Strategies
Progress Monitoring
Student progress needs to be monitored across instructional targets including counting range, ability to subitize sets and the quantity of those sets, flexibility and fluency with various number sets, success with problem types and level of solution strategies. I will monitor individual progress, and use my knowledge to determine when to move onto a problem type with a more complex structure, when to increase the types of number families utilized in the problems and when students can benefit from explicit strategy instruction.
Teaching to Mastery
Children need ample opportunity to develop mastery of each problem type. The pacing is to be determined by the needs of the students and individual progress monitoring data will guide the instructional path. Student work and pacing will be adapted accordingly to allow for re-teaching as needed.
Problem Analysis
Often incorrect answers to word problems are the result of correct mathematical calculations performed using incorrect problem representations and equations. This indicates students may misunderstand the conveyed meaning or language of the word problem. In response to this, I will work with my students until they learn to examine the problem and identify critical information. I will focus their attention on what the problem is asking, and prepare an estimate of the type of answer they may come up with so they can judge the reasonableness of their approach and solution. Some questions I will ask include: What’s happening? What are we trying to find out? Do you think the solution will be a big number or a small number?
Teaching with Manipulatives
I intend to utilize developmental approaches to promote conceptual understanding of the problem solving process through the use of manipulatives including: unifix cubes, ten rods, unit cubes, Cuisenaire rods, rod tracks and ten frames. It has been widely accepted that manipulative based instruction promotes understanding of a variety of mathematics concepts. Use of manipulative materials in teaching mathematics has been documented for hundreds of years. The use of concrete manipulatives to promote conceptual understanding of the problem solving process is supported by the learning theory work of Piaget and Bruner. I will assist my students to explicitly connect the manipulatives with the symbolic work. This is explored further in the discussion below regarding the use of the concrete-pictorial-abstract approach.
Concrete-Pictorial-Abstract Approach
The concrete-pictorial-abstract approach, while proven beneficial to all students, has been noted to be particularly effective with students with academic deficits, speech and language needs, as well as with ESL students. Hands-on manipulatives or real life objects are used to demonstrate the concept, then students create pictorial representations and represent their thinking with a number sentence. This visual, pictorial modeling is often missing from many programs traditionally used in the U.S. It provides a transition from the mathematical process, to words, to the more abstract equation. This process serves both as a learning process and a scaffold because as students build mathematical understanding they can later rely on more efficient abstract algorithms once the conceptual understanding is developed. This is of course closely related with using manipulatives.
The concrete-pictorial-abstract approach, based on research by psychologist Jerome Bruner, proposes that there are three steps essential for pupils to develop conceptual understanding of a mathematic concept. Students must be actively engaged in learning and move through 3 stages: enactive, iconic, and symbolic. The enactive stage involves allowing a student to investigate a new skill or idea by acting it out with real objects such as manipulatives, involving themselves in tactile and kinesthetic experiences. In the iconic stage a student demonstrates understanding of the hands-on experiences by relating them to or constructing visual models, drawings and/or pictures of the problem. In the symbolic stage learners are able to abstractly model mathematics through the use of numbers and symbols to represent their process with mathematical equations.
Explicit Strategy Instruction
Explicit instruction will be provided on specific solution strategies as students move through the developmental conceptual levels. Progress monitoring data will be utilized to identify student instructional needs and students will be flexibly grouped accordingly. I will model strategies in a teacher led format, lead students through various strategies step by step, explaining concepts and calling out critical connections to ensure learners are making connections, as new vocabulary is introduced. During these sessions, as the instructor, I will determine the manipulatives used, the format in which a pictorial drawing is laid out, and the sequence of the steps. Students will be walked through the step-by-step process while I or classroom staff explains the rationale behind the strategy.
Think-Alouds
Think-alouds can be used to begin to teach metacognition strategies in children. (Van de Walle, 2013) I will demonstrates the steps needed to solve a problem while verbalizing the thinking process, using strategies that mirror student entry points and conceptual levels. I will model a problem using concrete materials and then build a pictorial model, while talking through the steps and identifying the reasons for each step in my strategy. During the process I will use questioning strategies to begin to engage students and transfer the responsibility to the student until they are discussing aloud their thinking and process.
Early Development of the Measurement Model
Knowledge of addition and subtraction can be constructed through the context of measurement. The introduction of linear measurement has several connections to counting and cardinality. Often students are first introduced to measurement through the use of non-standard tools. For example students are asked to measure items with objects. They are asked to line up cubes or paperclips and indicate how many units long it is. This builds on their knowledge of counting and cardinality. They are tasked with applying 1-to-1 correspondence to count objects representing the attribute of a unit of length. Measurement assigns a number to this less concrete attribute. The introduction of standard measurement tools through the use of objects of standard size assist in the understanding of the space as a unit. This leads to the ability to use standardized measurement tools such as a ruler with understanding and promotes the understanding of number line models and fractions. The measurement model can be developed by work with cubes and rods of various lengths. Students can work with concrete tools modeling addition and subtraction through the use of 1inch cubes, tiles, rulers, base ten blocks, Cuisenaire rods and tracks. Measuring arbitrary objects raises the issue of non-whole numbers. Cuisenaire rods avoid this, but I will need to be careful to deemphasize the colors.
Language Support
Language development has a significant effect on the accessibility of instruction. I will utilize many research-based scaffolds to provide additional support for students facing language barriers that impact student achievement regarding the problem solving process. I will: utilize concrete and visual models, utilize graphic organizers, explicitly teach academic vocabulary, utilize teacher modeling and explanation, utilize multimedia to enhance comprehension, build background, clarify key concepts by using present tense, shorter sentences, fewer clauses; using examples related to school contexts; using graphics and arrows to illustrate points; and using white space and color to accentuate important information, integrate oral and written instruction into content area teaching; incorporate structured opportunities to speak with partner or small group; provide structured opportunities to write, utilize sentence frames and sentence starters. (August, 2014.)
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