Warm up: Two machines one job

1. Warm up: Two machines one job • Ron’s Recycle Shop was started when Ron bought a used paper-shredding machine. Business was good, so Ron bought a new shredding machine. The old machine could shred a truckload of paper in 4 hours. The new machine could shred the same truckload in only 2 hours. How long will it take to shred a truckload of paper if Ron runs both shredders at the same time?

2. How I did it Shredder 1 • 1 truck in 4 hours = 1 truck in 240 min. • =1/240 load/ min Shredder 2 • 1 truck in 2 hours = 1 truck in 120 min • 1/120 load/ min. • 1/240 x + 1/120 x = 1 truckload • 3/240 x = 1 truckload • 1/80x = 1 truckload • X = 80 min.

3. How my wife did it • Shredder 1 • ¼ of a load done in 1 hour • Shredder 2 • ½ of a load in 1 hour • Combined • ¾ of a load in 1 hour • .75(x hours) = 1 load • X = 1/.75 • X = 4/3 hours = 1 1/3 hours • X = 1 hr 20 minutes

4. 1 truckload + 1 truckload = 1 truckload 4 hrs 2 hrs x hrs • ¼ + 2/4 = 1/x • ¾ = 1/x • 3x = 4 • X = 4/3 = 1 1/3 hr • X = 1 hr 20 min = 80 min.

5. PROBLEM #2 • 1 mile track • Car goes 40 mph for 1 mile • How fast does the car need to go to average 80 mph for 2 miles?

6. Math Assessment & Instructional Planning Chris Borgmeier, PhD Portland State University

7. Assessing Math

8. Example of an Informal Math Assessment of Computation Skills • 7th grade math class • Behavioral & Academic challenges in class • Where to start…. • Assess the problem(s)…

13. Making Curriculum Based Assessments • Fluency with telling time • Fluency with Counting Money • Worksheets • How many Correct in a minute? • # of errors • What types of errors did the student make?

14. During & After Instruction • Evaluate work to identify specific error patterns • In class this can be done through monitoring and looking at work • Look for common mistakes across students, which may signal the need for clearer, more explicit instruction • Look for individual student mistakes & provide 1:1 support while class during individual seatwork time • We don’t want students practicing misrules

15. Evaluating the data & making Instructional Decisions • Math Facts • Build in time for students to practice flash cards or double digit problems as a part of the day, matching task difficulty and task focus with skill level and specific skill deficits • Beyond Math Facts – more complex problems • Identify specific errors in strategies • Watch students work out problems • “Show me your work” – look at work to see what errors occur • Have students talk you through the work they do

16. Math Facts • Important to spend a significant amount of instruction on math facts • Alternatives: • Pocket sized facts chart • Calculator • But, need to teach student to use these alternatives to mastery

17. Instructional Strategies • Distributed Practice – much practice in small doses • for example, two 15-minute sessions per day, rather than an hour session every other day • Small numbers of facts per group to be mastered at one time • ...and then, frequent practice with mixed groups • Interactive and intensive practice (motivational materials such as games can help) • ... attentiveness during practice is as crucial as time spent

18. Worksheets are not Instruction Assessment v. Instruction • Math worksheets/packets are not instructional • If student is lacking skills, or doesn’t know facts, simply giving them work is not going to solve that problem – need to provide instruction • Worksheets will allow me to ID where students need instruction • i.e. student struggles with (x 6) and (x 8), but seems to have the rest of his multiplication facts… Observation – prefers x1, x2 and x5… he goes through and answers all of those first

19. Good InstructionTeach effective & efficient Strategies • Increasing task efficiency through effective strategies can greatly increase likelihood and student tolerance to do assigned tasks • This is where research based curriculum and strategies are important • Having students talk through strategies or watching their work can help to ID ineffective or inefficient strategies • Examples • 7+5 • Take 2 from 7 • Add 5 +5 = 10 • Add 2 taken away previously = 12

20. Be Wary of “Discovery” Learning Classrooms • For students with learning needs, it is important that we do not leave learning to chance or student “discovery” • For most tasks there are more and less effective & efficient ways of coming to the correct answer • For students with learning needs, and most students, it is beneficial to explicitly teach them the most effective & efficient strategies to use – you can still teach students there is more than 1 way to come to the right answer, but instruct the most effective & efficient way • Once students start using and practicing less efficient, or effective strategies – they will hold on to them and become more and more difficult to break, even if they lead to student frustration

21. Direct Instruction & Teach Effective Strategies • Don’t make students figure out how to do it on their own • Model-Lead-Test – using the most efficient and effective strategies • A great resource for Math Instruction is • Designing Effective Mathematics Instruction: A Direct Instruction Math • Stein, M., Silbert, J. & Carnine, D.W.

22. Math Operations

23. Error Analysis • Need to take time and analyze what errors the student is making, while either: • Looking at student’s written answers OR • Observing or having student “talk through” the problem as they do it • Noting what the student is doing right & what they Can Do is also important in an error analysis • If errors are consistent, there is often a logical thinking error, or logical misuse of rules that the student is applying • If there is no rhyme or reason to the errors, the student may be guessing

24. Common Mistakes to Look for • Unreliable at paying attention to the operational sign • Additions – carrying • Subtraction – borrowing & what to do with 0’s • Multiplication/Division - sequencing steps in complex operations

25. Math Operations - Addition • Common Errors w/ Multi-step mathematics: • Look at their work & try to understand -- Does the student understand the rules for: • Addition • Carrying What errors might look like: (no ticks) 776 776 776 +128+128+128 904 8914 894 1 1

26. What error pattern do you see?1st grade student

27. What error pattern do you see?1st grade student

28. Do this problems 301000 -118736 291 0 9 91 18 2 26 4

29. Subtraction • Borrowing 894 716 776 7006 -547 -128-108-1257 347 588 668 5749 • ERRORS 894 716 776 7006 -547 -128-108-1257 353 694 650 6251 8 1 6 101 61 6991 1 5 8 Made it work by putting bigger number first Borrowed by own rules Made it work by putting bigger number first Confused carrying & borrowing

30. Multiplication • There are many more opportunities for errors in more complex multiplication problems. • Student doesn’t know: • multiplication facts • Carrying in multiplication • Place holders • Sequence of Steps • Not neat in lining up numbers • Learned an inefficient strategy, with as many or more opportunities for error • 14 X 7 = 14+14+14+14+14+14+14

31. Multiplication • Common Errors 453 534 305 178 x112x345x 43x420 456 879 915 356 1220692 - 2135 6776 2 2 1 1 1 1 1 Added instead of multiplied Multiplied straight down Doesn’t know what to do w/ 0 on top; carrying mistake Placeholder problem

32. Division • There are even more opportunities for errors in long division problems. • Most often need to use addition, subtraction, multiplication and division to solve long division problems • Student doesn’t know: • Basic Facts in any of above areas, or doesn’t know operations for multiplication • Order of Operations for division • Place Holders • Not neat in lining up numbers • Doesn’t understand remainders

33. Lots of Room for Error Look for Patterns across problems 229 r 12 36 8756 36 36 -72x2x3 105 72 108 -72 336 36 -324x9 12 324 Multiplication Subtraction Sequence of Steps Alignment 5

34. Overview • This presentation is a summary of major findings from three syntheses of research on effective practices for students with mathematics difficulties including over 50 studies. • The practices are essential for developing interventions for students who require more than what typical classrooms can provide.

35. 8,727 papers Elementary or secondary study in mathematics? 956 studies Experimental study? 110 studies High quality research designs? Identifying High Quality Instructional Research

36. Method • Included only studies using experimental or quasi-experimental group designs. • Included only studies with LD or LD/ADHD samples OR studies where LD was analyzed separately. • Expanded to include studies with students with low achievement through 2003 (n=52)

37. Who can benefit from these findings? • Students who: • enter school with very limited knowledge of number concepts and counting procedures • receive inadequate instruction in previous years of schooling and fall behind their peers • regardless of motivation, quality of former mathematics instruction, and number knowledge and number sense when entering school still continue to experience problems

38. How were the effects of particular practices compared? • These syntheses compared the relative effects of instructional practices using “effect sizes.” Effect sizes are a proportion of a standard deviation. >.80 large Educationally Significant >.40 moderate >.20 small <.20 extremely small/negligible

39. Areas of Major Findings • Visual and graphic depictions of problems • Student think-alouds • Explicit instruction • Peer-assisted learning • Formative assessment

40. Effect Sizes for Instructional Variables

41. Findings: Visuals and Graphic Depictions of Problems • Graphic representations of problems and concepts are widely used in texts both in the U.S. and in nations that perform well in international comparisons • Teaching students to use graphic representations of the underlying concepts of a problem results in moderate effects.

42. Findings: Visuals and Graphic Depictions of Problems • Effects were larger when teachers provided students with multiple opportunities to apply graphic representations to specific problems • Effects were also enhanced when teachers taught students to select appropriate graphic representation and why a particular representation was most suitable

43. Visuals to depict different problem types. Jitendra, A. K., Hoff, K., & Beck, M. (1999). Teaching middle school students with learning disabilities to solve multistep word problems using a schema-based approach. Remedial and Special Education, 20(1), 50-64.

44. Findings: Visuals and Graphic Depictions of Problems • When teachers used graphic representations to demonstrate problems, results were much less consistent. • Visuals were not particularly useful unless students were provided opportunities to practice using them. • Concrete-Representational-Abstract (CRA) approach seems promising based on 3 studies. Teachers model problems with concrete manipulatives to ensure students understand before moving to more abstract representations

45. Findings: Student Think-Alouds • Encouraging students to verbalize their thinking and talk about the steps they used in solving a problem – was consistently effective • Verbalizing steps in problem solving was an important ingredient in addressing students’ impulsivity directly

46. Findings: Student Think-Alouds • Verbalizing appeared to be most effective when multiple approaches to solving problems were demonstrated and students were encouraged to think-aloud as they solved multiple practice problems.

47. Findings: Explicit Instruction • Explicit instruction consistently resulted in large effects both for learning single skills as well as multiple related skills in complex problem solving. • These findings must be tempered by the fact that the measures on which the effect sizes were calculated were all researcher-developed.

48. Findings: Formative Assessment • Formative assessment is the process of collecting data on a randomly selected array of relevant topics at regular intervals (e.g. once per week or twice a month) • Evidence has shown that this approach is superior to the typical weekly or biweekly unit tests that appear in many texts

49. Findings: Formative Assessment • Formative assessment use has consistently lead to low or moderate effects on mathematics achievement • Feedback based on formative assessment coupled with specific suggestions for intervention strategies (e.g. problems for practice, alternate ways to explain a concept) improved effects • This type of feedback was consistently effective for special education teachers.

50. Findings: Feedback to Students about their Performance • Providing students with feedback about their performance resulted in moderate effects. • For students with disabilities, these effects were much smaller.