The Division Process: Division with Remainders: AbstractLevel

More Teaching Plans on this topic: Concrete, Representational

PHASE 3: Evaluation

Monitor/Chart Performance

Purpose: to provide you with continuous data for evaluating student learning and whether your instruction is effective. It also provides students a visual way to “see” their learning.

Materials:

Teacher -

• Appropriate prompts if they will be oral prompts
• Appropriate visual cues when prompting orally

Student -

• Appropriate response sheet/curriculum slice/probe
• Graph/chart

Description:

Steps for Conducting Continuous Monitoring and Charting of Student Performance:

1) Choose whether students should be evaluated at the receptive/recognition level or the expressive level.

2) Choose an appropriate criteria to indicate mastery.

3) Provide appropriate number of prompts in an appropriate format (receptive/recognition or expressive) so students can respond

At the abstract level of understanding, the most efficient format for a curriculum slice/probe is written (e.g. student responds in writing to written prompts).

4) Distribute to students the curriculum slice/probe/response sheet.

5) Give directions.

6) Conduct evaluation.

7) Count corrects and incorrects/mistakes (you and/or students can do this depending on the type of curriculum slice/probe used – see step #3).

8) You and/or students plot their scores on a suitable graph/chart. A goal line should be visible on each students’ graph/chart that represents the proficiency (near 100% accuracy with two or fewer incorrects/mistakes) and a rate (# of corrects per minute) that will allow them to be successful when using that skill to solve real-life problems and when using the skill for higher level mathematics that require use of that skill.

9) Discuss with children their progress as it relates to the goal line and their previous performance. Prompt them to self-evaluate.

10) Evaluate whether student(s) is ready to move to the next level of understanding or has mastered the skill at the abstract level using the following guide:

Abstract Level: demonstrates near 100% accuracy (two or fewer incorrects/mistakes) and a rate (# of corrects per minute) that will allow them to be successful when using that skill to solve real-life problems and when using the skill for higher level mathematics that require use of that skill.

11) Determine whether you need to alter or modify your instruction based on student performance.

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Assessment

Purpose: to assess where student understanding of the division process is “breaking down.”

Flexible Math Interview/C-R-A Assessment

Materials:

Teacher -

• Appropriate concrete materials for dividing (See Concrete Level Instructional Plan – Explicit Teacher Modeling.).
• Appropriate examples for assessment (division problems)
• Paper to record notes.

Description:

Have students solve division problems using concrete materials, by drawing, and without concrete materials or drawings. Ask students to explain their answers as they respond. Note where in the division process students “break down;” both at what level they begin having difficulty and at what point within that level of understanding they demonstrate misunderstanding/non-understanding. Based on where students demonstrate difficulty, provide explicit teacher modeling at that level of understanding and for the particular sub-skill they are having difficulty with. As the student demonstrates understanding, scaffold your instruction until they are ready to practice the skill independently. As students demonstrate mastery of the skill at that level of understanding, then provide explicit teacher modeling at the next level of understanding. Follow this process until students demonstrate mastery at the abstract level.

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Key Ideas

Students who demonstrate difficulty at the abstract level of understanding may have “gaps” in their understanding that can be traced back to their representational/drawing level of understanding or even their concrete level of understanding. By providing additional teacher modeling at the level their “gap” in understanding began and then moving them from a concrete-to-representational-to-abstract level of understanding, you can assist students to become successful at the abstract level of understanding.

Sometimes students demonstrate difficulty at the abstract level because they did not receive enough practice opportunities at the concrete and representational/drawing levels. The drawing level is a very important step for these students. Some students need continued practice drawing solutions and associating their drawings to the abstract symbols and the mental processes necessary to perform at the abstract level.

Some students understand the concept, but have difficulty remembering the steps involved to perform the skill at the abstract level. Providing students with cues they can refer to as they practice at the representational/drawing and abstract levels of instruction is very helpful (e.g. DRAW Strategy). Such cueing provides them the independence to practice. Multiple practice opportunities translate into repetition, and repetition enhances memory. The use of instructional games and self-correcting materials are an excellent way to provide students with multiple opportunities to solve division problems.

Helping your students build their fluency for solving division facts can also increase their abstract level problem-solving efficiency. Providing daily one-minute timings and charting student performance is an effective way to do this. It is important to communicate with students what their “learning pictures” (charts) mean and to set short-term achievable goals. Seeing “what” they are striving for and seeing their progress as they move toward a goal is very motivating for children! (See the description of the instructional strategy “Continuous Monitoring and Charting Student Performance” located in the Instructional Strategies site for more information. This description can be found by clicking on “Instructional Strategies” on the main menu bar found on your left panel.)

Enhancing the “meaningfulness” of abstract equations can also aid students who are having difficulty achieving mastery at the abstract level both by providing them a deeper level of conceptual understanding and by enhancing their memory of the problem-solving process. One approach you might try is to reinforce what the numbers and symbols mean using language. By modeling language (and encouraging students to use their own language) that describes what each number and symbol represents, students can gain a deeper level of understanding of the “abstract process” they are struggling to master.

For Example:

12 ¸ 4 = ___

12 “CD’s” shared among 4 friends becomes ? CD’s per friend

Provide your students multiple opportunities to use their language as they practice solving equations. As students practice, they have the opportunity to associate “meaning” to the abstract process.

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