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.

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

Jan Richardson

Dr. Jan Richardson has made accessible her nationally renowned, full-featured, nuts-and-bolts instruction manual for setting up and maintaining a world-class guided reading program.  I don't think Dr. Richardson would be at all offended by the comparison of her masterpiece to an automobile repair guide, complete with a trouble-shooting guide.  She has provided a proven toolkit of reading assessments for a full range of learners along a literacy continuum.  Moreover, Dr. Richardson has made accessible a full kit of developmentally appropriate activities, lessons, and unit planning guides.  Dr. Richardson has made guided reading doable.


Although my primary focus now is to learn all I can about Dr. John A. Van De Walle's approach to teaching mathematics developmentally, it makes sense to begin my process of assembling a Master Teacher's toolkit based on Dr. Richardson work.  My goal is to apply Dr. Richardson's approach to guided reading to the task of developing a full-featured, nuts-and-bolts guided math program.


Last year, I enjoyed the privilege of receiving Math Investigations training, which is rooted in the philosophies described by Dr. Van De Walle.  Math Investigations, which received terrible press last year in Prince William County, reminds me of where guided reading was after Fountas and Pinnell wrote a very thick book which became a literacy bible in Fairfax County.


The scope of materials provided by Math Investigations can be overwhelming, just like guided reading. When I taught 3rd grade Language Arts in my first year of teaching, I was handed Fountas and Pinnell's book on the first day of school.  Knowing what I know now, I probably should have plowed through it.  Without guided reading training, I wasn't fully prepared for the full range of literacy issues my students were facing.  I had more courage then sense - in a Title I school, guided reading training should be mandatory.


Fortunately, in my first year, I was mentored by the Title I math teacher.  Although we didn't do guided math, we did do guided discoveries.  My third grade students made remarkable progress in math in learning how addition, multiplication, division, and fractions are all related, because of the planning, the coaching, and co-teaching of my mentor, and the way a team of us planned key units.  But that kind of rapid growth was difficult for me to duplicate as a 4th grade teacher in a new school system, because I lacked a full-featured, nuts-and-bolts instructional template.


I predict that Dr. Richardson's approach to guided reading will become a national template.  A similar template is needed for guided math.


Below is a chunk of a plan for applying Dr. Richardson's guided reading program.  In later blogs, after I digest more of Dr. Van De Walle's thinking, I'll return to the task of assembling my guided reading kit, based on the work of Dr. Richardson.


Guided Reading: 1^st 6 weeks (inspired by Jan Richardson, The Next Step in Guided Reading)

Week 1:

•          Objective:  Build community & collaboration routines (small group, team building activities)

•          Supplies:  5 themed tubs (reading to self, puzzles/games/computers, content explorations, simple art projects,  reading and talking about books); non-verbal signals

•          Skills/Procedures to teach:

o        Practice rotating and working in teams – every 10 minutes on a timer / non-verbal signals

o        Practice sound, look, and feel

§         Sound (silent, whisper, inside voices):

·                     We have times when its not okay to talk

o                    when anyone is addressing the class everyone listens

o                    readers and writers are silent during independent reading/writing

o                    we raise hands and take turns during discussions

·                     we have times when talk is permitted

o                    whisper: turn and talk (partner strategies); buddy read

o                    inside voice: think-pair-share (team); group discussions (class)

·                     we use different sound levels for different activities

o                    independent reading: silent

o                    partner:  whisper

o                    group / team discussions and activities:  inside voices

§                     Look:

·                     We have times when its okay to move

o                    We expect movement in some activities

§                     we get our materials and supplies before the music stops

§                     we stay within our work bubbles

§                     we go places quickly and quietly – no wandering

o                    We have procedures for bathroom / water breaks / sharpening pencils

§                     not okay during direct instruction or independent work time

§                     use hand signal; get a friend (boy/girl); sign bathroom log; bring timer (2 minutes or less)

§                     water bottles remain on the floor, not on work surfaces

§                     sharpen pencils in AM; not during LA (class supplies)

·                     Independent/Partners/Teams/Groups have jobs to do: do not disturb!

o                    should not be interrupted and should not interrupt others

o                    should keep materials and supplies within work area bubble

o                    should clean up and transition quickly and quietly

·                     We use non-verbal signals

o                    Bathroom/water hand signals

o                    Bell / chime / music for quick, quiet, and safe transitions

o                    We use non-verbal signals as reminders to adjust

§                     Feel:

·                     We must feel good in order to do our best work (use calm, business-like tones)

·                     Every day we gather at a central place to reflect + / -

·                     Good Faith Participation is required:  if off task, student gets a check; at end of week, add checks, multiply by 5, and subtract from 100 (Richardson, 2009)

•          Procedure:

o        Daily:  Put 10 minutes on timer; observe, praise, and chart cooperative behavior

o        Week:  Vary the activities; chunk; chart the routines & expectations

•          Schedule:

T	Monday	Tuesday	Wednesday	Thursday	Friday
1	Tub 1	Tub 2	Tub3	Tub 4	Tub 5
2	Tub 2	Tub 3	Tub 4	Tub 5	Tub 1
3	Tub 3	Tub 4	Tub 5	Tub 1	Tub 2
4	Tub 4	Tub 5	Tub 1	Tub 2	Tub 3
5	Tub 5	Tub 1	Tub 2	Tub 3	Tub 4

Week 2:

•          Objective:  Introduce 1^st center; teach to one new group per day

•          Supplies: 5 themed tubs (reading to self, puzzles/games/computers, content explorations, simple art projects, reading and talking about books); non-verbal signals

•          Skills/Procedures to teach:

o        Practice rotating and working in teams – every 10 minutes on a timer / non-verbal signals

o        Center 1 Procedures:  Reading to self/Writing in Reader’s Notebook

§         Model & practice all parts

·         Book choice:

o        Just-right from book box (modeled/practiced in mini-lesson)

o        Variety of genres (modeled/practiced in mini-lesson)

·         Look, sound, and feel

o        Look:  students read independently at their desks or take AR tests

o        Look:  students write independently in their Reader’s Notebook

§         Must log title, author, date, genre (Richardson, page 24)

§         Must respond to reading – answer one response task cards in RN

o        Sound:  silent

o        Feel: (gather and reflect + / -)

§         Our books may be too difficult or too easy – we may need help with  our book choices (conference with teacher or librarian)

§         Someone or something is distracting us (conference with the teacher or a counselor)

§         Good Faith Participation procedures

§         Observe & clarify confusions

•          Schedule:

T	Monday	Tuesday	Wednesday	Thursday	Friday
1	Center 1	Tub 2	Tub3	Tub 4	Tub 5
2	Tub 2	Center 1	Tub 4	Tub 5	Tub 1
3	Tub 3	Tub 4	Center 1	Tub 1	Tub 2
4	Tub 4	Tub 5	Tub 1	Center 1	Tub 3
5	Tub 5	Tub 1	Tub 2	Tub 3	Center 1

Week 3:

•          Objective:  Introduce 2^nd center; groups continue to practice 1^stcenter; extend time to 15 minutes per center

•          Supplies: 5 themed tubs (same 5 themes); non-verbal signals

•          Skills/Procedures to teach:

o        Practice rotating and working in teams – every 10 minutes on a timer / non-verbal signals

o        Center 2 Procedures:  Partner reading/retell

§         Model & practice all parts

·         Book choice:  (help our partner)

o        Just-right (book box)

o        Variety of genres

·         Look, sound, and feel

o        Look:  knee to knee/shoulder to shoulder in work bubble

o        Look: One reads/one retells what other read; switch roles

o        Look:  5-finger retell organizer (character, setting, problem, events, ending)

o        Sound:  whisper voices; asking and answering questions

o        Feel:  best thinking expected; teamwork (gather and reflect + / -)

o        Feel:  Good Faith Participation Procedures

o        Observe & clarify confusions

•          Schedule:

T	Monday	Tuesday	Wednesday	Thursday	Friday
1	Center 1	Tub 2	Tub 3	Tub 4	Center 2
2	Center 2	Center 1	Tub 5	Tub 3	Tub 4
3	Tub 2	Center 2	Center 1	Tub 5	Tub 1
4	Tub 3	Tub 4	Center 2	Center 1	Tub 2
5	Tub 4	Tub 5	Tub 3	Center 2	Center 1

Week 4:

•          Objective:  Introduce 3^rdcenter; groups continue to practice 1^st and 2^nd centers; extend time to 20 minutes per center

•          Supplies: 5 themed tubs (same 5 themes); non-verbal signals

•          Skills/Procedures to teach:

o        Center 3 Procedures:  Word Study (see Richardson, p. 28

§         Model & practice all parts

·         Sort (category/abc)

·         Partner

·         Search (find patterns in independent reading books or around room)

·         Antonyms / Synonyms

·         Magnetic letters, white boards

·         High frequency word lists

§         Look, sound, feel

o        Sound:  whisper voices; asking and answering questions

o        Feel:  best thinking expected; teamwork (gather and reflect + / -)

o        Feel:  Good Faith Participation Procedures

•          Schedule:

T	Monday	Tuesday	Wednesday	Thursday	Friday
1	Center 1	Tub 2	Tub 3	Center 3	Center 2
2	Center 2	Center 1	Tub 5	Tub 3	Center 3
3	Center 3	Center 2	Center 1	Tub 5	Tub 1
4	Tub 3	Center 3	Center 2	Center 1	Tub 2
5	Tub 4	Tub 5	Center 3	Center 2	Center 1

Week 5:

•          Objective:  Introduce 4^th and 5^th centers; groups continue to practice 1^st, 2^nd, and 3^rd centers

•          Supplies: 1 themed tub; non-verbal signals

•          Skills/Procedures to teach:

o        Center 4 Procedures:  Writing

§         Model & practice all parts (Writing Cycle Chart)

·         Planning (4sq,  web, outline)

·         Drafting (hot pencils)

·         Revising (Revising prompts/color codes)

·         Editing (peer editing prompts/codes/checklist; word wall / high frequency words)

·         Publishing (expectations/rubrics)

§         Look, sound, feel

·         Look:  Folders/Notebooks

·         Sound:  Whisper voices

·         Feel:  best thinking expected; teamwork (gather and reflect + / -)

·         Feel:  Good Faith Participation Procedures

o        Center 5 Procedures:  Content (vocabulary & explorations in Science, Social Studies, and Math)

§         Model and practice all parts

·         Vocabulary

·         Explorations (sequence, cause/effect, compare/contrast, procedures/methods)

§         Look, sound, feel

·         Look:  INB’s,  experimentation, NF reading strategies

·         Sound:  whisper voices; asking and answering questions

·         Feel:  best thinking expected; teamwork (gather and reflect + / -)

·         Feel:  Good Faith Participation Procedures

•          Schedule:

T	Mon		Tue		Wed		Thu		Fri
	10	10	10	10	10	10	10	10	10	10
1	T1	C1	C2	C3	C4*	C1	C2	C3	C4	C5*
2	C1	C2	C3	C4*	T1	C2	C3	C4	C5*	C1
3	C2	C3	C4*	T1	C1	C3	C4	C5*	C1	C2
4	C3	C4*	T1	C1	C2	C4	C5*	C1	C2	C3
5	C4*	T1	C1	C2	C3	C5*	C1	C2	C3	C4

Week 6:

•          Objective:  Lengthen center time to 40 minutes (20 minutes per center); introduce guided reading in whole class meeting; meet with groups

•          Non-verbal signals

•          Supplies:

Tubs (1-5)

Center 1:  Read to self

Center 2:  Partner reading/retelling

Center 3:  Word Study

Center 4:  Writing

Center 5:  Content Vocabulary and Explorations

•          Skills/Procedures to teach

o        GR Procedures (no interruptions)

o        Look, sound, feel

•          Schedule:

T	Mon		Tue		Wed		Thu		Fri
	20	20	20	20	20	20	20	20	20	20
1	C5	C1	C2	C3	C4	C5	C1	C2	C3	C4
2	C1	C2	C3	C4	C5	C1	C2	C3	C4	C5
3	C2	C3	C4	C5	C1	C2	C3	C4	C5	C1
4	C3	C4	C5	C1	C2	C3	C4	C5	C1	C2
5	C4	C5	C1	C2	C3	C4	C5	C1	C2	C3
TC	GR1	GR 2	GR 3	GR 4	GR 1	GR 2	GR 3	GR 4	GR 1	GR 2