By Allen Broyles and Tom Pittard
What looks like a struggle with math may actually be a deficiency in the underlying cognitive processes required to compute math problems.
Bring up the topic of math to a group of adults and you will hear the predictable number of groans. But ask the same adults about the importance of math in today’s world, and they will all agree that it’s a critical skill for every student to have. We all acknowledge that math is difficult but essential. For the child who struggles with a learning difference, the difficulties with math are amplified.
Four kids and a math problem
In order to teach math responsively, we must first acknowledge the actual neuro-developmental demands that math places on a student. It is not useful to speak in terms of a student being good or bad at math. Four students may get the same problem wrong, but for four completely different reasons. The first student may understand the process perfectly, but make a simple fact error. The next may understand the process, but has working memory deficits that prevent keeping the problem’s individual steps in mind long enough to apply them. The third may inaccurately transcribe on paper the correct number he is holding in his mind. The fourth student may not understand the concept underlying the problem. To address their errors, each of these students needs a different approach to teaching, which requires an awareness of the many neuro-developmental processes required for successful math performance, as well as how learning differences can cause those processes to break down.
Math and the brain
Stanislas Dehaene, in Origins of Mathematical Intuitions (PDF) suggests that humans, and some animals, are born with a sense of magnitude or quantity that is hard-wired into the brain, which permits quick evaluation of about how many objects are in a scene, whether this number is more or less than another, and how this number is changed by simple addition and subtraction. This system in action has been observed in infants as young as four months. It is a general sense, though, not exact computation. To calculate, humans have developed an abstract system of numerals that maps onto this built-in sense of quantity, each function probably being represented by separate brain systems. As children get older, they move from an intuitive sense of quantity, to a more formal system of ordered numbers, which requires a much more complicated set of neuro-processes to be brought to bear.
Mel Levine, author of A Mind at a Time and other books on learning differences, identifies many of the brain’s processes that math requires:
Higher Order Cognition
Applied reasoning
Rule understanding and use
• Concept formation and linkage
• Analytical thinking
• Mental representation
• Metacognition |
Language
Semantics
Symbolization
Sentence comprehension
• Phonological awareness |
Memory
Pattern recognition
• Procedural recall
• Sequential memory
• Working memory |
Spatial Sense
Visual/Mental imagery
• Graphic representation
• Geometric perception
• Grapho-motor control |
Attentional Processes
Detail processing
• Production control
• Sustaining attention |
What does this mean for the student sitting in math class? The following example provides a quick glimpse into the complexities of understanding math. Consider the long division problem at right. The red numbers show the order in which students approach and work the problem. The first five of those steps are listed with a short description of that subtask.
The process repeats twice, through step 15. If you count the substeps of the embedded subtraction and multiplication problems within long division, a student must complete more than 20 steps, which require sustained attention to process, procedural recall, language processing, detailed paper organization, scattered visual tracking, and strong working memory. Together there are at least 13 of the above subprocesses at work in long division. If teachers just teach the order and process of the algorithm, instead of investigating at what point the process is breaking down, the student will likely continue to struggle.
Finding the weak spot
Teachers need to be attentive to the many facets of a student’s learning profile that may make math difficult, and design or alter instruction accordingly. While not all of the neuro-developmental processes mentioned above are discussed here, the following provides some general ideas to keep in mind while investigating what might be most effective for a struggling student:
- The computation is not the concept. It is important to distinguish between skill in computation and skill in mathematical thinking. Many students who struggle with pencil and paper computation are strong spatial thinkers and mathematical problem solvers. If you verbally ask the question, “If you put nine balls evenly in three baskets, how many go in each?” and a student can answer it verbally, then he understands division. If the same student can’t do long division, then some process besides conceptual understanding is breaking down. In this case, calculators could be allowed during problem solving tallow the teacher a true gauge of the student’s mathematical thinking.
- Weak working memory needs support. A weak working memory has a tremendous impact on math performance. Even simple computation requires working memory to complete, and as math’s complexity increases, so does the demand on working memory. In the broadest sense, the primary strategy to compensate for this is to represent on paper what the brain is having trouble retaining. This may include detailed writing of each step, jotting notes along the way, and having the steps of a procedure available as a checklist.
Manipulatives are not just for younger students. Students need to understand the physical concepts that the pencil and paper algorithm represents. All students move through a developmental sequence from concrete to abstract (see below), and many need the physical representations for a longer period than is often provided in schools. Also, many students may need the concrete-to-abstract sequence represented for each new concept.- Switching from quantity systems to procedural or semantic systems may be difficult. It is possible that some students’ main difficulty is in linking their quantitative sense and the procedural steps. These students may be able show full understanding using manipulatives to model problems, and also be able to solve them on paper using rote steps, but aren’t able to link the two processes. This difficulty comes to play in complex algorithms like long division, which require shifting from semantic systems to quantitative systems throughout the problem.
- Dyslexia makes everything a reading test. If a child struggles with decoding, then word problems are not math problems. Those who struggle with mapping letter
symbols onto phonological sounds may also have trouble mapping number symbols onto quantity sense, even if they excel in quantitative reasoning. These students
should have problems read to them if teachers want to gauge their math abilities.
- Language matters. We mediate our understanding of math through language. If there are known language difficulties, there will almost certainly be difficulties in receiving
instruction through language-heavy methods. Students with phonological processing issues may make errors like writing 16 for 60, since the closing “n” in “sixteen” is the only phonetic difference between the two numbers. Others may write 31 for 13, since the teens are the only numbers not written in the order they are said (i.e., 63 is sixty-three, 179 is one hundred seventy-nine, but 16 is said “sixteen” with the last digit spoken first).
- Math facts are not the same as the process. If a student does not know his facts, then any problem attempted will be incorrect, regardless of whether or not he understands
the process. It is important to separate the two. Until a student has memorized the facts, he should use a fact chart while working on the process of algorithms.
- Word problems can be language problems. If language or decoding is a struggle, then word problems are a language processing issue. Consider the following word problem: A can of paint covers 200 square feet. Each can costs $23.50. Your room is 12 feet tall, 15 feet long and 14 feet wide. How many cans of paint will you need? How much will it cost? This problem will create difficulties for those students who have language processing issues. But if those students are placed in an actual room with verbal instructions, a can of paint, and a tape measure, then the students’ natural spatial abilities will allow them to solve the problem with much greater ease. Word problems should be approached with many of the same strategies used for reading comprehension, with special attention to the “math words” that cue the student to perform a particular operation (more than, difference, one-third of). Once the word problem is fully analyzed, students should receive explicit instruction in strategies for sketching out or modeling the problem to translate the semantic into the quantitative.
- Grapho-motor difficulty makes accurate computation difficult. Lining up all the numbers in a complex algorithm is extremely hard for some students. Give these students graph paper, turn the notebook paper sideways so the lines form columns, or allow them to dictate and talk through the problems.
- Fifty minutes a day is not enough. If math abilities are to develop in the sense of use and application, then math should be part of all subjects when appropriate. Social studies teachers should ask students to compare elements such as land area, GDP, population and the passage of time. Science teachers should have students collect and process data. Language arts teachers should have the students work with numbers that occur in literature for instance distances, dates and time.
Math is hard for many students, but it is very important. Most of the math struggles arise from breakdowns in cognitive processes that are not necessarily related to mathematical thinking. However, many math programs aimed at students with learning differences are stilted heavily toward the pencil and paper algorithm, and they do not address the neuro-developmental processes involved. This perpetuates the misconception that these students can only learn rote steps. Unfortunately, graduating students enter a job market that values mathematical thinkers, not algorithm solvers. As teachers and schools become better educated about the cognitive processes involved in math, all students, including those with learning differences, can become mathematical thinkers.
Allen Broyles is the principal of the middle school of The Howard School, a school for students with language-based learning differences. He has taught math at the lower, middle and high school levels, and has worked with students who have a wide range of processing struggles that impact their mastery of mathematics. He can be contacted at abroyles@howardschool.org.
Tom Pittard is a 14-year faculty member at The Howard School and serves lower and middle school students as a math lab teacher. His instruction is personalized to complement individual learning styles, address student needs, and help students understand their unique learning process. He can be contacted at tpittard@howardschool.org.
RSS feed for comments on this post.
TrackBack URI
