Ohjective2 Days!J-13

ac:tlvities

Objective 1 = Recall addition facts (sums to 18).

Objective 2 = Understand the efficiency of memorization (in certain circumstances).

Objective 3 = Gain a working knowledge of various memorization strategies.

Assess 1

=

Observations of students.

Assess 2 = Questions to students in class.

Assess 3 = Mad Math Minutes.

Assess 4 = Weekly quizzes.

Dark shading indicates the strongest aligrunent-an objective, an instructional activity, and an assessment are all present in the same cell. Li.ghter shading indicates two of the three are present.

168 Section IIl The Taxonomy in Use

COMMENTARY

Ms. Hoffman's questions focus on applying procedural knowledge. Through these assessments she is able to determine which procedures students are using. The changes in Mad Math Minute scores over time provide evidence of student im- provement in remembering factual knowledge. Unlike the Mad Math Minutes, which are organized around a single addend, exercises on the weekly quizzes are drawn somewhat randomly from the universe of addition facts. Also, un- like the Mad Math Minute, the quizzes have more liberal time allocaöons. As a consequence, students have sufficient time to use a variety of approaches.

Nonetheless, the emphasis remains on remembering factual knowledge.

The results of our analysis are shown in Table 10.3. Once again, initial analysis of the stated objectives is shown in bold type and analysis of the in- structional activities is shown in italics.

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PART 4: CLOSING COMMENTARY

In this secöon we examine the vignette in terms of our four basic questions: the leaming question, the instruction question, the assessment question, and the alignment question.

THE LEARNING QUESTION

In terms of the learning question, we distinguish between what we term

"focus" and "emphasis." The focus is clearly on rememberingfactual knowledge.

This is quite clearly the desired end result of the three-week unit. The focus is evident in both the stated objectives and the assessments. In contrast, the em- phasis is on understanding conceptual knowledge. With the brief exception of the Mad Math Minute, virtually all the activities in which students engaged dur- ing the first two weeks (approximately two-thirds) of the unit emphasize un- derstanding conceptual knowledge. This discrepancy between focus and emphasis can perhaps best be explained by the difference between means and ends. For Ms. Hoffman, the end (her focus) is dear: students are to remember factual knowledge. On the knowledge dimension, Conceptual, Procedural, and to a cer- tain extent Metacognitive knowledge are means to this end. Similarly, on the c~g- nitive process dimension, Understand and Apply are the means. Thus, the emphasis in the unit reflects the means by which the end will be achieved.

THE INSTRUCTION QUESTION

Primarily because of the Mad Math Minute activity, some instructional activi- ties related to the major objective (remember factual knowledge) took place every day. Activities related to the two longer-term objectives were reserved for the end of the unit {i.e., Days 9-13). As shown in Table 10.2, numerous activities are placed in cells of the Taxonomy Table that do not contain the stated objectives.

In her description of these activities, Ms. Hoffman suggested that they were in-

Chapter 10 Addition Facts Vignette 1 69

tended to help students develop a framework for efficient memorization. The activities during the first two weeks, for example, focused largely an under- standing conceptual knowledge. lnherent in the structure of the Great Addition Wall Chart, for example, were pattems and connections that could make mem- orization easier.

Similarly, Ms. Hoffman introduced a variety of memorization strategies to her students. Her intention was for students to (1) choose the one or ones most useful to them, and (2) come to realize that memorization is more efficient than alternative ways of arriving at an answer. These activities had a dual focus: ap- ply procedural knowledge and understand metacognitive knowledge.

Finally, what is interesting here is what Ms. Hoffman did not do. She did not give students a steady diet of "drill and practice." Rather, she made use of five cells of the Taxonomy Table (see Table 10.2) even though her intended leaming for her students feil into a single cell.

THE ASSESSMENT QUESTION

Ms. Hoffman used both informal and formal assessments. She observed her students and asked them questions in dass to gather information about the procedures they used to remember the addition facts. She used Mad Math Min- utes and weekly quizzes to get at the "bottom line" -had students memorized the addition facts? Thus, the informal assessments were intended to get infor- mation about the process; the formal assessments were intended to get infor- mation about the outcome.

THE ALIGNMENT QUESTION

As we show in Table 10.3, the alignment of assessments and instructional ac- tivities with the stated objectiv~s is fairly strong. Cells Aland C3 include an objective, several activities, and assessments. As described above, the assess- ments in cell Al (remember factual knowledge) were more formal; those in cell C3 (apply procedural knowledge) were more informal.

Only a few examples of misaligrurtent occur. Ms. Hoffman has no formal assessment of understanding metacognitive knowledge, although she did infor- mally assess how students were arriving at answers and inferring processes. It is not clear if she evaluated (or taught) whether students saw using analogies as applicable to other than addition facts. Several activities in cells B2 (under- stand conceptual knowledge) and C2 (understand procedural knowledge) have no as- sociated objective or assessment. The latter supports the distinction between emphasis and focus that we made in our discussion of the leaming question.

PART

5:

CLOSING QUESTIONS

As with the analysis of all our vignettes, we were left with a few unanswered questions. We raise three of the most important in this closing section.