12.1 Selecting Representations & Instructional Tools for All Students

12.1 Selecting Representations & Instructional Tools for All Students

Skill 5 of Competency 5 lists representations explicitly — verbal statements, pictures, graphs, algebraic expressions — and adds instructional tools, with the phrase "for all students." Items describe a concept and ask which representation or tool best supports it.

The five representations

Each hides what another reveals. A graph makes a trend obvious but an exact value hard to read; a table gives exact values but obscures the overall shape; a symbolic rule generalizes but says nothing about the context it came from.

[!IMPORTANT] The instructional goal is translation among representations, not merely exposure to several. A student who can produce a table, a graph, and an equation separately but cannot explain how the table's constant difference appears as the graph's slope and the equation's coefficient has three disconnected skills rather than one understanding.

Standard translation tasks: given a graph, write the equation; given a story, build a table; given an equation, describe a situation it could model. That last direction — symbolic back to verbal — is the least practiced and the most diagnostic.

Manipulatives matched to middle-grades topics

+---------------------------------------------------------------------------+
|  TOOL                     BEST FOR                                        |
+---------------------------------------------------------------------------+
|  Algebra tiles            expressions, combining like terms, distributing, |
|                           factoring trinomials, completing the square      |
|  Two-color counters       integer addition and subtraction, zero pairs     |
|  Fraction strips/circles  equivalence, comparison, addition of fractions   |
|  Base-ten blocks          place value, decimal operations, regrouping      |
|  Number line              integers, rational number order, absolute value  |
|  Geoboards                area, perimeter, coordinate figures, similarity  |
|  Nets and solids          surface area, volume, faces/edges/vertices       |
|  Pattern blocks           fractions of a whole, angle relationships, tiling|
|  Spinners, dice, cubes    experimental probability, sampling               |
+---------------------------------------------------------------------------+

Two examples of a tool doing real conceptual work:

Two-color counters and the zero pair. A red-yellow pair sums to zero, so any number of zero pairs can be added without changing a value. To compute −3 − (−5), start with three red counters, add five zero pairs so that five red counters are available to remove, remove them, and seven yellow remain: −3 − (−5) = 2... Working the model carefully: begin with 3 red (−3). To subtract −5 you must remove five red counters, but only three are present, so add two zero pairs to make five red available. Removing the five red leaves two yellow, giving +2. The model shows why subtracting a negative increases the value, which the slogan "two negatives make a positive" never does.

Algebra tiles and factoring. To factor x² + 5x + 6, arrange one x² tile, five x tiles, and six unit tiles into a rectangle. The only arrangement that works has dimensions (x + 2) by (x + 3), which are the factors. The area model of section 11.4 is the same idea, and the connection is worth making explicit to students.

[!WARNING] A tool must match the structure it models. Base-ten blocks embody grouping by tens and are excellent for place value and decimals, but they cannot represent adding 1/3 and 1/4, because thirds have no place-value representation. Fraction strips, which can be subdivided to twelfths, model that sum directly. Items offer a mismatched tool as a plausible-looking distractor.

Selecting technology

Choose technology for what it makes visible, not for novelty.

The judgment items test is whether the tool adds understanding or merely automates. Using software to compute the mean of a data set students should be able to find by hand automates; using a spreadsheet to show how the mean shifts when one value is changed to an extreme reveals the mean's sensitivity to outliers, which is a genuine insight.

Dragging a triangle in dynamic geometry software while its angle sum stays 180° is the paradigm case: no number of static hand-drawn triangles makes the invariance as clear.

"For all students"

The phrase in the skill statement points to designing instruction that works across the range of learners in a classroom.

• Multiple entry points. A task with a low floor and a high ceiling lets every student begin and lets advanced students extend.
• Multiple representations offered simultaneously, so students can enter through the one that makes sense to them.
• Language support. Explicit vocabulary instruction, word banks, and sentence frames matter for multilingual learners, because mathematics vocabulary includes everyday words with technical meanings — table, mean, product, volume, similar, rational.
• Scaffolds that preserve the goal. A partially completed table or a simpler set of numbers in the same problem structure supports access without lowering the mathematics, unlike substituting an easier topic.

The recurring correct answer in these items maintains the mathematical goal while varying the route to it.