Mathematics as Problem Solving and Reasoning: Strategies, Approaches, and Cognitive Development

1. Math Standards

2. Standard 1: Mathematics as Problem Solving In grades k-4, the study of mathematics should emphasize problem solving so that students can: • Use problem solving approaches to investigate and understand mathematical content. • Formulate problems from everyday mathematical situations. • Develop and apply strategies to solve a wide variety of problems. • Verify and interpret results with respect to the original problem. • Acquire confidence in using mathematical meaningfully.

3. Standard 1: Mathematics as Problem Solving The mathematics teacher understands and uses numbers, number systems and their structure, operations and algorithms, quantitative reasoning, am technology appropriate to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]) in order to prepare students to use mathematics..

4. Standard 3: Mathematics as Reasoning In grades k-4, the study of mathematics should emphasize reasoning so that students can: • Draw logical conclusions about mathematics. • Use models, known facts, properties, and relationships to explain their thinking. • Justify their answers and solution processes. • Use patterns and relationships to analyze mathematical situations. • Believe that mathematical make sense.

5. One step problems For one step problems, students can be asked the following questions as a way to discuss their work: • What are you trying to find? • Which data in the story were needed to find the solution? Were there unnecessary data? • What action in the story suggested the operation you used to find the answer? • Can you give the answer in a complete sentence? • Have you checked your work and your answer?

6. Piaget’s Levels of Cognitive Development

7. Piaget’s Levels of Cognitive Development Children evolve through specific stages in which cognitive structures become progressively more complex. Cognitive development refers to the changes that occur in an individual’s cognitive structures, abilities, and processes. Cognitive development is the transformation of the child’s undifferentiated, unspecialized cognitive abilities into the adult’s conceptual competence and problem-solving skill. Piaget believed children’s schemes, or logical mental structures, change with age and are initially action-based (sensorimotor) and later move to a mental (operational) level.

8. Sensorimotor Stage (0-2 years) Intelligence develops through sensory experiences and movement. During the sensorimotor stage, infants and toddlers "think" with their eyes, ears, hands, and other sensorimotor equipment. Piaget said that a child’s cognitive system is limited to motor reflexes at birth, but the child builds on these reflexes to develop more sophisticated procedures. They learn to generalize their activities to a wider range of situations and coordinate them into increasingly lengthy chains of behavior. Learning involves pulling pushing, turning, twisting, rolling, poking, and interacting with many different properties of objects.

9. Preoperational Stage (2-6/7 years) Intelligence includes the use of symbols such as pictures and words to represent ideas and objects. At this age, according to Piaget, children acquire representational skills in the area of mental imagery, and especially language. They are very self-oriented, and have an egocentric view; that is, preoperational children can use these representational skills only to view the world from their own perspective. Learning involves discovering distinct properties and functions of objects as they compare, sort, stack, roll, distinguish triangles from squares, and begin to use abstractions to communicate

10. Concrete Operational Stage (6/7-11/12 years) Cognitive development includes logic but requires physical examples to which the logic can be applied. They require experiences with touching, smelling, seeing, hearing, and performing. They must use hands-on tools to investigate. As opposed to preoperational children, children in the concrete operations stage are able to take into account another person’s point of view and consider more than one perspective simultaneously, with their thought process being more logical, flexible, and organized than in early childhood.

11. Concrete Operational Stage (cont.) They can also represent transformations as well as static situations. Although they can understand concrete problems, Piaget would argue that they cannot yet contemplate or solve abstract problems, and that they are not yet able to consider all of the logically possible outcomes. Children at this stage would have the ability to pass conservation (numerical), classification, seriation, and spatial reasoning tasks.

12. Formal Operational Stage (11/12+ years) Thinking includes abstract concepts. This allows analytical and logical thought without requiring references to concrete applications. Persons who reach the formal operation stage are capable of thinking logically and abstractly. They can also reason theoretically. Piaget considered this the ultimate stage of development, and stated that although the children would still have to revise their knowledge base, their way of thinking was as powerful as it would get.

13. Piaget’s Levels of Cognitive Development Piaget suggest that there are four broad factors that are necessary and that affect the progression through these stages of cognitive development. They are (1) maturation (2) physical experience, (3) social interaction, and (4) equilibration. Clearly, learning experiences for children through the age of 12 must involve objects, tools, interaction, reflection, and social interaction with materials for optimal cognitive growth. The EC-4 teacher knows that mathematical concepts are best learned by children by manipulating materials and observing what happened-individually and collaboratively. A key to EC-4 mathematics is planning concrete experiences that facilitate learning. A sound mathematics classroom learning environment and curriculum reflect this cognitive approach to learning.

14. Mrs. Jones was teaching her prekindergarten class about volume, but she could not get students to understand that the liquid in a cup was the same as when she poured the liquid into a flat bowl. What was happening? Select the best answer. • The children are in the preoperational stage and do not have the ability to conserve yet. • The children are in the preoperational stage and do not have the ability to transform yet. • The children are in the preoperational stage and do not have the ability to classify yet. • The children are in the preoperational stage and do not have the ability to seriate yet.

15. Ms. Mehrman wishes to foster critical thinking skills in her first-grade students. Using overhead attribute blocks, Ms. Mehrman places an attribute block on the overhead and has her students identify the four attributes (size, shape, color, thickness) of the block. She then asks the students to select a block that is "one different" from the beginning block (that is, it has only one attribute different from the original one). She places the new block next to the first one and begins to form a "train." The class identifies the four attributes of this new block and justifies that it is "one different" from the first block. The students are now asked to find a block that isdifferent from this second block, and the process continues. This activity would be most appropriate for developing students' understanding that:

16. There can be more than one answer to a question. • It is necessary to recognize the four attributes of each block. • The size of the attribute blocks may be large or small. • The thickness of the block is very important.

17. Ms Ismail asks her fourth-grade students to discus then package color tiles into individual bagas. There are 420 color tiles in the class set, and 24 students in her class. How many color tiles will be in each student’s bag? How many are left over?The students decide to divide 420 by 24 and get an answer of 17.5. They are not sure what the answer means. The best response for Ms. Ismail would be to: • Have the students recheck the answer • Ask the students to discuss why they divided and what 17.5 means to the answer • Cut some color tiles in half • Ask the students to round the answer

18. Ms. Baumbach is working with the entire first-grade class. She says, “I have put some pennies, nickels, and dimes in my pocket." She places her hand in her pocket. "I have put three of these coins in my hand. How much money do you think I have in my hand?" Many children are confused and begin voicing guesses. In order to facilitate problem solving in this situation, the teacher should: • Suggest an approach using trial and error. • Suggest an approach that uses real coins. • AlIow students to verify their guesses. • AII of the above.