See It, Do It, Imagine It The Power of Cause and Effect in..

See It, Do It, Imagine It The Power of Cause and Effect in Problem Solving By Karen Ross-Brown

See It, Do It, Imagine It The Power of Cause and Effect in Problem Solving mathematics

Conversations with my college roommate, before and after we graduated, turned out to be some of the most valuable ex- periences of my career. We talked for hours in our dorm room about our teaching philosophy and the rights of adults with dis- abilities. Many of these conversations were launched by our ex- periences working in adult group homes and by our special edu- cation professors who seemed to have glimpsed into the future, guided us along a path that was progressive and long-lasting - a formative thread as we began to teach students. These conversations led me to pause and think before I pressed ahead with ideas in my classroom and in my current position in curriculum development. Although they took place with a growing contingency of special and general educators and support staff over the years, the topic has remained steady, beginning with ‘What do we teach students with significant dis- abilities?’ and evolving into ‘How do we teach students with sig- nificant disabilities?’ Later in my teaching career, the idea that learning depends on motor and language access was discussed in multiple conversations with the speech and language clini- cian and occupational therapist at my school. I knew it applied to my students despite their many challenges in those areas, sometimes alongside cortical vision impairments or hearing loss. The bottom line was I was getting paid to teach reading and math to my students regardless of their disabilities and chal- lenges. My primary job was to figure out what that looked like for each student. The concept of cause and effect came to mind last year as I was preparing Equals math training for teachers whose students

have significant challenges with cognition, motor skills and lan- guage. My thoughts wandered around actions and outcomes as I was thinking about making the process of problem solving more concrete and immediately accessible. As I searched the Internet for basic math concepts and prob- lem solving that were viewed as manifestations of cause and ef- fect, I found very little beyond causation and correlation. None of it was connected to basic problem solving in math. Why isn’t there a clear focus on cause and effect in math class? After all, cause and effect is utilized in problem solving models by man- agement gurus in the business world. It is very valuable in shap- ing our knowledge of how the world works and the basis for ap- proaching a problem. I began to think about typically developing students and their experiences. Then it hit me: cause and effect life experienc- es have been essentially inaccessible to students with significant disabilities. It is fantastic that software, computer interfaces, switches and adapted joysticks have been created to make com- puter use more accessible, but these are not always the optimal solution for a student with multiple challenges and do not re- place hands-on experiences in the real, physical world. Students without disabilities come to Kindergarten with a ton of real-life, hands-on moments from spontaneous play to instant creation when they grab whichever toys are handy. There isn’t a barrier on the way to the table of Legos or motor skill challenges that prevent pushing plastic bricks together with a pincer grasp. They can begin to make something as soon as they think of it. Is it assumed that students come ready to learn, with the advan-

KAREN ROSS-BROWN has a BS degree in special education and MLS degree with an emphasis in assistive technology. She taught in special education for over 23 years in Minnesota and Wisconsin, specializing in teaching students with significant disabilities. Karen joined the curriculum team at AbleNet, Inc. in 2008. She was the primary author of Equals Mathematics curriculum and co-author of Equals Math curriculum revision (2017) with Jennifer Emanuele and James LaRocco. Karen has trained teachers in AbleNet curricula across the United States. She has presented at National Council of Teachers of Mathematics (NCTM) National and Regional Conferences, National Council for Exceptional Children (CEC) National Conferences, Closing the Gap (CTG) Conference, and Assistive Technology Association (ATIA) Conference.

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first steps in that process: 1) make sure students are receiving instruction that covers and connects basic foundational math skills, 2) build cause and effect background knowledge with models and exploration opportunities using math tools and strategies present in the lesson, 3) support problem solving with concrete objects, 4) show concrete objects and lesson materials in two or three choices as a way for students to communicate, 5) expand cause and effect by asking questions with well-planned choices and 6) support students’ collaboration and risk-taking by accepting each answer to a question, then check and discuss. 1. Make sure students are receiving instruction that cov- ers and connects basic foundational math skills by providing math curriculum that is connected meaningfully across all math content areas, such as those recommended by the NCTM: Num- bers and Operations, Geometry, Algebra, Data and Probability and Measurement. (It is worth noting that other sources, such as Common Core State Standards, show different configurations of content areas. However, the same math content is covered in each source, just sorted differently.) Typically, the idea of con- nected math across all math content areas has been ignored in traditional special education math programs, especially those described as functional math. When you consider that we all learn by connecting new information to relevant known infor- mation, the idea of pre-requisite skills becomes monumentally important. As we learned from Equals math curriculum data, students with disabilities learn very well with general educa- tion math instruction methodologies that are based on brain and learning research, including making connections in learn- ing, as long as access, appropriate pacing and adaptations are provided. When you think about it, isn’t teaching math connect- ed within and between all math content areas an example of promoting generalization that we have embraced as proof of a learned concept? Why, then, have we been teaching in function- al math silos for so long, with limited scope in math content and no connections between concepts? In talking with many educators over the years across the United States, I found that the principles of math instruction ne- glected in pre-service teacher education tend to play a role in teachers clinging to long-held, ineffective practices in math. Es- sentially, students with moderate or significant disabilities have been left without quality math instruction altogether, result- ing in special education teachers improvising. Unless teachers have had some background in curriculum or the good fortune to be handed an appropriate curriculum for their students, they wouldn’t have necessarily known what was missing. Now that we know what is missing, it is essential to look for a math curriculum that is robust, connected and complete, deftly utilizing quality math instruction based on best practice princi- ples of learning and brain research with access that is necessary for optimal engagement and understanding.

tage of countless cause and effect experiences needed to solve math problems? Students with significant disabilities may not have had those experiences. So, it makes sense to arm students with information about how our physical world works, whatever the impact may be. Given that students with significant disabilities typically come to school with less free-flowing or independent explora- tion in their play, less access to use of toys and tools, and few- er hands-on experiences overall compared to same-age peers without disabilities, it makes sense to turn to the answer that lies within math itself: teach cause and effect that is present in math concepts and use of math tools in the process of problem solving. Using tools and strategies with learned, identified (and adapted) actions to solve problems makes problem solving more concrete. Consider the concept of addition from the National Council of Teachers of Mathematics (NCTM) math content area Numbers and Operations, for example, which easily comes to mind when thinking about cause and effect: add two sets (cause) which re- sults in one larger set (effect). All four operations are excellent examples of cause and effect, however, there are more examples from each remaining NCTM math content area: Data Analysis and Probability: When you compile data and graph it (cause), a visual comparison is displayed (effect). Geometry: Drawing two intersecting lines (cause) shows an angle (effect). Measurement: Place a bag of apples on a dial scale (cause) and the pressure makes the needle move on the dial (effect). Algebra: Write and multiply input amounts: 1, 2, 3 and 4 times 2 on an input/output table (cause) to reveal a pattern: 2, 4, 6 and 8 (effect). By modeling and providing hands-on experiences with math tools and actions, students have firsthand views of cause and ef- fect. With access to language, students can talk about what hap- pened. This, in turn, can be used when problem solving. When students are familiar with an action and its effect, they are more likely to choose that action when it makes sense in solving a par- ticular problem. For example, if I know adding means to join two sets together for a total, I will think of it when I solve a problem with two sets that are put together. Teachers must make cause and effect experiences accessible for students with disabilities by modeling and adapting actions with math tools and materials so students can understand and have as much hands-on experience and independence as possi- ble. With access to language, students can then talk about what is happening or what has happened right in front of them. These ideas fit nicely with the math community’s persistence that hands-on math is very valuable, especially when paired with models, exploration, and discussion. For students with disabil- ities, there are steps to take to provide access so they, too, can watch, do and talk about it, specifically with concrete objects and actions that make clear what is happening. Here are basic

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2. Build cause and effect background knowledge with models and exploration opportunities using math tools and strategies present in the lesson prior to problem solving. This is a must in good instruction and should be part of the math curriculum itself. Background knowledge, with an eye towards cause and effect, must include structure for students to rely on, such as familiar vocabulary, a related game, and concepts brought forward from past lessons. Modeled use of tools and strategies paired with math concepts and action words, e.g. add, place, sort, etc. help students see what is happening. When they have access to those same materials to explore, they can experience cause and effect firsthand. In Equals math, the Ac- tion Dictionary provides access by way of multiple adaptation suggestions for each action present in the lessons. Teachers choose which adaptation works best for a student and tweaks it to fit, or perhaps comes up with a new idea inspired by the adaptation ideas. Whichever is chosen, the idea is to basically provide enough support so students are able to watch, do and talk about what is happening as independently as possible. It may be necessary to show explicitly what the problem or the cause and effect relationship is about so students know what you are talking about, even if it feels like you are ‘giving away the answer’. It doesn’t matter, however, because if a student doesn’t know what you are getting at, wildly guessing the answer has no significance or merit. 3. Support problem solving with concrete objects. Through the process of problem solving, we usually think about several things, in varied order, such as naming the facts in the problem, choosing an action to take, identifying what the out- come might look like and which tool and strategy to use. Stu- dents with significant disabilities typically require concrete ma- terials for understanding this way of thinking, one step at a time. (This is not to say that students with significant disabilities never see semi-concrete and abstract representation. They require ex- posure to these as well, since numerals, operations symbols, pic- tures and X’s on a graph or array are a part of our world). Problem solving in a story format makes good sense for its relationship to the real world. However, the story can become a barrier when trying to navigate a math problem buried in a ton of words. This is where concrete objects cut through to the essence and in- crease interest as well. Well-placed objects fastened to the prob- lem above matching key words can assist students in focusing on the facts. Reducing the language down to basic facts and concepts of the problem is also helpful as long as math terms are maintained and the essence of the problem is preserved. When it comes to math terms, concrete representations of vocabulary, as seen in Equals math, support the meaning in the problem. The pic-symbol supported and demonstrated action words embedded in Equals lessons as background knowledge re- emerge when students are asked, “What do you do?” to solve

the problem. Making an estimate about an amount or a predic- tion of an outcome can be shown in a display of possible out- comes as depicted by objects. For example, in a problem adding three insects and two insects, I can ask my students to estimate how many total insects there will be when I add them, giving three choices displayed with insect counters: two insects, six insects or one insect. The students can estimate, comparing the insect sets of three and two attached to the problem to the three choices in the display. When solving a problem, students are asked to choose a tool or strategy. Often, the strategy can be linked to a tool or counter so the question has concrete answer choices. Using tools/strategies that were previously explored is key. In Equals math, you will find problem solving hinges on a well-planned exploration of background concepts, vocabulary and lesson objects and materials that are carried into the prob- lem-solving process. Choosing a tool and strategy, then, is easy to provide as the students already have had experience with them in the beginning of the lesson. 4. Show concrete objects and lesson materials in two or three choices to narrow the field after asking a question or expecting students to comment. Providing choices for stu- dents who do not have access to a communication device pro- grammed to represent what is happening in math at the mo- ment is a necessary support. Of course, students need access to their own communication systems at all times. However, in math class, students also need to have relevant choices to answer questions about the problem at hand. Providing two or three choices will give students power in the process while limiting the scope of possibilities from all the math tools in the world to just the math tools on the table. It gives the student a chance to focus on and use familiar objects that were explored earlier in the lesson to answer a new question. Students have access to a display by eye gaze, point or touch. If a student requires a different response, recording “That’s the one I want,” on a Step- by-Step communicator, placing it near the student, and pointing to each choice works well. 5. Expand cause and effect by asking questions with well- planned choices. By expanding on questions about the lesson concept and what happened in the problem, students have an opportunity to engage in basic reasoning with materials they know. For ex- ample, students learned about using three dimensional shapes (cubes) to make a new three-dimensional shape (rectangular prism tower). Three objects are placed in a display: a butter- fly counter, a cube and a hexagon with piles of each object on the table nearby. The teacher asks, “What do you use to make a tower?” When a student chooses, the teacher follows through and demonstrates building a tower with the chosen object, ask- ing students to assist or build it themselves (with adaptations to support students building a tower as needed). Discuss what

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EQUALS MATH ACTION DICTIONARY - SAMPLE ADAPTATIONS These sample adaptations are provided to illustrate the types of ideas that can be considered for students with significant dis- abilities to increase their independence in using math tools and materials as they learn about cause and effect and solving prob- lems. The Equals Action Dictionary is provided in its entirety as part of the Equals math curriculum from AbleNet, Inc. 1. Students begin to learn about addition when they join sets and see what happens. To help a student complete that task when a student need support in handling manipulatives, anchor one set on a surface with sticky tac, Velcro or tape. The student then sweeps a second amount to join sets with a large craft stick, ruler, or hand.

happened, e.g. Did stacking the butterflies make a standing tower? What happened? Repeat with the remaining choices and vote on the one that worked best. This type of activity increas- es opportunities to observe and/or experience cause and effect with different materials, resulting in a simple discussion about what works well and what does not. 6. Support students’ collaboration and risk-taking by ac- cepting each answer to questions, then check and discuss. Whether asking a question during a lesson or when expanding cause and effect, students should be given the opportunity to make a choice and follow through with it. Not only do they ex- perience the cause and effect of their choice, it opens up stu- dents’ thought processes to try new things and make comments without fear of judgment. It’s wise in these instances to sidestep the idea of right and wrong answers, and instead vote for the most likely solution that will solve the problem. Decide as a group which choice works best or whether or not the problem is solved accurately. There are many benefits for students with significant disabili- ties when solving problems with a focus on cause and effect us- ing concrete materials. Opportunities for language use increase in a small group when students have a way to talk about what is happening in the moment during math class. Students expand their language when they use and understand math vocabulary, tool names and action names. Research has shown that com- municating about a math concept increases achievement. Ad- ditionally, making choices is a staple of language access for any purpose. We all make choices when we speak, whether it occurs using a verbal voice or a communication system. Also, working and communicating in an interactive, collaborative group as a valued member satisfies the need for belonging within a sup- portive atmosphere. Viewing and engaging in cause and effect activities is a prac- tical and concrete way to see how the world works. Use of the tools in making choices bring students into the here and now to directly connect with the task at hand. Becoming familiar with the function of math tools is useful, making it more likely the appropriate tool for a specific problem will be chosen now and in the future. Students can experience and begin to understand the pow- er of making a choice, completing an action, and observing the outcome, especially when adaptations are made for the stu- dent to act with more independence. There is value in setting up students for success by offering ways for them to observe and engage in thinking about cause and effect, using concrete objects, seeing how it all plays out right in front of them and talking about it, which brings together cognition, motor skills and language in a learning environment. This kind of learning puts students with significant disabilities on track for reaching their potential, whatever that might be.

Student adds two sets with a tool.

2. For students who need support holding a pencil to draw points and lines in creating a line graph, use circle counters and AngLegs or Wikki Stix. The student places circle counters on or near the correct line and connect the circles with Wikki Stix or AngLegs as pictured here.

Student placed larger circle counters and Ang-Legs to create a graph.

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3. When a student measures a line in whole inches and re- quires support to differentiate between 1” and ½” lines, adapt the ruler by fastening Wikki Stix to each 1” line.

5. To support building a rectangular prism with cubes, a straight edge is used to steady the figure so it remains vertical as the student is stacking the cubes. The straight edge also helps the student keep the cubes straight so it more closely resembles a whole three-dimensional shape.

4. When a student is learning to use an array to multiply and the two-step process needs to be broken down into single steps, simplify the task. For solving 2 x 5 = ___, the student counts the number of columns (5) for each row and fastens them togeth- er into two rows (or asks a partner to help). Student places two rows of five into the corner of the right angle frame (made with AngLegs). Wikki Stix placed on each line of the ruler extends the length and provides a tactile line.

6. The Step-by-Step communicator is a sequential message device for recording, for example, numerals 1-10 for counting objects. It can also be used as a single message device for stu- dents to choose an answer from a display as the teacher points to each choice. Record “That’s the one I want” or a similar phrase for the student to activate when the teacher points to the de- sired choice. The ruler is placed vertically to act like a barrier, holding the rectangular prism in place.

Single or multiple messages can be recorded on the Step-by- Step communicator.

A three-dimensional array is easier to create and move.

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