Hands-On Math, KP® Ten-Frame Tiles, Math Content
I don’t know about you…but whenever I get on an airplane, the first thing I usually do is put in my headphones to subtly let everyone around me know that I intend to keep to myself during the flight. However, on those occasions when I strike up a conversation with the person next to me, it doesn’t take long for them to find out that I work with math teachers. And then is happens…I get their math autobiography. And it’s rarely a happy story!
Far too many adults had bad experiences with school math and have carried that negative perception into adulthood. We cannot afford for this to continue!
Way too many adults in the U.S. are eager to tell their horror stories about word problems and fractions and formulas and how math just never made sense. Far too many adults had bad experiences with school math and have carried that negative perception into adulthood. We cannot afford for this to continue! We cannot afford for the next generation of students to go into adulthood without the tool set that everyone recognizes as critical to future career success. We owe it to our students to prepare them for school and for life. They don’t all have to love math – but they all deserve to be comfortable and competent with the logic and thinking skills that mathematics provides.
So, what are we going to do to break this generational cycle of fear of mathematics and the bragging rights that come with it?
For starters, all across the United States, educators are using powerful mathematics standards that aim high for student thinking and understanding. These standards, designed by the best and the brightest mathematics educators, mathematicians, and practicing teachers and leaders, reflect what we know to be in the best interest of our students.
Here’s something you need to know about these standards – they’re really nothing new! Great math teaching and learning always has always been and always will embody the content of these standards. These standards include what math educators and teachers have known for decades about what our students need to know and understand to be successful with mathematics…not just in school, but in life.
To take things further, KP Mathematics offers tools and strategies that help bring state standards to life! Using revolutionary manipulatives that focus on the basic principles and structures of mathematics, students gain conceptual understanding that leads to procedural fluency. The rigor found in the new state math standards is met using these simple tools and the strategies they encompass. Check out our previous posts to learn more about what these revolutionary tools can do for the students in your life!
Hands-On Math
In my work with teachers and their students, I embrace opportunities to develop my own understanding of the profound fundamentals of elementary mathematics. Here’s what I noticed about my practice … I caught myself teaching decimal fraction operations conceptually and then reverting to procedures-only instruction when it came to decimal operations. The questions I had to ask myself helped me identify ways to scaffold instruction of decimal procedures with strategies that make the operations easier to understand. Below you will find the questions I asked and the answers that evolved.
Why Are Decimal Operations So Difficult for Students?
Why do so many students struggle with decimal operations? Why is it that they get to middle school and continue to struggle with what we consider to be simple of arithmetic? What is it about dealing with decimal points that gets in the way of real problem solving? And what can we do about this?
To understand the conceptual underpinnings of decimals, students must connect their thinking to fractions. After all, decimals are fractions based on iteratively dividing one unit into ten parts (Lamon, 2005). Empson and Levi write that “…the development of children’s understanding of decimals simultaneously draws on their understanding of fractions and our base-ten place value system” (Empson & Levi, 2011). A study conducted with college undergraduate students revealed that even at that level, students preferred to use fraction notation rather than decimal notation in most contexts, stating that it helped them better conceive the relationships inherent in fraction understanding (DeWolf, Bassok, & Holyoak, 2014).
To understand the conceptual underpinnings of decimals, students must connect their thinking to fractions.
Because students often lack full understanding of decimals as base-ten notation for fractions, they tend to learn decimal operations in a procedures-only format. Added to the problem is the tendency for teachers’ misconceptions to parallel that of their students, often causing the teacher’s misconceptions to become a source of students’ faulty reasoning (Kastberg & Morton, 2014).
Place Value – How are decimals and fractions related?
In fourth grade, the common core standards provide guidelines that distinctly connect decimal numbers to fraction notation (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). These standards are appropriately placed in the Number and Operations – Fractions domain, further promoting the strong connection. For example, fourth graders should demonstrate decimal place value using fraction notation for the digits to the right of the decimal point:
435.62 = 400 + 30 + 5 + 6/10 + 2/100
This connection promotes understanding and coherence between base-ten numbers and fractions.
Connections to fractions provide great opportunities for fourth-graders to develop conceptual understanding of the digits to the right of a decimal point. However, the fifth- and sixth-grade standards shift decimal work back into the Numbers and Operations in Base Ten domain. In fifth grade students develop place value and operation concepts for multi-digit decimal numbers, and by the end of sixth grade, they should achieve fluency with all four operations.
“…the development of children’s understanding of decimals simultaneously draws on their understanding of fractions and our base-ten place value system” (Empson & Levi, 2011).
Although the fifth- and sixth-grade standards do not explicitly express the connection between base-ten and fraction notation, full understanding of the operations may be more fully attained if the spirit of the fourth-grade standards is carried through all operations. Let’s now turn our attention to the operations and how one might assist students in connecting the fourth-grade concepts to operational work in the next two grades.
Addition and Subtraction – Why does “lining up the decimal point” matter?
Far too often, we teach students to “line up the decimal points,” and in their compliance, they do so, albeit with little understanding. One way to help students see the need for “lining up the decimal point” is to use fraction notation to help them identify the value of each digit. After all, lining up the decimal points ensures that students add like terms (tenths to tenths, hundredths to hundredths, etc.). Using fraction notation to represent the numbers in expanded form, as indicated in the example below, may assist students in grasping this idea of adding like terms.
2.75 + 23.2
(2 + 7/10 + 5/100) + (20 + 3 + 2/10)
In this example, students can see that “the ones go with the ones and the tenths go with the tenths,” leaving stand-alone values in the tens and hundredths places. Connecting this notation back to the original expression assists students in seeing that by lining up the decimal points, the like terms are lined up and are ready to be added.
Multiplication – Why does the “digit shift” work?
Once conceptual understanding of fraction multiplication is achieved (e.g., attending to principles such as scaling and applying the identity property), students can use their understanding of fraction multiplication to scaffold their understanding of multiplying multi-digit decimals. Once again, to build on previous understandings, using fraction notation to represent the decimals can help students understand why it is that the “digit shift” works in the standard algorithm.
0.42 x 0.6
42/100 x 6/10 = 252/1000 = 0.252
As seen in the example above, multiplying the denominators, 100 x 10, results in a product that has a denominator of 1000. This provides the foundation for understanding why one must “shift the digits” three places to the right in the product of 0.42 x 0.6 – multiplying the “hidden” denominators in the decimal fractions, which are always powers of ten, results in a new denominator that is also a power of ten.
Division – Why does the “digit shift” work?
I recently encountered a problem that asked why the following process works:
2.63 ÷ 0.72 = 263 ÷ 72
The correct response was that when you multiply both the dividend and the divisor by the same amount, the multiplicative relationship between the two (and, therefore, the quotient) remains the same. Because I had spent so much time reflecting on the relationship between decimal and fraction notation in the past few months, this was a mind-blowing moment for me. Never before had I thought of using fraction notation and the identity property to help students understand the digit shift in decimal division. In my mind’s eye I was envisioning the following:
2.63 ÷ 0.72 = 2.63/0.72 x 100/100 = 263/72 = 263 ÷ 72
This example combines the fifth-grade standard of representing division as a fraction and ongoing work with equivalent fractions. By using this notation, students can use their understandings of the relationship between fractions and division and of the identity property to scaffold their understanding of multi-digit decimal division.
In Closing…
Are any of these ideas new to you? What discoveries have you made along the way that brought insight into your teaching? We’d love to hear about it – please leave your comments below.
Hands-On Math
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.
