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 3rd 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 3rd graders made the fraction decimal connection seamlessly.
In my 3rd 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 4th 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 3rd grade class with a similar demographic profile.
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