is suggested that teachers should model the procedure and then ask students to work independently.  The old method could hinder student sense-making of subtraction with rational numbers or connections to related subtraction ideas.  When students can think about subtraction in terms of distance or vector length they are provided with chances to connect their actions in a visual space to operations on numbers.  This also should allow the teacher to elicit student reasoning that will indicate their connections and meaning that they assign to mathematical procedures.  In this instance it allows students to develop their own algorithms for rational number addition and subtraction. This takes more time and requires teacher commitment to content coverage in order to arrive at the student goal.  However, it requires students to have conversations about the patterns they see related to adding and subtracting rational numbers in a variety of different problem contexts.  The teacher will observe that this activates students to form conjectures about the patterns they have noticed and discussions will allow students to express these conjectures to their peers.  During the discussions, students will need to use precise mathematical language, establish standards of evidence, and focus on patterns that will be useful when they later develop algorithms.  The teacher will also notice that some of their conjectures are ambiguous or incorrect because some students are unable to distinguish between relative and absolute values of the numbers.  Teachers can use this opportunity to formally define absolute value.  Students can acquire deeper understanding of operations on rational numbers, by proceeding through these steps and working as a group to generate a collective algorithm, which they decide, is accurate, sensible or efficient.  Following this student should justify the accuracy of the algorithms in different ways such as number lines, chipboard models, or a few examples that the algorithm works. The student understanding developed in this manner is important since it allows students to flexibly experiment and use rough-draft mathematical language to convey their final outcome. 

Reflection

Although this was a time consuming activity, the outcome allowed the students to build flexibility in thinking.  It gives them control of the learning process and allowes the teacher to facilitate while allowing the students to experiment and reflect, discuss, and discover different ways to reach the goal.  This kind of learning is very important to expansion of individual cognitive development.  It also encourages students to work in teams and share information, rationalize, and debate the different ideas, and come to provable conclusions.  Teachers need to plan how they would implement this type of student learning, being mindful of the time constraints.  Even so, this is an excellent way to move student thinking onward and upward and let them work on projects, individually or in teams, the same as they would have to in the real world

Choppin, J. M., Callard, C. H., & Kruger, J. S. (2014). Interpreting standards as sense-making opportunities. Mathematics Teaching in the Middle School, 20(1), 24-29.