David Stevens, Ed.D.Developmental Psychologist

How Math LearningDifficulties Develop

01 Developmental Readiness

Some students arrive inkindergarten and first grade without having the necessary foundation for math instruction.

Robbie Case & Sharon Griffin

Number Knowledge Test

0 25 50 75 100

100

100

85

72

64

96

92

93

79

24

38

18

At Risk Not at Risk

Have a candy. Here are 2 more. How many do you have?Which pile has more?(Show two piles of chips)

How many triangles are there?(Show mixed array)If you had 4 candies and received 3 more, how many would you have?

What comes two numbers after 7?

Which number is bigger/smaller? (Show two arabic digits)

Robbie Case & Sharon Griffin

02 Mile Wide Inch Thick

The large number of required topics does not allow some students the time they need to understand foundational concepts.

Schmidt, McKnight, & Raizen 1997

Grade

1st 2nd 3rd 4th 5th 6th 7th 8th

3 3

8

15

2017 16

18

24 24 25 25

34 3537

39

Top Achieving Countries NCTM

Number of Topics per Grade

Center for Research in Math & Science Education, Michigan State University

Center for Research in Math & Science Education, Michigan State University

80%

38%

Percent Correct

“There are 600 balls in a box, and 1/3 of the balls are red.

How many red balls are in the box?”

Grade 4 International Test Question

03 Teacher Training Programs

Teacher training programs traditionally emphasize Language Arts instruction.

%

25%

50%

75%

100%

13%

Colleges Providing Sufficient Training

www.nctq.org

04 Memorization over Understanding

Some students move forward in the early grades only by using counting and memorization strategies.

How Many cookies?Which is larger 8 or 9?5 + 3 = ?4 x 6 = ? 328 + 486

2 + 3 = ?3 + 2 = ?2 + ? = 5 5 = 3 + ?5 - 3 = ?5 - 2 =?

05 Curriculum Becomes More Complex

As the curriculum becomes more complex the counting and memorization strategies are not effective.

The parts are of equal size

There are a specific number of parts

The whole is divided

The parts equal the whole

The complexity of fractions make it more likely that students will forget that fractions represent quantities.

This leads to memorization without understanding:

“Find the common denominator, then add”“Flip it and multiply”“The bigger the denominator the smaller the fraction”

06 Outcomes

Many Students are not developing the foundation they need for long-term success in mathematics.

%

25%

50%

75%

100%

34%

Grade 8 Passing Rate

National Center for Educational Statistics -- 2009

07 Solutions

Recent research documents recognize these problems and agree on proven solutions.

Changing Course

“Teachers face long lists of learning expectations to address at each grade level, with many topics repeating from year to year. Lacking clear, consistent priorities and focus, teachers stretchto find the time to present important mathematical topics effectively and in depth.”

www.nctm.org/focalpoints

NCTM Now Recommends

Instruction should devote “the vast majority of attention” to the most significant mathematical concepts.

Focus on developing problem solving, reasoning, and critical thinking skills.

Develop deep understanding, mathematical fluency, and an ability to generalize.

www.nctm.org/focalpoints

National Math Panel Report

“The manner in which math is taught in the U.S. is "broken and must be fixed."

www2.ed.gov/about/bdscomm/list/mathpanel/reports.html

National Math Panel Recommendations

Be "streamlined and should emphasize a well-defined set of the most critical topics in the early grades."

Emphasize "the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts."

Teach with "adequate depth."

Have an "effective, logical progression from earlier,less sophisticated topics into later, more sophisticated ones."

Have teachers regularly use formative assessment.

Math curricula should:

www2.ed.gov/about/bdscomm/list/mathpanel/reports.html

1. Screen all students and provide interventions to students identified as at-risk.

2. Instructional materials for students should focus intensely on in-depth treatment of whole numbers.

3. Instruction during the intervention should be explicit and systematic.

4. Include instruction on solving word problems that is based on common underlying structures.

5. Students should work with visual representations.

6. Devote ten minutes in each session to fluent retrieval of basic arithmetic facts.

7. Monitor student progress.8. Include motivational strategies.

ies.ed.gov

For over a decade, research studies of mathematics education in high performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country.

To deliver on the promise of common standards, the standards must address the problem of a curriculum that is “a mile wide and an inch deep.”

These Standards are a substantial answer to that challenge.

www.corestandards.org

www.corestandards.org

Kindergarten Standards

The Best Practices

Streamline the curriculum to emphasize the big ideas of mathematics.

Individualize learning experiences to each student’s developmental level.

Adapt to the learning style of each student.

Bridge from the concrete to the abstract.

Provide concrete feedback and levels of scaffolding to provoke thinking.Develop knowledge of concepts AND math facts.

Promote cognitive flexibility and problem solving.

Draw on intrinsically motivating aspects of mathematics to inspire learning.

Teach for mastery.

Introduce concepts at the simple level and build to the complex level.

Help students make connections between concepts.

Guide instruction with formative assessment.

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