How Investigations supports
learning in higher level math
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Skip to discussion of mathematics beyond
Algebra I
Algebra I
Most
algebra teachers will vouch for the fact that many, many students struggle
through Algebra I (and as a result, later courses) because they do not have the
conceptual background to make sense out of the symbolic notation that dominates
a traditional Algebra I textbook.
When students, even very high achieving students, who are used to
memorizing and using certain procedures to solve problems find that those
procedures do not always apply in higher mathematics (or at least not in
exactly the same way to which they are accustomed), they often bottom out (so
to speak) because they have little or no experience reasoning through problems
for which there is no set procedure that will always work. Factoring polynomials is a skill that
is an excellent example of this issue.
Factoring can be extremely frustrating for typically high-achieving
students who are used to memorizing one way of doing a problem because there is
no one way to factor that will always work for every problem. Students must be able to try different
strategies, think about the problem in different ways, and be flexible in their
thinking. Factoring requires
mathematical intuition that is not encouraged in traditional programs.
Investigations supports success in a
traditional or non-traditional Algebra I curriculum because students gain the conceptual
foundation for understanding why the symbols that they are using make sense
(and why what they do with those symbols makes sense). Variables represent numbers, and when
students are used to thinking about numbers in a variety of ways, they are much
more able to understand how variables can be manipulated in a variety of
ways. The following are some
examples of this:
There
are 12 legs in the Smith house.
There are people and cats in the house. How many people and cats could there be?
This
task has multiple solutions. In
Algebra I, if we were to set up a table, equation, and graph of these
solutions, the equation would be 2x
+ 4y = 12 (x = number of people, y = number of cats), which would form a
line on the graph. If we were to
specify a number of creatures in the house, say 5, we could set up a system of
equations and solve it graphically or algebraically to determine the
answer. That is:
2x
+ 4y = 12
x
+ y = 5
Solving
for x and y, we would find that x = 4 and y = 1, meaning that there are 4
people in the house and 1 cat. On
the graph, these two lines would cross at the point (4, 1). Most students who work with systems of
equations in a traditional setting do not understand the connections between
the tables, graphs, and equations, and they have even more trouble connecting
the variables to any real-life context.
Coming from this program, students will have a much better foundation
for understanding the concepts represented by the symbols and thus be able to
make sense out of the system of equations.
A
cookie jar in the teacherıs lunchroom has 42 cookies in it at the start of the
day. By the end of the day, 17
cookies are left. How many were
eaten?
Students
will solve this problem in a variety of ways when given the opportunity. Generally the most common solution for
students who have not learned the American standard regrouping algorithm is to
take 10 away from 42, then take another 7 away. Some will take 5, 5, 5, and then 2. Some will take 20 and then add 3 back
on. Some will even take 10 from 40
(30), 7 from 2 (-5), and then combine the answers (30-5 = 25). Of course, there are other methods
students might use as well.
Encouraging students to decompose and recompose 42 and 17 in a variety
of ways develops number sense, which many Algebra I teachers will note that
their students from traditional programs do not have. Moreover, this decomposing and recomposing of number helps
students to understand similar processes used with symbols because they have
come to understand that the symbols represent quantities. For instance:
42
- 17 = 42 - (10 + 7) = 42 - 10 - 7 = 25
Algebra
I: a - (b+c) = a - b – c
This
is an example of the distributive property.
(40
+ 2) - (10 + 7) = (40 - 10) + (2 - 7) = 30 + (-5) = 25
Algebra
I: (3x+7) - (2x+8) = (3x-2x) +
(7-8) = x-1
This
is an example of combining like terms using the distributive, commutative, and
associative properties.
Why do students who only know the traditional American regrouping
algorithm often struggle with these procedures?
Simply put, students struggle because the traditional American
regrouping algorithm generally does not work with problems such as those from
Algebra I listed above. (Another way to say this: it is a nice
trick that happens to work for numbers in the same base system, in this case,
base 10). For instance, the second
problem above [(3x+7) – (2x+8)] is not stated in a way that looks exactly
the same as a multi-digit subtraction problem, so students may be stuck even at
the initial step. Even if a
student tried to write it vertically with the 2x+8 underneath the 3x+7:
3x + 7
-
2x + 8
and
tried to cross out the 7 and the 3x to regroup, what would he/she regroup? The 3x can stand for any number! But students who understand the meaning
behind these symbols will understand that the 3x and 2x can be combined because
they are multiples of the same value (x), and the 8 can be subtracted from the
7 to get -1.
Students must be able to make sense out of the symbols, and how
they can be combined and separated, in order to succeed in higher math.
In
two different ways, determine how much money is needed to take 27 students on a
field trip if the field trip costs $12 per person.
Again,
given the chance, students will use a variety of strategies to solve this
problem. Some will multiply 27 x
10 and then add 27 x 2. Some will
find 30 x 12 and then take away 3 x 12.
Some will use 10 x 12, 10 x 12 again, and then 7 x 12 and add the
answers. Some might find 27 x 3
and then multiply by 4. Others
might know that 25 x 12 is 300 because they can think about this in the context
of quarters and dollars, and then they might add 2 x 12. What is
noteworthy about all of these strategies is that they are all efficient,
accurate, and relatively easy to use mentally. Facility with mental computation is one of the goals of this
program.
Why will this flexibility be more helpful in Algebra I than knowing
only the standard American algorithm?
Once
again, knowing only the standard American algorithm
for multiplying does not help students to gain the flexibility that is
essential for success in Algebra I and higher mathematics. Consider this problem expressed more
symbolically:
27x
(10 + 2) = (27 x 10) + (27 x 2)
Algebra
I: a(b + c) = ab + ac
This
is an application of the distributive property.
(30
- 3) x 12 = (30 x 12) - (3 x 12)
Algebra
I: (a - b)c = ac – bc
This
is another application of the distributive property.
27
x 12 = 33 x (1.5)(23) = (1.5)(3x2)3
Algebra
I: a3 x (1.5)b3
= (1.5)(ab)3
This
is an application of the commutative property and the property stating that the
product of numbers each raised to the same power is equal to the same power of
the product of these numbers.
Moreover, trying to use the American traditional algorithm to
multiply two polynomials will be futile (try to do this problem with the American
standard multiplication procedure!):
(4x
+ 2) multiplied by (y – 7)
HOWEVER,
there is a definite relationship between the traditional American algorithm and
the so-called FOIL procedure that most adults learned in Algebra I, though many
are not aware of it at all! Try
the problem 27 x 12 with FOIL (first, outer, inner, last): (20 + 7)(10 + 2) = 200 + 40 + 70 + 14 =
324. Typically, we would write 27
with 12 below it:
27
12
and
then multiply 7 by 2 to get 14.
What if, instead of regrouping, we multiplied the 2 in 12 by 20 (not ³2²) in 27 to
40. Then, go to the 1 in the 12
(representing 10). Multiply this
by the 7 in 27 to get 70. Finally,
multiply the tens values, 20 and 10, to get 200. Notice that we get the same four products that we did above
with FOIL! The troublesome fact,
though, is that most adults never have had the opportunity to make this connection,
so it is no wonder that most struggle with the idea that (4x + 2)(y – 7)
= 4xy – 28x + 2y – 14.
Symbols and procedures are meaningless without the ability to use
them flexibly and with conceptual understanding.
Since
there is especially a concern for fifth grade parents about whether Investigations will adequately prepare
students for algebra and for the new achievement tests, we have put together an
extensive packet comparing tasks from the previous textbook, Investigations, the sample fifth grade
achievement test, and Algebra I. This packet is available here. (PDF format)
For a free download of Adobe Acrobat Reader, click
here.
In
all of higher mathematics, students need conceptual knowledge, the ability to
work with unfamiliar problems in a variety of forms (graphs, tables, equations,
words, symbols, and other representations), and the ability to see
relationships among ideas. None of
these elements are present in any substantial form in a traditional program,
but they are very much present in Investigations. Similar to the example of factoring
cited at the top of this page, finding the integral is a skill required in a
great many problems in calculus, and once again, there is no one way to do so
that will work in every problem.
In fact, there are many ways to do so that will only work in specific
problems, and often it is not clear at the start of a problem which one will
work. Students must be able to
approach a problem in different ways in order to have success, and many
students who have always done well in math do not have success in higher math
because they are so unaccustomed to having to think in several ways about a
problem.
A
group of about 20 mathematicians from all around the United States gathered in
July 2005 to discuss how to make college mathematics more accessible and
meaningful to their students. They
generated the following list of ideas they would like their students and the
general public to know about mathematics:
1.
It
was developed to make sense of the world around us.
2.
It
is abstraction.
3.
It
is dynamic - not a closed body of knowledge. Old ideas are modified; new ideas come to light.
4.
It
is a human invention (teachers and students are human).
5.
Good
mathematics is math that creates new problems. Math feeds on itself (the idea of inquiry).
6.
An
author recently stated that 40,000 new theorems were proved last year –
this is probably an underestimation – it is probably more like 200,000.
7.
Math
is not just arithmetic.
8.
Mathematics
is not the responsibility of the super-intelligent - it can (and is) done by
anyone as long as he/she has the right tools.
9.
Math
is not a look-up table.
10.Mathematics is good for (and about) solving
problems.
11.
Math
is interconnected and a study of relationships.
Then,
as a group, they condensed these ideas into three key ideas:
1.
Mathematics
is an understanding of proof and how we know what is true. (using definitions, proof, conjecture)
2.
Math
is a human endeavor; it takes thought.
3.
Math
is about problem solving, discerning patterns, and connecting ideas.
The
philosophy of Investigations and its design (based on years of research on how
students learn mathematics) support all of the above ideas. This program will prepare students for
the challenges of higher mathematics better than any traditional program would
prepare them. Stow-Munroe Falls
City Schools intends to give our students the best learning opportunities
possible, and this program will allow us to do so.
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