EDMA Final

How can you use base 10 blocks to teach addition, including regrouping.

Students can use base ten blocks to show a direct representation of the ten digit and how they can be broken up and added to create different numbers A student can have a place value chart with ones, tens and hundred columns. They can start by putting the first number of blocks and then add the next number. The student must decide how to regroup the ones or tens to create a new number. Direct modeling provides a necessary background of ideas and can transfer direct modeling to their own ideas and no longer have to rely on materials or counting

Reflect on the "big ideas" of this course and how you will utilize them in your own classroom.

"Doing mathematics" Flexible and conceptual thinking Manipulatives Learn by doing- games and stations Create comfortable environment with open discourse and discussion Multiple approaches/methods Model different representations Understanding the "why" before the "how" of math Collaboration Math is fun!!

Know about understanding of number as it relates to pre-place value

(Pg. 223) Before learning place value, kids understanding is based on a count-by-ones approach. Example: if they count to 18 to them it just means 18 ones, they can't separate that into place-value groups. If they count 18 teddy bears, they might tell you that the 1 stands for 1 teddy bear (instead of 10) and the 8 stands for 8 bears

Assessments which can be used to determine students' understanding of base-ten development (not sure if this is correct!!)

(Pg. 231) Use a class observational checklist How do students count out the objects? Do they make groupings of ten? Do they count to 10 and then start again at 1?

Be able to model a multiplication problem in at least 4 different ways

Arrays Array is any arrangement of things in rows and columns, such as a rectangle of square tiles or blocks Area Model Partial Product Repeated addition

Use and benefits of a hundred chart

At kindergarten and first grade levels, students can count and recognize two digit numbers with the hundreds chart. In second grade and beyond, students use the hundred chart to develop a base-ten understanding, noticing that jumps up or down are jumps of ten, while jumps to the right or left are jumps of one Place-value-related patterns students may notice using the hundreds chart The numbers in a column all end with the same number, which is the same as the number on top In a row, the numbers "count" from left to right or the "second" number goes up by 1, the first number stays the same You can count by 10s going down the right hand column

Know about compatible numbers in addition

Compatible numbers are numbers that go together easily to make landmark numbers Numbers that are tens or hundreds are the most common example Compatible sums also include numbers that end in 5, 25, 50, 75 because these numbers are easy to work with as well The teaching task is to get students accustomed to looking for combinations that work together and then looking for these combinations in computational situations

Appropriate tools used to measure attributes

Decide in the attribute to be measured Select a unit that has that attribute Compare the units- by filling, covering, matching or using some other method- with the attribute of the object being measured. The number of units required to make the object is the measure

Know the three components of relational understanding of place value

From book How groups are recorded in our place value scheme How numbers are written How they are spoken From notes- Students must know 3 different representations of place value: Model Written number name Written symbol

Compare and contrast standard algorithms and invented strategies.

Invented strategies are number oriented rather than digit oriented Using the standard algorithm for 45+32, students think of 4+3 instead of 40+30. This can be confusing for students because it can "unteach" place value Invented strategies are left handed rather than right handed For 263+126 invented strategies will begin with 200+100 is 300, providing some sense of the size of the eventual answer in just one step, standard algorithms tend to "hide" the answer until the end Invented strategies are a range of flexible options rather than "one right way" The standard algorithms suggests using the same tool on all problems. The standard algorithm for 7000-25 typically leads to student errors, yet a mental strategy is relatively simple

Know the levels and characteristics of Van Heile's Geometric Thinking.

Level 0: Visualization The objects of thought are Shapes and what they "look" like Recognize and name figures based on the global visual characteristics of the figure Appearances can overpower properties of shape because appearance is dominate in this stage Students will sort and classify shapes based on their appearance Students are able to see how shapes are alike and different Students can create and begin to understand classifications of shapes The product of thought is Classes or grouping of shapes that seem to be "alike" Shapes that students can observe, feel, build, take part or work with in some manner General goal is to explore how shapes are like and different and to use these ideas to create classes of shapes Properties of shapes, such as parallel sides, symmetry, right angles and so on are included at this level by only in an informal, observational manner Level 1: Analysis The object of thought is Classes of shapes rather than individual shapes Able to consider all shapes within a class rather than just the single shape on their desk Begin to appreciate that a collection of shapes goes together because of properties May be able to list all the properties of squares, rectangles, and parallelograms but may not see that these are subclasses of one another The products of thought is Properties of shapes Students will continue to use models and drawings of shapes, they begin to see these individual shapes as representatives of classes of shapes Their understanding of the properties of shapes-such as symmetry, perpendicular and parallel lines and so on Level 2: Informational Deduction The object of thought are the properties of shapes Begin to think about properties of geometric objects without focusing on one particular object they are able to develop relationships between these properties With greater ability to engage in if-then reasoning, students can classify shapes using only a mini Mum set of defining characteristics Observations go beyond properties themselves and begin to focus on logical arguments about the properties Informal deductive arguments about the shapes and properties The product of thought are relationships between properties of geometric objects Level 3: Deduction The object of thought are relationships between properties of geometric objects Students move from thinking about properties to reasoning or proving related to the properties Is able to work with obstruct statements about geometric properties and make conclusions based more on logic than intuition The product of thought are deductive axiomatic systems for geometry Students build on a list of axioms and definitions create theorems Level 4: Rigor The object of thought are deductive axiomatic systems for geometry The objects of thought are axiomatic systems themselves, not just the deductions within a system Distinctions and relationships between different axiomatic systems

Demonstrate partial sums, partial product, partial quotient, and subtraction using an invented strategy.

Partial sum Add tens, add ones then combine (46+38+ ? 40+30=70 and 6+8=14 so 70+14 is 84) Add on tens, then add on ones shown on a number line Move some to make ten Use a nice number and compensate Partial product By decades (27x4=? 4x20=80 and 4x7=28 28+80= 108) Partitioning the multiplier By tens and ones Partial quotient Equal groups (fair share) 12 ones to 3 people----> each person gets 4 Subtraction Add tens to get close, then the ones Add tens to overshoot, then come back Add ones to make a ten, then tens and ones

What are the four major content goals of geometry?

Shapes and Properties Includes a study of the properties of shapes in two dimensions and three dimensions, as well as a study of the relationships build on properties Transformation Includes a study of translations, reflections, rotations (slide, flips and turns) the study of symmetries and the concept of similarity Location Refers primarily to coordinate geometry or other ways of specifying how objects are located in the plane or space Visualization Includes the recognition of shapes in the environment, developing relationships between two dimensional three dimensional objects, and the ability to draw and recognize objects from different viewpoints

Van heile theory of geometric thought graph

Shapes-classes of shapes-properties of shapes-relationship between properties- deductive systems of properties-anaylsis of deductive systems

Describe fair share or sharing tasks, and how you could use them to teach fractions.

Students in the early grades partition by thinking about fair shares (division) Students initially perform sharing tasks by distributing items one at a time. When this process leaves leftover pieces, students must figure out how to subdivide so that every group (or person) gets a fair share Problem difficulty is determined by the relationship between the number of things to be shared and the number of shares Ex: 5 brownies shared with 2 children 2 brownies shared with 4 children 5 brownies shared with 4 children 7 brownies shared with 4 children 4 brownies shared with 8 children 3 brownies shared with 4 children When students who are using a halving strategy try to share five things among four children they will eventually get down to two halves to give to four children It is important to meet the needs of the range of learners in your classroom Guess and check Fraction bars and fraction circles can be subdivided Some students who struggle may need to cut out and physically distribute the pieces on a number of plates Students can use connecting cubes to make bars that they can separate into pieces

What are misconceptions related to fractional parts?

Students think that the numerator and denominators are separate values and have difficulty seeing them as one single value Finding fraction values on a number line or ruler can help students develop this notion Avoid the phases "three out of four" or "three over four" and instead use "three fourths" In thinking of the numbers separately, students may think that ⅔ means any two parts, not equal-sized parts Students think that a fraction such as ⅕ is smaller than a fraction such as 1/10 because 5 is less than 10 Many visuals and contexts that show parts of the whole are essential in helping students understand Students mistakenly use the operation "rules" for whole numbers to compute with fractions, for ex: ½ + ½ = 2/4 Students who make these errors do not understand fractions

How can you create an environment that promotes invented strategies?

Students who are attempting to investigate new ideas in math need to find their classroom a safe and nurturing place for expressing naive or rudimentary thoughts Establish the climate for taking risks, testing conjectures and trying new approaches Avoid immediately identifying the right answer when a student states it gives other students the opportunity to consider where they think it is correct Expect and encourage students to student interactions, questions, discussions and conjectures Encourage students to clarify previous knowledge and make attempts to construct new ideas Promote curiosity and openness to trying new things Talk about both right and wrong ideas in a non evaluative or non threatening way Move unsophisticated ideas to more sophisticated thinking through coaxing, coaching and strategic questioning Use familiar contexts and story problems to build background and connect to students experiences

What is a lesson you could do to help students with equivalent fractions? Know ways to illustrate equivalent fractions.

The general approach to helping students create an understanding of equal alert fractions is to have hem use contexts and models to find different names for fractions Filling in regions with fraction pieces Paper folding Grid paper Dot paper Length models Area models are a good place to begin understanding equivalence Different Fillers Using an area model for fractions that is familiar to students, prepare a worksheet with two or three outlines of different fractions Do not limit it yourself yo unit fractions For example, if the model is circular fraction pieces, you might draw an outline for ⅔, ½, and ¾. The student starts is to use their own fraction pieces to find as many equivalent fractions for the area as possible After completing three examples, have students write about the ideas or patterns they may have notices in finding the names Apples and Bananas Have students set out a specific number of counters in two colors Ex: 24 counters with 16 being red (apples) and 8 being yellow (bananas) the 24 counters make up the whole The task is to group the counters into different fractional parts of the whole and use the parts to create fraction names for the fractions that are apples and fractions that are bananas 24=1, 16 apples and 8 bananas (16/24 apples and 8/24 bananas) Make the 16 apples into 4 groups of 4 (4/6) and 2 more sets of 4 makes 24 (2/6) 16 is 2 groups of 8 (⅔ apples ⅓ bananas) 8 groups of apples (8/12) 4 groups of bananas (4/12) Apples are 16/24=4/6=⅔=8/12 of the fruit Bananas are 8/24=2/6=⅓=4/12 of the fruit

Know what a student must understand to be proficient in division.

There are two concepts of division: There is the partition or pair sharing idea The measurement or repeated subtraction concept

Know subtraction strategies (most effective)

Think-addition: use known addition facts to produce the unknown quantity or part of the subtraction Example: 13 - 5 = ___ Think: "Five and what makes thirteen?" → 8 and 5 is 13 Down Under 10: this strategy focuses on thinking about the distance between two numbers on a number line Example 14 - 8 = ___ First take away 4 to get to 10 (benchmark number), then take away 4 more to get the answer of 6 Or you can think about it like "How far apart are 14 and 8? Jump down four to get to 10 and then two more to 8 and they are 6 jumps apart

Be able to discuss what Diller suggests about general considerations when using math manipulatives

Whenever possible, limit the amounts of manipulatives you give students to simplify management Let children explore the manipulatives when you first introduce a new type Be explicit about you expectations for how to handle manipulatives. Show children what they do, make it very clear what they may not do Model, Model, Model Be considerate: Once you've established how to use materials, hold students accountable. If a child is playing around using materials incorrectly, remove the manipulatives form that student immediately If you have foam manipulatives use them they are so much quieter than hard plastic Don't use paper manipulatives unless you have no other alternatives If you don't have much money and need counters, consider getting small fun items from a dollar store or ask parents to clean out the toy box at home and send small items that will fit in a sandwich size bag Use math mats. The boundaries set by them will help a lot with management because they let children know what you expect Set a purpose. Let children know why they are using a type of manipulative and how it will help them learn a concept in math After children have worked with manipulatives, ask for their feedback whether this manipulative helped or if there's another that would have helped better Don't take away manipulatives because you feel they are a crutch

Describe invented strategies and their benefits.

invented strategies are any strategy other than standard algorithm or that does not involve the use of physical materials or counting by ones Benefits The development and use of invented strategies deliver more than computational proficiency Students make fewer errors (less frequent and almost never systematic) Less reteaching is required The extended struggle in the early stages results in a meaningful and well-integrated network of ideas that are robust and long lasting The increase in development in time is made up for with a significant decrease in reteaching and remediation Students develop number sense "More than just a means to produce answers, computation is increasingly seen as a window on the deep structure of the number system" Students development and use of number-oriented, flexible algorithms offer them a rich understanding of the number system Invented strategies are the basis for mental computation and estimation Flexible methods are often faster than standard algorithms Algorithms invention is itself a significantly important process of "doing mathematics" Develop confidence in their ability Students using invented strategies either are on a par with students using standard algorithms or outperform them