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
• 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
23: Differentiation Ch.
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
with Investigations and the backlash Prince William County
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 3rd 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 4th 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 4th 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)
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
• 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 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
• 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