Similar to the ripples caused by throwing a rock into a pool of water, meaning expands and "simple elements of thought [are] transformed into a complex whole."
(Miller 2002, pg. 227)
According to Laney Sammons, synthesizing may be the most complex of the comprehension strategies with having to merge together the other strategies in order to generate an entirely new and original idea, perspective, or line of thinking (227). Synthesizing is the catalyst for the construction of mathematical meaning (229).
Sammons goes on to explain what students need to know about synthesizing:
- Mathematicians may change their mathematical thinking with each new mathematical experience.
- Mathematicians construct new mathematical meaning through the synthesis of new and existing mathematical knowledge.
- Mathematicians know that mathematical knowledge is constantly evolving.
- Mathematicians can explain how synthesis helps to create new understandings in math. (230)
How can we make this rather abstract strategy more concrete for students? One recommendation made by Sammons was to use nesting dolls. After lining them up, Sammons recommends asking students how the dolls represent their thinking. Then after stacking them from smallest to largest, students should be asked to reflect how the dolls now represent their thinking. The goal here is for students to recognize that big ideas are made up of smaller ideas that build and change over time. (235-236) After reading this, I thought, this does make sense!
Conjectures: When students synthesize they take new mathematical ideas along with what they already know to create new understandings. After observing patterns and relationships that appear to be true but have not been tested, students can form conjectures, or informed guesses and predictions (237). Laney Sammons goes on to explain that students rarely make conjectures unless the process has been modeled and encouraged. Sammons goes on to recommend the following categories that can offer opportunities to model and create conjectures.
- Properties of number operations (identity, zero, inverse, commutative, associative, and distributive)
- Characteristics of special types of numbers (odd/even, prime, improper fractions)
- Procedural rules (regrouping, multiplying a decimal) (239)
Support or Disprove This Conjecture Stretch (240): Sammons recommends this stretch where students are asked to add evidence that would support or disprove a given conjecture. You can begin this stretch by posting a chart and having students share evidence, verbal or numerical. Using sticky notes to post ideas is one recommendation Sammons makes. I thought the next step could be small group/independent work where students can ponder a conjecture, record their thinking, then share out with others using the template below. I chose the puzzle template to help remind students that synthesis requires students to put together different mathematical ideas in order to create new understandings or revise mathematical thinking. Click on the image below to see some examples.
I am going to leave you with this statement from Sammons:
Students may be "taught" the mathematics curriculum, but unless they are able to recognize the big mathematical ideas, see how the details relate to the big ideas, and apply these to real-life situations, they are not mathematically literate (246).
How do you help your students synthesize the big ideas in math?
