2nd Graders get Algebra 1 Problem not on college SAT test

A Math Standards Revision is Harder Than College SAT Algebra for 2nd grade

If you hope to have the math standards approved by the
legislature, you really MUST fix these errors in the standard
which you have published on the OSPI site.


Arthur Hu Feb 2, 2008


This is a review of problems with the Charles A. Dana
Center's Washington Mathematics Standards Revision. 
While in general many have pointed out the lack of rigor
and low general level of expectations, there are some
shocking inclusions of above grade level math concepts 
maquerading as kindergarten, 1st and 2nd grade level 
math. Kindergarters are essentially asked to multiply
without using multiplication, 2nd graders are asked to
do what adults would using long division to divide 39
3 ways with a remainder. 2nd graders are also asked to 
solve an unknown an unknown in two variables where one
is related to the other - solving a linear Algebra 1 
level equation. That is a higher level of skill than
required by the College Board's SAT test.

The inclusion of these expectations reveals a shocking
degree of sloppiness in insuring that grade level 
expectations are developmentally appropriate, and do
not simply overlay expectations of what adults can do
onto very young children under the title of "problem
solving". Problem solving needs to be restricted to 
appropriate, efficent mathematical tools, not brute 
force substitutes for more powerful concepts taught 
in later grades such as multiplication, division, and 
solving linear equations which are expected of students
at the end of a K12 education.

None of the problems were removed after I had sent 
an email commenting on the first draft, and also
stated in person at a PTSA meeting with Terry Bergeson
attending in Federal Way. The 2nd grade algebra problem appeared
in the 2nd revision of the standards.

Project page
http://www.utdanacenter.org/wamathrevision/

Standards documents
http://www.utdanacenter.org/wamathrevision/standards.php

Send feedback to
wastandards@austin.utexas.edu

Summary of problems

* Kindergarten multiplication 2 x 5 x 3
* 1st Grade Division 10 divided by 2
* 2nd Grade Long Division 39 divided by 3 = 12 R 2
* 2nd Grade Long Division 39 divided by 3 = 12 R 2


* Kindergarten multiplication 2 x 5 x 3

Kindergarten
5. Core Processes: Reasoning, problem solving, 
and communication

K.5.A Identify questions to be answered when
solving a problem.
K.5.B Solve problems, choosing from a variety of
problem-solving strategies such as drawing
pictures, manipulating objects, using
numbers, or acting out the situation.
K.5.C Determine whether a solution makes sense.
K.5.D Tell what the student did to solve a problem.

Example:
• Grandma went to visit her three grandchildren
and discovered that the gloves they were
each wearing had holes in every finger. She
will fix their gloves. How many glove fingers
need to be fixed?

=== Problem ===

This is a multiplication problem 2 x 5 x 3 = 30, even
if done by modeling.

Addition in kindergarten is only through modeling. 
K.2.C Model addition by joining sets of objects
with 10 or fewer total objects when joined;
model subtraction by separating a set of 10
or fewer objects.
Get 4 counting chips. Now get 3 counting chips.
How many counting chips are there altogether?

Multiplication is not introduced until grade 2
2.4.C Model, create, and describe multiplication
situations in which sets of equal size are
joined.
- This pretty much describes this alleged 
kindergarten level problem.

This is 2 years ahead of grade level.

==============================================
* 1st Grade Division 10 divided by 2


1.6.B Solve problems, choosing from a variety of
problem-solving strategies such as drawing
pictures, manipulating objects, using
numbers, or acting out the situation

Example:

There are ten feet living in my house. Who
could be living in my house?
Think about how many feet a person has.
How many feet does a cat have? How many
feet does a snail have? How about a fish or a
snake?
Come up with a variety of ways you can have
a total of ten feet living in your house. Use
pictures, words, or numbers to show your
answer.

== Problem == 

This not a problem that has one or even 
neccesarily a small number of possible solutions.
It is MORE complicated than a division problem,
since not all creatures may have the same 
number of feet.

Division is not introduced until the 3rd grade,
so this is 2 years ahead of grade level.

=================================================
* 2nd Grade Long Division 39 divided by 3 = 12 R 2

2.5.D
Solve problems, choosing from a variety of
problem-solving strategies such as drawing
pictures, manipulating objects, using
numbers, looking for a pattern, or making a
list.

Suzy, Ben, and Pedro have found 1 quarter, 1
dime, and 4 pennies under the sofa. Their
mother has lots of change in her purse, so
they could trade any of these coins for other
coins adding up to the same value. She says
they can keep the money if they can tell her
what coins they need to end up with so they
can share the money equally. How can they
do this?

== Problem ==

This is clearly a division problem, easily solved
by division. Dividing 39 by 3 requires long division
with a remainder which is not in these standards
until grade 5. 4.1.: "division algorithms, including long
division, are developed in fifth grade"


This problem is 3 years ahead of grade level.

=================================================
* 2nd Grade Algebra = X + X + 7 = 20, x = 6.5

2.2.D Solve a variety of addition and subtraction
problems and justify the solutions.

Problems should include those involving takeaway
situations, missing addends, and
comparisons.

Hazel and Kimmy each have stamp
collections. Kimmy’s collection has 7 more
stamps than Hazel’s collection. Kimmy has 20
stamps. How many stamps are in Hazel’s
collection?
A student may justify a solution orally, with
pictures, or in writing. For instance,

20 - 7 = 13

Hazel + xxxxxxx = 20
   Kimmy's 

=== Problem ===

* This problem is misstated, the solution, by 
algebra, is 6.5

x + x + 7 = 20  state problem
2x + 7 = 20     factor x
2x = 13         subtract 7 both sides
x = 6.5         divide both sides by 2

* This cannot be solved by mere addition or 
subtraction, the given example gives no hint
of the solution method, and it is incorrect.
If the difference was 6, then 7 would be correct
for X.

This is actually solving for a linear equation
of the form ax + b = c, traditionally taught in
Algebra 1 in grade 8 or grade 9. 

The college SAT does NOT require algebra 1, so
the level of this alleged 2nd grade problem is
MORE difficult than required in the SAT.

If Algebra 1 is to be taught at grade 9, then this
is 7 years ahead of grade level, and IT IS NOT
EVEN REQUIRED FOR COLLEGE ENTRANCE for most 
majors.

Washington's New 2nd Grade Algebra and Long Division standard

From Math Standards Revision Process Overview and Discussion
Wednesday, January 16, 1:30 - 3:00 p.m.
http://www.k12.wa.us/communications/k-20conferences.aspx

http://www.k12.wa.us/communications/pubdocs/MathStandardsGrade2Sample.doc
This appears to be the new 2nd grade problem solving sample.

2.2.d ProblemSolving/Reasoning/ Communication Solve a variety of addition and subtraction problems and justify the solutions.

Including take away, missing addend, and comparison.

Example of Comparison: Hazel and Kimmy each have stamp collections. Kimmy’s collection has 7 more stamps than Hazel’s collection. Altogether, they have 20 stamps. How many stamps are in Hazel’s collection

Solution: 20 - 7 kimmy = 13 hazel , with drawing.

=======================================================
This solution is not correct because 13 - 7 = 6, not 7.=======================================================

How would a grown up solve this problem? Let x = Hazel and x + 7 = Kimmy. x + x + 7 = 20. Simplify 2x + 7 = 20. Subtract 7 from both sides gives 2x = 13, divide both sides by 2 gives x = 6.5, so Hazel has 6.5, and Kimmy has seven more, or 13.5. Together they have 20 stamps. Do they have fractional ownership of a stamp, of half-stamps?

To check, 6.5 + 6.5 + 7 = 20.

None of the 2nd grade textbooks that I have (and I have several) have a solve for the unknown like this one. Their solution evidently relies on guess and check, but this is another classic example of a upper grade problem (solve an equation of the form ax + b = c) to be solved without a proper solution method.

Also note that most graduates of Core Plus probably don't know enough algebra to solve this problem either since this basic level of algebra isn't taught at high school level, and I don't think it's covered by connected mathematics either (can somebody verify that?) The college prep SAT does NOT require knowledge of algebra and does not contain problems like this! It WOULD be appropriate for the 10th grade WASL, but that's only if we assume algebra is required for all students.

They still have the 2nd grade long division problem, though it has been changed so that they have to compute the total of the change, and it still says "struggling students" (those that can't do long division?) can count objects. This entire sample must be tossed, as well as the language "struggling" 38 cents divided by 3 is solved by long division with a remainder, which isn't covered in the curriculum until grade 5. The previous version of this mentioned that the students would encounter a remainder.

Example: Suzy, Ben, and Pedro have found a quarter, a dime and 4 pennies under the sofa. Their mother says they can keep the money if they can tell her what coins they need to share it equally. How can they do this?
• It has multiple entry points: — struggling students can count out objects and move them around.— children at different levels of number facility can combine and separate the values using their skills.
• This problem is understandable to some children through using models, to others by drawing a picture or by using numbers to make a list.
• This problem addresses important grade two mathematics such as place value, grouping, sharing, and money concepts.
• Children can easily verify whether their solution works.

Come on folks, this is total garbage, I shouldn't have been the first one to spot this.

********* ARGH ************