Work organization is an important technique in Math. Students need to be able to classify and correlate parts of mathematical questions. They need to understand and comprehend the structural meaning of what is being asked.
As we take a closer look at structure in Math, Routines for Reasoning provides us best practices to establish productive mathematical thinking in scholars.
So, what does it mean to “think structurally” in math class? (p.74)
Structural thinking is connecting a visual aide to an expression or question. The representations may not be similar but can still lead to solving a problem. Three useful actions that aid in thinking structurally are chunk, change and connect.
Chunking allows students to put pieces together in order to solve a problem easier. Chang(e)ing the form of the representation also allows for similarity of a problem. For example, 2(2x4) is the same as two (2x4)’s.
Making a mathematical connection between similar problems is another action that helps one to think more fundamentally.
Connecting Representation Routine
To help students be successful while developing structural thinking, teachers must choose representations that will build on their strong points yet provide them with a challenge.
The following chart is an aide in building the organizational development that students need:
1. Launch a routine – The
teacher establishes the
target problem and goes
over the steps of the
routine, keeping in mind
students need to be
thinking of how to solve –
not so much find the
solution.
2. Students make connections
and use this to match
representations. If there is no
match, then students can
create a depiction. In a whole
class discussion, students are
encouraged to share their
connections about the structure.
3. Students work together and discuss the representation. Students also consider
representations with no match.
4. After allowing individual think time, students work together to create a common
structure of the illustration. Think-pair-share is a great way for students to articulate their
mathematical thoughts.
5. Self-reflection is a crucial last step. This allows students the opportunity to reflect on what they have learned that will aid in future problem solving.
To provide assistance in the students thought process, sentence frames and sentence starters are necessary for focus and the continue approach to structural thinking. Sentence frames such as “I noticed _____ and connected it to _____ because _____” allow students to notice how they made the connection. Sentence starters, for example, “They chunked ______” or “They changed ______ to _____” moves the student to make use of the mathematical process through structural reasoning. Teachers must constantly model the language for the students to use.
Annotation is another strategy in developing structural thinking. Students share and make sense of the problem. They are given multiple examples with the opportunity for discourse to occur. Again, teacher patterning is so important for student growth.
Let’s look at an example a teacher could use in an elementary classroom. The teacher would show the diagrams and have the students match the word problem to the representation. Students would then create a word problem for the leftover representation, following the chart for reasoning.
The following is an example using graphs. The student would match the statements about the data to the graph. Once the connections have been made, students would then make one or two statements about the data in the graph that was not used.
Teachers must be decisive in choosing representations that allow students to thrive in a math classroom. Representations should build on their mathematical thinking to foster growth in their strengths while allowing their weaknesses to slowly rescind. As well, representations should be challenging but allows student to grow using their familiarity with mathematical structure to be successful.
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