Teacher - Suzanne Alejandre
Comments - Ihor Charischak
|The teacher poses the question: how many moves would it
take if there were four pairs?
The teacher asks the students if they can extend the first two columns.
This turns out to be an easy one for them. The numbers in the third column
are another story. The teacher encourages the students to come up with
some ideas or conjectures.
Several interesting responses follow. One girl predicts 20 moves for
four people on a side. She explains her thinking. The numbers go up by
5, then 7. So the next number should go up by 5 and then you add 5 and
add 7 again. The teacher responds with, "That's a possibility" and then
asks another girl what she thinks.
|A student says "19" after he completes what
he thinks is a puzzle with four people on each side. Actually the Java
applet set up three people on one side and four on the other, so for that
situation the number of moves is indeed 19. Students realize they can not
use the Java applet to answer this part of the question.
The teacher prompts the students by commenting that the number of pairs
goes up by one and the number of people goes up by two. What about the
|Another girl (below) says "24" (which is the right answer) but she
doesn't know why. The teacher who does not reveal that this is the correct
answer, asks whether it is an educated guess. The girl nods affirmatively.
(Was it really educated?)
Another student says "it's a skipping pattern." The teacher tries to
refocus students on horizontal conjectures rather than vertical ones.
Some students misinterpret the teacher's meaning of sideways. They
think it means to physically turn the page sideways.
This is like getting the pattern using the local train method -- you
need the previous value to get the next value. (In other words, you have
to make all the local stops.)
|The scene shifts to kids in groups sitting at
tables. They get up and say what their findings are.
The first group spokesperson reports their results for the first two
columns. They see the pattern that the first column numbers go up by 1
and the second column number goes up by 2.
Group 2 actually comes up with a working version of how to determine
column 3 from columns 1 and 2. For three pairs, the girl said "3 times
3 plus 6."
With some teacher prompting, the students agree that if you multiply
the first number by itself and add the second number, then you get the
third number. The rest of the period is spent getting the students from
this point to an understanding of the algebraic generalization: n*n + 2*n
which yields the minimum number of moves.
Using algebra to determine the minimum number of moves is like taking
the express train. You don't need to know the previous number in the pattern.
If you know the number of pairs then you can determine the moves even if
you don't know how many moves the previous number of pairs yields.
|Some interesting conversation followed about
Here are some of the teacher's questions and students' responses which
are not necessarily correct.
T: If the first column is n, then what should you call the second
S: Double N.
T: How do you write a double n?
T: "What does double n mean? Is that different than n dot n or
n plus n? You are making this way too hard."
The teacher writes n+n on board. Not n squared, but it can be also 2n
not n2. n2 means n*n
If n is the number of pairs and 2n is the number of people then the
minimum moves is n*n + 2n. She writes it again as students are copying
in their notebooks.
Mrs. Alejandre is entering a tough area. It is more about language
and the nuances of the language of algebra. The students could easily get
lost. My question is, what did the students get out of this experience?
Do they really understand what just happened?