*Discovering Geometry*. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.

A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:

3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)

The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.

Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.

Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.

My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an

*index card*, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text. My worksheet with the modified rules will appear at the very end of this blog post.

Now I've been thinking about Wu's proposal from yesterday -- that there should be specialized math teachers at elementary schools. Even though I want to focus on high school math on this blog, I do want to give my opinion on Wu's proposal.

As I mentioned yesterday, back when I was in the upper elementary grades, I had a different teacher for math from the teacher I had for English/language arts. Actually, with separate teachers for math comes the opportunity to separate students into classes by mathematical ability. And this is exactly what my elementary school did, by implementing a "Path Plan." I'll describe a modified version of this "Path System" that incorporates some of Wu's ideas as well.

My elementary school ranged from kindergarten to sixth grade, but instead of dividing the classrooms into grade levels, they are divided into "paths." Students are placed into one of the various paths based on their ELA (

*not*their mathematical) ability. The approximate grade levels corresponding to each path were:

Primary: Grades 1-2

Transition: Grades 3-4

Preparatory: Grades 5-6

but it was possible to be placed into a different path than the nominal path for the grade level -- so there could be an above-grade-level second grader in the Transition Path, and likewise a below-grade-level third grader in the Primary Path. Officially kindergartners, along with four-year-old children in Headstart, were placed in the Early Learning Path, but in practice these classes were simply called "kindergarten."

In my version of the Path Plan, students stay in their homerooms to study ELA, and this lasts until every group has had recess. Then the students change classes. There are 40-minute blocks when the students spend time with a different teacher. How many such blocks there are depends on the path:

Primary: 1 block

Transition: 2 blocks

Preparatory: 3 blocks

For Primary Path students the block is used for math, of course. For Transition Path students, I recommend that the second block be a fun class like art or possibly an exploratory wheel of classes that rotates every semester or trimester. In the Preparatory Path, the third block could be for science -- especially since fifth graders take a standardized test in science under the No Child Left Behind requirements. It may be better to reverse the Transition and Preparatory Paths here, with science in the Transition Path, since many fifth graders might be in that path (and besides, I have no idea how testing requirements will change under the Next Generation Science Standards).

This is where teacher specialization can take place. Since Primary Path students have only one block, specialization isn't possible, but starting in the Transition Path, stronger math teachers can teach math during both of the blocks while others teach science (or whatever subjects were chosen for the blocks).

After the proper number of blocks, the students return to homeroom to pick up their sack lunches, lunch money, or lunch cards for free/reduced lunch. After lunch recess, the students spend the remainder of the day in homeroom to pick up the missing subjects, including P.E. and others. Kindergartners spend the whole day in one classroom -- once again, I strongly disagree with the idea of making five-year-old children move from class to class during the day.

There is a very similar plan to this Path Plan, well-known as the "Joplin plan." But the major difference between the Joplin and Path Plans is that for Joplin, all grade levels spend

*ELA*(and sometimes math as well) in another classroom during a block of time, whereas in the Path Plan ELA is part of

*homeroom*(since the homerooms are already divided by reading ability). Notice that if math is included in the Joplin plan, teacher specialization may be possible if some teachers teach math during both blocks while others teach reading during both blocks. Also, under both plans, all teachers would still need a multiple-subject credential to teach, since they all still teach more than one subject during the day. It's just that only some teachers will have to teach math -- ideally, the strongest math teachers.

My plan gives a gradual increase in the number of teachers a student has in a day as that student progresses in age. In particular, kindergartners have one teacher, Primary Path students have two (homeroom and math), Transition Path students have three, and Preparatory Path students have four. In my district, this pattern continues in middle school, since seventh and eighth graders have only five teachers -- ELA and history being combined into a two-period "Core" class. Not until high school do the students have six teachers, a different teacher for every period. (As an aside, notice that after that, the number of teachers a student has at one time begins to decrease again. Juniors and seniors in my district only had to take five classes, many community college students take four classes a semester, upper division students take three classes, Masters candidates take two, and Ph.D. candidates have only one adviser for their dissertation...)

Of course, plans such as this are controversial. This plan is very similar to academic tracking, and the problem is that no one wants to be the parent of a student placed on a lower track, or a third grader on the Primary Path, or a fifth grader on the Transition Path. One thing I noticed about the Path Plan at my school was that although students were placed on lower paths for reading -- no student was placed into a lower math class than their actual grade level. So a third grader might be placed in third grade math, or possibly fourth grade math, but never in second grade math -- not even a third grader on the Primary Path. And so the classes were not completely homogeneous -- notice that a

*third*grader in a fourth grade math class is at a fourth grade level, but the

*fourth*grader in the class might not be. But actually, I don't believe that completely homogeneous classes are even possible -- after all, whenever a lesson is taught, some students will understand it while others won't. So a class that was homogeneous

*before*the lesson is no longer homogeneous

*after*the lesson.

And now I present my worksheet for the Daffynition Game. Remember that only one of these worksheets need to be given to each group -- in particular, to the scorekeeper in each group. The students write their guesses for Rounds 1-4 (or 5) on their own separate sheet of paper. I recommend that it be torn into strips so that they are harder to recognize. And the teacher provides the index cards, one for each student. Make sure that the students give back the index cards so you can reuse them for the next period. The students may keep their "guess cards," so there should be one for every student in every period.