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The house that math built

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The house that math built

National Council of Teachers of Mathematics, 2008 offers some thoughts to ponder when creating a classroom (e.g. house) for mathematical instruction and learning. NCTM principles support building a “culture of equity” in the classroom, where everyone is welcome. All means all. Let’s think about how mathematics instruction can be designed so that everyone can reach his highest potential. Consider this analogy; building a classroom with strong mathematics instruction with opportunities for learning is like building a house that is fully accessible with a strong foundation; a frame; an accessible entry point; and levels. How might we build a “house” for mathematics that allows for everyone to “enter the door” and “climb the stairs” to the second, third, fourth floor and beyond?
Before construction begins
What will the blueprint look like? What is needed for good mathematics instruction? What are the most essential pieces to building a house that everyone can enter? One approach might be to consider the “math house” and to take the mathematics lesson and “script it.” Think about and identify the:
  • steps of the lesson
  • big math concepts
  • math vocabulary
  • responses expected by the students
  • individual accommodations
  • individual modifications
  • assessment options and revisions
Building a foundation
In the same way we know that a house cannot be built on a foundation of sticks, abstract concepts and relationships of mathematics cannot be learned without children having direct and concrete interactions with mathematical ideas (Burns, 2007). We know that children who learn mathematics prior to kindergarten are greatly influenced by how they learn math throughout their education (UDSOE, 2008).  In Virginia, the foundations of mathematics encompass number and number sense, computation, measurement, geometry, data collection and statistics, and patterns and relationships. If young children do not have meaningful opportunities to explore these concepts, there will be holes or cracks in their foundation, which will impact them as they attempt to “climb the stairs” to a higher level of mathematics.
Setting the frame
The curriculum provides the “framework” for the math. Like a house, the curriculum provides the guidelines and boundaries, as well as the path to follow. In the Standards of Learning of Virginia, mathematics is divided into 5 strands, from grades k- 12, each extending from the foundation up into the “attic” of higher level math skills. As instruction occurs, teachers may discover that students are at different levels within the framework and realize the need to differentiate lessons so that supports can be built and construction (e.g. instruction) can continue.
Let me in!
Houses have doors and access to the doors vary within a neighborhood. Some houses have steps on the front, which may prevent some people who have physical challenges from entering. Modifications are made so that there is a ramp to the door, the house might be built on a level surface, the doors are wider and the handles may be changed. In mathematics instruction, some students cannot perform on the same grade level as their peers, even with accommodations, but they can learn the same concepts at a more basic level. Students can be in the house (e.g. classroom) with their peers, perhaps on a different level of learning, but included and engaged in all of the activities.
Spiral staircase
A common instructional term used to differentiate is spiraling. In a house we use a spiral staircase to get from one level of the house to another. Well in the mathematics instruction, spiraling is used to review previous skills learned and connect them with the current procedures and concepts. Don’t spiral if you can accommodate! Identify an alternative method of learning the skill. Stay on that level of the house! However if your students need more practice or review, use the stairs, but be sure to come back up!
The attic
Underneath a solid roof is the attic where all kinds of interesting items are stored: toys, decorations, old clothes and jewelry. In the same way the attic can be used to discover old memories, it can also be used in math to explore how facts and concepts have been learned and how to apply them to new and different solutions. It’s like redesigning an old dress, put on a new necklace add a sash and pumps and you have another great outfit. That’s the beauty of mathematics, it builds on itself overtime and you as you acquire new skills (necklace, sash, pumps) you have a more challenging, sophisticated way of solving mathematics.
Let’s decorate
A scaffold, like you might use for painting high ceilings in your family room, might be used on a temporary basis. Once your finished painting you take it down. In the same way, scaffolds are used in instruction on an “as needed” basis. When introducing fractions, students might use fraction strips to help make the concept more concrete, but once the students understand that a whole is made up of a certain number of parts, the strips are no longer needed. Design your instruction to have the scaffolds available to assist with making the abstract concepts concrete.
How about some wallpaper? Think about the math vocabulary that will be addressed during the lesson. As you are designing and scripting the lesson, write down the words that are used during instruction. Do the students know what they mean? Which ones are new and which ones are for review? Wallpaper is difficult to remove, as is ignoring the math vocabulary that is essential for understanding mathematical concepts. Often students with intellectual disabilities are challenged in this area, because they may not have the access to these words in their memory and experience nor within their communication system. Take this into account when decorating your house and explore what words are needed to express mathematical concepts and how all students might have access to them.
Game room
In a house, the typical game room might include a ping-pong table, video game system, board games, and a stereo. It is a place designed for individuals to relax, socialize and discover different ways to play games and discuss solutions to problems. For mathematics instruction a game room might be used to provide different ways to explore mathematical concepts with manipulatives, technology, and problem solving in small groups. The game room provides opportunities for effective and meaningful instruction, so that students move beyond memorizing facts and formulas to applying and reapplying mathematical relationships within their own minds (Burns, 2007).
Is the power on?
Are all of the wires connected properly? If I flip the switch will the lights come on? Have the students grasped the concepts of the lesson? How am I going to assess their knowledge? These are questions that you may ask as you are moving through the lesson. You might not want to wait until the entire house is built. If you put the drywall up, it will be difficult to find where the faulty wire is in the electrical system. Plan for periodic assessment and redesign as needed. With good blueprints, major renovations in instruction will not be necessary.
Who can help?

Well that’s it! Building a house takes time, but the end result is worth it! Ask yourself, “Is my mathematics instruction really designed for everyone or do I need to renovate?” Students with intellectual disabilities often are provided a limited scope of mathematics. Teachers are often encouraged to think about what functional skills the student will need to be successful after graduation (Browder, 2006). For many students, simply because of their “label” access to higher level math content is limited (NCTM, 2008). General and special education teachers need to work together to build a mathematical curriculum that is strong and challenging. They can’t build the house alone! 


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