No Math is an Island

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No Math is an Island
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No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
wholeIf I can build a bridge of connections for
my studentsBy using meaningful explorations to
give students opportunities to actively build
new knowledge from experiences and prior
knowledge (NCTM Principles and Standards
2000)Providing opportunities that develop
mathematical reasoning for concepts such as
ratios. Ratios will lead to rates. Rates will
lead to proportions. Proportions will lead to
rate of change leading directly to slope. This
perspective of slope will lead students to think
of a functions rate of change in concrete
settings in ways that are consistent with ideas
of rate of change over some interval To know
Calculus helps me to see what bodies of middle
school mathematical knowledge will lead to
Calculus. In knowing this I can choose
activities and implement instruction that gives
students a deeper understanding of these middle
school concepts. Thus providing them a bridge of
connectionThese questions will guide the
activities I choose Will the activity provide a
mental image? "Use of mental imagery is a
characteristic of a meaningful mathematical
activity. (Wheatley 1997)Will it involve at
least three of standards of NCTM? As we explored
multiplication through base 10 blocks it hit me
when Dr Pratt stated, So although my focus for
the day may be multiplication, I will need to
find a way to incorporate as many content area as
possible to accomplish rich and deeper
understanding (Pratt 2008),Will it provide
opportunities for modeling, logical analysis,
making inferences, optimization and abstraction?
Activities that provide these opportunities
allow for me to develop questions to encourage
mathematical thinking.Will it interest and
connect to the students I teach? When I can tap
into the students world they become interested
and vested in their own learning.Darlenes
comment if we as educators impart clear
understandings to the many concepts we teach in
math it will be much easier for the students to
understand future concepts, especially calculus
and analytical geometry. made it all clear that
the bridge must be constructed through rich
explorations should become part of mathematical
instruction opposed to the quick method, quick
answer, one algorithm boring repetitive math (
Jardine 2006) that some students have come to
hate.An excerpt in the Calculus wiki solidified
my thoughts. What you should concentrate on
depends on why you're taking the course. If
you're going to be a physicist, for example, you
would actually _use_ calculus on a day-to-day
basis, in which case it's worth actually
memorizing various formulas for derivatives and
integrals on a long-term basis. If you're just
supposed to get an 'appreciation' for calculus,
then you should make sure that you understand all
the definitions, and that you can set up
integrals   Why do I, as a middle grades
mathematics teacher, need to know about calculus
and analytic geometry? It really depends on the
goals I have for my students. I want to provide
my students with meaningful mathematical
explorations that will help construct the bridge
to higher mathematic courses.
5
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
wholeIf I can build a bridge of connections for
my studentsBy using meaningful explorations to
give students opportunities to actively build
new knowledge from experiences and prior
knowledge (NCTM Principles and Standards
2000)Providing opportunities that develop
mathematical reasoning for concepts such as
ratios. Ratios will lead to rates. Rates will
lead to proportions. Proportions will lead to
rate of change leading directly to slope. This
perspective of slope will lead students to think
of a functions rate of change in concrete
settings in ways that are consistent with ideas
of rate of change over some interval To know
Calculus helps me to see what bodies of middle
school mathematical knowledge will lead to
Calculus. In knowing this I can choose
activities and implement instruction that gives
students a deeper understanding of these middle
school concepts. Thus providing them a bridge of
connectionThese questions will guide the
activities I choose Will the activity provide a
mental image? "Use of mental imagery is a
characteristic of a meaningful mathematical
activity. (Wheatley 1997)Will it involve at
least three of standards of NCTM? As we explored
multiplication through base 10 blocks it hit me
when Dr Pratt stated, So although my focus for
the day may be multiplication, I will need to
find a way to incorporate as many content area as
possible to accomplish rich and deeper
understanding (Pratt 2008),Will it provide
opportunities for modeling, logical analysis,
making inferences, optimization and abstraction?
Activities that provide these opportunities
allow for me to develop questions to encourage
mathematical thinking.Will it interest and
connect to the students I teach? When I can tap
into the students world they become interested
and vested in their own learning.Darlenes
comment if we as educators impart clear
understandings to the many concepts we teach in
math it will be much easier for the students to
understand future concepts, especially calculus
and analytical geometry. made it all clear that
the bridge must be constructed through rich
explorations should become part of mathematical
instruction opposed to the quick method, quick
answer, one algorithm boring repetitive math (
Jardine 2006) that some students have come to
hate.An excerpt in the Calculus wiki solidified
my thoughts. What you should concentrate on
depends on why you're taking the course. If
you're going to be a physicist, for example, you
would actually _use_ calculus on a day-to-day
basis, in which case it's worth actually
memorizing various formulas for derivatives and
integrals on a long-term basis. If you're just
supposed to get an 'appreciation' for calculus,
then you should make sure that you understand all
the definitions, and that you can set up
integrals   Why do I, as a middle grades
mathematics teacher, need to know about calculus
and analytic geometry? It really depends on the
goals I have for my students. I want to provide
my students with meaningful mathematical
explorations that will help construct the bridge
to higher mathematic courses.
6
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
wholeIf I can build a bridge of connections for
my studentsBy using meaningful explorations to
give students opportunities to actively build
new knowledge from experiences and prior
knowledge (NCTM Principles and Standards
2000)Providing opportunities that develop
mathematical reasoning for concepts such as
ratios. Ratios will lead to rates. Rates will
lead to proportions. Proportions will lead to
rate of change leading directly to slope. This
perspective of slope will lead students to think
of a functions rate of change in concrete
settings in ways that are consistent with ideas
of rate of change over some interval To know
Calculus helps me to see what bodies of middle
school mathematical knowledge will lead to
Calculus. In knowing this I can choose
activities and implement instruction that gives
students a deeper understanding of these middle
school concepts. Thus providing them a bridge of
connectionThese questions will guide the
activities I choose Will the activity provide a
mental image? "Use of mental imagery is a
characteristic of a meaningful mathematical
activity. (Wheatley 1997)Will it involve at
least three of standards of NCTM? As we explored
multiplication through base 10 blocks it hit me
when Dr Pratt stated, So although my focus for
the day may be multiplication, I will need to
find a way to incorporate as many content area as
possible to accomplish rich and deeper
understanding (Pratt 2008),Will it provide
opportunities for modeling, logical analysis,
making inferences, optimization and abstraction?
Activities that provide these opportunities
allow for me to develop questions to encourage
mathematical thinking.Will it interest and
connect to the students I teach? When I can tap
into the students world they become interested
and vested in their own learning.Darlenes
comment if we as educators impart clear
understandings to the many concepts we teach in
math it will be much easier for the students to
understand future concepts, especially calculus
and analytical geometry. made it all clear that
the bridge must be constructed through rich
explorations should become part of mathematical
instruction opposed to the quick method, quick
answer, one algorithm boring repetitive math (
Jardine 2006) that some students have come to
hate.An excerpt in the Calculus wiki solidified
my thoughts. What you should concentrate on
depends on why you're taking the course. If
you're going to be a physicist, for example, you
would actually _use_ calculus on a day-to-day
basis, in which case it's worth actually
memorizing various formulas for derivatives and
integrals on a long-term basis. If you're just
supposed to get an 'appreciation' for calculus,
then you should make sure that you understand all
the definitions, and that you can set up
integrals   Why do I, as a middle grades
mathematics teacher, need to know about calculus
and analytic geometry? It really depends on the
goals I have for my students. I want to provide
my students with meaningful mathematical
explorations that will help construct the bridge
to higher mathematic courses.
7
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
wholeIf I can build a bridge of connections for
my studentsBy using meaningful explorations to
give students opportunities to actively build
new knowledge from experiences and prior
knowledge (NCTM Principles and Standards
2000)Providing opportunities that develop
mathematical reasoning for concepts such as
ratios. Ratios will lead to rates. Rates will
lead to proportions. Proportions will lead to
rate of change leading directly to slope. This
perspective of slope will lead students to think
of a functions rate of change in concrete
settings in ways that are consistent with ideas
of rate of change over some interval To know
Calculus helps me to see what bodies of middle
school mathematical knowledge will lead to
Calculus. In knowing this I can choose
activities and implement instruction that gives
students a deeper understanding of these middle
school concepts. Thus providing them a bridge of
connectionThese questions will guide the
activities I choose Will the activity provide a
mental image? "Use of mental imagery is a
characteristic of a meaningful mathematical
activity. (Wheatley 1997)Will it involve at
least three of standards of NCTM? As we explored
multiplication through base 10 blocks it hit me
when Dr Pratt stated, So although my focus for
the day may be multiplication, I will need to
find a way to incorporate as many content area as
possible to accomplish rich and deeper
understanding (Pratt 2008),Will it provide
opportunities for modeling, logical analysis,
making inferences, optimization and abstraction?
Activities that provide these opportunities
allow for me to develop questions to encourage
mathematical thinking.Will it interest and
connect to the students I teach? When I can tap
into the students world they become interested
and vested in their own learning.Darlenes
comment if we as educators impart clear
understandings to the many concepts we teach in
math it will be much easier for the students to
understand future concepts, especially calculus
and analytical geometry. made it all clear that
the bridge must be constructed through rich
explorations should become part of mathematical
instruction opposed to the quick method, quick
answer, one algorithm boring repetitive math (
Jardine 2006) that some students have come to
hate.An excerpt in the Calculus wiki solidified
my thoughts. What you should concentrate on
depends on why you're taking the course. If
you're going to be a physicist, for example, you
would actually _use_ calculus on a day-to-day
basis, in which case it's worth actually
memorizing various formulas for derivatives and
integrals on a long-term basis. If you're just
supposed to get an 'appreciation' for calculus,
then you should make sure that you understand all
the definitions, and that you can set up
integrals   Why do I, as a middle grades
mathematics teacher, need to know about calculus
and analytic geometry? It really depends on the
goals I have for my students. I want to provide
my students with meaningful mathematical
explorations that will help construct the bridge
to higher mathematic courses
8

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994)To know Calculus helps me to see what
  bodies of middle school mathematical knowledge
  will lead to Calculus. In knowing this I can
  choose activities and implement instruction that
  gives students a deeper understanding of these
  middle school concepts. Thus providing them a
  bridge of connectionThese questions will guide
  the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning.Darlenes
  comment if we as educators impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry. made it all clear that
  the bridge must be constructed through rich
  explorations should become part of mathematical
  instruction opposed to the quick method, quick
  answer, one algorithm boring repetitive math (
  Jardine 2006) that some students have come to
  hate.An excerpt in the Calculus wiki solidified
  my thoughts. What you should concentrate on
  depends on why you're taking the course. If
  you're going to be a physicist, for example, you
  would actually _use_ calculus on a day-to-day
  basis, in which case it's worth actually
  memorizing various formulas for derivatives and
  integrals on a long-term basis. If you're just
  supposed to get an 'appreciation' for calculus,
  then you should make sure that you understand all
  the definitions, and that you can set up
  integrals   Why do I, as a middle grades
  mathematics teacher, need to know about calculus
  and analytic geometry? It really depends on the
  goals I have for my students. I want to provide
  my students with meaningful mathematical
  explorations that will help construct the bridge
  to higher mathematic courses

9

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 )Knowing Calculus helps me to see what
  bodies of middle school math will lead students
  to a better understanding of Calculus. In
  knowing this I can choose activities and
  implement instruction that gives students a
  deeper understanding of these middle school
  concepts. Thus providing students a bridge of
  connection.These questions will guide the
  activities I choose Will the activity provide a
  mental image? Use of mental imagery is a
  characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning.Darlenes
  comment if we as educators impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry. made it all clear that
  the bridge must be constructed through rich
  explorations should become part of mathematical
  instruction opposed to the quick method, quick
  answer, one algorithm boring repetitive math (
  Jardine 2006) that some students have come to
  hate.An excerpt in the Calculus wiki solidified
  my thoughts. What you should concentrate on
  depends on why you're taking the course. If
  you're going to be a physicist, for example, you
  would actually _use_ calculus on a day-to-day
  basis, in which case it's worth actually
  memorizing various formulas for derivatives and
  integrals on a long-term basis. If you're just
  supposed to get an 'appreciation' for calculus,
  then you should make sure that you understand all
  the definitions, and that you can set up
  integrals   Why do I, as a middle grades
  mathematics teacher, need to know about calculus
  and analytic geometry? It really depends on the
  goals I have for my students. I want to provide
  my students with meaningful mathematical
  explorations that will help construct the bridge
  to higher mathematic courses

10

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 )Knowing Calculus helps me to see what
  bodies of middle school math will lead students
  to a better understanding of Calculus. In
  knowing this I can choose activities and
  implement instruction that gives students a
  deeper understanding of these middle school
  concepts. Thus providing students a bridge of
  connection.These questions will guide the
  activities I choose. Will the activity provide a
  mental image? Use of mental imagery is a
  characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning.Darlenes
  comment if we as educators impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry. made it all clear that
  the bridge must be constructed through rich
  explorations should become part of mathematical
  instruction opposed to the quick method, quick
  answer, one algorithm boring repetitive math (
  Jardine 2006) that some students have come to
  hate.An excerpt in the Calculus wiki solidified
  my thoughts. What you should concentrate on
  depends on why you're taking the course. If
  you're going to be a physicist, for example, you
  would actually _use_ calculus on a day-to-day
  basis, in which case it's worth actually
  memorizing various formulas for derivatives and
  integrals on a long-term basis. If you're just
  supposed to get an 'appreciation' for calculus,
  then you should make sure that you understand all
  the definitions, and that you can set up
  integrals   Why do I, as a middle grades
  mathematics teacher, need to know about calculus
  and analytic geometry? It really depends on the
  goals I have for my students. I want to provide
  my students with meaningful mathematical
  explorations that will help construct the bridge
  to higher mathematic courses

11

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and
  Standards)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 article)To know Calculus helps me to see
  what bodies of middle school mathematical
  knowledge will lead to Calculus. In knowing this
  I can choose activities and implement instruction
  that gives students a deeper understanding of
  these middle school concepts. Thus providing
  them a bridge of connectionThese questions will
  guide the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning.Darlenes
  comment if we as educators impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry. made it all clear that
  the bridge must be constructed through rich
  explorations should become part of mathematical
  instruction opposed to the quick method, quick
  answer, one algorithm boring repetitive math (
  Jardine 2006) that some students have come to
  hate.An excerpt in the Calculus wiki solidified
  my thoughts. What you should concentrate on
  depends on why you're taking the course. If
  you're going to be a physicist, for example, you
  would actually _use_ calculus on a day-to-day
  basis, in which case it's worth actually
  memorizing various formulas for derivatives and
  integrals on a long-term basis. If you're just
  supposed to get an 'appreciation' for calculus,
  then you should make sure that you understand all
  the definitions, and that you can set up
  integrals   Why do I, as a middle grades
  mathematics teacher, need to know about calculus
  and analytic geometry? It really depends on the
  goals I have for my students. I want to provide
  my students with meaningful mathematical
  explorations that will help construct the bridge
  to higher mathematic courses

12

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 article)To know Calculus helps me to see
  what bodies of middle school mathematical
  knowledge will lead to Calculus. In knowing this
  I can choose activities and implement instruction
  that gives students a deeper understanding of
  these middle school concepts. Thus providing
  them a bridge of connectionThese questions will
  guide the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks, some using
  the area model, some using arrays and yet some
  using grouping, it was summed up for me by Dr.
  Pratts statement. So although my focus for the
  day may be multiplication, I will need to find
  ways to incorporate as many content standards as
  possible to gain a more rich and deeper
  understanding for my students .Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning.Darlenes
  comment if we as educators impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry. made it all clear that
  the bridge must be constructed through rich
  explorations should become part of mathematical
  instruction opposed to the quick method, quick
  answer, one algorithm boring repetitive math (
  Jardine 2006) that some students have come to
  hate.An excerpt in the Calculus wiki solidified
  my thoughts. What you should concentrate on
  depends on why you're taking the course. If
  you're going to be a physicist, for example, you
  would actually _use_ calculus on a day-to-day
  basis, in which case it's worth actually
  memorizing various formulas for derivatives and
  integrals on a long-term basis. If you're just
  supposed to get an 'appreciation' for calculus,
  then you should make sure that you understand all
  the definitions, and that you can set up
  integrals   Why do I, as a middle grades
  mathematics teacher, need to know about calculus
  and analytic geometry? It really depends on the
  goals I have for my students. I want to provide
  my students with meaningful mathematical
  explorations that will help construct the bridge
  to higher mathematic courses

13

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 article)To know Calculus helps me to see
  what bodies of middle school mathematical
  knowledge will lead to Calculus. In knowing this
  I can choose activities and implement instruction
  that gives students a deeper understanding of
  these middle school concepts. Thus providing
  them a bridge of connectionThese questions will
  guide the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  and develop mathematical thinking.Will it
  interest and connect to the students I teach?
  When I can tap into the students world they
  become interested and vested in their own
  learning.Darlenes comment if we as educators
  impart clear understandings to the many concepts
  we teach in math it will be much easier for the
  students to understand future concepts,
  especially calculus and analytical geometry.
  made it all clear that the bridge must be
  constructed through rich explorations should
  become part of mathematical instruction opposed
  to the quick method, quick answer, one
  algorithm boring repetitive math ( Jardine 2006)
  that some students have come to hate.An excerpt
  in the Calculus wiki solidified my thoughts.
  What you should concentrate on depends on why
  you're taking the course. If you're going to be a
  physicist, for example, you would actually _use_
  calculus on a day-to-day basis, in which case
  it's worth actually memorizing various formulas
  for derivatives and integrals on a long-term
  basis. If you're just supposed to get an
  'appreciation' for calculus, then you should make
  sure that you understand all the definitions, and
  that you can set up integrals   Why do I, as a
  middle grades mathematics teacher, need to know
  about calculus and analytic geometry? It really
  depends on the goals I have for my students. I
  want to provide my students with meaningful
  mathematical explorations that will help
  construct the bridge to higher mathematic courses

14

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 article)To know Calculus helps me to see
  what bodies of middle school mathematical
  knowledge will lead to Calculus. In knowing this
  I can choose activities and implement instruction
  that gives students a deeper understanding of
  these middle school concepts. Thus providing
  them a bridge of connectionThese questions will
  guide the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning. This provides
  an experience of meaningful imagery and allows
  students to fold back (Kieren 1994) when
  needed.Darlenes comment if we as educators
  impart clear understandings to the many concepts
  we teach in math it will be much easier for the
  students to understand future concepts,
  especially calculus and analytical geometry.
  made it all clear that the bridge must be
  constructed through rich explorations should
  become part of mathematical instruction opposed
  to the quick method, quick answer, one
  algorithm boring repetitive math ( Jardine 2006)
  that some students have come to hate.An excerpt
  in the Calculus wiki solidified my thoughts.
  What you should concentrate on depends on why
  you're taking the course. If you're going to be a
  physicist, for example, you would actually _use_
  calculus on a day-to-day basis, in which case
  it's worth actually memorizing various formulas
  for derivatives and integrals on a long-term
  basis. If you're just supposed to get an
  'appreciation' for calculus, then you should make
  sure that you understand all the definitions, and
  that you can set up integrals   Why do I, as a
  middle grades mathematics teacher, need to know
  about calculus and analytic geometry? It really
  depends on the goals I have for my students. I
  want to provide my students with meaningful
  mathematical explorations that will help
  construct the bridge to higher mathematic courses

15

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 article)To know Calculus helps me to see
  what bodies of middle school mathematical
  knowledge will lead to Calculus. In knowing this
  I can choose activities and implement instruction
  that gives students a deeper understanding of
  these middle school concepts. Thus providing
  them a bridge of connectionThese questions will
  guide the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning. This provide
  an experience of meaningful imagery and allows
  students to fold back when needed.Darlenes
  comment, If we ,as educators, impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry., made it clear that the
  bridge must be constructed through rich
  explorations. These explorations should be
  implemented in my instruction opposed to the
  quick method, quick answer, one algorithm boring
  repetitive math(Jardine 2006) that some students
  have come to hate.An excerpt in the Calculus
  wiki solidified my thoughts. What you should
  concentrate on depends on why you're taking the
  course. If you're going to be a physicist, for
  example, you would actually _use_ calculus on a
  day-to-day basis, in which case it's worth
  actually memorizing various formulas for
  derivatives and integrals on a long-term basis.
  If you're just supposed to get an 'appreciation'
  for calculus, then you should make sure that you
  understand all the definitions, and that you can
  set up integrals   Why do I, as a middle grades
  mathematics teacher, need to know about calculus
  and analytic geometry? It really depends on the
  goals I have for my students. I want to provide
  my students with meaningful mathematical
  explorations that will help construct the bridge
  to higher mathematic courses

16

• No math is an island, entire of itself Middle
  school math a piece of Calculus, a part of the
  wholeIf I can build a bridge of connections for
  my studentsBy using meaningful explorations to
  give students opportunities to actively build
  new knowledge from experiences and prior
  knowledge (NCTM Principles and Standards
  2000)Providing opportunities that develop
  mathematical reasoning for concepts such as
  ratios. Ratios will lead to rates. Rates will
  lead to proportions. Proportions will lead to
  rate of change leading directly to slope. This
  perspective of slope will lead students to think
  of a functions rate of change in concrete
  settings in ways that are consistent with ideas
  of rate of change over some interval (Thompson
  1994 article)To know Calculus helps me to see
  what bodies of middle school mathematical
  knowledge will lead to Calculus. In knowing this
  I can choose activities and implement instruction
  that gives students a deeper understanding of
  these middle school concepts. Thus providing
  them a bridge of connectionThese questions will
  guide the activities I choose Will the activity
  provide a mental image? "Use of mental imagery
  is a characteristic of a meaningful mathematical
  activity. (Wheatley 1997)Will it involve at
  least three of standards of NCTM? As we explored
  multiplication through base 10 blocks it hit me
  when Dr Pratt stated, So although my focus for
  the day may be multiplication, I will need to
  find a way to incorporate as many content area as
  possible to accomplish rich and deeper
  understanding (Pratt 2008),Will it provide
  opportunities for modeling, logical analysis,
  making inferences, optimization and abstraction?
  Activities that provide these opportunities
  allow for me to develop questions to encourage
  mathematical thinking.Will it interest and
  connect to the students I teach? When I can tap
  into the students world they become interested
  and vested in their own learning. This provide
  an experience of meaningful imagery and allows
  students to fold back when needed.Darlenes
  comment, If we ,as educators, impart clear
  understandings to the many concepts we teach in
  math it will be much easier for the students to
  understand future concepts, especially calculus
  and analytical geometry., made it clear that the
  bridge must be constructed through rich
  explorations. These explorations should be
  implemented in mathematical instruction opposed
  to the quick method, quick answer, one
  algorithm boring repetitive math(Jardine 2006)
  that some students have come to hate.An excerpt
  in the Calculus wiki solidified my thoughts.
  What you should concentrate on depends on why
  you're taking the course. If you're going to be a
  physicist, for example, you would actually _use_
  calculus on a day-to-day basis, in which case
  it's worth actually memorizing various formulas
  for derivatives and integrals on a long-term
  basis. If you're just supposed to get an
  'appreciation' for calculus, then you should make
  sure that you understand all the definitions, and
  that you can set up integrals   Why do I, as a
  middle grades mathematics teacher, need to know
  about calculus and analytic geometry? It really
  depends on the goals I have for my students. I
  want to provide my students with meaningful
  mathematical explorations that will help
  construct the bridge to higher mathematic
  courses.

17
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
whole I will build a bridge of connections for
my students using meaningful opportunities to
develop mathematical reasoning.
Middle school math
18
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
whole I will build a bridge of connections for
my students using meaningful opportunities to
develop mathematical reasoning.
Calculus and Analytical Geometry
Rich explorations
Middle school math
19
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
whole I will build a bridge of connections for
my students using meaningful opportunities to
develop mathematical reasoning.
Calculus and Analytical Geometry
Middle school math
20
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
whole I will build a bridge of connections for
my students using meaningful opportunities to
develop mathematical reasoning.
Calculus and Analytical Geometry
Meaningful activities
Incorporating more than one standard in activity
Middle school math
21
No math is an island, entire of itself Middle
school math a piece of Calculus, a part of the
whole I will build a bridge of connections for
my students using meaningful opportunities to
develop mathematical reasoning.
Develop questions that encourage mathematical
reasoning
Middle school math
Calculus and Analytical Geometry
22
No Math is an Island based on No Man is an Island
by John Donne Music by Groove Masters