PECT-Test PreK-4
Mathematics
A. Problem-Solving Skills:
Being able to solve problems is fundamental to all other components of mathematics. Students learn the concept that a question can have more than one answer and a problem can have more than one solution by participating in problem-solving activities. To solve problems, a student must be able to explore a problem, a situation, or a subject; think through the problem, situation, or subject; and use logical reasoning. These abilities are needed to not only solve routine, everyday problems but also novel or unusual ones.
Using problem-solving skills not only helps students think mathematically but also promotes their language development and their social skills when they work together. Learners are naturally curious about how to solve everyday problems. Teachers can take advantage of this inherent curiosity by discussing everyday challenges, asking students to propose ways to solve them, and asking them to explain how they arrived at their solutions. Teachers can also invite students to propose problems and ask questions about them. This helps them learn to analyze different types of problems and realize that many problems have multiple possible solutions.
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B. Common Steps that Prepare Students to Learn Math:
The process of solving problems often involves the following steps:
1. Understanding the problem
2. Coming up with a plan to solve the problem
3. Putting that plan into action
4. Observing the outcome
5. Reflecting on whether the solution was effective and whether the answer arrived at makes sense.
6. They maintain flexibility and experiment with alternate methods.
7. They also demonstrate self-regulation skills.
Solving problems not only involves learning this series of steps but also requires learners to develop the qualities needed to solve problems. Students who are able to solve problems have a number of characteristics. For example, students who are effective problem solvers are able to focus their attention on the problem and its individual component parts.
They can formulate hypotheses about the problem/situation and then test them for veracity. They are willing to take risks within reason. They are persistent if they do not solve a problem right away and do not give up if their first attempt at solving a problem is unsuccessful. They maintain flexibility and experiment with alternate methods. They also demonstrate self-regulation skills.
C. Using Problem-Solving Skills in Daily Life:
Young learners continually explore their environments to unravel mysteries about how things work. For example, preschoolers use math concepts to understand that they have three toys, to comprehend that three fingers equal toys, or to understand that two cookies plus one more equal three cookies. To do abstract mathematics in the future, young learners will need two major skills that are also used to solve problems: being able to visualize a scenario and being able to apply common sense thinking. Thinking and planning to achieve goals within the constraints of the properties of the surrounding environment is a natural behavior for young students. They will persist in their efforts to get an older sibling to stop another activity to play with them, to repair broken toys with tape or chewing gum, to manipulate a puzzle or plastic building blocks to get one uncooperative piece to fit, etc. The great 20th-century mathematician and teacher George Polya stated that problem-solving is "the most characteristically human activity." He pointed out that problem-solving is a skill learned by doing, and that developing this skill requires a great deal of practice.
D. Games/Activities That Encourage the Use of Problem-Solving Skills:
One method that has been found to enhance students' reasoning skills is using adult-child conversations to play mental mathematics games. For example, once students are able to count beyond five, adults can give them basic oral story problems to solve (e.g. "If you have two plums and I give you two more, how many will you have?"). Using children's favorite foods in story problems, which takes advantage of their ready ability to envision these foods, is a good place to start.
Teachers can sometimes insert harder tasks (e.g. problems involving larger numbers, problems involving division with remainders, or problems with negative number answers). Even toddlers can solve problems such as how to divide three cookies between two people. The division may not be fair, but it will likely be efficient. Teachers should use the Socratic method, asking guiding questions to allow children to arrive at a solution to a problem themselves, rather than telling them a "right" answer.
E. Beneficial Practices of Playing Mental Math Games:
Teachers can use students' favorite foods and toys to pose story problems to students that involve addition and subtraction. For example, they can ask them questions like "If I give you more, how many will you have?" or "If we take away, how many are left?" It is better to ask students questions than to give them answers. It is important to use turn-taking. In this method, the students pose a story problem, even if the student, and then the child gets to pose one to the adult. Teachers must try to solve the problem, even if the child makes up numbers like "bazillion" or eleventy."
Games should be fun, not strictly factual like math tests. Teachers can introduce age-appropriate story topics as students grow older. At the end of early childhood/around school age, learners can handle the abstract algebraic concept of variables or unknown numbers and use this concept in games. Teachers can pose riddles where x or n is the unknown number, and students must use an operation to solve the riddle.
B. Reasoning Skills:
1. Communicating with Children to Promote Mathematical Reasoning Skills:
Teachers should reciprocally talk to and listen to students during communication that is focused on using mathematical skills like problem-solving, reasoning, making connections, etc. To promote young students' understanding, teachers can express mathematical concepts using pictures, words, diagrams, and symbols. Encouraging students to talk with their peers and teachers helps them clarify their own thoughts and think about what they are doing. Communicating with students about mathematical thinking problems also develops their vocabularies and promotes early literacy and reading skills.
Teachers should listen to what students want to say and should have conversations with them.
Communicating about math can also be accomplished through reading learners' books that incorporate numbers and repetition or rhyme. In addition to talking, teachers can communicate math concepts to students by drawing pictures or diagrams and using concrete objects to represent numbers and solve problems. Children also share their learning of math concepts through words, charts, drawings, tallies, etc. Even toddlers hold up fingers to tell others how old they are.
2. Using Reasoning Skills to Understand and Apply Early Mathematical & Scientific Concepts:
A major component of problem-solving is reasoning. Students reason when they think through questions and find usable answers. They use reasoning skills to make sense of mathematical and scientific subject matter. Students use several abilities during the reasoning process. For example, they use logic to classify objects or concepts into groups. They follow logical sequences to arrive at conclusions that make sense. They use their analytical abilities to explain their own thought processes. They apply what they have learned about relationships and patterns to help them find solutions to problems. They also use reasoning to justify their mental processes and problem solutions.
To support students' reasoning, teachers can ask students, give questions, give them time to think about their answers and listen to their answers. This simple tactic helps students learn how to reason. Students can also ask students why something is as is- letting them think for themselves rather than looking for a particular answer - and listen to the ideas they produce.
3. Role of Representation Skills in Student's Learning:
Young students develop an understanding of symbolic representation - the idea that objects, written letters, words, and other symbols are used to represent other objects or concepts - at an early age. This is evident in their make-believe/pretend play and in their ability to learn written language and connect it to spoken language. As students develop early math skills, representing their ideas and the information they acquire helps them organize, document, and share these ideas and facts with others.
Students may count on their fingers; create tallies using checkmarks, tick marks, and words; draw pictures or maps; grow older, and make graphs. Teachers must help students apply mathematical process skills as they use learning center materials. For example, when a child enjoys sorting rocks by color, the teacher can state that the child is classifying them bridging informal math activities with math vocabulary. Asking the child after he or she finishes what other ways the rocks could be classified encourages problem-solving.
4. Making Connections & Helping Students Transition From Intuitive To Formal Math Thinking:
Students informally learn intuitive mathematical thinking through their everyday life experiences. They naturally apply mathematical concepts and reasoning to solve problems they face in their environment. However, one frequent problem among students when they begin formal education is that they can come to see academic mathematics as a collection of procedures and rules, instead of viewing it as a means of finding solutions to everyday, real-life problems. This view will interfere with students' ability to apply the formal mathematics they learn to their lives in a practical and useful way.
Teachers can help prevent this outcome by establishing the connection between students' natural intuitive math and formal mathematics. They can do this by teaching math through the use of manipulative materials familiar to children. They can use mathematics vocabulary words when describing students' activities, which enables students to develop an awareness of the natural mathematical operations they use in their daily lives. When a teacher introduces a new mathematical concept to students, he or she can give illustrative examples that draw upon the students' actual life experiences.
Cardinal, Ordinal, Nominal & Real Numbers:
Cardinal numbers are numbers that indicate quantity. For example, when we say "seven buttons" or "three kittens", we are using cardinal numbers.
Ordinal numbers are numbers that indicate the order of items within a group or a set. For example, when we say "first", "second", third, etc., we are using ordinal numbers.
Nominal numbers are numbers that name things. For example, we use area code numbers along with telephone numbers to identify geographical calling areas, and we use zip code numbers to identify geographical mailing areas. Nominal numbers, therefore, identify categories or serve as labels for things. However, they are not related to the actual mathematical values of numbers and do not indicate numerical quantities or operations.
Real numbers include all rational and irrational numbers. Rational numbers can always be written as fractions including all rational and irrational numbers.
Rational numbers can always be written as fractions that have both numerators and denominators that are whole numbers. Irrational numbers cannot, as they contain non-repeating decimal digits. Real numbers may or may not be cardinal numbers.
Modue-3 Math-Q/A
Q.1. An understanding of which of the following mathematical concepts is most helpful when working with probabilities?
Ans: ratios & proportions
Q.2. A Venn diagram would be most appropriate for visually representing which of the following problems?
Ans: What are the common multiples of 2, 3, and 5 that are less than 40?
Q.3. A teacher wants to help students develop their understanding of the relative magnitude of numbers. Which of the following types of mathematics learning materials would be most effective for the teacher to use for this purpose?
Ans: number lines
Q.4. Which of the following strategies is most appropriate for solving the problem below?
(One week Liam baked twice cookies and gave half of them away. The next week Liam baked twice as many cookies as the week before and gave half of them away. In the third week, Liam again baked twice as many cookies as the week before and gave half of them away. If Liam gave away 24 cookies in the third week, how many cookies did he give away altogether?)
Ans: working backward
Q5. To correctly use the ordinal numbers first through fifth with a group of five objects placed in a row, a young child must first be able to:
Ans: recognize that the last number counted is the number of objects in the group.
Q.6. A preschool child measures the length of a long rectangular block by lining up several smaller blocks along the length of the longer block. This demonstrates that the child has an understanding of which of the following measurement concepts?
Ans: unit iteration
Q.7. With respect to mathematics instruction, a standardized developmental screening test would be most appropriate for helping a second-grade teacher:
Ans: identify whether any student is at risk of a possible learning problem in mathematics
Q.8. A kindergarten teacher wants to promote students' ability to interpret data. The teacher provides stickers of various colors and asks students to select the sticker that best matches the color of the shirts they are wearing. The students then place the stickers in columns based on the color on a large sheet of paper. To meet the objective of the lesson, the most appropriate step for the teacher to take next would be to:
Ans: Ask students to identify which columns have more stickers and which have fewer.
Q.9. A second-grade teacher is having students work on adding double-digit numbers. The teacher writes a problem on the board, asks students to estimate the answer, and then asks them to explain how they arrived at their answers. As students listen to their peers' explanations, some students are able to revise their answers correctly while others are not. Which of the following follow-up activities would be most appropriate?
Ans: working with students in small groups to help students at different ability levels construct their understanding of the concept of addition.
Q.10. Which of the following mathematics activities would be most appropriate for promoting preschool students' skills in measurement?
Ans: sequencing three different events that occur in the daily classroom routine
Q.11. A third-grade teacher is developing a criterion-referenced test for a mathematics unit on multiplication. During the unit, students use arrays to solve multiplication problems involving one-and-two-digit numbers. To produce a valid criterion-refer5enced test for this unit, the teacher must:
Ans: including test questions that allow students to solve multiplication problems using arrays as they did during the unit.
Q.12. A preschool teacher wants to promote children's skills related to counting and comparing numbers. Which of the following activities is most appropriate to use for this purpose?
Ans: Letting the class vote on which book the teacher should read next and then asking the class which book won based on the votes
Q.13. A third-grade student is having difficulty solving word problems in mathematics. Which of the following strategies would be most helpful for the teacher to suggest the student use first?
Ans: resting the problem in his or her own words
Q.14. A teacher provides mathematics instruction using manipulatives, worksheets, games, and class discussions. The primary benefit of providing instruction in multiple formats is that it:
Ans: addresses a variety of learning styles
Q.15. A teacher gives groups of students a number of triangles, squares, and semicircles and asks the students to combine these shapes to make as many different figures as they can. This activity will help build a foundation for student's future development of which of the following skills?
Ans: determining the area of irregular shapes
Mathematics, Science, and Health:
PreK–4
Resources
Abruscato, J., & DeRosa, D. (2010). Teaching children science: A discovery approach (7th ed.). Boston: Allyn & Bacon.
Council of Chief State School Officers (CCSSO)/National Governors Association (NGA). (2010). Common core state standards for mathematics.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Drolet, J., & Wycoff-Horn, M. (Eds.). (2006). Health education teaching strategies for elementary and middle grades. Reston, VA: American Association for Health Education.
Michaels, S., Shouse, A. W., & Schweingruber, H. A. (2008). Ready, set, science! Putting research to work in K–8 science classrooms. Washington, DC: The National Academies Press.
Musser, G. L., Peterson, B. E., & Burger, W. F. (2008). Mathematics for elementary teachers: A contemporary approach (8th ed.). Hoboken, NJ: John Wiley & Sons.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author.
National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Committee on Early Childhood Mathematics, Cross, C. T., Woods, T. A., & Schweingruber, H. A. (Eds.). Washington, DC: The National Academies Press.
National Research Council. (2000). Inquiry and the National Science Education Standards: A guide for teaching and learning. Washington, DC: National Academy Press.
Pangrazi, R., Beighle, A. & Pangrazi, D. (2009). Promoting physical activity and health in the classroom. San Francisco: Pearson Benjamin Cummings.
Sorte, J., Daeschel, I., & Amador, C. (2011). Nutrition, health, and safety for young children: Promoting wellness. Upper Saddle River, NJ: Prentice Hall.
Van de Walle, J. A., Karp, K., & Bay-Williams, J. M. (2009). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston: Allyn and Bacon.