\doc\web\99\12\average2.txt
Exemplars was supposedly based on national standards, but every math
standard I've ever seen does not introduce this concept until 4th or
5th grade. 2nd graders typically are lucky if they get integer
multiplication, decimal division is typically after 4th grade.
Division with fraction exact answers is also 6th grade fraction math.
In my opinion, average and median don't belong until after 5th grade.
This problem would be difficult for most 7th or 8th graders, assuming
no direct instruction in how to solve the problem.
This is very typical of "high standards". The rule is to demand "high
standards" but not bother to teach kids how to get there. As I
recall, the 4th grade 1964 book had a page asking kids to figure out
out the forumula sum/ number but only the teachers edition gave the
correct formula! In other words, if you didn't figure it out, the
textbook didn't tell you the answer.
Did you have a chance to discuss this with the teacher, and ask where
she got the idea that these skills are taught in 2nd grade math? Of
course considering my 1st grade teacher expected my kid to read
"estivation" and write sentences his first week homework, I shouldn't
be surprised.
Mean = Average = sum / number
2nd Exemplars
4th MCAS assesment
4th Vermont (begin simple concepts)
4th Addison Wesley 1991
--------- assessment appropriate -------
5th Addison Wesley 1964
5th Holt 1970
5-8 Vermont - appropriately use
From: Nanny714@aol.com
Date sent: Tue, 17 Aug 1999 14:45:35 EDT
Subject: [education-consumers] Math Problem Solving/Exemplars
To: "ClearingHouse"
Send reply to: Nanny714@aol.com
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Here is one of the "so-called" problem solving tasks that was
"developed" in my daughter's first grade classroom. Keep in mind that
the majority of the children had not mastered addition and subtration
and received little, if any direct instruction. Our K-6 school
(Marion Cross School, Norwich, VT) has been extremely progressive for
years, heralds constructivism, frowns on direct instruction, promotes
political ideology, and doesn't use textbooks. This one put a
hundred bucks in the teachers pocket and is now sold by Exemplars and
is linked by Exemplars to some of your state standards. (I have
others which are children were exposed to which the teacher was not
so fortunate in earning money from). I have posted the teacher
comments exactly as they appeared relative to why individual students
were scored in a particular group. Question: How do we "solve" the
"problem" of our children being subjected to such claptrap?? Your
comments will be MOST welcome and appreciated! :
Grade K-2
Task
Average Number of letters in a group of five names.
As a group, discuss strategies to figure out how to find the average number
of letters in a group of five names.
Remember that there will be five groups in the end because you started with
five names. Some of you will have even groups of five and others will have
uneven groups and will have to decide how to solve the problem.
Context
We have been working on averages in several areas including graphing averages
in a science unit on bubbles. We have discussed and practiced averages of
lower numbers and small groups of objects. this gave my students the
opportunity to try averages in a different context.
What this task accomplishes
This problem gives the students the opportunity to apply their knowledge of
making even groups out of objects they can see and move around. For kids with
uneven groups, they have a chance to physically divide whole numbers and
fractions.
What the student will do
Several kids who have had experience with averages will have suggestions like
using a list of the names and/or cutting out the letters. Provide them with
both so they have options to use their own strategies.
* Make 5 even groups with cut out letters and split the leftover letters up
into equal pieces for each of the five groups (halves, quarters, fifths,
tenths) and call the average number rough but close.
* Make 5 even groups and split the leftover letters into the same number as
the groups so that each group gets a different size piece of the letter and
call all pieces one.
* Make 5 even groups of letters (if they have a number divisble by 5).
* Make six or seven groups so that the numbers divide up evenly.
* Make 5 groups of 3 or 4 and then split the last letters and put them in
groups even though splitting up wasn't necessary.
* Cross off letters on a list and rewrite them in blank spaces until the
groups are even.
* Cross off letters on a list and rewrite some in other spaces.
Concepts to be assessed and skills to be developed
Division
Grouping
Counting
Problem Solving
Reasoning
Fractions
Organization/Planning
Rubrics and Benchmarks
Novice
(left 22 letters unused)
Student: "I put the y next to the h from Brittany, the d from Clifford next
to the y. I put the t from Catherine next to the d, r came from Brittany. I
think the a came from Brittany
(The student didn't check to make sure she had used all the letters with this
list. Instead she used this list. First, she put some letters on and folded
about five of them when she moved them to blamk spaces. When I asked her what
the reason was, she said "to move them to another group.")
Teacher: "Are all your groups the same size?"
Student: "I guess"
Teacher: "What is the average number of letters in a group?"
Student: "8"
This student had limited awareness of the problem. His/her organization of
the problem was random and weak. S/he applied inappropriate concepts and
inappropriate procedures therefore no solution was reached. S/he put a few
cut letters on the list (her name and a few random letters), folded five
others and placed them in blank spaces and left about 22 letters off the
list. This student didn't have a clear or connected explanation of his/her
solution. S/he was unable to explain his/her strategy.
Apprentice
This student made 5 even groups of five. Then he tried seven groups but had
an extra. He put them back into 5 groups. He placed 3 letters in groups.
He divided the other half and a half in quarters. He placed 5 halves and 2
quarters in each group and called the average 7 because there was seven
pieces in each group.
This student was able to use the correct mathematical procedure to solve this
problem but couldn't carry out the procedure to find a correct solution.
His/her strategy was partially useful. S/he lined up the different size
pieces vertically in seven columns and came to the conclusion that the
average was seven. This student has a basic understanding of the problem but
a lack of understanding of fractions and was unable to get a solution.
Practitioner
The student was frustrated when he reached 5 goups of four with 3 left over.
He cut two in half and placed 4 halves in four groups and 1 whole in the 5th
group. I asked him if the groups were even. He said they weren't and tried 6
groups of four but was short one to be even. (He forgot that 5 groups was
what he needed to keep). He went back to his first try and decided to go with
it. Some groups had 4 1/2 and one had five. He said
4 3/4 was the average and wrote 4 1/3 (as 4 3/4).
This student has a broad understaning of the problem. S/he was able to come
close to the correct solution using fractions. S/he had three extra pieces to
put in five groups. S/he placed a whole in a group and split the other two in
half and placed them in the remaining groups. S/he labeled each group with
either 4 and 1/2 or 5 and called the average 4 and 3/4 written as 4 and 1/3.
S/he has a solution that reflects effective mathematical reasoning because
s/he came up with a roughly correct average.
Expert
The student made 5 even groups of four whole pieces. She asked if she could
cut the pieces. With the ok she cut the three into halves and put 5 of them
in the 5 groups. She told me she had an average of 4 1/2 with a little extra
piece that fit into 5 groups. She cut it into 1/10 slivers and asked how much
it was. I told her to. She wrote her average as 4 and a half and one tenth.
This student has a deep understanding of the problem. S/he divided up the six
cut halves into the five groups and was able to express the amount in each
group using correct terminology. S/he referred to the leftover half as "a
little extra in each group." S/he proceded to cut the last half into five
equal pieces. S/he concluded that the average was 4 and 1/2 and 1/10 which is
true. S/he applied his/her understanding of fractions and the procedures
necessary to find a correct solution. His/her explanation was clear. S/he
generalized from previous mathematics experience.
Author
CINDY PIERCE graduated from the Upper Valley Teacher Training Program, an
alternative certification program that matches teaching interns with strong,
innovative teachers. She is in her third year of teaching
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