Problem Solving and Technology Implementation in an Inclusion Classroom

1. Problem Solving and Technology Implementation in an Inclusion Classroom Annie Fetter, The Math Forum @ Drexel

2. The INCLUDE Grant The grant is designed to ensure that all students in the general education classroom, including those with mild to moderate disabilities, struggling students, and English language learners, are provided the necessary accommodations in the general mathematics classroom that will support their achievement of the Core Curriculum Content Standards. The basis for the INCLUDE grant is to improve academic achievement in mathematics by using educational technology effectively.

3. Woodlynne Public SchoolWoodlynne, NJ (Camden County) • Grades Pre – K through 8 • 497 students in district • Ethnicity: Latino – 49% Black – 35% Asian - 9% White - 7% • Special Education – 21% • Free Lunch – 57% Reduced Lunch – 13% • Median Income = $41,000 • Teacher : Student Ratio is 1 to 17

4. What is The Math Forum? ... the leading online resource for improving math learning, teaching, and communication since 1992. We are teachers, mathematicians, researchers, students, and parents using the power of the Web to learn math and improve math education. We offer a wealth of problems and puzzles; online mentoring; research; team problem solving; collaborations; and professional development. Students have fun and learn a lot. Educators share ideas and acquire new skills. mathforum.org

5. Today we may… • Explore activities focused on problem solving and communication • See samples of students’ work • Learn how we used technology to support and enable problem solving • Get ideas for designing activities with multiple levels of support • Find out how problem solving helped all our students make connections between different areas of mathematics

6. UnderstandingtheProblem

7. Greta’s Garden(Scenario Only) • Greta has a vegetable garden. She sells her extra produce at the local Farmer’s Market. One Saturday she sold $200 worth of vegetables — peppers, squash, tomatoes and corn. • Greta received the same amount of money for the peppers as she did for the squash. • The tomatoes brought in twice as much as the peppers and squash together. • The money she made from corn was $8 more than she made from the other three kinds of vegetables combined.

8. Greta’s Garden What do you notice?

9. Greta’s Garden What do you wonder?

10. The Procedure What do you notice? • Independent Thinking (1 min) • Collaborative Sharing in Heterogeneous Groups (1 min) • Made Class List of Noticings - input from all groups What do you wonder? • Independent Thinking (1 min) • Collaborative Sharing in Heterogeneous Groups (1 min) • Made Class List of Wonderings - input from all groups

12. Students practiced solving the problem using guess and check.

13. Guess and Check Vocabulary Constraints - Something that is true about the quantities in the problem Quantities – Something in the problem that is measured

14. Kaytee’s ContestThe Constraints: 1.) Two cows, Bertha and Billy, together weigh a total of 1, 696 pounds. 2.) Bertha weighs 848 pounds more than Billy All work from this problem was done by 6th graders. Each grade has one general education class and one inclusion class.

15. Name: __________________________ Identify the constraints. Guess and Check Use the constraints & show your work for at least 3 guesses below. Guess 1Guess 2Guess 3 Keep going until you find a guess that works.

16. Kaytee’s Contest • 1st problem where we had students identify the constraints instead of noticings & wonderings • Most students in 6th grade were successful in identifying the constraints but they didn’t know how to use them to guess and check • We discovered we needed to model how to use the constraints to make a guess then check for an answer. (created four steps for guess and check) • Students did not know how to work together in groups and stay on task.

17. Work shows limited organization, challenging to understand General Education Students – Both are Proficient

18. It was a common occurrence for students to erase their guesses and work if it was not correct. This happened regardless of whether they were a general or special education student.

19. No work or explanation, no guesses, student only wrote one answer which they thought was correct Resource Student – Proficient

20. No work or explanation, only guesses – Student did not know how to use their guess to check for an answer General Education Student – Advanced Proficient

21. Student understood the constraints and how to use their guess to determine if it was accurate. General Education Student – Not Proficient

22. This student explained why his guess did not satisfy a constraint in the problem. NOTE: This student does very little in math class most days, but on this day he had one of the best papers in 6th grade. General Education Student – Not Proficient

23. Guess & Check Steps 1.) Identify the Constraints 2.) Make a Guess 3.) Check Your Guess 4.) Repeat as Necessary

24. A Cranberry CravingThe Constraints: 1.) Each day Carissa ate 7 more cranberries than she ate the day before, beginning on Thursday, Thanksgiving Day. 2.) Carissa ate a total of 161 cranberries in 1 week. Work from this problem was done by 6th,7th and 8th graders. Each grade has one general education class and one inclusion class.

25. A Cranberry Craving • We found that students were able to understand the constraints and some were even able to begin guessing and checking without a teacher having to model the first guess. • This problem allowed students to focus on using one constraint to solve for an answer. • Our goal in this problem was to have students use their previous guesses to make more accurate subsequent guesses. • We started to see more organization from students when guessing and checking their responses. • Some 7th graders in the general education class attempted to solve this problem using their knowledge of algebra.

26. With this problem, students began to show some type of organization in their responses when guessing and checking. In fact, many students chose to make a table. General Education Student – Not Proficient

27. Student’s first guess was too high to satisfy one constraint, yet he still guessed a larger number for his second guess. Our aim was to have students use previous guesses to make more accurate future guesses. General Education Student – Not Proficient

28. This student went beyond our expectations of using previous guesses to make more accurate subsequent guesses. The student’s first guess was very close to satisfying the second constraint. Since the first guess had a total of 154 cranberries and she needed 161, she was only short 7 cranberries. Since a week consists of 7 days, the student determined that Carissa ate one more cranberry each day than she originally thought in her first guess. General Education Student – Proficient

29. This class consists of general education students, most are proficient in math. Some 7th graders were very eager to try solving the problem using their knowledge of algebra. Although they needed prompting, many students were able to write an expression to show the number of cranberries that were eaten each day. Some students were able to write an equation from the expressions and there were even a few that were able to solve the equation using algebra skills.

30. The student writes an expression for each day that relates to the number of cranberries Carissa ate each day. The student then collects like terms and writes an algebraic expression in simplified terms. General Education Student – Proficient

31. This work was completed after he solved it using guess and check. In my experience, it is a challenge for resource students to show each step when solving an algebraic equation. For this reason I accept any work that these students write as long as they also have the correct answer. The work this student demonstrates shows a clear understanding of the problem in algebraic terms. By analyzing the work closely, I can deduce what steps the student took to solve the algebraic equation. ESL and Resource Student – Not Proficient

32. This student had an unique approach to writing an equation and collecting like terms. General Education Student – Advanced Proficient

33. Ostrich Llama CountThe Constraints: 1.) There are 47 heads. 2.) There are 122 legs. The Question How many ostriches and llamas are on the farm? Work from this problem was done by 7th and 8th graders. Each grade has one general education class and one inclusion class.

34. Ostrich Llama Count • We chose this problem as one that would deter students from using algebra to solve for the answer. • Students had to satisfy the first constraint, then check to see if the second constraint was also satisfied. • Most students demonstrated some type of organization when solving for the answer. • Students started to feel successful with problem solving. We started to hear comments such as, “I get it now!” • We saw some unique problem solving strategies – in one case, students had two very different approaches to solving this problem even though both drew a picture.

35. The student demonstrates knowledge of the problem and constraints. Although it is hard to follow, there is some type of organization of ideas. Grade Resource Student – Proficient

36. Student demonstrates knowledge of the problem and constraints. The paper shows organization of guesses and checks. General Education Student – Not Proficient

37. FRONT BACK In certain situations, this student can be a challenge because of poor choices regarding his behavior. On this day, the student isolated himself at a desk and worked diligently for 20 minutes. He produced guess after guess and yet did not give up. He did not stop, not even to talk to his peers. When finally solved, he commented, “I was always a little more or a little less and had trouble getting the right answer.” Since he clearly shows ability to use the constraints to check his guesses and the patient to do the thorough checking, he’s ready to focus on making better next guesses. General Education Student – Not Proficient

38. Since this student solved the problem before most of his peers, we asked him to write an explanation of how he solved the problem. General Education Student – Advanced Proficient

39. This was written on a student’s paper. General Education Student – Proficient

40. Resource Student Collaborative Group Consisted of one resource, one self-contained and one general education student. The self-contained student is the only group member proficient in math. Self-Contained Student (except for math, science & specials)

41. Student drew a picture and, in doing so, was able to solve the problem. This was a different approach from other students. General Education Student – Not Proficient

42. Here is a student who drew a picture as a strategy to solve the problem. Interestingly, the student’s drawing became the guess and she then needed to check her drawing to see if her guess was correct. Resource Student – Not Proficient

43. Wheels R UsThe Constraints: 1.) ¼ are tricycles & the rest are bicycles 2.) total of 45 wheels Work from this problem was done by 6th & 7th graders. Each grade has one general education class and one inclusion class.

44. Wheels R Us • The student must understand the constraints in order to solve the problem. • This problem also requires an understanding of number sense and fractional parts. • Students who previously hesitated when asked to problem solve, quickly engaged themselves in finding the answer using guess and check. • We started to see reasonable guesses from students that were close to the actual answer, resulting in students attempting a fewer number of guesses to solve the problem.

45. This student did not understand the constraints in the problem or have a clear understanding of parts of a whole. ESL Student – Not Proficient

46. The student concluded that… This student either did not understand the constraints or does not have a concrete understanding of fractions. General Education Student – Proficient

47. This is the same student who drew a picture to solve a previous problem yet still used a guess and check approach to solve the problem. The student is clearly a visual learner. Resource Student – Not Proficient

48. Attempt 1 Attempt 2 One of the 8th grade students who resisted the strategy guess and check and thought he could rely on his algebra skills to problem solve, could not write an algebraic equation to solve for the answer. General Education Student – Proficient

49. This 7th grade student has a strong foundation of number sense. Therefore, he was easily able to use his knowledge of number sense and the problem solving strategy guess and check to solve for the answer. Resource Student – Proficient

50. Eating ContestThe Constraints: 1.) Josh ate 8 more beans than Caleb. 2.) Elsie ate 15 fewer than twice as many as Caleb. 3.) Sol ate 30 more than ½ of Caleb. 4.) The children ate a total of 176 beans. Work from this problem was done by 6th, 7th and 8th graders. Each grade has one general education class and one inclusion class.