Handwriting in the mathematics classroom promotes a deeper understanding of concepts and procedures. Writing helps students extend their critical thinking abilities as well as the ability to link a new idea to relevant prior knowledge (e.g., Bicer, Capraro, & Capraro, 2013; Craig, 2011; Powell, 1997; Pugalee, 2004, 2001).
Handwriting Encourages students to create their own problem solving knowledge (Carr & Biddlecomb, 1998; Steele, 2007)
Promotes a metacognitive frameworks that extends students' reflection and analysis (Pugalee, 2004; Pugalee, 2001; Boscoloand Mason, 2001)
Students' mathematical knowledge is extended as a result of their handwriting in mathematics (Reilly, 2007) and reflects changes in their understanding of mathematical concepts changed over time.
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Acquisition Stage
In the acquisition stage of the instructional hierarchy, a student has begun to learn how to write numerals correctly but is not yet accurate or fluent in the skill. The goal in this phase is to improve accuracy, making sure that students are forming the numerals correctly. You will notice that as students progress in this stage, errors go down and the number of correct responses go up. In this stage, teachers actively demonstrate how to form the numbers correctly using models of correct numeral formation. Teachers use ‘think-aloud’ strategies while modeling how to write the numbers using wording about formation and directionality. Students get feedback about how they are doing and receive praise and encouragement for their effort and accuracy.
A Focus on Acquisition
Fluency Stage
In the fluency stage, students are able to write numerals accurately and legibly but work slowly. The goal of this phase is to increase the student's speed of responding. It is this accurate speed of responding writing by hand that helps to free up working memory to attend to higher-level cognitive tasks of computation and the application of those operations and procedures to word problems and other mathematical tasks. If students cannot efficiently form numerals, they cannot calculate efficiently. If the child cannot write numerals automatically, speed of performing written math tasks could be very slow and math assignments may not be completed on time or accurately.
The teacher structures learning activities to give students opportunities for active (observable) practice of writing numerals with direct repetition. Students get feedback on fluency and accuracy of performance and receive praise and encouragement for increased fluency. Fluency routines fall on a continuum of support from a teacher. These fluency practice routines are not independent practice; teachers guide the practice at each step. Generally, fluency practice can take the form of a 3-7 minute warm-up and do not replace the direct, explicit instruction that comes prior in acquisition lessons. (Chart used with permission, Dr. Amanda VanDerHeyden)
A Focus on Moving from Acquisition to Fluency
TCCC is a modification of the evidenced-based practice, Cover, Copy, Compare. In this routine, teachers scaffold students' transition from acquisition of writing numerals to fluency of writing numerals by modeling how to trace and write the numerals. Students trace the numeral with their finger and pencil, make the numerals with the model, and then cover the models and write the numeral using their mind’s eye to reproduce the number. Students also practice other numbers that they have been writing in previous fluency lessons.
Generalization Stage
In the generalization stage, the student is accurate and fluent in writing numerals but does not typically use it in different situations or settings. The goal of this stage is to for the student to use the skill in the widest possible range of mathematical work. Teachers give academic tasks that require that the student use the handwriting of numerals regularly in assignments.
Students receive encouragement, praise, reinforcement for using the skill in new settings and get periodic opportunities to review and practice target skills to ensure maintenance.
A Focus on Generalization
Teachers can use activities that focus on students writing the missing number in a sequence, ex. 3, _, 5 or 6, 7, _, 9. In addition, to help with generalization, the teacher can ask students to write the numeral(s) that come before and after other numerals. Examples, What number comes after 7? What number comes before 5?
•Write in personal definitions of terms, rules, theorems, etc.
•Write explanations of mathematical concepts and ideas
•Write a summary of a lesson or task
•Write explanations of errors (What went wrong? How can the error be addressed?)
•Offer examples and justify selection
•Describe rules, their application, and mathematical importance
•Write problems, applications, and provide solutions Describe how to solve a problem
•Compare and contrast alternative approaches to a problem
•Describe how technology helped in finding a solution (Describe the mathematical processes required for the output.)
•Write a formal report for approaching a problem situation.
•Describe what made a problem or task easy and/or difficult
•Explain why an answer or solution is reasonable •Identify and react to questions one may raise about your work (or respond to a question someone raised about your work)
•Analyze the quality of one’s work (process, methods, mathematical soundness, communication)
•Describe how different decisions might impact an answer
•Describe how problems are similar and/or different
•Write an autobiography about a mathematics experience
•Write about the role mathematics plays in your life or might playin the future.
•Describe how mathematics changes or changed one’s life.
•Explain what helps or hinders you in understanding mathematics
•Describe how you feel about your performance on a task or problem
•Write a note or notes to the teacher for additional information
•Specify lesson components which were not understood or components which you understood well
•Write a journal entry about some aspect of the day’s class
•Summarize and interview with a peer or other individual about a topic, problem, or other mathematically related idea
•Write a response as a group or team to a problem or task
Using and writing an equals sign correctly can be a challenge. Whenever an equals sign is used, it signifies that the expressions on either side are equal. A sequence of expressions separated by equals signs should all be equal to each other.
However, the equals sign often gets used and written in an operational way, meaning “give the answer”. Our calculators do not help at all, as we press the “=” sign as an instruction to calculate. Many students have an operational view of the equals sign. The problem with this is that the operational use of the equals signs leads to incorrect equality statements.