Van De Walle, Ch. 3 on Reflective Thinking

I am using a process called "writing to learn" which is the way Dr. Van De Walle intended Elementary School Mathematics to be read.  Since each question asked by Van De Walle for this chapter has  been so thought provoking, it hasn't made sense for me to simply move on from chapter 3 without a fuller exploration of the essential questions.  After three days, I'm ready to elaborate on the six ways in this chapter Dr. Van De Walle suggested "reflective thought" could be promoted.


Here is how Dr. Van De Walle defines reflective thought:

Active thinking about or mentally working on an idea can be called reflective thought.  (Van De Walle, 29)

Here are the six ways suggested to promote reflective thought: (Van De Walle, 30)

1. Create a problem-solving environment.
2. Use models:  manipulatives, drawings, calculators.
3. Encourage interaction and discussion.
4. Use cooperative learning groups.
5. Require self-validation of responses.
6. Listen actively.

Create a problem-solving environment

Creating a problem-solving environment is the first thing Dr. Van De Walle suggests that is needed to promote reflective thought.  Dr. Van De Walle is not proposing that by chaining students to their desks and forcing them to "solve" lots of problems they will become better mathematicians.  In fact, fewer problems may be better.

In The Culture of Education, Jerome Bruner writes,

One of the great triumphs of learning (and of teaching) is to get things organized in your head in a way that permits you to know more than you “ought” to.  And this takes reflection, brooding about what it is that you know.  The enemy of reflection is the breakneck pace – the thousand pictures.  (Bruner, 129)

The key differences Van De Walle has with traditional math instruction in the United States involves the relationship of the learner to new information, the relationship of the teacher to the learner, plus a focus on developing relational understandings versus instrumental understandings.  Here, we avoid problems associated with teaching knowledge in a way that is "a mile long and an inch deep," or teaching math procedures in an unconnected way.  How would Dr. Van De Walle or Bruner evaluate the pacing guide / data-driven classroom?  

The 5 other ways of promoting reflective thought are components of a problem-solving environment.  My question is where in pacing guides is there time for instructing math the way Dr. Van De Walle intended?

Each of the 5 other ways of promoting reflective thought are components of a problem-solving environment.

Use Models:  manipulatives, drawings, calculators

In Jean-Jacques Rousseau’s Emile, Rousseau writes:

In any study whatsoever, unless one has the ideas of the things represented, the representative signs are nothing.  However, one always limits the child to these signs without ever being able to make him understand any of the things which they represent.  (Rousseau, Book II, p. 109)

Models are math tools which represent math concepts physically; they enable students to internally grasp relationships embodied in the tool. (Van De Walle, 30) A flat, for example, can represent one hundred or one, depending on the concept being modeled. 

Here are a few exemplars Dr. Van De Walle uses to demonstrate how models can be used to help a learner construct relational knowledge in his or her own mind. (Van De Walle, 31)

Model	Relationship
Counters	The concept of six can be demonstrated with counters and number mats: it relates to counting words one, two, three, etc.; also, it relates to the idea of one more than five.
Base-10 blocks, sticks and bundles	The concept of hundred relates to the relationship of the flat to the unit, not to the flat in isolation.  Sticks and bundles can also show place-value relationships.
Rods	The concept of length can be shown as a comparison between two objects.  Rods can be used to measure the spaces within objects.
Spinners	The concept of probability can be shown as a relationship of the spinner to comparative outcomes
Dot grid paper	The concept rectangle can be illustrated with dot paper to show it as a relationship between length and space.
Calculator	Concepts of pattern can be shown using calculators:  we can test relationships between starting and jumping numbers using repetitive entries.
Number line and arrows	The concept of integers can be shown on a number line with arrows to direction and length.

Learners need opportunities to handle these tools and “change, move, count, compare, draw, measure” with them to build mental schemas. (Van De Walle, 30)  Thus, Dr. Van De Walle suggests they be used in three parts of a lesson: (Van De Walle, 32)

1.   To introduce unit concepts.

2.   To help children connect concepts to symbols as they record their observations and defend their answers.

3.   To assess understanding – if math tools were used in instruction, student should be able to use them when demonstrating understanding.

Encourage Interaction and Discussion

When a child explains an idea, several things happen.  First, a child organizes her thoughts, thinking about what she knows.  Next, she considers how she will explain her understanding to others:  Will she draw a picture or some other model?  What words will she use? What does her listener understand?  What questions remain to be answered?  In writing, people are forced to clarify their thinking, which is why “writing to learn” is so important.   (Van De Walle, 30) Likewise, student talk is an essential part of a problem-solving environment, because it enables students to build connections.  A classroom management plan that enables these kinds of discussions to regularly take place  is critical.

Cooperative Learning Groups

Dr. Van De Walle suggests that students learn best in groups of 3 or 4 because it maximizes the level of interaction and discussion that can occur in a classroom.  Similarly, my experience with guided reading was that the program worked best when the class was organized in groups of 4.  Dividing the class into groups enabled me to differentiate instruction, and provided regular scheduled opportunities for small group instructional opportunities.  When I taught a 4^th grade class in Prince William County, my Literacy Coach and I used Guided Reading Levels as a starting place, but groupings were flexible.

Math students need similar kinds of opportunities for structured small group learning.  Within a Guided math format, students could rotate through different “centers”, discussing the nature of math problems, using math tools to help develop understandings, and explain their answers while I listen in and facilitate discoveries one group at a time.  Another specialist or Instructional Assistant could set up shop in a different part of the room.  For this to happen, routines would need to be established over the first 6 weeks.

Require Self-validation of Responses

A good explanation requires evidence.  A requirement that students use representations and logical arguments to defend their answers in class should help eliminate bad habits such as guesswork and mechanical answer finding.  The reading strategy QAR (question answer relationships) provides a framework for defending an argument.  Is the answer right there in the text (a simple addition problem)?  Do I need to think and search for clues (find key ideas in a story problem, collect data, and draw a picture)?  Is it a relationship that I just know in my head (already mastered math facts)?  Is it a relationship that involves the author and me (do I need to think about what the author does and does not tell me and need to make inferences)?  Self-validation is a skill that can be taught and reinforced in small group instruction.

Listen Actively

In a problem-solving environment with lots of reflective thinking going on, the relationship of students to their learning and to their teacher must necessarily change.

When children respond to questions or make an observation in class, an interested but very nonevaluative response is a way to ask for an elaboration.  “Tell me more about that, Karen,” or “I see.  Why do you think that?”  Even a simple “Um-Hmm, followed by silence is very effective, permitting the child and others to continue their thinking.  (Van De Walle, 31)

Van De Walle, Ch. 3 The importance of understanding

Reflection:

Why does it matter whether or not someone understands mathematics on a conceptual or relational level (lots of linkages) versus a procedural or instrumental level (an emphasis on rote memorization and drill)?

The broader a person’s mental schema, the greater the number of neural connections, the quicker the processing speed.   This is fact!  Neuroscientists have observed that neurons that fire together wire together. (The Brain Which Changes Itself, Norman Doidge. MD, pp. 63-70) 

Too often, teachers do not systematically link procedural knowledge to underlying concepts.  When the brain processes new information in isolation, new information doesn’t readily transfer to long-term memory.

Neurons that fire apart wire apart – or neurons out of synch fail to link.  (Doidge, p.64)

Dr. Van De Walle’s analysis is consistent with my own personal observations of “slow learners”.  For the field of education, the ramifications of Dr. Van De Walle’s argument and what modern neuroscience is telling us are mind-blowing:

A negative effect of the behaviorist influence on education was the fragmenting of mathematics into seemingly endless lists of isolated skills, concepts, rules, and symbols.  Each has to be mastered before moving on.  The lists grow so large that teachers and students become overwhelmed.  When items are learned rationally, they become part of a larger web of information.  Frequently, the network is so well constructed that whole chunks of information are stored and retrieved as single entities rather than isolated bits.  (Van De Walle, 27)

Recently, I spent a day as an Instructional Assistant Substitute teacher, and worked with a 3^rd grade math student with a learning disability.  She had learned procedures for multiplying 3-digit numbers by a 1-digit number and  used a multiplication table for math facts she didn’t know to solve problems.  When given story problems, however, she frequently added the factors instead of multiplying them, she didn’t consider whether her result should be a big number or a small number, and lacked strategies for visualizing the problem.  In one problem, I modeled a problem as a repeated addition problem.  Rather than multiplying the digits, four 4’s for example, she counted on her fingers – a highly inefficient strategy – which led to errors in how she applied the borrow-and-trade procedure.  I modeled another problem involving how far a plane traveling 367 miles per hour would fly in four hours by drawing a diagram, which made it evident that we had another repeated addition problem.  The little girl was capable of using strategies for understanding, but it seemed to me that she may not have been exposed to them.  By the 4^th problem, she bypassed the repeated addition step on her own and applied the multiplication procedure appropriately.

In education, perhaps more-so than any other field, people are demanding results.  What can be more important than the future of our country?  Dr. Van De Walle cites research that seems to indicate that despite a back-to-basics, renewed emphasis on traditional computation skills since the 1970’s, student problem solving skills and concept knowledge are not significantly improving.  (Van De Walle, 27)  Despite huge educational budgets, and an ever-increasing emphasis on “data-driven” instruction, despite general acknowledgement that differentiation is an educational right, huge existential questions remain unresolved.

What are our mathematics assessments truly measuring?  Is our educational pacing developmentally appropriate to individual learners?  Is our mathematics instruction preparing learners for the thinking requirements of the 21^st century? How badly does a bias towards "answer finding" skew our educational data?

For years, we were told the stock market could only go up.  Then, we discovered that our gains were built upon a phony edifice.  An entire banking system came to rely on derivatives from an inflated real estate market, where properties were "flipped several times over".  Then our banking system collapsed, only to be rescued by a multi-billion dollar bailout.  Education has come to rely on thinking from the results-driven business world, and I wonder whether the problems identified by Dr. Van De Walle have gotten worse because of a misplaced emphasis on this thinking, not better.

Van De Walle, Ch. 3 What it means to understand mathematics

Notes and Reflections on Elementary School Mathematics by John A. Van De Walle

Ch. 3:  On the nature of mathematical knowledge and what it means to understand mathematics

Knowledge consists of internal or mental representations of ideas that our mind has constructed (Van De Walle, 21)

Piaget’s three types of knowledge (Van De Walle, pp. 21-22)

1.      physical knowledge

2.      logico-mathematical knowledge

3.      conventional / social knowledge

Physical:  derived from tangible / solid real-world objects

Logico-mathematical:  derived by association (relationships & connections)

Conventional:

           arbitrarily agreed upon by society

           names of things

           meanings we attach to symbols

Conceptual and procedural knowledge of mathematics

Conceptual:

Conceptual knowledge consists of relationships constructed internally and connected to already existing ideas.  (Van De Walle, 22)

A children’s understanding of base-10 blocks provides an example we can use to elaborate critical distinctions between 3 kinds of knowledge (physical, logico-mathematical, and conventional)

           Physical:  base-10 blocks provide a model

           Logical-Mathematical:  the relationships between units, rods, flats, and cubes (ten “ones” equals one “ten”, ten “tens” equals one “hundred”, same as, more than, less than, etc.)

           Conventional:  we name a unit “ones”; a rod “tens”; and a flat “hundreds”; we can just as easily name a flat “ones”, a rod “tenths”, and a unit “hundredths”

Ideas such as seven rectangle, ones/tens/hundreds (as in place value), sum, product, equivalent, ratio, and negative are all examples of relationships. (Van De Walle, 22)

Rectangles help us see relationships of parts to wholes (fractions):  the objects we see (rectangles) do not physically change; we construct logical relationships such as halves and fourths in our minds – it’s our understanding that changes.

We can show things or objects to children where the concept consists of relationships within or among the objects, but we can only be sure that the children are seeing objects.  They must create the relationships.  It is critical that we get children to be active mentally, to reflect on the ideas we present.  That is the only way that the mind can construct a relationship.  A passive learner will only see the object, not the relationships. (Van De Walle, 23)

Procedural: task-oriented knowledge which may or may not connected to conceptual knowledge (Van De Walle, 23)

A struggling learner might identify symbols (conventional knowledge) and might manipulate algorithms (procedural knowledge) without a deep enough understanding of how ideas (conceptual knowledge) are connected to what the child already knows.

To the extend that procedural knowledge is intimately connected with conceptual knowledge, procedures and symbolism become powerful tools in the construction of new knowledge. (Van De Walle, 23)

Reflection:

A reliance on skewed testing data can lead educators down the primrose path.  Using unconnected conventional and procedural knowledge, children can pass tests and progress without developing essential understandings.  Children are routinely misled to believe that procedural knowledge is good enough.  Unless struggling learners are immersed in an environment where they are allowed sufficient time to develop understanding, in my experience, they tend to develop “swiss cheese” understanding which leads to later confusion.

There doesn’t seem to be the same level of agreement about the kinds of assessments needed to guide math instruction as exists in the reading discipline where we have DRA levels, Guided Reading levels, Accelerated Reading levels, etc.   Unlike the way assessments are routinely matched to individual learners along the literacy continuum, which occurs in guided reading, math assessments are often one-size-fits all, with too much emphasis on getting answers right.

Understanding mathematics

Understanding (long-term memory):  mathematical concepts or procedures must be connected to or integrated with prior knowledge (schema).

Different levels of understanding:  based on the strength of the connections and how concepts and procedures are integrated with what we already know.

Networks of synaptic connections can be observed, using fMRI brain scans and other methods, as the brain literally changes shape.

Relational vs. instrumental understanding (a continuum) (Van De Walle, 24)

           Relational:  concepts and procedures are networked

           Instrumental:  procedural and conceptual knowledge are not integrated (learned through rote memorization and drill)

Conceptual Knowledge:  involves a large network of relationships and connections; procedures, semantic features, vocabulary, ideas, etc., are all connected

Procedural Knowledge: too often devolves into rote memorization, rules-without understanding, and often leads to frustration when not connected with concepts – pacing guide pressure tends to drive  educators away from the kind of explorations where teachers allow students  time to develop understanding.

Too many children are using procedures with fractions without an understanding of the concepts behind them. (Van De Walle, 25)

Reflection:

When I was being mentored by a Title I Math specialist in Fairfax County, another 3^rd grade teacher and I followed a unit plan based on “Speaking Fractioneze, ” an article by Rachel McAnallen, or Ms. Math (Wonderful Ideas, Volume XIV, Number 1, Sept./Oct. 2002)  We progressed from the set model of fractions to the area model, then deftly connected fractions to decimals and decimals to Fractioneze with base-10 blocks.

We started by explaining the rules of Fractioneze and handed the children piles of junk (spiders, rings, cars, paperclips, etc.) We used Ms. Math’s invented language to develop the concepts of wholes, parts of a whole, the meaning of the numerator and the denominator.  We progressed to other physical models and other kinds of connections.

The rules of Fractioneze are simple:  in Fractioneze, a set of parts is always described in relationship to a whole; also, we always describe how much of the whole is not part of the set.  For example, we might say 4 out of 6 are paperclips.  We must also say, 2 out of 6 are not paperclips.  We might say 2 out of 6 are triangles, and 4 out of 6 are not triangles.

Everyday, we used the overhead with various related shapes.  We started with a yellow hexagon and stated, “Today, we’re going to call the hexagon one; if the hexagon is one, how would we describe the red trapezoid?  What about the green triangle?  What about the blue rhombus?” We noticed that two red trapezoids made one whole.  In describing the red trapezoid, it followed that denominator was two because it took two parts to make a whole.  We also noticed that six green triangles made up a whole, so it followed that the denominator was six.  We noticed that three blue rhombi made a whole, so it followed that denominator was 3.

Later in the unit, we would ask, “Who is one today?  If the red trapezoid is one today, how much is the green triangle today?”  Ultimately, we connected the base-10 flat to one.  It followed naturally that the denominator for a rod would be 10 because ten rods make a whole; it also followed that the denominator for a unit was a hundred because a hundred units made a whole.  My 3^rd graders made the fraction decimal connection seamlessly.

In my 3^rd grade class, rapid advancement followed readiness.  For our culminating activity, we made construction paper pizzas with different toppings and discussed them in Fractioneze.  Then, we had a pizza party!

Few children are taught fractions in a way that develops understanding.  The 4^th grade Math Investigation fraction unit that I used in Prince William County built-in time for students to develop conceptual knowledge, but my team skipped key lessons to keep pace.  Sadly, we missed opportunities to build the kinds of connections that led to the rapid advancement I witnessed with a 3^rd grade class with a similar demographic profile.

Van De Walle, Ch. 2, What it means to know and do math

Notes and Reflections on Elementary School Mathematics by John A. Van De Walle

Chapter 2:  Knowing and doing math

Reflection: contrast the new and old paradigm:

Dr. Van De Walle describes Math as a science of pattern and order (a reliance on reason, relevant tasks, students defend their reasoning)

•          “The world is full of order and pattern” (Van De Walle, 7) – all of the great ancient cultures discovered that we are immersed in a world of pattern and order.

•          My favorite example of mathematics revealed as a science of pattern and order comes from the movie, Donald Duck in Mathmagic Land.  I remember how I felt about math when I first saw it in elementary school in the 1970’s.  Students continue to ooh and ah at patterns revealed in the movie whenever I show it.

Mathematics discovers this order and uses it in a multitude of fascinating ways, improving our lives and expanding our knowledge.  School must help children with this process of discovery.  (Van De Walle, 8)

•          Consider the story of Friedrich Gauss, who was tasked by his teacher to compute the sum of a sequence of numbers.  After solving the problem in mere seconds, Gauss was accused of cheating rather than complimented for his brilliance (http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss).

•          Math Investigations explores the idea of volume in a developmentally appropriate way; unfortunately, it doesn’t match the pacing guide.

The traditional view of mathematics as a collection of rules and procedures (a reliance on authority, students are taught arbitrary rules, and students learn to rely on memorization)

•          No school wants to be held before the public as a failing school; students who can’t pass standardized tests reflect poorly on the teacher and the school; thus, lesson plans and testing data are scrutinized to verify that students are keeping pace.

•          Computers compute better than people do; by 2030, according to renowned futurist, Ray Kurzweill (http://www.kurzweilai.net/index.html?flash=1), $1,000 of computing equipment will have the same computing power as the human brain – old skills are indeed becoming obsolete.

•          America was built on a “frontier” mentality which valued thrift, hard work, sobriety, and a can-do spirit.  America was known for its can-do practicality.  Where is that same frontier mentality today?  Due to pacing guide pressure, when jobs are on the line, educators flee activities that involve deep level processing for the kind of one-size-fits-all procedure/rule driven instruction opposed by Van De Walle.

•          Do we limit opportunities for higher level thinking to the gifted?  Or do we allow children of poverty the same opportunities to discover order and patterns in mathematics?

Reflection:  becoming mathematical problem solvers is, in my opinion, the most important of the five Curriculum Standards goals for students highlighted in this chapter

1.       learn to value mathematics

2.      become confident in their ability to do mathematics

3.      become mathematical problem solvers

4.      learn to communicate mathematics

5.      learn to reason mathematically

•          An environment conducive to problem solving involves worthwhile tasks, builds confidence, requires students to elaborate, and builds reasoning skills – thus, goal # 3 incorporates the four other goals.

•          In this chapter, Dr. Van De Walle provides 5 examples of worthwhile tasks, and provides a play-by-play analysis of how these tasks might play out in a classroom environment.

•          Activities such as Start and Jump, Combining Tiles, In-between numbers, Finding Areas, and One equation (Van De Walle, 11-17) encourage students to notice patterns, record observations, and reflect upon what happened when they varied their approaches – student processing time is built in to the process, which should facilitate the development of long-term memory.

•          Dr. Van De Walle contrasts true problem-solving with mechanical answer finding; the difference is in the level of thinking being tasked; problems without a single clear answer involve analysis and reasoning; students become active learners instead of passive recipients of base-line computational knowledge

•          Are we creating a generation of struggling learners who are being advanced through too quickly and in a manner that doesn’t allow enough time for core understandings to develop in their long-term memories?

Reflection:  overcoming a reliance on the “math god” – what is the single most important thing that can be done to prevent this concept from developing?

•          Students are taught to be passive learners of mathematics from a young age; rules and procedures can seem arbitrary, with answers, rules, and procedures passed down to students from intelligent authority figures

So where do these rules come from?  Is there a “math god,” someone who has all the rules and figures the out and to whom the teacher is somehow connected?  In such an environment, what else could children come to believe? (Van De Walle, 9)

•          Teaching for understanding, on the contrary, empowers students to discover order and pattern for themselves

•          When students are habitually expected to elaborate their thinking as problem solvers, only then can learned helplessness be overcome

Reflection:  Why does Lauren Resnick think mathematics might be characterized as an ill-structured discipline rather than a well-structured one? (Van De Walle, 17)

•          Most teachers are control freaks by nature; this element of Math Investigations especially infuriates many teachers and parents of struggling learners – “but how will little Johnny pass the test?”

•          A lack of structure causes frustration, which shuts down learning, especially for a generation of students who have grown accustomed to having and endless succession of rules and procedures handed to them

By fifth or sixth grade there are many children who simply refuse to attempt a problem that has not been first explained:  “You haven’t shown us how to do these.”  This is a natural consequence of the bits-and-pieces, rules-without-reasons approach to mathematics.  Children come to accept that every problem must have  a method or solution already determined, that there is only one way to solve any problem, and that there is no expectation that they could solve a problem unless someone gave them a solution method ahead of time.  Mathematics viewed this way is certainly not a science of patterns and order.  In fact, it is not mathematics at all.  (Van De Walle, 9)

•          The sentiment expressed by Resnick is unfortunately worded and unnecessarily inflammatory, in my opinion.  I agree that a reliance on mechanical answer finding is precisely what’s wrong with mathematics instruction, but that is not the same as saying there is too much structure structure.  Learning through explorations of open-ended questions, probably requires more up-front structure that traditional methods.  A well-designed guided math component, analogous to what Jan Richardson has done for guided reading, would enable the teacher to assume the strategic question-asking, discourse promoting, evaluative role Van De Walle envisioned.

John A. Van De Walle, Elementary School Mathematics

Yesterday, I used some of the ideas of Jan Richardson, someone who has mastered guided reading, to introduce the idea of developing a guided math program.  I posted a reasonably detailed first six week unit plan for installing critical routines and habits necessary to run a successful guided reading program.  Fountas and Pinnell also had a 6 week plan, which I tried to follow as a 3rd grade teacher after being handed their very thick book.  As much as I admire the Fountas and Pinnell plan, I found it unworkable because I was following it without understanding key elements elaborated upon by Jan Richardson.

Today, as promised, I begin dissecting and reflecting upon the ideas of John A. Van De Walle  You'll be able to follow along as I read and develop understanding.  In his preface, Dr. Van De Walle requested:

Consider reading this text not just with a highlighter or a pencil to take notes, but with some simple materials on hand - counters, grid papers, a calculator, blocks, and so on....Reflecting on how children learn from activities is the best way to grow as a teacher.

Dr. Van De Walle challenged the traditional ways of teaching mathematics, and wrote this as a revolutionary mastermind who had achieved many of his goals in changing the way mathematics is taught in the US (Van De Walle, 1).  There is an ongoing backlash against Math Investigations in Prince William County, where I taught last year.

Dr. Van De Walle describes mathematics as a science of patterns and order, as opposed to a process of mechanical answer finding.


Dr. Van De Walle challenges educational practices where students are taught to passively defer to the math god, whose arbitrary rules must be blindly followed (Van De Walle, 8 and 9).  Teaching children to discover patterns; represent and defend their thinking by drawing pictures, using counters, base-10 blocks; and communicate mathematical ideas with their peers,  is designed to shift the balance of power away from authority figures.  By empowering students with reason, students learn to think for themselves.

What kind of educator would question the value of encouraging higher level thinking for all students?  Why would educators and parents whose students have the most to gain, in my opinion, be leading the backlash?

The dividing line between revolution and counter-revolution seems to be the issue of learner readiness, the amount of time it takes learners to explore ideas and develop understanding and number fluency, pacing guide pressure, and highly ingrained patterns of learned helplessness.  The question remains, what are the best methods for preparing students to pass their SOL tests and for schools to meet AYP (Adequate Yearly Progress) benchmarks?  Do we want to teach children to pass tests, or do we want to prepare them for a lifetime of learning?  Do we want to keep feeding children information, or do we want to teach them how to fish for information?  Like most things, time and money are at the bottom of the discussion.

I'm inclined to believe that until someone like a Jan Richardson develops a full-featured "automobile repair guide" for teaching mathematics, complete with a troubleshooting section, and developmentally appropriate assessments, Math Investigations will remain under investigation.  To set up the learning environment envisioned by Dr. Van De Walle, a first 6 weeks plan to establish the routines and expectations of guided math must be incorporating into pacing guides.  These, I believe, are areas insufficiently elaborated upon in Math Investigations.

Below, I've shared my notes from the first chapter and preface, along with my reflections woven in.

Notes and Reflections on Elementary School Mathematics by

John A. Van De Walle

Preface:

•          Chapters 1-5:  Foundation (key ideas)

o       Ch 1:  NCTM  Standards / change in way mathematics are being taught (why / where)

o       Ch 2:  What it means to know and do mathematics (developmental perspective)

o       Ch 3:  Teaching developmentally

o       Ch 4:  Helping children become problem-solvers

o       Ch 5:  Assessment

•          Chapters 6-20:  Activities, Learning, and Children

•          Chapters 21-23:

o       Ch 21:  How to incorporate technology

o       Ch 22:  Lesson-planning, use of HW and text book

o       Ch. 23:  Differentiation

Chapter 1:  Key ideas

What images and emotions do I personally associate with the idea of teaching mathematics?

•          Agony and despair

o       Frustration:  children who have serious gaps in their understanding, i.e., don’t see patterns and relationships, don’t know simple facts, lack essential strategies, and can’t keep pace

o       Learned helplessness:  children who have been taught from a young age to follow procedures but lack solving skills and give up quickly when faced with uncertainty (learned helplessness)

o       Adult Anger: last year’s countywide curriculum sequence disaster in Prince William County with Investigations and the backlash

o       Shame:  my personal experience as a second grader with borrow and trade – I couldn’t get it without one-one-one instruction

•          Excitement (the thrill of victory)

o       Gestalt:  My 3^rd grade class seamlessly discovered relationships between multiplication, division, and fractions – it was beautiful the way the understanding came together; in the hall, a teacher from Annandale HS noticed my class doing related multiplication and division problems while we waited to have our class picture done.

o       Discipline: I had a 4^th grade student who didn’t get it initially, but stuck fought through his lack of understanding and ended up at the top of the class

o       Creativity:  In my 4^th grade class last year, a few discovered powers of 10 and were basically doing scientific notation during the Investigations multiplication unit)

What should it look like? (personal reflections)

•          Understanding is developmental (critical factors / short cuts are problematic)

o       Students must be able to concentrate, remain on task, and work cooperatively (students must be engaged in their own learning)

o       Assessments should guide instruction

o       Readiness comes before rapid advancement; students who require remediation must be quickly identified and given extra support / time

§         Must recognize patterns (visual / shape, auditory / rhythm, kinesthetic / timing, same / different)

§         Must recognize sequence on number line / hundreds chart (left /right, up/down, before/after, odd/even, greater/less-than, forwards/backwards, near/far)

§         Must recognize connections between hands-on observations and simple symbolic representations (=, <, >, +1)

§         Must achieve automaticity with benchmark numbers (0, evens, 3, 5, 10, 10, halves, quarters, eighths, thirds, fifths, tenths)

§         Must achieve automaticity with close-to numbers

§         Must achieve automaticity with math facts (repetition)

•          Students need to be fully engaged in a learning environment where higher level thinking is the norm rather than the exception:

o       More efficient strategies must be discovered, noticed, named, compared and practiced (grouping, number decomposition, factors, parts/wholes)

o       Less efficient strategies need to be unlearned (counting by ones).

o       Elaboration and analytical skills must be modeled and practiced

o       Students must represent problems in context (visualize, draw, explain)

o       Students must apply appropriate strategies and procedures.

o       Students must evaluate whether or not the answers make sense.

4 themes of NCTM standards (page 3)

•          Problem solving (students must develop a repertoire of strategies)

•          Communication (students must be actively engaged in discussing, writing, and visually representing mathematical ideas)

•          Reasoning (students must extend patterns, apply logical reasoning, and evaluate reasonableness of hypotheses, data, and conclusions)

•          Connections

o       Students need to discover connections within ideas

o       Symbolic representations must be clearly connected to concepts

o       Math must be connected to the real world and other content areas

5 Shifts

•          Toward classroom communities, away from individualism

•          Toward reasoning / logic / evidence, away from authority

•          Toward reasoning, away from memorization of procedures

•          Toward problem solving and reasoning, away from mechanical answer finding

•          Towards connecting mathematics to world and other disciplines, away from treating mathematics as isolated concepts and procedures

Inclusion (Van De Walle feels many have historically been excluded)

o       Minorities

o       Females

o       Struggling learners

4 categories of professional teaching standards (learning environment)

•          Providing worthwhile tasks (quality activities)

•          Encouraging student / teacher discourse

•          Enhancing learning (evidence of growth)

•          Reflective teaching and learning practices

Reflections

•          Societal factors (p. 2)

o       Technological change: shift away from paper / pencil computing in work environment

o       Modern workforce demands workers who can interpret data (graphs / charts) – applied, evaluative level thinking

•          2 most significant technological trends / factors (p. 2)

o       Calculator & computer have reduced the need for low-level pencil / paper computation skills – job obsolescence

o       Calculator & computer have created new instructional opportunities (activities for teaching number sense, estimation, relationships, visuals, audio, etc.)

•          Gist of 4 thematic standards discussion: teaching mathematics as mechanical answer finding is outdated; learners must be empowered to reason, communicate, and use math to solve relevant problems (what about readiness?)

•          Evaluation section (p. 4): assessment should reflect 4 themes; thus, it should involve rubrics, portfolios, and authentic tasks; however, last year I experienced a worst case scenario where the pacing guide, the scope and sequence of  SOL’s, the time involved in teaching Investigations, and administrative data collection requirement all seemed to be working at cross purposes

•          To support the shift in emphasis, math instruction needs to change in three ways:

o       first, just as a literacy continuum has been adopted, we educators need to adopt a continuum of mathematics learning model

o       second, the framework and nuts-and-bolts procedures for Guided Math need to be perfected, just like Jan Richardson, following the work of Fountas and Pinnell, has perfected the Guided Reading model

o       third, just as Jan Richardson has matched assessments and instructional focus to learner needs along the literacy continuum, assessments and instructional focus must be matched to student needs along the continuum of mathematics