5 Keys to Math Narration for Improved Fluency and Conceptual Understanding
Math narration is an excellent tool for developing deep conceptual understanding. Narration in itself is a natural human process.
Science has demonstrated that we think in stories, we remember better through stories and we demonstrate understanding through the act of storytelling. Yet, narration is not a discipline applied often to the educational field of mathematics.
Even if it is a part of the mathematical discipline, it is often done haphazardly. Students generally read equations one way: "one plus one equals two." With limited and simplistic math narration, students struggle to get beyond basic word problems.
So how should math narration be conducted? What are the best tools? How can math narration be used to improve fluency and conceptual understanding?
Here are 5 Keys to Deep Conceptual Math Narration for Improved Fluency.
Hands-on Math Narration
Without understanding, reciting a mathematical idea or notation is a waste of time. The key to understanding is to provide a student with context. A hands-on manipulative is important for establishing context.
Use caution when choosing a manipulative. Three Popular Math Manipulatives that Fail to Build Number Sense is a suitable blog post for those who cling to the everyday items of buttons and pebbles for mathematics. Everyday items provide context but they also hinder the student from perceiving several important mathematical ideas.
With Cuisenaire rods, students simply begin to build trains of rods. Then the teacher reads the train for the student. The student reads the train back. "A red, an orange, a light green, a blue, and a black."
This is the beginning of math narration with Cuisenaire rods. It is simple yet more complex than two plus one. Without numbers, students have the opportunity to perceive general ideas of mathematics like transformations, graduation, odd and even, etc. Students begin to perceive the relationship of sums and missing rods.
Over time new language is added. "A red plus an orange plus a light green, plus a blue plus a black." Then, relationships of equivalent lengths are added. "A red plus an orange is the same length as a tan plus a purple."
Early on, the student needs the context of the manipulative all the time. Math is a language but language has no meaning without context. Gattegno provides varying tasks to provide contextual language development in math throughout Textbook 1 (FREE HERE). You can read more about a few of those tasks HERE and HERE.
Oral Math Narration
Oral math narration must preceed written. The use of the rods engage the senses of sight and touch. The educator adds the additional sensory input by verbalizing observations and asking for verbal responses. That is a lot of sensory input/out for a young student during a single session.
It is best to avoid burdening the student with required written work. Educators may act as the scribe for the student early on in mathematical development.
Gattegno asks the student to describe what they see. He asks for the students to notice relationships. Activities in Chapter 2 of Gattegno's Textbook 1 are mainly for oral discussions. The focus is to hear, see and manipulate the new language in the most general sense.
Varied Math Narration
Varied math narration is vital to fluent understanding. The equal sign doesn't always go at the end of an equation. Yet, open a elementary math textbook and this is what you get.
It is no wonder when a student is asked, "What does an equal sign mean?" They answer, "It means to put the answer here." The student demonstrates they do not understand the meaning of the equal sign.
To prevent this issue, it is important to vary math narration. Variation includes varying vocabulary and sentence structure. Here is a variation of simple math narration for Cuisenaire rods.
- The length of the red and light green train is the same as a yellow rod.
- The yellow rod is the same length as the red and light green train.
- The sum of the red and light green train is equal to yellow.
- Yellow is equal to the red and light green train.
- Light green is the difference of yellow and red.
- The difference of yellow and red is light green.
- Red is the difference of yellow and light green.
- If I add a light green to the red it is equal to the yellow rod.
- If I combine the red with the light green, it is the same length as the yellow rod.
- A white plus a white plus a white is the same length as a light green.
- 3 whites are the same length as a light green.
- A light green is made up of 3 whites.
- Three whites fit into the length of a light green.
- A red plus a white fit into the length of a light green.
- Light green is the length of 3 whites.
- Two reds plus a white is the length of a yellow.
- My light green rod needs a red to make the same length of a yellow rod.
- I need a white to make my 2 red rod train equal to the yellow rod.
Not an exhaustive list, but it is one to get the educator started. It is important to practice such variation every day. Also have the student practice expressing their math narration in various ways.
Original Math Narration
The importance of original composition of mathematical structures and notation must be stated. Students are too often provided with simplistic notation. That is simplistic worksheets and activities filled with one answer equations like 5 + 3 = ?, 7 - 2 = ?, etc .
Students need a manipulative that makes it easy to compose and decompose mathematical structure with more variety. This is why Cuisenaire Rods have a superior value in mathematics. They offer the student endless opportunity to compose and decompose numbers with more depth.
Jo Boaler, in Mathematical Mindsets, writes this about the openness of numbers.
"Teachers can create such mathematical excitement in classrooms, with any task, by asking students for different ways they see and can solve tasks and by encouraging discussions of different ways of seeing problems."
Gattegno Textbook 1 provides this excitement of number openess through number studies. After developing language and conceptual understanding through simple tasks, students compose and decompose of numbers through operations of addition, subtraction, multiplication, fractions and so on.
Collaborative number building is best done through the substitution game. Students work together to extend and lengthen equations using the openness of numbers. Read more about that game here.
Cuisenaire rods also offer students opportunity to observe patterns, test ideas (whether good or bad) and uncover new relationships. This is done through structure studies. This cultivates a healthy curiosity for math early on as well.
Original math narrations provide students opportunity to play with the language of math and deepen understanding. With practice, students uncover the many ways to describe a math observation. This is not true in a procedural based math education in which the young student is generally limited by one procedure or one operation.
Written Math Narration with Story
The student should not be left in the realm of the concrete but must be pushed onward into the realm of the abstract. Use math narrations with stories to push students to the mental realm.
Another important reason to use Cuisenaire rods is the provision of a blank slate. Through the imagination, Cuisenaire rods become cupcakes, chemicals, sandwiches and ladders to be fixed. Yet, the rods remain mathematical by their nature.
It is important to engage the student to imagine the rods as sandwiches to share, chemicals to mix and ladders to be fix. This allows the simple math narrations to develop a deeper conceptual meaning outside the realm of the Cuisenaire rods.
The student recognizes the 3 whites equal to a light green as a sandwich made of 3 parts equal to the whole. The budding scientist sees the composition and decomposition of a flask as an expression of many relationships of differences, fractions, multiples and sums.
The search for the missing parts of the ladder develops an intuition in the student to look for existing relationships to build towards the unknown. Stories engage the imagination and take the student to the abstract. The student perceives the mathematics all around them.
The mental exercise of imagination builds a strong relationship for the student to use mathematics abstractly.
Fluency through Math Narration
Number fluency is not strictly math facts. It is the ability to see the connections between the many mathematical relationships present in numbers.
Math facts come naturally when the emphasis is on relationships rather than number facts. The student steeped in mathematical relationships use those relationships to strategize and streamline math facts. Check this post out for more on that.
Math narration needs to be cultivated like any narration. We do not expect a student to write a novel their first day. We recognize the student needs a model of good writing and to be surrounded by good literature.
Model varied math narration, engage the imagination and provide opportunity for original mathematical composition both orally and written. In time, students will grow deeper conceptual understanding and fluency.
Resources for Math Narration
For more resources on general math narration activities, check out Math Task and Play Mats. For resources on providing playful context for math narration, check out PDL's Interactive Math Notebook Bundle. It is filled with playmats to inspire your students' imagination while providing context for math narrations.
For more resources on simple constraints to inspire creative mathematical notations, check out PDL's Number Studies and PDL's Math Stories for Cuisenaire Rods.