Teaching Math to Students with Learning Differences and/or ADHD

Supporting Struggling Students in Math Ellen Engstrom, M.A.

This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Workshop Goals  Understand the cognitive / developmental process of acquiring mathematical learning  Understand the cognitive processes that contribute to math difficulties  Review the beneficial practices for teaching math to struggling students – Graphic organizers – Learning strategies – Math study skills – Promoting math metacognition

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Teaching Math to Students with Learning Differences and/or ADHD

Development of Mathematical Concepts and Skills

This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Early Childhood  Infants < 1 year are sensitive to groups of 3 and 4  Toddlers (18 months) show and understanding of simple and ordinal relationships (< >)  Preschoolers show and understanding of counting, ordinality, and cardinality  Preverbal number system becomes integrated with emerging language through the use of number words This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Teaching Math to Students with Learning Differences and/or ADHD

Elementary School  By the end of elementary school children should have… – Memorized basic facts: addition, subtraction, multiplication – Solve complex multi-column problems

 Poor understanding of base 10 leads to errors in borrowing This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Issues that affect learning in elementary school  Base 10 structure of the Arabic number system  Learning base 10 system is dependent on – Educational practices – Number words used in the language – English and French words do not correspond to the base 10 representation of the quantity

 Failure to learn base 10 results in difficulty counting past 10 and learning to rename number groups This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Teaching Math to Students with Learning Differences and/or ADHD

Secondary School (12-18)  Major task is to learn to solve arithmetical and algebraic word problems  Factors that affect this process: – Wording of the problem – Ease with which sentences can be translated from verbal to mathematical representations – Number of sentences which must be translated and d integrated d

 High quality instruction can act as a buffer against memory and language deficits This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Cognition g and Math Learning g

This material is based upon work supported by the National Science Foundation under Grant No. HRD072652. Any opinions, findings, and conclusions of Recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation

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Teaching Math to Students with Learning Differences and/or ADHD

Relationship between innate and learned mathematical ability  Number sense is genetic  Exact calculation requires cultural tools – Symbols – Algorithms

 Calculation accomplished by parts of the brain that have evolved for other purposes  Nature provides number sense, and culture provides numerals and number words – Stanislas Dehaene

This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Challenge of memory  Human memory is associative – Associations cause bits of knowledge to interfere with each other – 7 X 6 OR 7 + 6? – Multiplication facts have to be stored in a form that doesn’t fit well with how human memory is organized – Adults make errors in single digit multiplication 10-25% of the time This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Teaching Math to Students with Learning Differences and/or ADHD

Challenge of Language  English has special words for the numbers from 11-19 and for the decades 20-90

 Language limits memory span of digits a person can hold

This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National

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Science Foundation

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Why do students struggle with math? Dehaene (1999)

 Exact calculation is language dependent  Approximation relies on non-verbal visuo-spatial networks of the left and right parietal lobes  Innate ability for approximation conflicts with requirements for calculation and problem solving, and both are mediated by language  Calculation and higher level math more difficult than other cognitive processes This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Teaching Math to Students with Learning Differences and/or ADHD

Why do students struggle with math? (Geary, 1993, 2000)

This material is based upon work supported by the National Science Foundation under Grant No. HRD0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

 Semantic memory deficits – Difficulty with math fact retrieval

 Procedural memory – Errors carrying out procedures – Difficulty sequencing multi-step procedures – Delays understanding concepts

 Visuo Visuo-spatial spatial deficits – Misalignment of numbers – Misinterpretation of place value – Great difficulty learning geometry Groves Academy Outreach

Why do students struggle with math? Swanson (2006)

 Memory problems may reflect poor coordination of information in the executive system  Working memory system is either: – Not accessing numerical information from the phonological system – Failing to provide adequate capacity for processing information This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Teaching Math to Students with Learning Differences and/or ADHD

Working ki Memory and d Inattention

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Working Memory

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Teaching Math to Students with Learning Differences and/or ADHD

Working memory and math deficits  Simple arithmetic depends on speedy and efficient retrieval from long-term memory  Temporary storage of numbers when attempting to solve a problem is crucial  Poor recall of facts leads to difficulties executing calculation procedures and immature problem-solving strategies This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Inattention  Inattention is present in many brain function disorders  Inattentive behavior predicts a poor response to reading and math instruction  Poor attenders do worse on tests of oral reading fluency. g risk for  Poor attenders are at high academic failure This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National

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Science Foundation

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Teaching Math to Students with Learning Differences and/or ADHD

Working Memory and Attention This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Working Memory (executive attention)

Academic Attainment

Inattentive Behavior

Risk Triad Tannock, 2008 Groves Academy Outreach

Executive Function and Math  What is the impact p of p poor executive function on a student’s math achievement?  In what ways does poor executive function impact math students?

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Teaching Math to Students with Learning Differences and/or ADHD

Increasing Students’ Attention  Increase active engagement – Use activators to start the lesson – Vary the pace – Change activities throughout the class period – Alter groupings

 Increase instructional supports – Model procedures – Teach strategies – Use cues and prompts This material is based upon work supported by the National Science Foundation under Grant No. HRD-0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Math Instructional Strategies g

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Teaching Math to Students with Learning Differences and/or ADHD

Beneficial practices for teaching math to struggling students  Graphic organizers – Teacher directed to support instruction – Student directed to reinforce concepts

 Strategy instruction – Verbal strategies – Visual strategies – Organizing strategies  Problem solving  Note taking

http://www.tricl.org/algebra

This material is based upon work supported by the National Science Foundation under Grant No. HRD0726252. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the Groves Academy Outreach views of the National Science Foundation.

Verbal Instructional Strategies  PEMDAS Acronym: Memory strategy for remembering the sequence of steps for simplifying lf algebraic l b equations – Parentheses – Exponents – Multiplication – Division – Addition – Subtraction

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Teaching Math to Students with Learning Differences and/or ADHD

Verbal Instructional Strategy  KNOW Acronym: Memory strategy for remembering important problem solving steps – Key words – Numbers – Operation – Work it out

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Visual Instructional Strategies  FOIL (First – outer – Inner – Last)

x23  2

=

3x  2 x 2

2

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Teaching Math to Students with Learning Differences and/or ADHD

Teaching a Problem Solving Strategy  1. Provide an advance organizer of the strategy  2. Provide teacher modeling of the strategy steps  3. Provide guided practice  4. Provide independent student practice  5. Provide feedback and correction  6. Provide regular review to promote generalization of the strategy Groves Academy Outreach

Example: Problem Solving Strategy  Reread the problem  What is known?  What is unknown?  What is the formula?  Enter the known information and solve for the unknown Groves Academy Outreach

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Teaching Math to Students with Learning Differences and/or ADHD

Distance, Rate, and Time  Carla and Jake leave town at the same time, traveling in opposite directions. If C l travels Carla l at 55 mph h and d Jake k travels l at 35 mph, how long will it take for them to be 330 miles apart? D = R x T  How can we improve the instruction of this type of problem? Groves Academy Outreach

Graphical Representation  Carla and Jake leave town at the same time, traveling in opposite directions. If Carla travels at 55 mph and Jake travels at 35 mph mph, how long will it take for them to be 330 miles apart?  Known Information: they leave at the same timethey have the same time, t. – Rate for Carla is 55 mph – Rate for Jake is 35 mph – Total distance traveled is 330

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Teaching Math to Students with Learning Differences and/or ADHD

Graphical Representation  Carla and Jake leave town at the same time, traveling in opposite directions. If Carla travels at 55 mph and Jake travels at 35 mph mph, how long will it take for them to be 330 miles apart?  Unknowns are: – How far each one traveled individually. – How o long o g they t ey were e e traveling. t a e g  Formula: D = R x T

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Graphical Representation  Carla and Jake leave town at the same time, traveling in opposite directions. If Carla travels at 55 mph and Jake travels at 35 mph mph, how long will it take for them to be 330 miles apart? 330 miles

Carla’s distance

Jake’s distance

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Teaching Math to Students with Learning Differences and/or ADHD

Graphical Representation

Rate

Time

Distance* *

Carla

55 mph

t

55 t

Jake

35 mph

t

35 t

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Graphical Representation  Now fill in what we know: – 55t + 35t = 330  Combine like terms – 90t = 330  Divide both sides by 90 t = 3.66 hours  They both traveled for 3 2/3 hours.  Carla went 55 (3.66) = 201 .66 miles.  Jake went 35 (3.66) = 128.33 miles.

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Teaching Math to Students with Learning Differences and/or ADHD

Heirarchical Graphic Organizer

Groves Maccini Academy & Gagnon, Outreach

2005

The Teacher Says… “Write five different examples of polynomials for Each column.”

1.

1.

1.

2.

2.

2.

3.

3.

3.

4.

4.

4.

5.

5.

5.

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Teaching Math to Students with Learning Differences and/or ADHD

Sequence Charts 1. Search the word problem 2. Translate the problem 3. Answer the problem 4. Review the solution Maccini & Gagnon, 2005 Groves Academy Outreach

Effective Use of Graphic Organizers  Consistent, coherent, and creative usage  Use of both teacher-directed teacher directed and studentstudent directed approaches  Use partially completed graphic organizers  Highlight information in text P Provide id cues att the th b bottom tt off a blank bl k graphic organizer  Provide group activities Groves Academy Outreach

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Teaching Math to Students with Learning Differences and/or ADHD

Math Study y Skills

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The Master Notebook

Course Syllabus

class notes

Semester Calendar

tests/quizzes Daily Calendar of Assignments & Appointments

completed homework

handouts

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Teaching Math to Students with Learning Differences and/or ADHD

Two Column Notes Main ideas Borrowing money

Supporting Details when you borrow money from a bank, credit union or other lending institution, institution you pay for the privilege of using the money until it is repaid. This privilege is called “interest.” When you deposit money into the bank or other savings institution, they pay interest to you for the privilege of using your money.

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Three Column Notes Concept/Main Idea Slope p of a line

Explanation

Example

p is the ratio of Slope the vertical change to the horizontal change in a line Slope = m

m = change g in y change in x m = y2 – y1 (rise) x2 – x1 (run)

Slope intercept form y = mx + b m = slope b = y intercept

y = -3x-2 m = -3 1 b = (0, -2)

y intercept

y = (0, b)

Where a given line crosses the y axis

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Teaching Math to Students with Learning Differences and/or ADHD

Templates Step 1

Slope Intercept Formula

y = mx + b

Step 2

m is slope

m=

Step 3

b is y intercept

b=

Step 4

point of y intercept

(0,b) = (0, ___)

Step 5

Graph Groves Academy Outreach

Using g Technology gy to Teach Math  Instructional Resources  Teacher Training

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Teaching Math to Students with Learning Differences and/or ADHD

Online Math Instruction and Support Review some of the math lesson plans available on the Internet-based Activities for Higher Education web page from RWLO's Resource Library web site at: http://www.ciese.org/pathways/rwlo/search.php?filter =math&show=menu Consider YouTube instructional videos: Khan Academy Free ee instructional st uct o a videos: deos www.khanacademy.org a acade y o g

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Real World Learning Objects

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Teaching Math to Students with Learning Differences and/or ADHD

Real World Resource Library

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Promoting g Math Metacognition g

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Teaching Math to Students with Learning Differences and/or ADHD

How Can We Promote Metacognition in Math?

 What ideas do you have for fostering metacognition in math instruction?

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Practices to Foster Metacognition  Homework process diagnosis worksheet  Math introductory questionnaire  Math notebook evaluation  Math test anal analysis sis

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