question

List the 5 strands that compose what we think of as learning and understanding mathematics.

answer

1. Adaptive reasoning 2. Strategic competence 3. Conceptual understanding 4. Productive disposition 5. Procedural fluency

question

Adaptive reasoning is:

answer

Capacity for logical thought, reflection, explanation and justification.

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Strategic competence is:

answer

Ability to formulate, represent and solve mathematical problems.

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Conceptual understanding is:

answer

Comprehension of mathematical concepts, operations and relations.

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Productive disposition is:

answer

Habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one's own efficacy.

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Procedural fluency is:

answer

Skill in carrying out procedures flexibly, accurately, efficiently and appropriately.

question

List 4 things a teacher should do to create a positive learning environment:

answer

1. Make sure the classroom arrangement is safe and comfortable and that it supports the lesson's learning activities. 2. Make sure the classroom is intellectually stimulating for learning mathematics. Encourage intellectual risk taking and help children feel secure about taking risks. 3. Make sure students understand that learning mathematics is a long term process and that they will not all learn the same things at the same time or be equally proficient but that everyone can become proficient. 4. Reward students for critical thinking and creative problem solving so that students learn to value and respect these approaches.

question

List 6 things a teacher can do to cope with anxiety about math:

answer

1. Emphasize meaning and understanding rather than memorization. Help students make connections between the concrete and the abstract to facilitate understanding better. (Communicating, manipulating and modeling) 2. Model problem solving strategies rather than presenting finished solutions. Focus on the process rather than the result. 3. Show a positive attitude toward mathematics. 4. Give students mathematical experiences that they will enjoy and that will interest and challenge them while allowing them to be successful. 5. Encourage students to tell you how you feel about mathematics. 6. Don't overemphasize speed tests or drills in your classroom.

question

Learned helplessness is:

answer

the belief that the individual cannot control outcomes and is destined to fail without the existence of a strong safety net. It includes feelings of incompetence, lack of motivation and low self esteem. Girls are particularly susceptible to this.

question

List 8 things teachers can do to eliminate bias in the classroom:

answer

1. Dispel myths about mathematics like "all mathematicians work in isolation or white males are the best at math" 2. Have high expectations for all students and clearly communicate them to students and their parents. 3. Communicate to parents the importance of encouraging their children. 4. Make sure boys and girls participate equally in the classroom and receive equal amounts of your time and attention. 5. Make relevant connections between mathematics and the students' lives. 6. Call attention to mathematical role models of all backgrounds. 7. Explicitly teach students about the idea of learned helplessness and develop ways to prevent it or remedy it. 8. Use a variety of ways to assess student performance.

question

Retention is:

answer

The degree to which students can hold onto and use what they have learned.

question

What is retention of information so critical to mathematics instruction?

answer

Because mathematics is cumulative.

question

Which type of learning is more stable over time and less susceptible to big declines-- skills or problem solving?

answer

Problem solving

question

List 4 ways that teachers can help improve retention:

answer

1. Teach in a way that promotes understanding and meaning. 2. Provide students with hands on investigation that help students discover for themselves the idea or concept. 3. Help students make connections between mathematical ideas and the real world. 4. Periodically review key ideas.

question

What is the overriding goal of mathematics instruction?

answer

For students to become mathematically proficient.

question

Procedural knowledge is:

answer

Reflected in skillful use of mathematical rules or algorithms. For example knowing how to do 2 digit long division.

question

Conceptual knowledge is:

answer

Understanding what mathematical concepts mean. For example knowing that one meaning of division means forming equal groups.

question

Research shows that conceptual knowledge does not ____ but may even help students____

answer

Diminish skills; recall and use skills

question

There is a _____(negative or positive) consequence of teaching procedural knowledge without conceptual knowledge.

answer

Negative

question

Rote learning is:

answer

Memorizing without meaning

question

A student who learns a procedure without meaning will have difficulty with:

answer

1) knowing when to use it 2) remembering how to do it 3) applying it in new situations 4) judging if the results are reasonable

question

Instruction that focuses on develop conceptual understanding can also:

answer

Yield efficient skills

question

To learn more abstract mathematical concepts students must:

answer

Have developed enough both physically and psychologically

question

This person asserted that learning comes through experience and active involvement by the learner

answer

John Dewey

question

This person argued that learners actively construct their own knowledge.

answer

Jean Pieget

question

Constructionist view of learning suggests that:

answer

Rather than simply accepting new information students interpret what they see, hear or do in relation to what they already know. In other words they build new knowledge from their personal experiences and prior knowledge. It concentrates on what happens in between the stimulus and response--that is the focus is on the thinking students do.

question

Behaviorism focuses on:

answer

Observable behaviors and is based on the idea that learning means producing a particular response (behavior) to a particular stimulus (something in the external world). In other words students learn specific skills (behaviors) by observing teachers demonstrating those skills in relation to specific stimulus. Learning depends upon not only the teacher but also on the students themselves. Behavior can be shaped through reinforcement--- i.e. rewards and punishment.

question

The main advocates of behaviorism were:

answer

Edward Thorndike, B.F. Skinner and Robert Gagne

question

True or false: Excessive practice, premature practice or practice without understanding is associated with negative effects.

answer

True. These often lead to a fear of, dislike of mathematics or an attitude that mathematics does not need to make sense.

question

A behaviorist approach can be helpful when:

answer

Teaching students a fixed set of skills in a fixed order (procedural knowledge)

question

The steps to use when planning a lesson from a behaviorist approach are:

answer

1. State precisely the objectives or goals of instruction 2. Identify the prerequisites for achieving the goal in order to use them as building blocks for instruction.

question

The benefits of a behaviorist approach are:

answer

1. Provides instructional guidelines 2. Allows for short term progress 3. Lends itself to the current focus on accountability

question

The drawbacks of a behaviorist approach are that:

answer

It leads to a focus on simple, short-term objectives that are easily measured and de-emphasizes long term goals and higher level cognitive processes such as problem solving.

question

Behaviorist oriented verbs for outcomes are:

answer

Explore, justify, represent, solve, construct, discuss, use, investigate, describe, develop and predict

question

Some useful ideas for teaching math that come from a behaviorist approach are:

answer

1. Behavior can be shaped by drill and practice 2. Students can be helped to learn specific skills in a fixed order 3. Clear statements of objectives help teachers design lessons directed at specific learning outcomes 4. Clear statements of objectives and learning outcomes give students a clear idea of expectations

question

Several advocates of constructivism are:

answer

William Bronell, Jean Piaget, Jerome Bruner and Zoltan Dienes

question

The levels of thinking by elementary school children characterized by Piaget are:

answer

1) Preoperational 2)Concrete operational 3) Formal operational

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The levels of thinking by elementary school children characterized by Bruner are:

answer

1) Enactive 2) Iconic (semi-concrete) 3) Symbolic

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The levels of thinking by elementary school children characterized by Dienes are:

answer

1) Free Play 2) Generalization 3) Representation 4) Symbolization 5) Formalization

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An elementary student in the preoperation stage:

answer

Represents action through thought and language but is prelogical in development

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An elementary student in the concrete operational stage:

answer

Thinking may be logical but is perceptually oriented and limited to physical reality

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An elementary student in the formal operational stage:

answer

Considers the possible rather than being restricted to concrete reality. Is capable of logical thinking that allows them to reflect on their own thought processes

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An elementary student in the enactive stage:

answer

Firsthand manipulating, constructing or arranging of real-world objects. Child is interacting directly with the physical world

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An elementary student in the iconic (semi-concrete) stage:

answer

Representational thinking based on pictures, images or other representations, Child is involved with pictorial and/or verbal information based upon the real world

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An elementary student in the symbolic stage:

answer

Manipulation of symbols. Child manipulates and/or uses symbols irrespective of their enactive or iconic counterparts.

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An elementary student in the free play stage:

answer

Interacts directly with physical materials within the environment. Different embodiments provide exposure to the same basic concepts but at this stage few commonalities are observed

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An elementary student in the generalization stage:

answer

Patterns, regularities, and commonalities are observed and abstracted across different models. These structural relationships are independent of the embodiments

question

An elementary student in the representation stage:

answer

Provides a peg on which to hang what has been abstracted. Images and pictures are used to provide a representation

question

An elementary student in the formalization stage:

answer

Provides an ordering of the mathematics. Fundamental rules and properties are recognized as structure of the system evolves

question

Students learn mathematics well only when:

answer

They construct their own mathematical understanding

question

Three basic tenets of constructivism are:

answer

1) Knowledge is not passively received; rather, knowledge is actively created or invented (constructed) by the students (made not found) 2) Students create (construct) new mathematical knowledge by reflecting on their physical and mental actions 3) Learning reflects a social process in which children engage in dialogue and discussion with themselves as well as with others (including teachers) as they develop intellectually

question

The lower and upper limits of what a child can learn at a particular developmental stage are determined as follows:

answer

Lower: The concepts and skills the child has already learned Upper: The tasks that they can successfully complete only with scaffolding or support by someone who is more skilled or knowledegable

question

Zone of proximal development (ZPD)

answer

Range described by Vygotsky of what a child can learn at a particular stage of development

question

Learning is:

answer

Active and internally monitored; it is a process of acquiring, discovering and constructing meaning from experience

question

Four general agreed observations about how children learn are:

answer

1. Several identifiable stages of thinking exist and children progress through these stages as the grow and mature 2. Learners are actively involved in the learning process 3. Learning proceeds from the concrete to the abstract 4. Learners need opportunities for talking about or otherwise communicating their ideas with others.

question

Teaching occurs only to the extent that______

answer

Learning occurs.

question

Three types of development that influence mathematics instruction:

answer

1. Cognitive 2. Physical 3. Social

question

Cognitive development refers to:

answer

How children think, reason and learn information

question

Physical development refers to:

answer

A child's muscles and motor skills

question

Social development refers to:

answer

How children interact with others and helps to describe their self-concept (how they feel about themselves.)

question

Active involvement sometimes requires physical activity but always involves:

answer

Mental activity

question

The two categories of actively involving students are:

answer

1. Exploration and sense making 2. Reflection and use of metacognition

question

Ways to encourage exploration and sense making:

answer

1. Through physical involvement

question

Children are more likely to remember something if:

answer

They have figured it out for themselves

question

Metacognition is:

answer

Thinking about one's own thinking

question

Competent problem solvers ask themselves three questions:

answer

1. What am I doing? 2. Why am I doing it? 3. How will it help me?

question

Ways to help students develop metacognitive awareness:

answer

1. Help students be explicit about how they work when solving problems. Ask: Why did they do that? How did you know not to use that information? 2. Share aloud your own thinking as a model. 3. Prepare students to accept and be aware of common problem-solving experiences by pointing out various aspects of problem solving. For example: Some problems take a long time to solve. Some problems can be solved in several different ways. 4. Encourage students to become more aware of and to think about their own mathematical thinking. For example: What math problems do you like best? What math problems are the most difficult?

question

How can we help students make sense of mathematics?

answer

1. Teach to the developmental characteristics of students. 2. Actively involve students 3. Move learning from concrete to abstract 4. Use communication to encourage understanding

question

What is concrete to one student may not be considered to be:

answer

concrete to another student

question

Children need opportunities to work with objects in the physical world before:

answer

they are ready to work with pictures and other representations

question

Mathematical ideas are abstract by their very nature so any model that embodies them is:

answer

imperfect and has limitations

question

Learning occurs best when students have:

answer

a meaningful context for the mathematical knowledge

question

Research has shown that lightly directing children's attention to the important attribute of the model:

answer

enhances learning

question

Multiembodiment or multiple embodiments is:

answer

The use of perceptually different models

question

Research has shown that transfer across contexts is difficult if children have:

answer

experienced the concept in only a single context rather than in multiple contexts. The more different models or contexts (examples and non-examples), the more likely the students are to focus only on the common characteristics and make the correct abstraction

question

Benefits of multiemodiment:

answer

1. Helps students abstract or generalize appropriately 2. Decreases the likelihood that children will associate a mathematical concept with a particular model

question

Research has shown that students learn more when presented with a combination of:

answer

Examples and non-examples

question

In general (too much or too little) time is spent with models?

answer

Too little

question

When children explain their approaches, talk about mathematics, make conjectures and defend their thinking orally as well as in writing their:

answer

understanding deepens.

question

Vygotsky believed that learning is social experience which challenges them to:

answer

make sense of new ideas

question

Students at all levels should take about mathematics before being asked to:

answer

talk about mathematics symbolically

question

Both student to student and _____ communication are important in the learning process.

answer

student to teacher

question

___________are a vital part in the learning process

answer

Questions

question

Using ________to represent mathematical concepts is a valuable part of being able to communicate mathematically

answer

Conventional symbols

question

One cultural characteristic of students that has been shown to have a large impact on learning is:

answer

Socioeconomic status (SES)

question

Students of low SES were more resistant to learning mathematics through:

answer

problem solving and discussion

question

Four recomendations for teaching students of low SES and minority students:

answer

1. Push for meaningful learning rather than rote memorization 2. Be certain that students are learning what was intended from the problems presented in the curriculum Provide scaffolding so that the students do not rely on explicit teacher instruction 3. Analyze achievement data and identify important topics that are causing difficulty for struggling students so that remediation may occur 4. Advocate for these students to have access to high quality teachers and curriculum

question

True or False Number sense is a finite entity that a student either has or doesn't have

answer

False. Number sense is not finite. It's development is a lifelong process

question

Characteristics of number sense include:

answer

1. An understanding of number concepts and operations on these numbers 2. The development of useful strategies for handling numbers and operations 3. The facility to compute accurately and efficiently, to detect errors, and to recognize results as reasonable 4. The ability and inclination to use this understanding in flexible ways to make mathematical judgements 5. An expectation that numbers are useful and that work with numbers is meaningful and makes sense

question

The stages of number sense development in early childhood and elementary are:

answer

1. Prenumber development, including classification and patterns 2. Early number development, including conservation, group recognition, one-to-one correspondence and comparisons 3. Number development, including: a) connecting groups with number names, including oral and written cardinal and ordinal numbers b) Counting forward and backward c) Skip counting d) Establishing benchmarks of quantities, such as 5 or 10 e) Place value

question

An important characteristic of a number is that it is:

answer

abstract

question

Research has shown that the more varied and different the experiences of a child the more likely it is that:

answer

they will abstract number concepts from their experiences

question

Prenumber experiences are:

answer

experiences with numbers that a child has in their early years. Examples are: telephone numbers, their age, television channels, how many sisters they have

question

Prenumber work is not all done before children do anything with numbers in school rather it typically occurs:

answer

simultaneously with activities involving numbers

question

Classification is fundamental to learning about the real world and it can be done:

answer

with or without numbers

question

Classification is naturally integrated with subjects such as:

answer

reading, science and social studies

question

Classification not only helps children make sense of things around them but also helps them:

answer

become flexible thinkers and fosters the development of reasoning and thinking skills

question

When children disagree about how an object should be classified it forces them to:

answer

defend their answers and clarify how the classification process was done. This helps them see the need for explaining their thinking

question

Cardinal number

answer

Answer to the question how many

question

Attribute blocks or logic blocks

answer

Are an excellent model for classification activities and help develop logical thinking

question

The combining or union of disjoint sets is a natural model for:

answer

addition

question

Disjoint sets are:

answer

Sets with no members in common

question

The logical connectives that can be used to help children classify pieces according to their attributes are:

answer

and, or and not

question

Many different experiences are needed in order to :

answer

sharpen children's observational skills and provide them with the basis on which to build the notion of numbers

question

Mathematics is the study of:

answer

patterns

question

Patterns can be based upon the following types of attributes:

answer

Geometric, relational, physical, or affective (like, happiness)

question

In the early grades patterns help children develop:

answer

Number sense, ordering, counting and sequencing

question

In the later grades patterns help children develop:

answer

thinking strategies for basic facts and algebraic thinking

question

Exploring patterns requires:

answer

active mental involvement and often physical involvement

question

Four different ways that patterns might be used in developing mathematical ideas are:

answer

1) copying a pattern 2) finding the next one 3) extending a pattern 4) making their own pattern

question

Conservation of number is the idea that:

answer

a given number does not vary. Typically young children think a number varies depending on it arrangement or configuration. Rarely do children conserve number before age 5 or 6

question

Subitizing

answer

A skill to instantly see how many there are in a group of objects. Comes from the Latin word for suddenly

question

Sight recognition of quantities up to five or six is important for several reasons:

answer

1) It saves time 2) It is the forerunner of some powerful number ideas 3) It helps develop more sophisticated counting skills 4) It accelerates the development of addition and subtraction

question

As children grow older their ability to recognize quantities continues to:

answer

improve but it is still limited

question

Certain arrangements are more easily:

answer

recognized or subitized. Children usually find rectangular arrangements easiest, followed by linear and then circular.

question

Few adults can recognize by inspection groups of more than:

answer

6 or 8. Even these groups must be in common patterns.

question

Research supports that subitizing is a prerequisite for:

answer

Counting

question

Another important part of learning to count which is also essential for developing number awareness is:

answer

Comparison of quantities

question

Placing of connectors provides a visual reminder of:

answer

one-to-one correspondence

question

The process of recording tally marks rests on the idea of:

answer

one-to-one correspondence

question

When making comparisons students must be able to:

answer

discriminate between important and irrelevant attributes

question

Some models used for making comparisons are:

answer

1) Counting 2) Physically comparing without counting 3) One-to-one correspondence

question

Children need to become aware of the descriptions of relationships such as:

answer

1) more than 2) less than 3) as many as

question

When comparisons are made among several different things:

answer

ordering is involved

question

Ordering often involves several different comparisons and these things help organize information:

answer

graphs

question

Children learn the first 12 number names by:

answer

imitating adults and older children

question

Discrete objects are materials that:

answer

lend themselves well to handling and counting

question

Continuous quantities are:

answer

measured rather than counted. Ex: water in a glass or the weight of a person

question

Four principles on which the counting process rests are:

answer

1) Each object to be counted must be assigned one and only one number name 2) the number-name list must be used in a fixed order every time a group of objects is counted 3) The order in which the objects are counted doesn't matter. 4) The last number name used gives the number of objects

question

Research shows that with encouragement and opportunities to count, young children will develop efficient counting strategies without:

answer

direct instruction

question

Counting Stages

answer

1) Rote counting 2) Rational counting

question

A child in the rote counting stage:

answer

1) Knows some number names but not necessarily the proper sequence. 2) May not always be able to maintain a correct correspondence between the objects being counted and the number names.

question

A child in the rote counting stage who knows some number names but not necessarily in the proper sequence needs to:

answer

Spend more time on the stable-order rule

question

In order to help a child in the rote counting stage who does not maintain a correct correspondence between the objects being counting and the number names the teacher needs to:

answer

1) Have the child slow down 2) Stress the importance of one-to-one correspondence. This is an important prerequisite to rational counting

question

A child in the rational counting stage:

answer

1) is able to answer questions about the number of objects being counted 2) exhibits all 4 counting principles: a) one-to-one correspondence b) stable-order rule c) order-irrelevance rule d) cardinality rule

question

One-to-one correspondence

answer

The idea that each object being counted has one and only one number name

question

Stable-order rule

answer

The idea that the number-name list is used in a fixed order every time as objects are being counted

question

Order-irrelevance rule

answer

The idea that the order in which the objects are being counted does not matter

question

Cardinality rule

answer

The idea that the last number name used when counting indicates the number of objects that were counted

question

Counting strategies:

answer

1) Counting on 2) Counting back 3) Skip counting

question

Counting on:

answer

1) Is proceeded by counting all, where a child counts all the objects again instead of counting all during an addition model 2) leads children to discover valuable patterns 3) is an essential strategy for developing addition

question

Research suggests that even children who reflect all the counting principles are confused about:

answer

The nested inclusion of previous numbers when counting on

question

Counting back:

answer

1) Provides another different experience in order to relate each number to another 2) Models subtraction

question

Research suggests that children are capable of understanding negative numbers:

answer

far earlier than was once thought

question

Skip counting

answer

1) Provides counting practice 2) Provides readiness for multiplication and division 3) Teachers should take advantage of every opportunity to encourage accurate and rapid skip counting

question

Counting money is a valuable skill which should be:

answer

introduced and extended as far as possible in the primary grades

question

Several predictable trouble spots for children when counting are:

answer

1) When they are about to enter the next decade i.e. 29 to 30 2) When they are about to enter the next century i.e. 199 to 200

question

Benefits of calculators:

answer

1) Help illustrate how numbers move in repetitive patterns 2) Show how transitions are made across the decades 3) Help students relate the size of a number by noticing the amount of time needed to count it 4) Promotes both critical thinking and problem solving

question

Number benchmarks

answer

Are perceptual anchors that become internalized from many concrete experiences, often accumulated over many years

question

Benefits of a ten frame:

answer

1) Encourage children to think flexibly about numbers thereby promoting number sense 2) Facilitate the development of addition, subtraction, multiplication and division as well as place value 3) Facilitate patterns 4) Developing group recognition of number

question

Development of the numbers 1-5 is principally done through:

answer

Sign recognition of patterns coupled with immediate association with the oral name and then the written symbol.

question

The most most valuable relationships between numbers are the notions of:

answer

1) one more 2) one less

question

The concept of zero should be:

answer

1) Taught as the absence of something 2) Should be introduced as early naturally possible 3) Should not be referred to as "nothing"

question

Different representations (embodiments, models) of the same number provide an opportunity for teachers to assess a students':

answer

conservation of number

question

Children should realize very early that ____ is a special number

answer

ten

question

Cardinal number

answer

Answers the question, "how many?"

question

Ordinal number

answer

Answers the question "which one?" Refers to order.

question

Nominal numbers

answer

Provide a label or classification; provide essential information for identification but do not necessarily use the ordinal or cardinal aspects of the number

question

Writing numbers

answer

1) Children can usually recognize a number before they can write it 2) Children should spend less time writing numbers than writing letters 3) Should practice writing number apart from mathematics instruction 4) If a child is having difficulty writing numbers a calculator is helpful 5) Children should be shown that numbers can sometimes be written in a different form such as with the number 4

question

Place value is:

answer

Critical to understanding and sense making and is one of the cornerstones of our number system

question

Hindu-Arabic number system

answer

The name of our number system

question

Characteristics of the Hindu-Arabic number system:

answer

1) Place value-- the position of a digit represents its value 2) Base of ten- base means collection so in our system ten is a value that determines a new collection 3) Use of zero- A symbol of zero exists and allows us to represent symbolically the absence of something 4) Additive property- Numbers can be written in expanded notation and summed with respect to place value i.e. 231 = 200+30+1 5) Can represent any number using only 10 digits (0-9)

question

Roman numbers differ from Hindu-Arabic numbers in that they:

answer

1) lack place value 2) have no symbol for zero 3) have no base

question

Development of place value promotes number sense and rests on 2 key ideas:

answer

1) Explicit grouping or trading rules are defined and consistently followed 2) The position of a digit determines the number being represented

question

Confusion or misunderstanding about place value can often be traced back to a :

answer

Lack of counting and trading experiences

question

Two types of materials used to help young children develop the idea of place value are:

answer

1) Ungrouped- provide value experience and prepare a child for using pregrouped materials 2) Pregrouped

question

Pregrouped materials are:

answer

Formed into groups before a child uses them

question

Research suggests that instruction should focus on concrete models that are simultaneously connected to:

answer

oral descriptions and symbolic representations of the models

question

Two types of place value models are:

answer

1) Proportional: beans, base-ten blocks, tongue depressors or measurement. Proportional models are more concrete and children need to understand them before moving on to nonproportional models 2) Not proportional: money, counters, abacus

question

Young children often focus on:

answer

Size proportionality

question

Asking a child to group by tens when they count a large number of items serves several purposes:

answer

1) If a child loses count, correction is easier 2) It also easier to check for errors 3) Shows a child how an unknown quantity can be organized into a form that can be interpreted by inspection

question

In developing place value and establishing number names it is better to:

answer

Skip beyond the teens and start with larger numbers. The lack of consistency between the oral and written form of numbers is confusing.

question

The ability to compose and decompose numbers reflects good:

answer

Number sense

question

The notion of representing a quantity with the least number of pieces for a particular model is critical in:

answer

place value

question

The idea of place value helps children proceed in from:

answer

1) Concrete physical models 2) Semiconcrete organizational models 3) Symbolic representational models

question

When children reverse digits children should:

answer

Use a ten frame to compare the modeled numbers and talk about them in an effort to better appreciate the magnitude of the differences

question

Encouraging children to name the same number in different ways promotes:

answer

Number sense

question

An example of being flexible when using and thinking about numbers is:

answer

Counting by tens and then dropping back or bumping up

question

The ability to focus on the lead or "front end" digits is an important part of:

answer

Number sense

question

Research reports that many children lack an understanding of the relative size of:

answer

numbers greater than 100. Possibly from the lack of opportunities to model

question

One way to help children appreciate the importance of representing place values accurately is by:

answer

using the same digits to represent different numbers

question

Skip counting helps to:

answer

Decrease bumps in the counting road

question

It is helpful for students to link different models to:

answer

larger numbers

question

Once children begin to develop an intuitive grasp of larger numbers and begin to use millions and billions intelligently they are ready to:

answer

read and write these larger numbers

question

To develop facility in reading larger numbers children need to:

answer

develop the correct vocabulary and practice actually naming them aloud

question

Rather than requiring children to read numbers a certain way children should be encouraged to:

answer

select the technique that is most appropriate for the given situation

question

As children develop rounding skills they should come to realize that:

answer

rounding rules may vary and are not universal

question

Teachers should encourage students to round to numbers that:

answer

are easier to work with and make sense to them

question

The precision of the rounded numbers should:

answer

make sense for the problem context

question

Rounding includes the idea of being:

answer

closer to

question

A roller coaster model is an effective model for teaching:

answer

rounding skills

question

Learning the basic number facts is one of the first steps children take as:

answer

they refine their ideas about each operation

question

Basic number facts are:

answer

An essential building block for arithmetic

question

These 2 things are needed in order to do pencil-and-paper computations proficiently:

answer

1) understanding of the operation (operation sense) 2) having immediate recall of basic facts

question

The ultimate instructional goal is that children not only know how to add, subtract, multiply and divide but also know:

answer

when to apply each operation in problem solving situations

question

Four prerequisites for learning about mathematical operations are:

answer

1) facility with counting 2) experience with a variety of concrete situations 3) familiarity with many problem-solving contexts 4) experience using language to communicate mathematical ideas

question

Research has indicated that children use counting to solve problems:

answer

long before they come to school

question

Because it is not always efficient to use counting to solving problems children need to learn:

answer

other procedures to cope with more difficult computations

question

In order to become proficient with computing children need:

answer

1) Experience with a variety of concrete situations 2) Familiarity with many problem contexts 3) Experience in talking and writing about mathematical ideas

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The use of materials should:

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Precede and then parallel the use of symbols

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In order to learn to write about mathematics children should:

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As soon as feasible put their ideas on paper- at first by drawings alone and then with writing

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The following general sequence is appropriate for helping children develop meaning for the 4 basic operations:

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1) Concrete modeling with materials 2) Semiconcrete representation with pictures, diagrams, drawings 3) Abstract representation with symbols, numeric expressions and number sentences

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Addition and subtraction: Separation problems

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"Take away" Involves having one quantity, removing a specified quantity and noting what is left. Ex: Peggy had 7 balloons. She gave 4 to other children. How many did she have left?

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Addition and subtraction: Comparison problems

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Finding the difference. Involves having 2 quantities, matching them one-to-one and noting the quantity that is the difference between them

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Addition and subtraction: Part-whole problems

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In this type of problem a set of objects can logically be separated into 2 parts. You know how many are in the entire set and you know how many are in one of the parts. You need to find out how many must be in the remaining part. Ex: Peggy had 7 balloons. Four of them are red and the rest are blue. How many are blue?

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Multiplication factors

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The first factor tells how many groups or sets of equal size. The second factor tells the size of each group or set.

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Difference between addition/subtraction and multiplication/division

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In addition and subtraction problems all the numbers must have a common label ( i.e. fruit, cats etc.) In multiplication/division problems all the labels are different

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Some of the most commonly used models for illustrating multiplication are:

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1) sets of objects 2) arrays 3) the number line

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Four distinct sorts of multiplicative structures are:

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1) Equal groups 2) Comparisons 3) Combinations 4) areas/arrays

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Multiplication: Equal groups

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Are the most common type of multiplicative structure. Repeated addition. The problem deals with a certain number of groups all the same size. Ex: Andrew has two boxes of trading cards. Each box holds 24 cards. How many cards does he have altogether?

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Multiplication: Comparison problems

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Ex: Hilary spent $35 on Christmas gifts for her family. Geoff spent 3 times as much. How much did Geoff spend? In this case Hilary's expenditures are being compared to Geoff's.

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Multiplication: Combination problems

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Ex: If you have 4 different ice cream flavors and 2 different toppings and each sundae can have exactly one ice cream flavor and one topping how many different sundaes are possible?

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Multiplication: The array model

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Can be especially effective in helping children visualize multiplication

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A good readiness experience for the concept of multiplication is:

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building and naming numerous rectangles with a variety of dimensions

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Division: Measurement problems (Repeated subtraction)

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In these problems you have equal sized groups and you know how many objects are in each group and you must determine the number of groups. Ex 1: Jenny had 12 candies. She gave 3 to each person. How many people got candy? Ex 2: How many 2 foot hair ribbons can be made from a 10-foot roll of ribbon.

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Division: Partition (sharing)

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In these problems a collection of objects is separated into a given number of equivalent groups and you seek the number in each group. Ex: Gil had 15 shells. If he wanted to share them equally among 5 friends how many should he give to each? Ex: Dealing cards

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Knowing mathematical properties is not a prerequisite for:

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working with operations. However it must be developed as part of understanding operations.

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Understanding mathematical properties implies knowing:

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when they apply

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Many children have difficulty with the mathematical property of:

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commutativity

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Basic addition facts are:

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those involving two one digit addends and their sum (100)

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Basic subtraction facts are:

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the inverse of the addition facts (100)

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Basic multiplication facts are:

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those involving two one digit factors and their product (100)

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Basic division facts are:

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the inverse of the multiplication facts (100)

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Researchers have found that, until very recently, there has been considerable variation in when

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students were expected to have mastered their basic facts

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Some students take years to master their basic facts due to a couple of reasons:

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1) A learning disability that makes it impossible for children to memorize basic facts 2) A child has not developed the ability to think multiplicatively 3) A child has not developed the underlying numerical understandings so that the memorization is rote and meaningless 4) The skill of retrieval has not been taught explicitly by teachers or understood by children

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Teachers can help children learn basic facts using a 3 phase approach:

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1) Start where the children are 2) Build understanding of the basic facts 3) Focus on how to remember the basic facts

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When determining where a student is in regards to learning basic facts it is the teachers job to:

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1) help children organize what they know 2) Construct more learning to fill in the gaps 3) develop more meaning for the basic facts and for the symbols we use to represent them

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In order to determine what a student already knows a teacher can:

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1) Use responses from group discussions 2) Observe how each child works with materials and paper-and-pencil activities and use 3) Individual interviews

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Teachers often use inventories at the beginning of the year to discover things like:

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1) Whether the children have the concept of an operation 2) What basic facts they understand 3) What strategies they use to find the solution to combinations 4) What basic facts they know fluently (within 3 seconds)

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When building understanding of basic facts a teacher's emphasis should be on:

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1) aiding the children in organizing their thinking and 2) seeing relationships among facts

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True or false No particular order for teaching the basic facts has been shown to be superior to another

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True.

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There are generally two types of thinking strategies:

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1) those that use concrete materials or counting 2) those (more mature strategies) that use a known fact to figure out an unknown fact

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Before students practice fact retrieval they should be able to:

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1) State or write related facts, given one basic fact 2) Explain how they got an answer or prove that it is correct 3) Solve a fact in 2 or more ways

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Key principles about basic fact drills

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1) Children should attempt to memorize facts only after understanding is attained 2) Children should participate in drill with the intent to develop fluency (Remembering should be emphasized) 3) Drill lessons should be short (5-10 minutes) and should be given almost every day 4) Children should be praised for good effort 5) Drill activities should be varied, interesting, challenging and presented with enthusiasm

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Teachers should take note of whether certain basic addition and multiplication facts occur:

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more or less frequently in their elementary school textbooks

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Although adding zero may seem easy to adults it is actually:

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one of the hardest strategies for children to learn. Because of this explicit work on facts involving zero should be postponed until after children have mastered some of the other fact strategies.

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Thinking strategies for addition facts are:

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1) Commutativity 2) Adding one and zero 3) Adding doubles and and near doubles 4) Counting on 5) Combinations to 10 6) Adding to 10 and beyond

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Thinking strategies for subtraction facts are:

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1) Fact families- this idea is used frequently 2) Think addition-the idea is the major thinking strategy used 3) Subtracting one and zero- similar to addition, most children find this rather easy 4) doubles- this strategy may need to be taught more explicitly than with addition; rests on the assumption that children know their doubles facts 5) Counting back-this strategy is the most efficient when the number to be subtracted is 1 or 0 6) Counting on- this strategy is used most easily when the difference is 1 or 2;research has shown that this is a very powerful strategy for many students

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Thinking strategies for multiplication facts are:

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1) Commutativity 2) Skip counting 3) Repeated addition 4) Splitting the product into known parts-- based upon the distributive property 5) Twice as much as a known fact 6) Working from know facts of 5 7) Patterns- Most useful with 9s; digits of the products always sum to 9 and the number in the tens digit is always one less than whatever factor was being multiplied by 9

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Thinking strategies for division

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1) Think multiplication- is the primary thinking strategy 2) Fact families 3) Repeated subtraction

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There appears to be a natural learning progression across all cultures for:

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single digit addition and subtraction

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The facts considered to be "basic" are not the same among:

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all countries

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Wise use of _____ and _______ are important instructional goals.

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calculators and written algorithms

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True or False Competence with different methods of doing computation is useful but not that important.

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False. Competence with different methods of doing computation is not merely useful it is essential.

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Historically, elementary school mathematics instruction has emphasized ____________more than other methods

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written computation

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There are 2 essential decisions in every computation:

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1) Deciding upon the type of result needed 2) Deciding upon the best method for getting the result

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When students are deciding upon which type of result is needed they are trying to decide if:

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1) an estimate is appropriate or 2) an exact answer is needed

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When students are trying to determine the best method for getting a result they may be considering:

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1) If a calculator may be helpful or 2) if mental computation is possible or 3) if paper and pencil calculation is more appropriate

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When teaching computation a teacher's goals should help students do the following:

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1) Develop competence with each of the computational methods 2) Choose a method that is appropriate for the computation at hand 3) Apply the chose method correctly 4) Use estimation to determine the reasonableness of the result

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More than ____ percent of all mathematical computations in daily life involve mental computation and estimation rather than written computation.

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80

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_____ to _____ percent of the instruction time in elementary school mathematics directed toward computation has focused on written computation

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70 to 90

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What percentage of time should be allotted for teaching the various methods of computation is:

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1) an open question 2) depends in part upon the developmental levels of the students

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Over the years the amount of instructional attention focused on written computation has:

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declined

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One benefit of spending less time focusing on written computation is that:

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more time is now available for other instructional purposes

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The NCTM position statement about calculators says that:

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appropriate calculators should be available to all students at all times

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The CCSSM says about mathematical tools that:

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mathematically proficient students should consider the available tools when solving a mathematical problem.

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Children outgrow________ just as they outgrow shoes

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calculators

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As a teacher you need to help students understand how to use calculators appropriately. This includes teaching them that:

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not all problems can be solved with a calculator

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Calculator Myth #1: Using calculators does not require thinking

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Fact: Calculators do not think for themselves. Students must still do the thinking.

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Calculator Myth #2: Using calculators lowers mathematical achievement

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Fact: Calculators can raise students' achievement

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Calculator Myth #3: Using calculators always makes computation faster

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Fact: It is sometimes faster to compute mentally

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Calculator Myth #4: Calculators are only useful for computations

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Fact: Calculators are also useful as instructional tools

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Calculators encourage thinking because:

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They free students from tedious calculations and give them more time to go through the important problem solving processes that precede or follow computation

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Calculators can raise student achievement because:

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1) Students who use calculators have a better attitude toward math and a better self concept 2) Research has shown that use of calculators can improve the average students' basic skills with paper and pencil both in basic operations and problem solving 3) Research has shown that the use of calculators encourages students to persist longer when faced with problem solving situations

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Calculators can be used as an instructional tool because:

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1) they may be used by older children to see patterns Ex: powers of ten

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A teacher should consider using a calculator when:

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1) The computational skills are not the main focus of instruction 2) It facilitates problem solving 3) Eases the burden of doing tedious computations 4) Focuses attention on meaning 5) Removes anxiety about doing computations incorrectly 6) Provides motivation and confidence 7) a learning-disabled student needs a safety net to allow them to participate in problem solving activities more equally

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A calculator should be used as an instructional tool when:

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1) It facilitates a search for patterns 2) Supports concept development 3) Promotes number sense 4) Encourages creativity and exploration

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Using the calculator only to play games or check there paper and pencil work may send the message that:

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a calculator is only for playing games or that using it is cheating

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Research has shown that the more teachers use calculators in the classroom the more they:

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develop creative and productive ways to use them

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There are 2 main uses for calculators in the classroom:

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1) as a computational tool 2) as an instructional tool

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Mental computation is:

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computation that is all done in the head without tools such as a calculator or pencil and paper

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Decomposing numbers means to:

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break them up so that they are easier to handle

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Compatible numbers are:

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"friendly" numbers that can be combined to make numbers that are easy to compute with Ex: 8 and 2 are compatible numbers because they combine to make 10

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The first step in developing proficiency with compatible numbers is:

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learning how to recognize them

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The benefits of mental computation are that it:

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1) encourages flexible thinking 2) promotes number sense 3) encourages creative and efficient work with numbers

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By encouraging children to talk with each other about the different ways they do problems in their heads you:

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help them learn to think more freely and flexibly

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Some things that you should encourage your students to do regarding mental computation are:

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1) Always try mental computation before using paper and pencil or a calculator 2) Use numbers that are easy to work with 3) Look for an easy way 4) Use logical reasoning 5) Use knowledge about the number system

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Why should teachers encourage mental computation?

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1) It is very useful. Adults do more than 3/4s of their computations mentally 2) Mental computation is the most direct and efficient way of doing many calculations 3) Mental computation is an excellent way to help children develop critical-thinking and number sense and to reward creative problem solving 4) Proficiency in mental computation contributes to increased skill in estimation

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When children about grade 2 tried mental computation their dominate strategy was to:

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apply written algorithms mentally. This is an international dilemma.

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Guidelines for developing mental computation skills:

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1) Encourage students to do computations mentally 2) Learn which computations students prefer to do mentally 3) Find out if students are applying written algorithms mentally 4) Plan to include mental computation systematically and regularly as an integral part of your instruction 5) Keep practice sessions short, perhaps 10 minutes at a time 6) Develop children's confidence 7) Encourage inventiveness 8) Make sure students are aware of the difference between mental computation and estimation

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The difference between mental computation and estimation is:

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Mental computations give exact answers and estimations give approximate answers.

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Computational estimation is:

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a process of producing answers that are close enough to allow for good decisions without performing elaborate computations. It is usually done mentally. It is a process that usually takes years to develop.

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Students can use estimation to monitor their computations at three different points in the process:

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1) Before starting exact computations-to get a general idea of what to expect 2) While doing the computation-- to see if the computation is moving in the right direction 3) After completing a computation-to reflect on their answer and decide if it makes sense

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Students who are proficient at written computations are not necessarily:

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good estimators

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You can significantly improve students' estimation skills by:

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paying systematic attention to estimation in your instruction and giving students repeated opportunities to estimate

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Some vocabulary words students use when learning to estimate are:

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about, almost, just over, nearly, approximate, reasonable, unreasonable, in the ball park, and close enough

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When teaching students how to estimate begin by making them aware of what estimation is about and that it is an important and useful skill so that they:

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develop a tolerance for error

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When teaching students to estimate it is important to:

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give them immediate feedback on their estimate and not to be too overly critical of their estimate

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One key to helping children develop good estimation strategies is to:

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encourage them to be flexible when thinking about numbers

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As with mental computation it is important that children develop a repertoire of:

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different strategies to use for estimation. They must learn to think about the problem, the operation, and the numbers involved and not rely on a fixed set of rules to produce an estimate.

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Ways to use behaviorist ideas in teaching math:

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1. Shape behavior with drill and practice 2. Help students learn specific skills in a specific order 3. Help teachers clearly define objectives during lesson planning that focus on desired outcomes. (How will student learning be manifested?) 4. Provide students with clear objectives and learning outcomes to give students clear idea of expectations

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Conservation of number is:

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That a number does not vary regardless of arrangement or configuration

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Patterns help to develop:

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Number sense, ordering, counting, sequencing, thinking strategies for basic facts, thinking strategies

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4 principles of counting are:

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1. Each object counted has one and only one number name (one-to-one correspondence) 2. The numbers names are used in a fixed order (stable order rule) 3. The order that objects are counted doesn't matter (order-irrelevance rule) 4. The last number named corresponds to the number of objects (cardinality rule)

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Being able to compose a number means that a child can tell:

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what the digits in a number represent i.e. in the number 24 that the 2 represents 2 tens or 20 and the 4 represents 4 ones

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Being able to decompose a number means that a child can tell:

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Regrouping: How to break a number up in different ways i.e that the number 24 can be broken up into 1 ten and 14 ones Money is a good example of this: How many ways can you make 50 cents? 5 dimes; 10 nickels; 50 pennies; 2 quarters; 1 quarter, 2 dimes and 1 nickel etc.

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Computational estimation is:

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a process of producing answers that are close enough to allow for good decision making. Typically done mentally

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Estimation helps students to monitor computation at several points in the process including:

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1. Before starting (to give them a general idea of what to expect) 2. While performing the computation (to verify that the computations are moving in the right direction 3. After completing the computation (to decide if their answer makes sense)

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When teaching estimation teachers need to:

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1. Help students realize that estimation and doing mental math are not the same things 2. Help students develop a tolerance for error 3. Help students realize that estimating is an essential and practical skill 4. Help students learn to be flexible when thinking about numbers 5. Help students learn and be able to choose from a repertoire of strategies when estimating

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Estimation strategies include:

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1. Front end estimation 2. Adjusting 3. Using compatible numbers based on the operation being performed 4. Flexible rounding 5. Clustering

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Regarding the use of calculators teachers should:

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1. Teach students that not all problems can be solved with a calculator 2. That calculators are not always the fastest way of doing computations

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Regarding the use of calculators some benefits/uses are:

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1. Helping facilitate problem solving 2. Ease the burden of tedious calculations 3. Help students focus attention on the meaning 4. Removing anxiety associated with computations (providing a safety net for struggling students) 5. Motivating students 6. As a tool to help students search for patterns 7. Encouraging creativity and problem solving

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A calculator should be used as a computational tool when:

answer

1. Helping facilitate problem solving 2. Ease the burden of tedious calculations 3. Help students focus attention on the meaning 4. Removing anxiety associated with computations (providing a safety net for struggling students) 5. Motivating students

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A calculator should be used as an instructional tool when it

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1. Serves as a tool to help students search for patterns 2. Encourages creativity and problem solving 3. Promotes number sense 4. Supports concept development

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The following is an example of what type of subtraction problem: Peggy had 7 balloons. She gave 4 to other children. How many did she have left?

answer

Separation

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The following is an example of what type of subtraction problem: Peggy had 4 balloons. Richard had 7 balloons. How many more balloons did Richard haven than Peggy?

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Comparison

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The following is an example of what type of subtraction problem: Peggy had 7 balloons. Four of them were red and the rest were blue. How many were blue?

answer

Part-whole

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