My Math Counts Too

Tim Whiteford PhDSaint Michael’s CollegeColchesterVermont 05439

- -8026542744twhiteford@smcvt.eduDepartment Pages Math website: http://academics.smcvt.edu/twhiteford/Math/Math.htm

First Let’s Challenge Some of Our Assumptions

We cannot assume a student’s apparent math difficulty is a function of her/his lack of English proficiency.

We cannot assume an ELL student is familiar with the important numbers in our US culture.

We cannot assume an ELL student’s different mathematical procedures lead to incorrect answers.

We cannot assume mathematics is the same the world over.

There Are Five Areas of Interest to explore.

1. Differences in the Procedural and Conceptual knowledge of mathematics.

2. The role of learning a different language in learning mathematics.

3. The mathematics of the culture. 4. How the student learned math in her/his country

of origin. 5. The nature of a individual student’s mathematics.

1. Conceptual and Procedural Knowledge

Conceptual knowledge is knowledge of ideas and concepts in mathematics; multiplication as repeated addition is a piece of conceptual knowledge.

Procedural knowledge is knowledge of symbols, conventions, and procedures. 23, +, $, 2 + 2 = 4 are examples of procedural knowledge.

Procedural knowledge differs more between cultures because it is arbitrarily derived. Conceptual knowledge relates more to the laws of nature and so tends to be constant the world over.

Examples of procedural knowledge differences

Multiplication algorithm

15 x12 150 30 180

Subtraction algorithm

612

- 218 3 4

Classroom Implications - OneClassroom Implications - One 1. Ask the student to tell you how she/he completes the 1. Ask the student to tell you how she/he completes the

different algorithm. different algorithm.

2. Does the student understand what she/he is doing? 2. Does the student understand what she/he is doing?

3. Decide if she/he needs to learn the standard US procedure.3. Decide if she/he needs to learn the standard US procedure.

4. Find out if the student knows when to use a particular 4. Find out if the student knows when to use a particular algorithm by presenting them with some word problems. algorithm by presenting them with some word problems.

5. If you need additional information about how the student 5. If you need additional information about how the student completes the algorithm arrange a conference with the completes the algorithm arrange a conference with the parents or refer to the parents or refer to the My Math Counts Too website website

Differences in the procedural knowledge of number systems

Number systems can be different. Most Asian systems use 10 and 1, 10 and 2 and so on

for the teen numbers. Maay Maay also uses this system.

Bases other than base ten. Some cultures such as those in the Amazon Basin use

only the words one, two and many for counting. Mayans used bases 5 (quinary) and 20 (vigesimal) . African cultures use a variety of bases, sometimes

more than one at a time.

Classroom Implications - TwoClassroom Implications - Two 1. 1. Take extra care when dealing with the numbers 11 Take extra care when dealing with the numbers 11

and 12 if they are new to the student. and 12 if they are new to the student.

2. Emphasize pronunciation of the n at the end of the 2. Emphasize pronunciation of the n at the end of the teen numbers for students who have difficulty teen numbers for students who have difficulty hearing or saying this phoneme at the end of a hearing or saying this phoneme at the end of a word. word.

3. Take extra time teaching the base ten system if 3. Take extra time teaching the base ten system if students are not familiar with it. students are not familiar with it.

4. Use 4. Use Arrow Cards to teach Base Ten place value– to teach Base Ten place value– this link will take you to a set of reproducible this link will take you to a set of reproducible arrow cards and the arrow cards and the Arrow Card Activities handout handout or teaching ideas and strategies. or teaching ideas and strategies.

2. The Role of Learning a Different Language in Learning Math.

Cognitive Academic Language Proficiency and the role of linguistic registers.

Phonetic characteristics of language. Semantic and syntactic differences.

Sometimes it’s not an ELL issue!

Cognitive Academic Language Proficiency

Meanings of new words are learned in specific contexts.

Cognitive academic language proficiency requires learning the meaning of a particular word in an academic context.

Many words need to be learned in more than one context – the role of linguistic registers.find, into, area, degree, counting, division, probability, obtuse, minus, express

Avoid the use of metaphorical words such as “reduce” with younger students.

Classroom Implications - ThreeClassroom Implications - Three Present new vocabulary in context with visual, non-Present new vocabulary in context with visual, non-

verbal, or contextual supports.verbal, or contextual supports.

Use concept maps, drawings, diagrams, pictures.Use concept maps, drawings, diagrams, pictures.

Use manipulative materials to support conceptual Use manipulative materials to support conceptual development.development.

Explore difficult words that have multiple meanings Explore difficult words that have multiple meanings based on different linguistic registers.based on different linguistic registers.

Phonetic characteristics of language

Different phonemes in different languages.Confusion between teen and decade number nameswhen there is no ‘n’ phoneme at the end of a word.(e.g. thirteen and thirty)

Pronunciation is a function of what you hear. Similarities between thirteen, fourteen and fifteen; students often only hear ”firteen” three times?

Semantic and syntactic differences

Words can be used figuratively. “How high can you count?”

Word meanings are different in other cultures. A Sum is any simple algorithm in the world of

Harry Potter (and the UK). The difference between 9 and 14.

Structure of sentences can differ. “12 goes into 4 three times”

Classroom Implications - FourClassroom Implications - Four

Listen carefully to what students say.Listen carefully to what students say.

Monitor the use of specific vocabulary.Monitor the use of specific vocabulary.

Try to become aware of any cultural Try to become aware of any cultural variations.variations.

Use “telephones” – two elbow pieces of 2 Use “telephones” – two elbow pieces of 2 inch PVC pipe glued together.inch PVC pipe glued together.

3. The Mathematics of the Culture

Cultures are defined by numbers. Subcultures have specific numbers. There are different referent units in different

cultures (e.g. Rhode Island). Numbers are frequently used without the

inclusion of the referent. Cultures are defined by pattern, shape and other

geometric entities. Referents can be “sensed”; a pound v a kilogram.

A mile v a kilometer, a centimeter v an inch.

Cultures are Defined by Numbers

3 is significant in Bosnia 4 can bring fear in Taiwan but not 13 26 is important in Ireland At 7s over 8s in many Asian countries 50 in the US No Pi day in the UK (14/3/11)

And in sub-cultures She rides a 750 Make a 360 Drop one purl 3

What are some of the numbers you use in your everyday life?

What are the referents of each of the numbers in this sporting event report?

Durham reached 124 for seven off 34 overs compared to Worcester's 128 for six, but the tail subsided... David Byas took his season's tally to 702, passing John Hampshire's 684 set in 1976, by hitting 54 as Yorkshire posted 214 for six. Then Darrin Gough took a competition‑best five for 13 on his 24th birthday as Sussex were dismissed for 177. Captain Alan Wells top‑scored with a battling 64 including five fours and a six off 70 balls. Mike Watkinson, with four for 32, led Lancashire to a 47 run win over Leicestershire despite a broken thumb".

Cultural numbers and referents also change over time; 11bu 2pk 3 qt, When did one and six equal 18?

There are different referent units in different cultures.

We think in terms of referent units; we reason quantitatively (estimate) with an implicit knowledge of the size of, say, an inch. Try it with centimeters.

Frequently encountered referents. Metric referents. Use of objects for trading in some cultures. Money coins and bank notes. Using familiar objects to convey size (e.g. a 5lb bag of

sugar) Doing a 360 It cost 4.99

Ethno-Geometry? Do the shapes and lines in a culture affect the way we think geometrically?

Classroom Implications - FiveClassroom Implications - Five Give student the opportunity to use the numbers, Give student the opportunity to use the numbers,

measuring systems and geometric foundations of measuring systems and geometric foundations of their culture by differentiating project work and their culture by differentiating project work and math problems.math problems.

Make sure referents are made explicit.Make sure referents are made explicit.

Include visual representations of referent units in Include visual representations of referent units in the classroom accessible to students such as on the classroom accessible to students such as on buletin boards or shelves so that students can buletin boards or shelves so that students can handle and see the see the referents.handle and see the see the referents.

4. How did the student learn math in her/his country of origin?

Does the student understand the math she/he has learned? What is the role of the parent in relation to the institution of

the school? Do schools exist for all students? Are there gender differences in educational opportunities? What type of manipulative materials and texts were used?

How are students grouped in schools?

Classroom Implications - SixClassroom Implications - Six Some classroom procedures may have to be taught Some classroom procedures may have to be taught

through direct instruction.through direct instruction.

Some expectations might need to be made explicitSome expectations might need to be made explicit..

5. The nature of the individual student’s mathematics.

Interview the student with the Cultural Math Interview to find out.

The interview is: A conversation with the student about her/his math. An exploration of the student’s math education

experiences. An opportunity to learn about and value the student’s

math. A chance to learn some new mathematics.

Their Math Counts Too - a web resource

A collection of resources for teaching math to ELL students.http://academics.smcvt.edu/twhiteford/Math/Cultural%20Math/Their%20Math.htmIf you would like to add to the collection please email your ideas/thoughts/resources to;

whiteford@smcvt.edu