Emergent Math Concepts and Accessible Activities

Emergent Math Concepts and Accessible Activities By Tiffanie Zaugg

mathematics

attributes as they learn to categorize or sort items. Categoriza- tion or sorting is a skill that is also used in literacy from sorting capital letters from small letters, vowels and consonants, etc. It is systematic. It is ordering. It is comparing and contrasting. It is also the very most basic form of algebra. Attributes are the char- acteristics of an object. The attributes of an object can be used to sort collections into sets. It is important that as teachers we instruct the attributes of objects. For example, if we are asking a student to sort silverware, we need to talk about the different attributes of each piece. We need to teach that it can be sort- ed as silverware or that the same collection can be sorted into different ways. This concept is also important for our learners who may be learning to use a category based communication system. If the learner does not understand sorting or categoriz- ing, he/she will not understand why certain items belong in cer- tain folders. Modeling this concept is important in mathematics and communication. Here are some activities or tools to try: A sound, ear, activity for sorting could be different sizes of bells. Each size makes a different sound. If you have a learner with a hearing loss, the learner could sort by size.

Do you struggle with what mathematical concepts you should start teaching first to emergent math learners? Do you wonder how to make some materials or tools accessible? With math, we know that mathematical concepts build on each other, so it is essential that we know what the foundational topics and concepts are and how to make them accessible for all learners. When we are designing math for our learners, we need to design with all the learner's needs in mind. We will be discussing the nine mathematical topic areas and different ways to accommo- date different learner needs in each area. These accommodation lists are not extensive, but are here to provide ideas of different ways to accommodate different learners needs. There are nine mathematical topics which highlight key math- ematical concepts that are the foundational skills needed to ensure a lifelong mathematical understanding. The nine math- ematical topics are sets and attributes, number sense, counting, composing and comparing numbers, patterns, measurement, data analysis, spatial relationships and shapes. Also this article will explain the topic area, and the math concepts, and provide ideas for making the activities accessible for the different types of learners. While designing for math instruction we want to de- sign learning activities that get mathematics into the learners eyes, ears, hands and feet. Learners who are nonverbal will need to have their math vocabulary available within his/her commu- nication system. For our learners who are blind, it is essential to have hands on materials that are different enough tactilely that the learner can determine the difference. For example, all hard plastic objects that have a smooth texture or pieces of material that have similar texture. Sets and Attributes are the first topic area. This topic area is very important for emergent learners to understand sets and

Different sizes of bells.

If you are looking for an iOS app for sorting, try Drawing Pad. You can split the screen up into however many categories you

TIFFANIE ZAUGG is the Assistive Technology Coordinator and Students with Significant Disabilities Lead at Prairie Lakes Area Education Agency. Tiffanie is also the State of Iowa’s Area Education Agency’s Assistive Technology Lead. She has been in education for over twenty years. Within those twenty years, Tiffanie has taught general and special education and has been a special education consultant.

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ory correctly. Rational counting is matching each number name to an item in a group. In order for rational counting to take place the learner must master one-to-one correspondence, stable or- der, cardinality and order irrelevance. While some people may feel that one-to-one correspondence is easy, it actually takes a while for learners to grasp it. Learners need to coordinate words to physical movement and the eye along a line of items, match- ing one number word to one item until all the items are used up. Here are some activities or tools to try: The use of number lines to teach stable order, board games where you count spaces going down a path to teach one-to-one correspondence, put- ting items in different groupings to teach order irrelevance, and lastly labeling a group after counting them to teach cardinality. The math learning center has a virtual number line that you can adjust the spacing, the number sets, and many other various tools. If you have students with visual concerns, you can create tactile number lines. To get your learners up and moving, create a number line on the floor for the learners to walk on. Another great resource for online manipulatives would be: National Li- brary of Virtual Manipulatives

are sorting and then use the stamps for the sorts. Learners could choose which stamp collection they would like to sort. This app also has the capabilities of changing the background color to black which may be beneficial for learners with a visual disability. The second topic area is Number Sense , which is a key building block of learning arithmetic. Number sense connects counting to quantities, solidifies the concepts of more and less, and helps learners estimate quantities and measurements. The first math concept within number sense is numbers are used in many ways. Numbers may be categorical or nominal, referential, cardinal or ordinal. The second concept is quantity is an attri- bute of a set of objects, and we use numbers to name specific quantities. The last concept is quantity of a small collection can be perceived without counting. This concept is also known as subitizing. As a learner I can look at a group of items and know “how many” without counting the items. Providing opportuni- ties, to observe and manipulate quantities and change in quan- tities builds on the learners natural sensitivities and interest. Providing language enhances these experiences. Here are some activities or tools to try: Life-size activity, feet, would be to take painters tape and make five frames or ten frames on the floor. Have the students be the manipulatives within the frames; make sure the frame sizes are large enough if you have a wheelchair user in your classroom. You can modify this to make a frame on a table and use paper plates to be the manipulatives.

A number line on the floor for the learners to walk on.

The fourth topic is Comparing and Composing Numbers this topic may also be considered Number Operations. The math concepts within this topic area are: sets can be changed by joining items or by separating items, sets can be compared using the attribute of numerosity and ordered by more than, less than, and equal to, and a whole can be decomposed into equal or unequal parts: the parts can be composed to form the whole. Number operations are the tools that we use to find the answers to the questions like: “How many fewer?”, “How many more?”, “How many now?”, etc. When we are first teaching math, there is no need to focus on the math symbols or arithmetic. The learners need many opportunities to think through the relation- ships between how quantities work in their everyday life before they can think of using addition or subtraction. When we hur- ry our learners through memorization of rote facts, we lose the time the learner has to understand why the fact is true. We need to provide time for the learners to problem solve and multiple opportunities to make sense of the problem. Allow the learn- ers to draw, use manipulatives, or act out the problem. Experi- ences encourage explorations and appropriate talk of parts and whole in many contexts. Manipulating and grouping all kinds of objects builds foundations for parts and wholes from which all

Frame on a table and paper plates.

The National Council of Teachers of Mathematics has an Illu- minations website where the learner can do online interactive Ten Frames. Interactive Ten Frame The third topic is Counting . Even before they can count, chil- dren learn that number, or counting words are special. Experi- ence hearing others count, or name quantities provides foun- dational experiences with number words and the idea that they refer to quantities and can be ordered. The math concepts with- in counting are: counting can be used to find out “how many” in a collection and counting has rules that apply to any collection. Counting is very complex as it includes rote counting and ratio- nal counting. Rote counting is reciting the numbers from mem-

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subsequent learning develops. Here are some activities or tools to try: The Rekenreck is a great tool to use for composing and decomposing sets. There are various ways to create a Reken- reck or purchase one. This one can be made and used right at a learner’s desk.

matical regularities and structures. They will learn to identify, du- plicate and extend sequential patterns such as ABCABCABC, but also to find regularities and structures in number and geometry. Repeating patterning skills predicts later math knowledge. Pat- terning is a critical math skill. The math concepts within pattern- ing are: patterns are sequences governed by a rule; they exist in the world around us and in mathematics. Identifying the rule of a pattern brings predictability and allows us to make gener- alizations and the same pattern can be found in many different forms. Here are some tools to try: If you are looking for an online manipulative and pattern options for students, Glencoe virtual manipulatives. Remember if you are using physical objects to teach patterning and you have a student with a visual impair- ment, you should use objects that have great diversity. The sixth topic area is measurement . Measurement encom- passes a lot of different learning trajectories. Measurement can encompass length, area, volume, angles and turns, but the be- ginning concepts that we will be discussing are that many dif- ferent attributes can be measured, all measurement involves a “fair” comparison, and quantifying a measurement helps us de- scribe and compare more precisely. As you see, it is important that the learner already knows about attributes prior to learning about measurement. The central challenge in helping emergent learners understand measurement concepts is to slow down our expectations for the learning process. It is important to know that the more authentic the measurement problem-solving sit- uation is, the more deeply the learners are to be engaged. Here are some activities or tools to try: There are various ways that you can adapt rulers to make them accessible. You can purchase rulers that have handles built on top for learners with fine motor concerns. For learners with vision concerns, you can glue bump dots or other objects on top of the ruler for each inch. Talking tape measures are also an option for our learners. The Tape Mea- sure app for iOS and Ruler App for Android are apps that you use augmented reality to measure the distance from one point to another point. The seventh topic area is data analysis . The purpose of col- lecting data is to answer questions that are not obvious. Collect- ing the data needs to be done for authentic problem solving. Data must be represented in order to be interpreted, and how data is gathered and organized is dependent on the question to be answered. Next, it is useful to compare parts of the data and draw conclusions about the data as a whole. As the learner is creating representations using objects, pictures, or graphs, the learner needs to know how to label and describe parts of their data displays. One is able to scaffold a learners charts by the use of different types of graphs. Some learners may need to start with using an object graph, while others may be able to use a pictograph, and others a bar graph or tally chart. Data analysis for emergent math learners is about quantifying information in order to answer a question. That requires children to organize data in some visible way so that comparisons and generaliza-

Can be used right at a learner's desk.

If you have a learner who is a wheelchair user, you could cre- ate this one.

Make one of these for a life size rekenreck or accessible for a learner who uses a wheelchair.

There is also Number Rack iOS app, web or Chrome that works for students who have fine motor concerns. The Cranmer Abacus was especially adapted for individuals who are blind to complete math operations. The fifth topic is Patterning . Patterns are any predictable se- quence and come in many different forms. It could be visual pat- terns, auditory patterns, movement patterns, temporal patterns or numerical patterns. Learners will learn to find and use mathe-

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tions are possible. Here are a few good books with good ques- tions to spark simple graph explorations with children: Which Would You Rather Be? by William Steig, Whose Shoes? by Stephen R. Swinburne, and Anno’s Flea Market by Mitsumasa Anno. The eighth topic area is spatial relationships . Learners de- velop the processes of generating, maintaining and manipulat- ing mental images of two- and three-dimensional objects, in- cluding moving, matching and combining them. It is a critical math topic. Spatial Relationships begin at birth. Most learners start to learn by reaching for and grasping objects that are dan- gled in front of them. Next, the learners are cruising around to negotiate a path through a room. Lastly, the learner can locate items or decide how best to get from here to there, which have begun to represent space. Learners will learn the relationships between different positions in space (includes maps and coor- dinates). The key concepts embedded in these activities that make them mathematical are their relational nature. Here are some activities or tools to try: Do games like Simon Says and allow learners take turns as the leader to give directions. Puzzles involve spatial sense. Have multi-piece puzzles, pattern blocks, and tanagrams should be available with different levels of sup- port and age appropriateness. The last topic is shapes , everything in the material world has a shape. Learners need to go beyond the use of superficial shape labels to recognizing and specifying the defining attri- butes of shapes. As learners sort shapes with others, they be- come aware of rules about shapes. Learners will also discover, while exploring three-dimensional solids, that the faces of solids look like circles, rectangles, triangles and other common two-di- mensional shapes. When learners are given the chance to com- bine, rotate and compare shapes this will help learners develop understanding of part/whole relationships within and among shapes. Learners can learn to identify structures within embed- ded figures (finding "hidden shapes" within more complex dia- grams). Here are some activities or tools to try: Using the web- site Illuminations website allows students to manipulate shapes. If a student struggles with fine motor skills, this site allows the use of a snap together tool. This tool snap shapes together once they are close. Educators can also purchase wooden shapes that have velcro on them, which allows the learners to connect real wooden shapes together. Busy Shapes app for iOS and Android has 150 incremental levels. In conclusion, it is vital for us to make sure that our learners have the foundational skills in mathematics before we move onto more complex concepts. Math concepts are around us in every aspect of our lives and filter into other content areas. The number sense, counting, comparing and composing numbers, sets and attributes, and patterns are the foundational building blocks to the other three topic areas which are measurement, shapes and data analysis. When we are designing our lessons we need to make sure that we are designing for all of our learn- ers’ needs.

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