- Nov 2019
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mathematicalmusings.org mathematicalmusings.org
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Students often ini-tially hold undifferentiated views of measurable attributes, sayingthat one object is “bigger” than another whether it is longer, orgreater in area, or greater in volume, and so forth. For example,two students might both claim their block building is “the biggest.”Conversations about how they are comparing—one building maybe taller (greater in length) and another may have a larger base(greater in area)—help students learn to discriminate and namethese measurable attributes. As they discuss these situations andcompare objects using different attributes,
Students start by using describing words to describe the attributes before learning how to measure these characters and represent them mathematically.
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Volumeis an amount of three-dimensional space that is con-tained within a three-dimensional shape. Volume measurement as-sumes that congruent shapes enclose equal volumes, and that vol-ume isadditive, i.e., the volume of the union of two regions thatoverlap only at their boundaries is the sum of their volumes
The idea of volume is much more complicated then area as a whole extra element is added making it much more difficult and taught after a strong grasp on area is achieved.
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Before learning to measure attributes, children need to recog-nize them, distinguishing them from other attributes. That is, theattribute to be measured has to “stand out” for the student and bediscriminated from the undifferentiated sense of amount that youngchildren often have, labeling greater lengths, areas, volumes, and soforth, as “big” or “bigger.
These attributes are often able to be visualized to help the learners.
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- Oct 2019
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mathematicalmusings.org mathematicalmusings.org
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Having drawn the number line diagram, the student can proceedthrough the data set recording each observation by drawing a sym-bol, such as a dot, above the proper tick mark. As with Grade 2 lineplots, if a particular data value appears many times in the data set,dots will “pile up” above that value. There is no need to sort theobservations, or to do any counting of them, before producing theline plot.Stude
Line plots are the first type of graphs students learn as it is one of the less complex types of graphs.
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The work shown in the figure is the result of an intricate pro-cess. At first, we have before us a jumble of specimens with manySorting categorical dataThe marks represent individual data points. The two categorycounts, 7 and 8, are a numericalsummaryof the data.attributes. Then there is a narrowing of attention to a single at-tribute (wings or not). Then the objects might be arranged intopiles. The arranging of objects into piles is then mirrored in thearranging of marks into groups. In the end, each mark representsan object; its position in one column or the other indicates whetheror not that object has a given attribute
Early representation of groups that will eventually develop into the ability to create different types of graphs.
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Students’ work with categorical data in early grades will supporttheir later work with bivariate categorical data and two-way tablesin eighth grade (this is discussed further at the end of the CategoricalData Progression)
Students start organizing objects very early in development this could be based on color or various other simple categories. Students build off this and eventually learn ways to visual represent these categories in math.
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Multistep problemsStudents extend their work to solving mul-tistep ratio and percent problems.7.RP.3Problems involving percentincrease or percent decrease require careful attention to the refer-ent whole.
As steps get added it becomes much more difficult for students to complete problem as there are more areas to get confused and make mistakes.
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Strategies for solving problemsAlthough it is traditional to movestudents quickly to solving proportions by setting up an equation,the Standards do not require this method in Grade 6. There area number of strategies for solving problems that involve ratios. Asstudents become familiar with relationships among equivalent ratios,their strategies become increasingly abbreviated and efficient.
Often many different strategies and ways to solve the same problem and as students progress they learn these multiple strategies.
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Ratios of two quantities have associatedrates. For example, theratio 3 feet for every 2 seconds has the associated rate32feet forevery 1 second; the ratio 3 cups apple juice for every 2 cups grapejuice has the associated rate32cups apple juice for every 1 cup grapejuice. In Grades 6 and 7, students describe rates in terms such as“for each 1,” “for each,” and “per.” In the Standards, theunit rateis the numerical part of such rates; the “unit” in “unit rate” is oftenused to highlight the 1 in “for each 1” or “for every 1.”
A base knowledge in fraction is very beneficial here as ratios are often common to ratios and the knowledge between the two can be interchangeable.
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Previously inGrade 3, students learned that53can be represented as thenumber of objects in5groups of3objects, describing this prod-uct as five threes. (As discussed in the Operations and AlgebraicThinking Progression, in other countries this may be described asthree fives.)
Grouping is a very effective and important strategy for students to understand and grasp early on and to build on as they progress.
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Progressions for the CCSS NF, 3–5 140Grade 3The meaning of fractions and fraction notationIn Grades 1 and2, students use fraction language to describe partitions of shapesinto equal shares.2.G.3In Grade 3, they start to develop a more gen-2.G.3Partition circles and rectangles into two, three, or four equalshares, describe the shares using the wordshalves,thirds,halfof,a third of, etc., and describe the whole as two halves, threethirds, four fourths. Recognize that equal shares of identicalwholes need not have the same shape.eral concept of fraction, building on the idea of partitioning a wholeinto equal parts and expressing the number of parts symbolically
It is easy to start students off by explaining fractions with shapes like pizza and a slice being 1/6. This helps provide a strong base to build from fractions.
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In learningabout base-ten notation, first graders learn to think of a ten as a unitcomposed of10ones, and think of numbers as composed of units, e.g.,“20is2tens” and “34is3tens and4ones.” Second graders learn tothink of a hundred as a unit composed of10tens as well as of100ones. Students decompose tens and hundreds when subtracting ifthey need to get more of a particular unit
Even when subtracting large numbers units are even more important as more are involved and a strong understanding from a young age is a necessity.
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Students fluently add and subtractmulti-digit numbers through 1,000,000 using the standard algo-rithm.4.NBT.4Work with the larger numbers allows students to con-solidate their understanding of the uniformity of the base-ten system
Students have now got a greater understanding of base ten system and know how to consolidate numbers into a simpler form to use their fundamental math skills.
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Students in Grade 4 work withfractions having denominators10and100.4.NF.54.NF.5Express a fraction with denominator 10 as an equivalentfraction with denominator 100, and use this technique to add twofractions with respective denominators 10 and 100.44Students who can generate equivalent fractions can develop strategiesfor adding fractions with unlike denominators in general. But addition andsubtraction with unlike denominators in general is not a requirement atthis grade.onestenstenthshundredths÷ 10÷ 10÷ 10÷ 10.1.1.1.1.1.1.1.1.1.1 1100 110each piece is .01 or each piece is .1 or ÷ 101$1dimepennydollarBecause it involves partitioning into10equalparts and treating the parts as numberscalled one tenth and one hundredth,
Students start to work with simple fractions in order to try and build a base on how to perform fraction problems.
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Students fluently add and subtractwithin 1000 using methods based on place value, properties of op-erations, and/or the relationship of addition and subtraction.3.NBT.23.NBT.2Fluently add and subtract within 1000 using strategiesand algorithms based on place value, properties of operations,and/or the relationship between addition and subtraction.They focus on methods that generalize readily to larger numbersso that these methods can be extended to 1,000,00
Students connect multiple ideas in order to do problems with larger numbers and the process in which to do so.
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In Grade 2, students extend their under-standing of the base-ten system by viewing 10 tens as forming a newunit called a “hundred.”2.NBT.1aThis lays the groundwork for under-2.NBT.1Understand that the three digits of a three-digit numberrepresent amounts of hundreds, tens, and ones; e.g., 706 equals7 hundreds, 0 tens, and 6 ones. Understand the following asspecial cases:a 100 can be thought of as a bundle of ten tens—called a“hundred.”standing the structure of the base-ten system as based in repeatedbundling in groups of 10 and understanding that the unit associatedwith each place is 10 of the unit associated with the place to itsright.
This is very important in math as the base 10 system is important to understand as it is the foundation of how numbers work and build on each other
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In problem situations, students must interpret and useremainders with respect to context.4.OA.3For example, what is the4.OA.3Solve multistep word problems posed with whole numbersand having whole-number answers using the four operations, in-cluding problems in which remainders must be interpreted. Rep-resent these problems using equations with a letter standing forthe unknown quantity. Assess the reasonableness of answersusing mental computation and estimation strategies includingrounding.smallest number of busses that can carry 250 students, if each busholds 36 students? The whole number quotient in this case is 6and the remainder is 34; the equation250636 34expressesthis result and corresponds to a picture in which 6 busses are com-pletely filled while a seventh bus carries 34 students. Notice thatthe answer to the stated question (7) differs from the whole numberquotient.
Problems with remainders are complex but are often related to real life scenarios allowing students to think about it differently and be able to solve the problem with more ease.
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Thereare three major types, shown as rows of Table 3. The Grade 3standards focus on Equal Groups and on Arrays.•As with addi-•Multiplicative Compare situations are more complex than EqualGroups and Arrays, and must be carefully distinguished from ad-ditive Compare problems. Multiplicative comparison first entersthe Standards at Grade 4.4.OA.1For more information on multi-plicative Compare problems, see the Grade 4 section of this pro-gression.4.OA.1Interpret a multiplication equation as a comparison, e.g.,interpret3557as a statement that35is5times as manyas7and7times as many as5. Represent verbal statements ofmultiplicative comparisons as multiplication equations.tion and subtraction, each multiplication or division situation in-volves three quantities, each of which can be the unknown. Becausethere are two factors and one product in each situation (productfactorfactor), each type has one subtype solved by multiplica-tion (Unknown Product) and two unknown factor subtypes solvedby division.
This is a very important concept to understand, even as problems get more complex it is often attempted to simplify them down to one of these three major types of multiplication and division.
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The Standards for Mathematical Practiceare central in supporting students’ progression from understandingand use of strategies to fluency with standard algorithms.
It is important to start students off with a strong base in mathematics as math is a progression of knowledge and to build of a strong foundation. As problems get more complex often can be simplified to the fundamentals.
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They relate their strategies to written methods and ex-plain the reasoning used (for addition within 100 in Grade 1; foraddition and subtraction within 1000 in Grade 2) or illustrate andexplain their calculations with equations, rectangular arrays, and/orarea models (for multiplication and division in Grade 4)
It is important to use real life examples for students as they are often more motivated to do work they can relate too as it has a clear use.
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These skills and understandings are3.OA.1Interpret products of whole numbers, e.g., interpret57as the total number of objects in 5 groups of 7 objects each.crucial; students will rely on them for years to come as they learn3.OA.2Interpret whole-number quotients of whole numbers, e.g.,interpret568as the number of objects in each share when56 objects are partitioned equally into 8 shares, or as a numberof shares when 56 objects are partitioned into equal shares of 8objects each.3.OA.3Use multiplication and division within 100 to solve wordproblems in situations involving equal groups, arrays, and mea-surement quantities, e.g., by using drawings and equations witha symbol for the unknown number to represent the problem.to multiply and divide with multi-digit whole number and to add,subtract, multiply and divide with fractions and with decimals
Multiplication and division are very crucial life skills but often take long to teach as there are many different strategies dependent on the numbers given.
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tandard algorithms•for base-ten computations with•The Standards do not specify a particular standard algorithm foreach operation. This progression gives examples of algorithmsthat could serve as the standard algorithm and discusses theiradvantages and disadvantages.the four operations rely on decomposing numbers written in base-ten notation into base-ten units.
Task 4
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- Apr 2019
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extendboundariesofliteracy.pbworks.com extendboundariesofliteracy.pbworks.com
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using it to extend our own ideas or to produce a criticism. Lessig (2005) says that every single act of reading and choosing and criticizing and praising culture is in this sense remix, and it is through this general practice that cultures get made.
Culture is made through changing and mixing an existing culture. This can also be applied to the classroom in a way by allowing students to "play" with an idea or object to remix it into something completely different that they created.
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wiobyrne.com wiobyrne.com
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Creation can be viewed simply as the act of producing, or causing to exist. Construction is the building or assembling of an infrastructure. Construction is equal parts inspiration and perspiration. Construction calls on creativity as well as persistence, flexibility, and revision. Construction asks our students and teachers to focus on the power and patience employed during work process…and not just the final resultant work product.
creation vs construction
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newlearningonline.com newlearningonline.com
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School was a universe of straightforwardly right and wrong answers, of authoritative texts and authoritarian teachers. The underlying lesson of the basics was about the social order and its sources of authority, a lesson which was appropriate for a society which expected its workers to be passively disciplined.
The old way of schooling must be rewritten to allow for more open minded questions with no real right or wrong answer.
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these new literacies are embodied in new social practices—ways of working in new or transformed forms of employment, new ways of participating as a citizen in public spaces, and even perhaps, new forms of identity and personality.
The way of life has changed so in return this must be taught in the classroom as students must be prepared.
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As a consequence, the traditional emphasis on alphabetical literacy (letter sounds in words in sentences in texts in literatures) would need to be supplemented in a pedagogy of Multiliteracies by learning how to read and write multimodal texts which integrated the other modes with languag
In a traditional school the way language was taught 15 years ago must change to allow for the teaching of multi modal texts and other modes of non traditional language.
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The world was changing, the communications environment was changing, and it seemed to us to follow that literacy teaching and learning would to have to change, as well.
As the world becomes more invested into multiliteracies and technology the classroom must as well.
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- Mar 2019
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www.ascd.org www.ascd.org
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During Phase 3, students work both individually and in small groups at using strategies and skills from the previous phases to develop lines of inquiry around curricular topics. This type of project requires clear questions, multiple reliable sources, citations, and a final product that communicates that information to others.
Finally students put together a final product that portrays all the information they've learned throughout the lesson.
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Students take responsibility for teaching their peers a variety of online reading comprehension strategies. Instruction also begins to move from search skills to critical evaluation and synthesis skills.
Student modeling starts to take place in Phase 2
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Phase 1 centers on computer basics, word processing skills, Web searching, navigation basics, and e-mail.
Phase 1 consists of teaching kids how to use the internet.
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strategies: predicting, questioning, clarifying, and summarizing. The teacher explains these strategies to small groups using a shared text, first modeling their use, and then asking students to lead the groups.
Model how to teach and then allow students how to teach each other.
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These Cs include such skills as creativity, communication, collaboration, critical thinking, and comprehension.
These 5 C's are widely seen in all classrooms today but were rarely said or exhibited a mere 25 years ago.
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The structure of the textbook was a map that Sarah could easily follow.
Using the internet to teach no longer allows to simply divide a textbook up and teach solely using a textbook for the year. Now students and teachers alike have a wider range of info and more problem solving to figure out what to use and not to use.
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wiobyrne.com wiobyrne.com
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As you become more familiar with Internet Inquiry Projects, you’ll find that you regularly use the web for teaching and learning every day.
It is clear that the web not only helps teachers teach but also allows them to teach even more then they previously could.
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It should also be noted that Internet Inquiry Projects are not only appropriate but also vital for use in classrooms from Pre-K up through higher ed. It is the responsibility of educators in all grades and content areas to modify as needed for learners.
Inquiry projects are a necessity in all modern classrooms.
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There are many variations as the project is student interest driven, and may last any amount of time. The design, focus, and length of the Internet Inquiry Project should be determined by your student learning objectives, as well as your own technological, pedagogical, and content area knowledge (TPACK) and objectives.
Great way to tie together an Inquiry project while also using TPACK.
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Another takeaway was that K-12 students don’t understand “credibility” and “relevance”…but they do understand words like “truthful” and “useful.” An entire program of study could be spent focusing on critical, media and information literacies.
By not over complicating simple tasks student understand and can do much better work.
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Students collaboratively (with the instructor) identify an area of interest and co-construct a driving question to guide inquiry. Students engage in online collaborative inquiry as they search and sift through online texts using digital tools to address their focus of inquiry. Students critically evaluate online information by considering the credibility (truthfulness) and validity (usefulness) of the information obtained. Students synthesize what they have learned during their online inquiry by actively curating and synthesizing information across multiple, multimodal sources. Student engage in online content construction by synthesizing what they have learned and selecting the best digital text or tool before sharing this answer.
these are the 5 phases involved in internet inquiry projects
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Internet Inquiry Projects are student interest driven, and are more authentic as a learning activity than traditional WebQuests
student driving learning is the best way to go about this as so many kids have a wealth of knowledge about technology that often educators even do not know.
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WebQuests play a vital role in the classroom by providing students with a scripted, guided examination of online resources in a topic. As students expand beyond the WebQuest, the next step is to engage in an Internet Inquiry Project. Internet Inquiry Project
Web Quests are a great way to have student explore the web while also doing so in a safe and guided way.
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soe.unc.edu soe.unc.edu
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The Wayback Machine records snapshots of a website's pages throughout its history. Those snapshots gather some or all of the pages on the website.
Wayback Machine is creating a database of snapshots of websites throughout history.
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newliteracies.uconn.edu newliteracies.uconn.edu
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ost importantly, it is reshap-ing the nature of literacy education, providing us with many new and exciting opportunities for our classrooms.
Very important as everyday new technology is coming out to improve both literacy and digital literacy.
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These changes will require each of us to always have one lens turned to the future so that we might continuously learn about even newer online tools that we can use in our classrooms, preparing our students for their future.
Very important today as teachers we do not know where technology will take us. This means we must constantly keep looking for changes in order to implement them in the classroom.
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- Feb 2019
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clalliance.org clalliance.org
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Understanding more about opportunities available to them after high school
Extremely important as the main point of education to me is for the ability to provide a stable life for yourself after school ends. When students leave school without the knowledge of their possible opportunities it becomes detrimental to their futures.
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Learners need support from peers and mentors to persist through setbacks and challenges.
A support system is very helpful in the classroom where students can trust peers and faculty for help when needed.
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A growing body of research indicates that interest helps us pay attention, make connections, persist and engage in deeper learning. For example, when reading about games they enjoy playing, teenage boys read at a much higher level than their reading level in school.
By being interested in something it instantly makes the wealth of knowledge you can retain and practice go up immensely.
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Learning is irresistible and life-changing when it connects personal interests to meaningful relationships and real-world opportunity.
I really want to implement this in my future classroom by allowing kids to work on what they find interesting and not forcing the whole class to do the same thing.
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Connected learning is realized when a young person is able to pursue a personal interest or passion with the support of friends and caring adults, and is in turn able to link this learning and interest to academic achievement, career success or civic engagement.
Very similar to Daniel Pink's ideas to have students be able to work on things they have a passion for.
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It advocates for broadened access to learning that is socially embedded, interest-driven, and oriented toward educational, economic, or political opportunity.
I think connected learning is very important to use as it is a way of learning that naturally motivates students as an intrinsic motivator. This will help the students be able to accomplish much more in the classroom.
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www.gettingsmart.com www.gettingsmart.com
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Teachers in the substitution and augmentation phase can use technology to accomplish traditional tasks, but the real learning gains result from engaging students in learning experiences that could not be accomplished without technology.
This shows the different levels of SAMR that a teacher may implement and achieve but that most beneficial way to implement the SAMR model is when you accomplish learning that wouldn't be possible without the technology you used.
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Researchers have determined that technology integration typically moves through specific levels. The higher the level of an activity the greater the educational benefit.
I think it is very interesting that the higher the grade level the more educational benefit technology integration has. This brings a lot of questions to mind about tech integration in young classrooms.
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- Jan 2019
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www.aeseducation.com www.aeseducation.com
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To become a global collaborator, students have to understand how their perspectives are different from others’ and work together to achieve a common goal.
very important for young students.
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