Tool choice matters! If a carpenter needs to pound a nail into a wall, h/she might use many tools to do so – the handle of a screwdriver, the blunt end of a pair of pliers, or the heel of a shoe. However, there’s no doubt that using a hammer provides the most effective and accurate results.
The same might be said about the math manipulatives students use to help them solve problems. Students may have access to any number of math tools to represent their thinking about a problem, but some tools are better suited than others for helping students arrive at an accurate solution in an effective manner. For example, when solving a multi-digit division problem, a student may use counters such as counting chips or square tiles. However, a base-ten manipulative that embodies the “nesting” principle to demonstrate powers of ten will provide structure for efficiently arriving at accurate results.
So here’s the question: which tools are most useful for helping students efficiently arriving at accurate results? And which tools have the greatest utility in helping students connect math ideas within and across grade levels? Of course, this depends upon the concepts being taught. Considering the following four ideas will help you determine which manipulatives provide the biggest bang for the “buck” when your buck is time, effort, and money.
Tool choice matters! If a carpenter needs to pound a nail into a wall, s/he might use many tools to do so – the handle of a screwdriver, the blunt end of a pair of pliers, or the heel of a shoe. However, there’s no doubt that using a hammer provides the most effective and accurate results.
1. Connecting Math Ideas within a Grade Level
Given the math concepts you teach in your classroom throughout the year, does the manipulative you use have utility across a variety of math ideas? For example, in a K-5 classroom, base-ten concepts and fluency are major concepts for students to master as they build toward mathematics proficiency. When students understand the connections between concepts such as place value, operations, estimation, money-counting, whole numbers, decimal fractions, and the like, they are more likely to see mathematics as a system rather than a set of isolated skills. If 3rd-grade students, for example, use a single representation to demonstrate the connections among place value, addition, subtraction, multiplication, and estimation, they gain the advantage of building understanding that is interconnected rather than isolated. A manipulative such as KP Ten-Frame Tiles (www.kpmathematics.com) lends itself perfectly to these base-ten ideas because it provides the structure of the ten-frame while embodying the nesting features of the place value system. Another powerful manipulative, one appropriate for non base-ten concepts, are pattern blocks, which can be used to connect many geometric principles, measurement ideas, and fraction concepts/operations within a grade level.
2. Connecting Math Ideas Across Grade Levels
Given the math concepts taught across grade levels, does the manipulative have capacity to “grow” across concepts? For example, KP Ten-Frame Tiles provide opportunities for pK and K students to work on simple concepts such as counting, subitizing, and beginning arithmetic. As students move up into grades 1-2, they use the same manipulative to represent place value ideas, multi-digit addition and subtraction, and problem-solving situations. In grades 3-4, students use KP Tiles to represent multiplication, multi-digit division, and rounding, and to recognize their profound connections to place value. Then, in grades 4-6, students use this same model to represent decimal-fraction concepts and operations. By using the same manipulative, students connect their learning not only from concept-to-concept, but also from one grade to the next.
3. Understanding Mathematical Structures over Time
Does the manipulative facilitate understanding of deep mathematical ideas such as the application of place value, properties of operations, and relationships among operations? Not only should students see the connections among concepts, they should also begin to see the deeper mathematical structures that create such connections. For example, place value is a connecting structure among many concepts including counting, whole numbers, number names and symbols, decimal fractions, comparing and ordering, addition, subtraction, multiplication, division, rounding, money counting, and written algorithms. Good manipulatives help students identify and rely on these structures for building ideas and explaining their thinking.
Good manipulatives help students identify and rely on these structures for building ideas and explaining their thinking.
4. Equipping Students for Problem Solving
Does the manipulative provide opportunity for students to make their reasoning visible in problem-solving contexts as well as in mathematical problems (e.g., “naked numbers”)? The best manipulatives allow students to make their thinking visible at all levels of rigor: conceptual understanding, procedural fluency, and application. Students should have access to a variety of tools (both manipulatives and paper-pencil strategies), know which tool to choose to help them solve the problem efficiently and accurately, and then use the tool appropriately and strategically. In essence, students must have opportunities to learn both how to use and how to choose the tools that best help them demonstrate their thinking.
With so many mathematical tools to choose from, your daily challenge is to select those tools that are most useful in building students’ mathematical understanding and proficiency within grade levels and from one grade to the next. Using the questions above to guide your selection will put you on the road toward equipping students with the best tools possible for developing the thinking and reasoning they need to succeed in school — as well as in life.
