Base_Blocks_-_NLVM.pngBase Blocks & Place Value

According to Flores (2009), "researchers have demonstrated that the use of the concrete-representational-abstract (CRA) sequence is effective for teaching mathematics to students who struggle with learning mathematics" (p. 145).

The CRA sequence has three phases:
  • during the concrete phase
    • students and the teacher use manipulatives to demonstrate the meaning of a particular math concept
      • The teacher models the math concept and the students use the manipulatives in the same manner as the teacher

  • during the representational phase
    • the teacher and the students use the manipulatives to represent math numbers

  • in the final phase, the abstract phase
    • the students do not use manipulatives to solve the math problem,
    • but instead use a mnemonic or memorization of steps to solve a math problem

In this task, you are going to use a math manipulative to:
  • practice place-value fluency
  • represent place values to develop the concept of 10 more, 10 less, 100 more and 100 less
  • teach place value math concepts using the CRA sequence

1.) Development of Pedagogical Content Knowledge

  1. Develop Pedagogical Content Knowledge
    • Content Knowledge
      • knowledge about the actual subject matter that is to be learned or taught[1]
      • Technological Content Knowledge
        • knowledge of the manner in which the subject matter can be changed by the application of technology

    • Pedagogical Knowledge
      • knowledge about techniques or methods to be used in the classroom
        • use of virtual manipulative to teach the math concept of 10 more, 10 less, 100 more and 100 less

      • knowledge that is involved with lesson plan development and implementation
        • reading of common core math module
        • practice procedural fluency— skill in carrying out procedures flexibly, accurately, efficiently, and appropriately (Bransford, Donovan, & others, 2004)
        • develop conceptual understanding— comprehension of mathematical concepts, operations, and relations (Bransford et al., 2004)

    • Pedagogical Content Knowledge
      • knowing how elements of the content can be arranged for better teaching
        • practice procedural fluency and then develop conceptual understanding

      • Technological Knowledge
      • Technological Pedagogical Knowledge
        • knowing how teaching might change as the result of using particular technologies
          • in order for students to use this virtual manipulative, they must:
            • use a computer and not an iPad
            • be provided a link to the manipulative so it is easy to get to
          • in order for teachers to use it, they must:
            • realize that the computer will distract students
            • understand that computers might not be charged
            • understand that computers might crash
            • know how to delete blocks
            • know how to create a new problem
            • know how to clear the manipulative

      • Technological Pedagogical Content Knowledge
        • an understanding of the representation of concepts using technologies
          • students can represent 1s, 10s, 100s and 1000s with the manipulative
          • students can show the concepts of 10 more, 10 less, 100 more and 100 less
          • the manipulative will represent a number in blocks and in standard form
            • the manipulative will change the standard form dynamically when blocks are added or removed

2.) Method

Read the entire method before you start doing the tasks. Then read and complete each of the 3 sections.

Rubric


I. Imaginary Teaching Scenario: Preparing for the a Math Lesson

This 1st section is a typical narrative any teacher who has to teach the Common Core Learning Standards might be able to tell! It is divided into 2 parts.
  • 1. Imagine this scenario
    • a. You are a second grade teacher;
    • b. You have to use the EngageNY curriculum to teach math modules;
    • c. You want to use a virtual math manipulative to develop the math concept of place value instead of using place value disks;
    • d. You choose to use a math manipulative developed by Utah State University called Base Blocks

  • 2. Prepare for Your Lesson as a 2nd Grade Teacher
    • a. You read the Common Core Math Lesson developed for you by the authors of EngageNY
      • (i.) You think about these questions as you read the module
        • A. What do the teacher and students have to do to practice Fluency? (p. 5.A.4 - 5.A.5)
        • B. What do the teacher and students have to do during the Application Problem? (p. 5.A.6)
        • C. What do the teacher and students have to during the Concept Development? (p. 5.A.6 - 5.A.7)
        • D. What do the students have to do during the Problem Set? (p. 5.A.8 & 5.A.11 - 12)
        • E. What do the teacher and students have to during the Student Debrief? (p. 5.A.9)
        • F. What do the students have to do for the Exit Ticket? (p. 5.A.13)
        • G. What do the students have to do for Homework? (p. 5.A.14 - 5.A.15)

    • b. You decide to:
      • (i.) Use a virtual (digital) math manipulative to do the 1st fluency practice (p. 5.A.4) for the lesson
      • (ii.) Complete the 2nd fluency practice (p. 5.A.5) as described in the module
      • (iii.) Skip the Application Problem because you are still shaky teaching the Read Draw Write (RDW) Procedure to solve problems
      • (iv.) Use a virtual math manipulative instead of place value disks in order to develop the math concepts of 10 more, 10 less, 100 more and 100 less

Cogito_And_Cogitas_-_Math_Stations.png
Figure 1: Example of setting up links to the math manipulative from a wikis page

    • c. Set up the Technology for the Next Day's Lesson
      • (i.) You start by posting a link to the Base Blocks virtual manipulative on your classroom web site (wikispace) - see Figure 1
        • (A.) You reserve the laptop cart for your class by posting your name on the paper calendar located on the laptop cart
        • (B.) Before you leave school, you make sure all the laptops are plugged in so that they can be charged

    • d. Tweak the Lesson Module instructions

II. Imaginary Teaching Scenario: Teaching the Lesson

This section is one way to teach this lesson by integrating technology.
  • 1. Setting up Technology for Learning -
    • a. Students grab the laptops from the cart
    • b. They login to the computer
    • c. They open a web browser which opens to the school district's web site
      • (i.) On the district's web site, they click a link to their elementary school teachers' web sites
        • (A.) On the new web site that opens, the students find Mr. Shively's name and click it
        • (B.) On the new web site that opens, the teacher directs them to the math section and they click on the math link
        • (C.) Students find the correct date for today's lesson, they find the fluency section and click on the Base Block Link (which opens in a new tab by design)
      • (ii.) To eliminate extraneous processing (Moreno & Mayer, 2007), the students setup the Base Block Virtual Manipulative by reducing the number of columns shown from 4 to 3

  • 2. Implement the Fluency Script
    • a. The teacher reads the revised fluency script to the students
    • b. The students use the virtual manipulative to represent numbers on the place value chart according to the directions on the script
    • c. The teacher finishes the script with the students
    • d. The teacher completes the other fluency script as written on the module

  • 3. Skip the Application Problem because you are not sure how to teach the Read Draw Write Problem Solving Section

  • 4. Concept Development Script
    • a. The teacher reads the revised concept development script
    • b. Students use the virtual manipulative to represent numbers on the place value chart
    • c. The teacher completes the script
    • d. The teacher completes the rest of the concept development section as written on the module

III. Your Tasks

You will complete teacher tasks and student tasks for this lesson
  • 1. Teacher Tasks for you to do
    • a. Re-create how I setup the links to the Base Blocks Link
      • createNewWikipage.png(i.) Create a wikipage on this wikispace
        • (A.) Procedure for creating a new wikipage
        • (B.) Name of wikipage: YOUR FIRST NAME + JTerm + 2015 (e.g. Chris JTerm 2015)
        • (C.)Tag: jTerm2015
        • (D.) Click Create
        • (E.) Once your wikipage has been started, type in a few words (they can be made up) and click SAVE. Typing words and saving the wikipage will create the wikipage.
        • (F.) - Click here - to create your wikipage (link will open in a new tab for your convenience)
        • (G.) To see if you followed the directions properly, - click here - to see all of the class' wikipages
        • (H.) If you do not see your wikipage, repeat steps A - D

    • b. Build what you see, on the image below, on your wikipage
teachertask_1.png
Image to build on your wikipage


  • 3. Teacher Tasks for you to do
As a teacher, it is important for you to reflect upon teaching
    • a. On your wikipage, which is here, please post a response to this writing prompt: Use one of the the assumptions below and describe how it IS or IS NOT addressed by the Base Block manipulative
      • (i.) According to the Cognitive-affective Theory of Learning with Media (CATLM), researchers have made the following assumptions:
        • (a) humans have separate channels for processing different information modalities (Baddeley 1992);
        • (b) only a few pieces of information can be actively processed at any one time in working memory within each channel (Sweller 1999);
        • (c) meaningful learning occurs when the learner spends conscious effort in cognitive processes such as selecting, organizing, and integrating new information with existing knowledge (Mayer and Moreno 2003);
        • (d) long-term memory consists of a dynamic, evolving structure which holds both, a memory for past experiences and a memory for general domain knowledge (Tulving 1977);
        • (e) motivational factors mediate learning by increasing or decreasing cognitive engagement (Pintrich 2003);
        • (f) metacognitive factors mediate learning by regulating cognitive processing and affect (McGuinness 1990); and
        • (g) differences in learners’ prior knowledge and abilities may affect how much is learned with specific media (Kalyuga et al. 2003; Moreno 2004; Moreno and Durán 2004) (as cited in Moreno & Mayer, 2007, p. 313)

    • b. On your wikipage, please post a response to this writing prompt: Moreno et al. (2007) wrote,"According to the feedback principle, novice students learn better with explanatory rather than corrective feedback alone. Explanatory feedback (EF) consists of providing a principle-based explanation for why students’ answers are correct or incorrect whereas corrective feedback (CF) consists of only communicating whether students’ answers are correct or incorrect" (p. 318). 1.) What type of feedback is provided by this virtual manipulative? 2.) Based on the feedback you described in question 1, what impact do you think it would have on student learning?
      • (i.) Use evidence from Moreno et al. (2007) to support your answer - see p. 318 - 319


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James Earle Fraser's Sculpture called End of the Trail
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References

Flores, M. M. (2009). Teaching subtraction with regrouping to students experiencing difficulty in mathematics. Preventing School Failure: Alternative Education for Children and Youth, 53(3), 145–152.

Moreno, R., & Mayer, R. (2007). Interactive multimodal learning environments. Educational Psychology Review, 19(3), 309–326.
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