Dr. Mari Caballero
Dr. Marj Bock
Dr. Catherine Ayantoye
Emporia State University
This issue of NASET’s Practical Teacher series was written by Drs. Mari Caballero, Marj Bock, and Catherine Ayantoye from Emporia State University. Their article describes the development and implementation of a mathematics lesson for an inclusive 3rd grade classroom. The 3rd grade teacher and the special education teacher collaborated to create this lesson. The lesson used a children’s literature book, direct instruction (DI) and Universal Design for Learning (UDL) to meet the needs of all students in the 3rd grade classroom, including three students with special needs whose IEPs contained mathematics goals and objectives. The article includes a discussion of what went well when this lesson was implemented. It concludes with a set of helpful tips for teachers who wish to create similar mathematics lessons.
Abstract
This article describes the development and implementation of a mathematics lesson for an inclusive 3rd grade classroom. The 3rd grade teacher and the special education teacher collaborated to create this lesson. The lesson used a children’s literature book, direct instruction (DI) and Universal Design for Learning (UDL) to meet the needs of all students in the 3rd grade classroom, including three students with special needs whose IEPs contained mathematics goals and objectives. The article includes a discussion of what went well when this lesson was implemented. It concludes with a set of helpful tips for teachers who wish to create similar mathematics lessons.
Keywords: mathematics, inclusion, children’s literature, DI, UDL
CHILDREN’S LITERATURE + DI + UDL + MATHEMATICS = SUCCESS
FOR STUDENTS WITH DISABILITIES
Joseph is a 3rd grade teacher. This year Joseph has 27 3rd grade students in his class. Three of his students receive special education services. Their Individualized Education Plans (IEPs) contain mathematics goals and objectives. Two of these students are identified as having specific learning disabilities in mathematics as well as reading and writing. In mathematics both students are able to rote count to 1000. Both know their addition and subtraction math facts to 20. Overall, they are approximately 1.5 years behind grade level; although, they present many peaks and valleys in their mathematics skills. Joseph’s other student who receives special education services has autism. This student speaks in single words and phrases. He has developmental delays and functions at a kindergarten grade level.
Joseph provides mathematics instruction for all of his students including those with special needs. In addition, he receives support from a special educator. He also has a paraeducator who works with his students with special needs during mathematics instruction. Nevertheless, Joseph often finds that as he teaches a mathematics lesson his students with special needs seem lost and confused. Joseph struggles to create mathematics lessons with learning activities that will benefit all of his students. He wonders if this is even possible. Similarly, Joseph wonders what teaching strategies work best for students with specific learning disabilities or autism. Are these teaching strategies appropriate for use with all of his students?
These challenges are not unique to Joseph. Many classroom teachers experience these same issues. Frequently students of all abilities are integrated in the general education classroom (Friend, 2018). Therefore, teachers have to make decisions regarding how to effectively teach mathematics to the wide range of learners in their classrooms (Maccini & Gagnon, 2006). Students with disabilities can have difficulty building conceptual and procedural understanding due to the abstract nature of mathematics (Mercer, Mercer, & Pullen, 2011).
The Standards created by the National Council for Teachers of Mathematics (NCTM) (NCTM, 2000) and the Common Core State Standards Initiative (corestandards.org, 2018) both help students with disabilities transition from concrete to abstract understanding (Mercer et al., 2011). Teaching students who struggle in mathematics by having them use manipulatives, problem solve, and make connections to real-life scenarios (Mercer et al., 2011) as well as to other subject content areas (e.g., science, social studies, and English language arts) (Allsopp et al., 2007) can be very beneficial in building mathematics understanding for students with special needs.
Table 1 (see below) shows the NCTM Principles, Content Standards, and Process Standards. The Process Standards help teachers understand how to teach the five Content Standards (NCTM, 2000). Students with disabilities are being included in the the general education classroom and are being taught using the same curriculum as their general education peers. “In the area of mathematics, teachers are expected to provide effective instruction on curriculum that address higher level math skills and encompasses open-ended problem-solving tasks as set forth by the National Council of Teachers of Mathematics Standards” (Maccini & Gagnon, 2006, p. 218). Therefore, students with disabilities are also being expected to learn these Standards, which makes it important for teachers to use effective instructional procedures and strategies to help all students learn the Standards (Maccini & Gagnon, 2006).
Table 1
NCTM Mathematical Teaching Principles and Standards
|
Principles |
Content Standards |
Process Standards |
|
Equity |
Numbers and Operations |
Problem Solving |
|
Curriculum |
Algebra |
Reasoning and Proof |
|
Teaching |
Geometry |
Communication |
|
Learning |
Measurement |
Connections |
|
Assessment |
Data Analysis |
Representation |
|
Technology |
|
|
Adapted from “Principles and Standards for School Mathematics.” National Council of Teachers of Mathematics (NCTM). Copyright 2000. Reston, VA: NCTM.
The standards-based mathematics curriculum recommended by NCTM and the Common Core State Standards Initiative promotes problem solving and reasoning in mathematics education. This should support conceptual and procedural mathematics understanding. This understanding can be achieved through problem solving, active learning with manipulatives, application to the real world, cooperative learning among peers, and teacher facilitation (i.e., teachers scaffolding student learning) (Maccini & Gagnon, 2006).
Children’s literature helps teachers create mathematical problems that promote authentic connections for students, i.e., connections to their daily lives, connections to their families, and connections to their neighborhoods (Larson & Rumsey, 2017). In addition, experts recommend that teachers provide mathematical manipulatives to accompany the children’s literature books (Larson & Rumsey, 2017) to further support the conceptual understanding of students including those with disabilities. Consequently, teaching students who struggle in mathematics by having them use manipulatives, problem solve, and make connections to real-life scenarios (Mercer et al., 2011), through children’s literature (Larson & Rumsey, 2017) can be very beneficial in building mathematics understanding.
Clearly, developing mathematics lessons derived from children’s literature books may be highly effective for all of Joseph’s students including those with disabilities. But what instructional strategy (or strategies) should Joseph use when to teach these mathematics lessons? Joseph’s students’ special education teacher suggests that Joseph use Direct Instruction (DI) (Stephan & Smith, 2012) and Universal Design for Learning (UDL) (Salend & Whittaker, 2017) to teach his mathematics lessons.
As the special education teacher noted, Direct Instruction (DI) (Stephan & Smith, 2012) is an instructional strategy that teachers like Joseph often use. It is an extremely effective instructional strategy used to teach all students including those with disabilities (Stephan & Smith, 2012). DI is a teacher-lead, whole-class instructional approach. Teachers start the lesson by activating the prior knowledge of their students. They then introduce new content for the lesson. They show their students how to do the new activity. This is often called the I do section of the lesson. After the teachers have modeled the new skill several times, they invite the students to do the skill with them. This guided practice section of the lesson is often called the We do section of the lesson. Following sufficient guided practice, students move to the independent practice, or You do, section of the lesson. This highly structured, whole-class instructional approach incorporates clear, concise language and a consistent learning routine. These, together with the modeling and guided practice sections of the lesson, lead to high levels of student success for all students including those with disabilities. DI assures that students do not practice errors. Rather, students practice correct, effective thinking and problem solving (Stephan & Smith, 2012).
Since no two students are alike, the special education teacher explained that effective teachers differentiate their teaching to accommodate their students’ learning differences. Universal Design for Learning (UDL) (Salend & Whittaker, 2017) is a framework developed to help teachers do this. UDL is derived from the architectural concept of universal design, i.e., the design of buildings, products, and services so that all persons can use them. For instance, a ramp is a universal design that provides access for many persons, e.g., those who use wheelchairs, those pushing strollers, and those making deliveries. Teachers who implement UDL are essentially educational “architects” who create learning structures that support all students’ success (Salend & Whittaker, 2017). UDL is based on brain research that applies universal design to teaching and learning (CAST, 2011). Consequently, to differentiate instruction for learners with a range of learning differences, UDL provides multiple means of:
- Representation – presenting content in a variety of ways;
- Action and Expression – varying the ways students are encouraged to respond and show their learning; and
- Engagement – using a variety of practices to heighten student motivation.
Just as architects create blueprints to design buildings that everyone can use, teachers use UDL lesson plans to develop learning activities for all students, including those with disabilities.
After visiting with the special education teacher, Joseph realized that he needed to create a lesson plan that uses children’s literature to teach mathematics. The lesson needed to draw upon the NCTM Principles, Content Standards, and Process Standards, as well as the Common Core Mathematical Teaching Practices. The lesson plan would also need to incorporate IEP objectives for each of his students with special needs. In addition, the lesson plan would need to conform to the DI format. And finally, it would need to integrate UDL strategies. Joseph asked the special education teacher to work with him to develop this lesson. Together they created the DI/UDL lesson plan (see Figure 1).
Joseph’s DI/UDL MATHEMATICS LESSON PLAN
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Figure 1. Lesson Plan Subject & Topic: Early Division with Equal Shares Developed by: Joseph Grade: 3rd Date: October 20, 2018 Unit: Numbers and Operations/Division |
NCTM Process Standards embedded in lessonRepresentationConnectionsCommunicationReasoning & ProofProblem SolvingCommon Core Mathematics StandardCCSS.Math.Content.3.OA.A.2: Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.IEP Objectives for John and Susan: By the end of the 2nd 9 weeks of the 2018-2019 academic year, John and Bill will divide one and two digit numbers (0-50) without remainders with 100% accuracy on 4 out of 5 consecutive trials as measured by curriculum based assessment.IEP Objectives for Brian: By the end of the 2nd 9 weeks of the 2018-2019 academic year, Brian will divide single digit numbers (0-9) without remainders with 100% accuracy on 4 out of 5 consecutive trials as measured by curriculum based assessment.Common Core Mathematical Teaching Practices embedded in lesson
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MaterialsThe Doorbell Rang by Pat Hutchins (big books)Small paper plates (12 per student)Circular cut outs (12 per student)White board for each student OR iPad with the whiteboard application (Educreations Interactive Whiteboard (this is free) or another whiteboard application)Examples of other children’s literature books with a focus on divisionOne Hundred Hungry Ants by Elinor J. PinczesOne Hungry Cat by Joanne RocklinDivide and Ride by Stuart MurphyThe Great Divide by Dayle Ann Dodds |
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TechnologyBold all that applyo Teacher laptopo SMART Boardo LCD projectoro SMART Senteoso Computerso iPad or tableto iPod or mp3 player(s) |
o Webcamo Digital camerao Document camerao Digital microscopeo Video camerao Scannero Color printero Calculatorso FM system |
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|
Direct Instruction Procedure |
||
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Introduction |
I Do We Do |
|
|
Lesson |
I Do After the doorbell rings and Tom and Hannah come in, bring out 2 more plates. |
UDL Procedures *Think aloud strategy Multiple Means of Engagement |
|
Summative Assessment |
Students can independently answer the following question using a whiteboard app on their individual iPad. The Educreations App allows students to write and draw their answers. When they are done answering the questions, they can email their answers to you or you can review them directly on their iPad. ***If students are successful with this first question, you may choose not to use the additional question. While waiting for other students to finish, students could work individually or in small groups with the other children’s literature books with a division theme that are located in the materials list. This time there are 64 cookies and 8 friends. How many cookies does each friend get? What is the division problem that represents the number of cookies each person gets? Show and explain 2 different strategies or representations for how you solved this problem. |
|
|
Review |
“Today we learned how to solve problems by creating equal groups of cookies. We were doing division! Division is breaking a number into equal groups, so that everyone has a fair share. If I had the division problem 56 ÷ 8 (write this on the board), what could be a situation that would describe this problem? How could we figure out the answer? (Solicit student responses). We will continue to learn different strategies for how to divide a number and we will also learn how we can use our understanding of multiplication to become better dividers!” |
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Joseph’s Lesson Plan Description
Joseph and the special education teacher created a lesson that integrates children’s literature and mathematics, specifically within the Numbers and Operations Content Standard. For all of Joseph’s lessons, he must include the Common Core Standard that aligns with the content he is teaching. Joseph’s lesson is based on CCSS.Math.Content.3.OA.A.2 which states,
“Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.” (core standards.org, 2018).
In Joseph’s lesson plan, he utilizes DI and UDL to help all students experience success while learning early division. See Joseph’s lesson plan “Early Division with Equal Shares”.
Joseph’s mathematics lesson incorporates all the Common Core Mathematical Teaching Practices and NCTM Process Standards. Joseph’s lesson plan shows that all of these Practices and Standards are included in order to help meet the range of student needs represented in his class and help his students learn division conceptually, rather than memorizing a procedure.
Below is a short example of how each NCTM Process Standard is embedded in the lesson.
- Representation: Joseph has student’s using cookie cut-outs to determine the correct equal shares. He also has student’s drawing pictures or using other forms of representation (e.g., counters) to figure out the equal share.
- Connections: Joseph connects English Language Arts and division throughout the lesson.
- Communication: Joseph has students working in small groups to figure out the solutions to the problems posed in the children’s literature book. He also has students communicate with him and share how they solved the problems.
- Reasoning and Proof: Joseph encourages students to share their strategies for solving the problems and prove that their strategy works.
- Problem Solving: Throughout the lesson, Joseph utilizes the problems posed in the children’s literature book, as well as poses additional real-life problems in the guided and independent portions of the lesson.
The Common Core Mathematical Teaching Practices are also embedded in Joseph’s lesson. The Common Core State Standards Initiative developed mathematical teaching practices to help educators teach mathematics in a more proficient manner (http://www.corestandards.org). Below is a short example how each of the 8 Common Core Standards for Mathematical Practice are embedded in the lesson.
- Make sense of problems and persevere in solving them: Joseph’s students experience errorless learning throughout the direct instruction lesson. They have a variety of opportunities to practice and be successful learning the objective.
- Reason abstractly and quantitatively: Joseph’s students learn to reason through the various problems posed in the children’s literature book. The students are encouraged to use manipulatives and gradually move to a semi-concrete and abstract representation of the division problems.
- Construct viable arguments and critique the reasoning of others: Joseph’s students work together to determine multiple strategies for solving the problems.
- Model with mathematics: Joseph’s students model the division problems with concrete objects (i.e., cookie cut-outs). They also have to express the division problems using technology (i.e., they are using their white board app for this).
- Use appropriate tools strategically: Joseph’s student’s use technology and concrete objects to solve the problems and represent their understanding. They have the opportunity to choose which tools they want to use while solving problems in the guided and independent practice portion of the lesson.
- Attend to precision: Throughout Joseph’s lesson, Joseph is checking for student understanding. He has options for students to continually practice if needed. Students are also encouraged to express their understanding verbally and using representations.
- Look for and make use of structure: Throughout Joseph’s lesson, students learn to find patterns and use repeated reasoning to solve the more difficult problems.
- Look for and express regularity in repeated reasoning: Student’s learn throughout the lesson to start generalizing their understanding to other problems and situations.
Joseph’s lesson plan contains numerous UDL modifications. Below are a few of the ways that Joseph provides UDL modifications.
- Multiple Means of Representation
- Using the think-aloud strategy
- Using of visual representations
- Pre-teaching mathematics concepts
- Multiple Means of Action and Expression
- Using practical, familiar items to improve focus
- Using tablets and apps for notetaking, procedural/conceptual review, frequent practice, etc.
- Providing frequent opportunities for cumulative and distributive review of rules, formulas, strategies, etc.
- Multiple Means of Engagement
- Making learning relevant/connect examples to student’s daily life
- Making connections between math and the real world
- Using flexible grouping (heterogeneous grouping to minimize the barriers of disability.
To see the full list of UDL modifications that Joseph has embedded into his lesson, refer to the bolded UDL modifications Joseph has on his lesson plan. These lists of UDL modifications come directly from https://sites.google.com/site/703gse/udl-for-math-calucation-and-problem-solving.
Joseph embedded DI in his lesson plan. He uses DI with his whole class in order to facilitate errorless learning. Joseph’s lesson utilizes clear, concise, structured language in order to help students understand his expectations, how to solve the problems, and to develop mathematics vocabulary. Although as the lesson moves on, Joseph allows students to reason through the problems and use their own invented strategies. Initially he models how his students can solve the problems successfully. This is particularly important for students receiving special education services in mathematics. Many of these students are either unable to generate their own problem solving strategies or generate only one very simplistic problem solving strategy. Thus they benefit from seeing Joseph as well as their peers generate and use multiple problem solving strategies.
Lesson Implementation
Joseph tried the lesson with his students. The lesson worked! All his students understood the lesson. John and Susan, his students with specific learning disabilities, understood equal share. They understood when and why they would use equal share. They learned how to represent equal shares as whole number quotients. Brian, Joseph’s student with autism, learned how to divide “real-life” manipulates into equal shares. Thus all three students completed a lesson that addressed one of their IEP mathematics objectives. In addition, Joseph’s other students achieved the objectives he identified for the lesson. Joseph differentiated the lesson to meet each of his student’s unique learning needs. He found that using DI did not “hold his more capable students back.” In short, those students moved to the independent work, or You do phase of DI, very quickly while his other students worked with him longer in the We do phase of DI. The paraeducator was able to provide additional DI for John, Susan, and Brian as needed during the You do phase of DI. She did this while working with these students at a small table in the corner of the room. Joseph provided additional We do instruction for a few of his other students as needed while the majority of his class completed the You do phase of the lesson.
Lesson Outcomes
Overall, Joseph’s lesson was very structured and thorough. Joseph modeled for his students, provided them opportunities to use their own invented strategies with cooperative learning groups, and experience multiple situations to practice the objective. He built this lesson to take advantage of his students’ strong reading and writing understanding. He utilized DI. He incorporated UDL strategies, as well as made sure the NCTM Process Standards and Common Core Mathematical Teaching Practices were embedded in his lesson. Further, Joseph developed a lesson that addressed IEP objectives from his students’ IEPs. In fact, this lesson helped his students with disabilities understand why they needed to learn to divide numbers and how they can use this skill on a daily basis in their real lives at school, at home, and throughout their neighborhoods. Therefore, this lesson benefitted all of Joseph’s students, including those with disabilities. He was able to implement it with all students in his class participating in the initial instruction.
Helpful Tips for Creating Mathematics Lessons Utilizing Children’s Literature Book’s
- It is important, specifically for students with disabilities, to provide them with a lot of repetition and practice on the content that is being taught. For this particular lesson on equal shares, the iPad apps and websites below could be useful for students who need additional practice and repetition:
- https://www.studyladder.com/teacher/resources/course/mathematics-division-508
- https://www.splashmath.com/division-games
- Thinking Blocks Multiplication (app)
- Number Pieces, by the Math Learning Center (app)
- When choosing a children’s literature book:
- Ensure it correctly portrays the mathematics content that you are teaching (Uscianowski & Ginsburg, 2017)
- Ensure it is grade level appropriate (Burns, 2015)
- Confirm that the book will help all students reason about mathematics and build a conceptual mathematics understanding (Burns, 2015)
- Ensure that the book illustrations support the mathematics content and are not confusing (Uscianowski & Ginsburg, 2017)
- Verify that mathematics is embedded in the content of the book (Uscianowski & Ginsburg, 2017)
- Ensure that the book has an interesting plot with relatable characters so that the students are engaged and able to make connections (Uscianowski & Ginsburg, 2017)
- Allow the students an opportunity to enjoy the book by having it available to them to reread or look more closely at illustrations (Burns, 2015)
- Remember that the most important thing about choosing a book is that it is engaging for the students and fun to read (Burns, 2015)
References
Allsopp, D. H., Kyger, M.M., & Lovin, H. L. (2007). Teaching Mathematics
meaningfully. Baltimore, MD: Paul H. Brookes Publishing Co.
Burns, M. (2015, September 9). Using children’s literature to teach Math.
[Blog post]. Retrieved from http://www.marilynburnsmathblog.com/using-childrens-
literature-to-teach-math/
CAST (2018). Universal Design for Learning Guidelines version 2.2. Retrieved from
Common Core Standard: Mathematics. (2018). Retrieved from www.corestandards.org
Council of Chief State School Officers & National Governors Association Center for Best Practices. (2010). Common core state standards for mathematics.
Friend, M. (2018). Special education: Contemporary perspectives for school professionals
(5th ed.). New York: Pearson.
Hutchins, P. (1986). The doorbell rang. New York: Greenwillow Books.
Larson, L. C., & Rumsey. C. (2017). Bringing stories to life: Integrating literature
and Math manipulatives. The Reading Teacher, 71(5), 589-96. https://doi-org.emporiastate.idm.oclc.org/10.1002/trtr.1652
Maccini, P, & Gagnon. J. C. (2006). Mathematics instructional practices and assessment accommodations by secondary special and general educators. Council for Exceptional
Children,72(2), 217-234. https://doi.org/10.1177/001440290607200206
Mercer, C. D., Mercer, A. R., & Pullen. P. C. 2011. Teaching students with learning
problems. Upper Saddle River, N.J: Pearson.
National Council of Teachers of Mathematics (NCTM). 2011. Achieving fluency: Special
education and Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (NCTM). 2000. Principles and standards for
school Mathematics. Reston, VA: NCTM.
Salend, S. J., & Whittaker. C., R. (2017). “UDL: A blueprint for learning
success.” Educational Leadership, 74(7): 59-63. Retrieved from eric.ed.gov
Stephan, M., & Smith, J. (Spring-Summer, 2012). Teaching common core Math practices to
students with disabilities. Journal of The American Academy of Special Education
Professionals, 162-175. Retrieved from files.eric.ed.gov/fulltext/EJ1135717.pdf
Uscianowski, C., & Ginsburg. H., P. (2017, August 30). How to choose a high-quality Math
picture book. Development and Research in Early Math Education. Retrieved from
https://dreme.stanford.edu/news/how-choose-high-quality-math-picture-book.
About the Authors
Dr. Mari Caballero is an assistant professor at Emporia State University. She earned her doctorate in mathematics education with a minor in special education from the University of Kansas. She is a member of Council for Exceptional Children and the National Council of Teachers of Mathematics. Her research interests include collaboration of professionals in schools in order to facilitate inclusion and utilizing effective instructional strategies for students with disabilities specifically in the area of mathematics.
Dr. Marj Bock is a full professor of special education at Emporia State University. She received a doctorate in special education from the University of Kansas. She is a member of the Autism Society of America, Council for Exceptional Children, Council for Children with Behavior Disorders, and Teacher Educator Division. Her research interests include metacognitive strategies for students with autism, inclusion, and co-teaching.
Dr. Catherine Ayantoye is an assistant professor at Emporia State University. She earned a Ph.D in special education from the University of Northern Colorado. She is a member of Council for Exceptional Children, and Teacher Education. Her research interest includes collaboration of professionals in inclusive settings, and students with disabilities in inclusive settings.
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