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Differentiating learning

English language learners (ELL) and children with Limited English Proficiency (LEP)

Are there particular methods or accommodations that Think Math! recommends for English Language Learners? What about children of poverty whose native language is English, but whose vocabulary and use of the language may be limited?

The most important principle is to recognize that limited English is not the same as limited cognitive ability. Think Math! has worked hard to make sure that guidance for strategies for ELL is not treated as a gradation of mathematical ability, but strictly as a gradation of proficiency with English. Keeping the cognitive demand appropriately high while reducing the linguistic demand as necessary is the best approach.

When we are learning new ideas or skills, we all benefit when other distractions and barriers are minimized. Giving clear and precise descriptions of mathematical ideas and actions, and of one's thinking about them, is extremely valuable to students, but it is also extremely hard. Learning new mathematical ideas and managing the language compounds the difficulty. For the child whose English is not yet fluent -- even for a child who appears, in casual conversation, quite competent -- this added difficulty may be enough to seriously impair the mathematical learning. The child is not bad at mathematics; it's just that the extra attention the child must pay to the language is being taken from the mathematics. Squeezing mathematics through the bottleneck of limited English hampers the learning.

Think Math! often separates the variables, using techniques like teaching without talking and visual formats (e.g., cross number puzzles and intersections and arrays) to help students through the task of learning the mathematics (and demonstrating that learning) without the additional cognitive load of mediating that learning through words. Visual presentations and methods like silent teaching reduce the language barriers to near-zero. Teachers who are comfortable with these methods can adapt them for use in lessons where they are not already used.

We must also develop and enrich/extend the children's ability to communicate clearly and mathematically, and to manage the language that they will need in classroom discussions and that they will face on state tests. But language needs to follow concept; until there is an idea to talk about, the talk is just in the way. One of the virtues of manipulatives is that, after they have been used for developing a concept or skill, they can support language development by being the concrete "stand-ins" for the ideas as those ideas are discussed in class. They help the child understand the language of the discussion and connect that language to the ideas that had been developed earlier.

Because the language is so important, Think Math! also provides a variety of specific, and tested, methods for developing that language. See a developmental approach to word problems and problem-solving and developing mathematical language for more details.

Students who need a narrowed range of methods, models, or strategies

Some theories of Autistic Spectrum Disorder (ASD) suggest that at least some of these students thrive better in mathematics when, instead of being offered multiple methods, they are taught one way of doing each thing they must learn. How does Think Math! accommodate the needs of such students?

Think Math! was designed with highly consistent mathematical models because the power and consistency of these models help all children. It has not been explicitly researched with ASD, but has been tested to be particularly effective with a variety of children who especially need that consistency and for whom it is advantageous to minimize the number of different strategies that must be mastered. Think Math! also incorporates numerous techniques that develop focus and attention, which can be an issue for the same group of students.

Although fluency with mathematics requires flexibility -- it is simply not possible to approach the subject (or the state tests of it) with "one way to do things" -- learning to do mathematics takes different directions for different students. Some do need to feel secure with a few tools before they can be flexible about using more. Because it is not possible to determine in advance which tool best suits which child, Think Math! doesn't pre-judge. Multiple approaches -- visual, verbal, manipulative -- are used, and if a child needs to have the strategies narrowed, a teacher or parent can then focus on what "works" best for the child.

But Think Math! is also highly focused. From the start, a very small number of powerful mathematical models carry through the entire program. Just three models are the basis for acquiring all arithmetic algorithms. This choice of only three powerful models is deliberate: it shows the internal order and relatedness of mathematics, and it simplifies the learning for students who might be overwhelmed with too many disconnected models and approaches. Lessons and materials vary -- visual, verbal, manipulative -- but the foundations are consistent.

  • The cross number puzzle is a tabular format that was adapted by TM to model the structure of the addition and subtraction algorithms. Cross number puzzles seem to be particularly clear to all students for this purpose. Because they are just tables, they serve, at the same time, to familiarize students with the structure of tables, which is also essential learning.
The cross number puzzle moves children though a lot of conceptual territory without having to learn new strategies. This single tool grows with children's understanding, taking them from addition and subtraction through (with the aid of the intersection and array models) multiplication and division with the same approach. This model is meant to be available to children always, and to support their understanding of a wide range of topics. Even when it does not formally appear in a lesson, it can and should be brought out to help children who are struggling with any of these concepts. It is built into the program K-5.
  • The number line is a tool of mathematicians that also works well with children. The number line represents the order of numbers and the distance between numbers. It builds measurement concepts as well as an understanding of addition and subtraction. A ruler is a chunk of number line.
Children use the number line for adding and subtracting, decimals, and fractions. Like the Cross Number Puzzle, the number line should also be an ever-present tool; it is not meant to come and go, but can and should be pulled out to help any child at any time anchor the mathematics. In K and grade 1, the children, themselves, move on a floor number line during lessons. Masking tape versions can allow that activity at home, too. In grades K-3, the Number Line Hotel -- a chunked and stacked version of the number line -- helps children understand the structure of adding tens. (A small version of the hotel is useful even with older students who are struggling with addition and subtraction.) In grades 3-5, the number line supports more advanced understanding of subtraction, and of the numbers-between-numbers that students must consider when they study fractions and decimals.
  • Intersections and arrays -- two related images that are both essential in mathematics -- have already been mentioned. They both model multiplication, and are both used in graphing.