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LD Teaching Strategies for Your Classroom: Math

Steve [email protected]

abspd.appstate.edu

Teach from Concrete to Representational to Abstract (CRA)

Concrete Representational

Volume = length x width x height

Volume = 3 x 3 x 3

Abstract


Use ManipulativesWhen students can feel, touch and experience math in action, they remember it! Manipulatives, objects students can move around that represent math concepts, help LD learners better understand math concepts.

Using their fingers benefits LD learners too. “Evidence from brain science suggests that far from being ‘babyish,’ [using fingers] is essential for mathematical achievement” (Boaler and Chen, 2016).

Manipulatives we can use in math class include:

Fraction Strips

Algebra Tiles

Use a Problem Solving ApproachTeach math using a problem solving approach with real world application. This will:

Increase student motivation and problem solving abilities

UPS ✔ MethodLD Teaching Strategies for Your Classroom: Math

1. Understand the problem

What are you asked to do?

Will a picture or diagram help you understand the problem?

Can you rewrite the problem in your own words?

2. Create a plan

Use a problem solving strategy:

Guess and check Solve an easier problemMake a list ExperimentDraw a picture or diagram Act it outLook for a pattern Work backwardsMake a table Change your viewpointUse a variable

3. Solve

Be patient

Be persistent

Try different strategies

4. CheckDoes your answer make sense?

Are all the questions answered?

What other ways are there to solve this problem?

What did you learn from solving this problem?

Source: Polya, How to Solve It

Understand


Plan

Solve

Check


Mathematical Mindset Research Apply mathematical mindset research in your classroom that says:

Believe in YourselfOur beliefs about our own abilities actually change the way our brains work when we do math. Researcher Dr. Carol Dweck has discovered people either have a growth or a fixed mindset. A growth mindset believes that anyone can learn and grow their brains through hard work. The fixed mindset believes either we are smart or not and that smartness is not changeable (it's fixed). Dr. Dweck describes the difference between the two mindsets as follows:

"In a fixed mindset students believe their basic abilities, their intelligence, their talents, are just fixed traits. They have a certain amount and that's that, and then their goal becomes to look smart all the time and never look dumb. In a growth mindset students understand that their talents and abilities can be developed through effort, good teaching and persistence. They don't necessarily think everyone's the same or anyone can be Einstein, but they believe everyone can get smarter if they work at it."

Everyone Can Learn Math to High Levels

Brain research shows that there is no such thing as a math person, so it is possible for everyone to learn math to any level they want. Synapses (the tiny gaps between nerve cells where electrical impulses are sent to other nerve cells) fire in our brains creating new connections all the time. Our brains are like a muscle. As we exercise our brain, it grows larger.

Our brains are plastic, meaning they can grow and change all the time based on our experiences. An incredible example of this is a nine year old girl, Cameron Mott, who had half her brain removed during an operation, including the part that controls movement. She amazed doctors by how quickly she regained her ability to walk! Scientists concluded that her brain regrew the missing connections.


Struggles and Mistakes are Very ImportantResearchers have discovered that when we make mistakes in math, our brains grow more than when we work problems correctly. Brain synapses will fire twice, once when we make mistakes and again when we become aware of mistakes. Even if we don't realize we have made a mistake, our brains will grow because they are struggling. Math researchers have called this "the productive struggle."

Recognize the value of mistakes as important learning tools:

1. Create the classroom norm that you love and want mistakes. Ask students to write their mistakes on the board so everyone can see them. Other students have probably made similar mistakes so everyone can learn from them.

2. Don't just praise mistakes but talk about why they are important. Share the research about how the brain grows more when mistakes are made than when work is done correctly.

3. If students are not making mistakes, the work they are doing is not challenging enough. Always strive to push the boundaries of student learning.

4. Praise students for their effort instead of being “smart” or getting the right answers.

Speed is Not ImportantActually, it may be better to be a slow thinker in math! Laurent Schwartz, who won the Fields Medal (the math equivalent of the Nobel Prize or an Oscar), always thought he was stupid because he was a slow math thinker. He said:

"I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time.

"I'm still just as slow . . . At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant."

Maryam Mirzakhani, the 2014 winner of the Fields Medal, also described herself as a "slow" mathematician. She was told by a seventh grade teacher that she could not do math, but she believed in herself and went on to win the highest recognition in mathematics!

Instead of speed, what is important is to deeply understand math ideas and connections. Seeing math visually is also important. Math is not about memorization or calculation, instead it is about ideas, visualizations and connections!


Use Low Floor High Ceiling TasksLow Floor High Ceiling Tasks (LFHC) are those learning activities that all students can do but that can be extended to high levels. LFHC tasks allow students to work at different paces and take work to different depths at different times. Strong LFHC tasks are those that are also visual and lead to rich mathematical discussion (Adapted from Youcubed.org).

Low Floor High Ceiling Tasks are found at:

Youcubed.org

Illustrativemathematics.org

Nrich.maths.org

Graph Paper, Space and More . . . Using graph paper with large blocks helps LD students with tracking issues

Put plenty of white space between problems to allow for LD students to put their work

Some LD learners do better with problems shown horizontally instead of vertically:

.42 x 1.3 = 4.3 x .5 =


Computation and Calculators Use calculators in applied math (problem solving) so students can focus on solving problems

without getting lost in computation.

Color code the calculator key-pad, especially the + and x keys

Talking calculators allow learners to both hear and see the numbers entered

Calculators that show the display history allow learners to make sure they have entered the correct information

Do not force learners to memorize their multiplication tables before allowing them to do math.

“Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging . . . I [Dr. Jo Boaler, Stanford Professor] learned math facts through using them in different mathematical situations, not by practicing them and being tested on them. I grew up in the progressive era of England, when primary schools focused on the ‘whole child’ and I was not presented with tables of addition, subtraction or multiplication facts to memorize in school. This has never held me back at any time or place in my life, even though I am a mathematics education professor. That is because I have number sense, something that is much more important for students to learn, and that includes learning of math facts along with deep understanding of numbers and the ways they relate to each other” (Boaler, 2015).

Math Humor


Teach Fewer Concepts with More Depth“Unfortunately, too many teachers feel like they don’t have the time to give students the foundation that would allow their students to actually understand what is being taught. They may teach students procedures and tricks, hoping that they will retain those procedures long enough to at least pass the test. However, without foundational understanding, students rarely remember those procedures. How many times have teachers shown students how to add fractions with unlike denominators, only to discover a few weeks later that students have already forgotten the procedure? Or, they watched students apply the procedure for adding fractions when faced with a proportion problem? As a result, teachers reteach the same procedures over and over again, rarely successfully getting their students to understand when to use those procedures. According to Givvin, Stigler, and Thompson (2011):

“Without conceptual supports and without a strong rote memory, the rules, procedures, and notations they had been taught started to degrade and get buggy over time. The process was exacerbated by an ever-increasing collection of disconnected facts to remember. With time, those facts became less accurately applied and even more disconnected from the problem solving situations in which they might have been used. The product of this series of events is a group of students whose concepts have atrophied and whose knowledge of rules and procedures has degraded. They also show a troubling lack of the disposition to figure things out, and very poor skills for doing so when they try. This leads them to call haphazardly upon procedures (or parts of procedures) and leaves them unbothered by inconsistencies in their solutions. (p. 5)”

“Teachers think that they don’t have the time to spend on conceptual understanding of core concepts.But, perhaps teachers need to reconsider what it means to be college and career ready, and what itmeans to have a core set of skills that allow learners to meet the demands of both academic and lifepriorities. The National Center on Education and the Economy (NCEE) asked: What does it really mean to be college and work ready? They conducted a two-and-a-half year study to try to answer thatquestion. What they discovered is most of the math that is required of students before beginning college courses and the math that most enables students to be successful in college courses is not high school mathematics, but middle school mathematics. Ratio, proportion, expressions and simple equations, and arithmetic were especially important (NCEE, 2013).

“And, according to Redefining College Readiness, a report published by the Educational PolicyImprovement Center (Conley, 2007), college success requires key cognitive strategies such as analysis, interpretation, precision and accuracy, problem solving, and reasoning. Students who are ready for college possess more than a formulaic understanding of mathematics. They are able to apply conceptual understandings in order to extract a problem from a context, use mathematics to solve the problem, and then interpret the solution back into the context.”

Source: Curry, 2017


ResourcesBoaler, J. (2016). Mathematical mindsets. San Francisco: Jossey-Bass.

Curry, D. (2017). Where to focus so students become college and career ready. Journal of Research and Practice for Adult Literacy, Secondary and Basic Education 6(1).

Garnett, K. (1998). Math learning disabilities. Retrieved May 17, 2008, from http://schools.nyc.gov/documents/d75/ais/Math_Learning_Disabilities.doc

National Institute for Literacy (2010). Learning to achieve. Washington DC: US Government Printing Office.

U.S. Department of Education, Office of Vocational and Adult Education. (2014). Math works! guide. American Institutes for Research. Retrieved from: http://lincs.ed.gov/sites/default/files/Teal_Math_Works_Guide_508.pdf


http://schools.nyc.gov/documents/d75/ais/Math_Learning_Disabilities.doc

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