# Reversing the Decline in SAT Math Scores

To reverse the decline in math scores and engender in our children an interest and expertise in mathematics, educators need to pay attention to teaching methods. In countries where students do exceptionally well and in U.S. schools where students excel, there are three key attributes:

1. Each grade has a focus. In other words, there are non-negotiable skills that students master in that grade;
2. Everyone in the school system understands and practices the common definition of knowing;
3. The key concepts and procedures are taught using efficient and elegant models—methods that are generalizable and develop mathematical ways of thinking in our students.

The Common Core State Standards in Mathematics (CCSS-M, 2010) have brought the nation’s attention to these three aspects of mathematics learning.  CCSS-M stands on three legs:

1. Every grade has a focus on a few key concepts to be taught and learned;
2. Those topics must be taught with rigor—understanding, fluency, and applicability (in other words, with mastery);
3. There should be coherence in teaching these topics—a teacher should know the trajectory of the development of the concept or procedure—where it begins and how it develops in different grades.

With effective methods of teaching, we will achieve the intended objectives of the CCSS-M: higher achievement and interest in mathematics on the part of our students and the general public.

One of the important components of effective teaching is achieving mastery and that is dependent on practice of the material taught—language, concepts, procedures, skills. In the previous post (SAT Math Scores Decline in 2015), I emphasized the importance of practice. In this post, I want to focus on how to do that practice.

To benefit from practice, there is need to dissect the main skill into sub-skills, constantly evaluating their status, and practicing them in isolation and in combination.

Take the long division algorithm, for example, which involves several sub skills—estimation (what do I expect as the answer?), multiplication tables (with fluency and understanding—a fact is recalled in less than two seconds or less), place value (in order to know the size of the partial quotient), spatial skills (organizing and aligning numbers), subtraction (with fluency and understanding—using decomposition/recomposition), and sequencing (following the order and sequence of actions involved in the procedure).

Step One
Break Down the Skill: Create a Graphic Organizer
First, write the skill to be improved in a central bubble. Make branches to recognize each sub-skill necessary for performing the broader skill well. For example, the long division algorithm is in the center and multiplication tables, place value, subtraction, spatial orientation/space organization, etc. are the spokes from the central bubble. Similarly, when adding fractions, the addition of fractions is in the center and multiplication tables, divisibility rules, prime factorization, short division, concept of fraction, such as a fraction is equal to one when the numerator and denominator are equal, etc. are in the spokes.

The process continues (creating branches from branches) until the leaves of the branches are simple skills. Depending on the skill and the ability to articulate it, one may have anywhere from a dozen leaves to many more.

Step Two
Evaluate Learner Performance on Each Sub-Skill
Mark each leaf on the branch by its importance with either critical, important, or of minor importance. Assess each branch using a rating for performance as good, okay, or poor for each leaf. Evaluate each sub-skill for understanding, fluency and applicability.

Identify branches and leaves that have high importance but low proficiency. If a leaf is critical but the student is unsuccessful in it, that represents the best candidate for immediate improvement and practice. Identify the top three sub-skills that require your most urgent attention.

Step Three
Develop Practice Exercises to Improve Sub-Skills
Develop an exercise to practice using efficient strategies. Even short intervals of supervised practice, first alone and then with others, can be powerful, if the skill was specified clearly and the strategy used is efficient. Independent practice should take place when one is fluent under supervision.

Deliberate practice isn’t easy, which is why most people plateau at a level far below their potential. It is hard to do concentrated, uncomfortable practice on previous skills on top of the demands of the current grade level of work. The planning of and focusing on the components of practice is therefore important. Successful practice is a function of the nature of the content to be mastered, number of practice events, time of practice, nature of practice, the nature and frequency of feedback, etc.

There are two ways of practicing skills to reach a level of automaticity: block practice—repetitive drilling on the same task in one block of activity and variable practice or interleaving—to practice several skills in succession. Interleaving works better in all but the youngest learners because it seems to fit the mind’s natural capacity to detect patterns and recognize differences.

Interleaving practice is particularly useful when students practice a complex, multi-step skill or process such as long division, operations on fractions, or solving equations.  If, for example, some aspect of the process is particularly troublesome for students, they might need to be given assignments that help them focus their practice on that one aspect only.

Achieving Mastery
Practice of a concept or skill begins with efficient note taking – an important academic skill students, particularly at the middle and high schools levels, need to develop to have higher achievement in mathematics. It includes how to take notes during teacher and student presentations and then how to summarize work and learning. Through note taking, the information transfers from the teacher to the student and from the student to the teacher. Because the development of note taking skills cannot be left to chance, the teacher has to teach these skills explicitly.

Practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information—which frees the working memory for more complex and higher order thinking and fosters the development of creative aspects of problem solving. With automaticity with understanding, the transfer—the ability to use skills learned in solving one class of problems to solving other problems—a vital part of mathematics learning is possible. Without understanding, students’ ability to make links between related domains is limited. When there is significant overlap in content areas, practice in one leaves an impact on the other. For example, in middle school, work on fractions, decimals, ratio, proportion, and percent should be integrated and practice exercises should include problems from all of them. If students practice pulling general concepts out of a collection of specific examples and give multiple different examples for the general idea, then their ability to transfer skills and procedures increases.

Individual and Small Group Practice
On the average, how many problems does an average student need to practice to achieve a level of competence? And when should students practice? To reach a fair level of competence in a skill takes a great deal of practice. Not until students have practiced a skill upwards of 12 times that they reach an 80 percent competency level. Of course, there will be variations depending on the nature of the skill being practiced and learner differences. Also, when a high level of mastery is reached, the increase in competence is less after each practice. This means the practice should not be in one big practice session but over a period of time. For this reason, part of each day must be devoted to practicing previous material—tool building.

The first four practice sessions result in a level of competence that is almost 50 percent of complete mastery. The next four sessions, however, account for only an approximate 14 percent increase. Therefore, learning new content does not happen quickly. It requires practice spread out over time. The results of this practice will be increments in learning that start out rather large but gradually get smaller and smaller as students fine-tune their knowledge and skill. Only after a great deal of practice, can students perform a skill with speed and accuracy.

Once an acceptable level of understanding and performance is evident, students should work individually because they can be expected to be reasonably close to high performance. They should practice problems and exercises to consolidate their knowledge and gain proficiency. Each student should attempt the same or similar sets of problems, and the differentiation should be achieved through the level of assistance offered for successful completion and the time it takes in achieving this.

Individual work may take multiple forms. For example, it may be conducted in silence to allow for quiet practice, contemplation, and self-assessment. The teacher can circulate to check that the students complete the work accurately, to diagnose problems, and to give strategies for remediation and improvements. During this time, the teacher monitors students’ work – how they are solving the problems, but also collects informal records on the students’ progress and decides which students require assistance and who will be asked to present solutions in the summarizing phase of the lesson.

Explicit instruction for students with and without mathematical difficulties has shown consistently positive effects on performance with both word problems and computation. By the term explicit instruction, I mean that teachers provide clear models for language, concepts, and procedures for solving a problem type using an array of examples. It also includes extensive practice in the use of newly learned strategies and skills and opportunities to think aloud (i.e., talk through the decisions students make, the strategies they plan and use, and the steps they take), and extensive feedback, in the use of strategies and relating strategies to outcomes.

Not all of students’ mathematics instruction should be delivered in an explicit fashion. However, struggling students should receive a significant part of their mathematics instruction explicitly. Some time should be dedicated to supervised practicing in automatizing foundational arithmetic skills and conceptual knowledge necessary for learning the mathematics at their grade level. While practicing, students should be helped to adapt and change strategies according to the context.

Understanding and Automaticity
To retain a skill and use it efficiently at will requires its mastery—understanding and automaticity. To master and use a skill or procedure, one needs to have either conceptual understanding so that one can reconstruct it as needed or automatized it. To be proficient in the use of a skill one should have both understanding and automaticity.

Conceptual understanding assures accuracy in its usage. However, accuracy alone is not enough. For instance, reconstructing a skill each time it is needed is inefficient. One can also automatize a skill or procedure by over learning it. However, if the conditions under which the procedure is to be applied are modified even minimally, one may not be able to apply the procedure. Automaticity is achieved more easily and faster if there is a clear understanding of the concept underlying the skill, first.  Then, practice is more productive.

In the initial learning phase of a skill, concept, or procedure, the teacher needs to focus on identifying its salient attributes and help students to understand it by modeling, demonstrating, and breaking it down into its components (task analysis). The aim is to accommodate and assimilate it within existing schemas or if necessary, expand and reorganize them in order to integrate the new concept with old concepts and procedures. The students must adopt and “shape” these new schemas. This learning phase is a “shaping process.”

During the shaping process, the teacher focuses on learners’ attention on their conceptual understanding of the concepts and skills.  Without this shaping, they are liable to develop misconceptions or use the procedure in ineffective ways.

The goal of practice is to master the skill—acquire both speed and accuracy, but the goal of the shaping phase is to gain accuracy. It is important to deal with only a few examples during the shaping phase of learning a new skill or process.

The shaping phase is not the time to press students to perform a skill with significant speed. Ineffective teachers tend to prematurely engage students in a heavy practice schedule and rush them through multiple examples. In contrast, effective teachers attend to the needs of the shaping process by slowly walking students through only a few examples.

Technology Assisted Drill and Practice
Tutorials using Computer Assisted Instruction (CAI) programs that are well designed and implemented can have a positive impact on mathematics performance, particularly at the middle and high school levels. Still, educators should critically inspect individual software packages and the studies that evaluate them. They should assess if the software has been indeed increased the learning in the specific domain and with students who are similar to those who will use the software.

Charting Accuracy and Speed
Skills should be learned to the level that students can perform them quickly and accurately. To facilitate skill development, students should be encouraged to keep track of their speed and accuracy. This might be best accomplished if they can chart both. For example, I use addition and multiplication ladders to automatize arithmetic facts. As the student completes an individual ladder for a number, he/she records the completion time. Next time, the student tries to beat his/her time.  When students can complete a ladder in 15 to 20 seconds without any error, at least three times, they have automatized that addition/multiplication table.

During the modeling phase, students can be given individual and collaborative targets in order to raise expectations and to keep the class working together. That is not to say that students are put under undue pressure, but that they are encouraged to take responsibility for making an effort and doing their very best. The constant monitoring provides the best form of diagnostic teaching and leaves the students little time for goofing off.

Tests as Aids in Mastery and Understanding
Tests have traditionally been used as a measure of learning. Properly used, tests can be and are themselves learning events. The retrieval processes triggered by tests enhance recall, sometimes to a much greater degree than do comparable opportunities to restudy the information. Restudying is better than testing in the short term, whereas an advantage for testing emerges at longer retention intervals. Testing also appears to have greater benefits for subsequent free-recall or cued-recall testing than it does for forms of testing that are less dependent on recall, such as tests of recognition or priming. Two factors that have been found to moderate the testing effect are final-test delay and final-test format:

1. Later recall profits from having an earlier test (or tests) of to-be-               remembered information versus not having a prior test, and
2. later recall profits, under some conditions, more from having an earlier test (or tests) than it does from having an opportunity (or  opportunities) to restudy the information in question.

From this perspective the test may have only one problem, but it calls for thinking and integration of knowledge. For example, rather than asking 20 multiplication facts, one can ask:

Problem 1: Make all the possible rectangles of area (a) 24, (b) 64, (c) 48 with whole number sides.

Problem 2: Write a six digit number between 400,000 and 500,000 that is divisible by 2, 3, 4, 5, 6, 8, 9, and 10.

Tests as Practice Aids
Tests tax a student’s fluency, recall/retrieval, and understanding. Tests provide training in retrieval in integrative ways. The testing situations add a level of desirable difficulty, recalling of various learning and problem-solving strategies used in the testing process enhance long-term memory. The more difficult the initial test, provided retrieval succeeds, the larger is the benefit of testing.

Restudying, on the other hand, involves much more fluent processing than taking a test and adds to fluency and a better subsequent memory.

The retrieval processes engaged by tests constitute practice for a criterion test later. The idea is that tests enhance memory on subsequent tests because they give learners the opportunity to engage in retrieval processes of the type that will be required on a later test. The benefits of practice tests increase as a function of their similarity to the final test.

Testing improves long-term memory relative to an equal amount of time spent restudying, even in the absence of feedback. However, testing is not always better than restudying, particularly when fluency and understanding are absent. Sometimes restudying is better, and sometimes the two conditions do not differ significantly.

# SAT Math Scores Decline in 2015

Math scores on the SAT have fallen to the lowest level (511points) not only since the college admission test was overhauled in 2005 (519 points) but in four decades (see recent articles in Inside Higher Ed and the New York Times). And stagnant results from high school students on state, federal and international tests are adding one more reason to worry about the nation’s high school mathematics teaching and achievements.

Elementary school children have made steady progress on mathematics on many state and national tests, including international tests such as: The Trends in International Mathematics and Science (TIMSS) and The Programme for International Student Assessment (PISA), but, a great deal of disparity exists between the achievement of minorities and poor students at all grade levels compared to those with privilege—economic, social, geographical, and educational. More importantly, improvement in scores is not translating into high school students’ higher achievements in mathematics.

Policy makers, educators, and social scientists identify a plethora of social and educational challenges that continue to pose hurdles in lifting high school mathematics achievement. Among them are social conditions: poverty, growing economic inequality, language barriers, low levels of parental education, and social ills that plague many urban neighborhoods. Others relate to educational conditions: poor quality of teaching and teacher preparation, diversity of teaching strategies and assessments, and lack of uniformity in marshaling resources for implementing common curricula such as Common Core State Standards in Mathematics (CCSS-M).

Per capita expenditures in urban schools may not be lower; however, unsatisfactory educational conditions—inadequate physical and administrative infrastructure, poor teacher quality, low accountability of services, lower expectations of students from Kindergarten to high school—plague schools.

There are other reasons for the decline in SAT scores in mathematics. It may very well be due to more students taking the test in response to new state and federal laws requiring or encouraging testing. For example, several states and school districts require that students take the ACT or SAT to graduate. It is possible that the types of students who weren’t previously taking the test don’t have the same level of ability on standardized tests as the students who consider selective four-year colleges. However, I believe that the most important factors are: low expectations in mathematics from students by parents, teachers, and administrators and lack of emphasis on mastering the material taught.

In the last twenty years, fourth graders have been doing substantially better in mathematics and eighth graders also have made some progress in mathematics; however, tenth graders have shown no progress and even some decline. Many elementary school principals point out that they are sending better students to middle school, but something happens to them there. Actually, it is a false sense of achievement because the improvement is not deep enough. A child can do well in elementary mathematics by just being a good counter and having no generalizable strategies and understanding.

To do well in high school mathematics, a student should have strong preparation in mathematics ideas. That means mastery of important components of mathematics ideas: (a) linguistic—rich vocabulary and understanding of the structure of mathematics language, (b) strong conceptual schema and flexibility of thought, (c) mastery of skills/ procedures—understanding, fluency and applicability.

At the elementary school level, methods of teaching mathematics that emphasize only the coverage of material, using inefficient strategies such as sequential counting methods and without acquiring real mastery (understanding, fluency, and applicability) and efficient strategies do not result in lasting effects. Without generalizable mastery, students’ achievements from the elementary grades do not translate to future achievement. The growth in mathematics at the elementary level, therefore, does not translate into increased interest in mathematics and achievement at the middle and high school level.

Teaching should result in learners demonstrating the expected skills and behaviors: from recall of information to performing tasks—routine and non-routine, applied in solving problems, demonstrating higher order thinking and creativity. With this demonstration of skilled performance, students will be able to transfer the skills to other domains of mathematics.

Research shows that characteristics of expert performance (mastery) are acquired through experience and that the effect of practice on performance is larger than earlier believed possible.

Elements and Conditions of Practice
Skilled performance, in any domain, is the result of interaction between environmental factors (instruction), genetic endowment (student characteristics), and sustained supervised and unsupervised practice. To understand expert performance, teachers need to know the nature of this interaction and the conditions that maximize it. Mastery of any content is individual, but there is a close relationship between practice and mastery. A fair amount of practice is necessary for learning and mastering concepts, skills, or procedures.

Many teachers and parents believe that practice makes perfect. To some extent, practice does show improvement. However, people may spend time and effort in working at a skill with little to no improvement after a certain point; in fact, they quickly reach a plateau in their skill level and performance. The pay off of just practice after that point is limited. The difference between those who plateau and those who go on to higher level of mastery with fluency depends on the type, frequency, and the intensity of practice used. One benefits from practice significantly when:

1. initial methods of learning a concept are efficient and strategy-based,
2. practice focuses on the elements that contribute to faster growth in expertise and proficiency,
3. practice is driven towards the improvement of a specific area of weakness or learning a particular skill or process, and
4. practice profits from the expertise of an effective coach/teacher.

Practice with immediate feedback from an expert/coach or the taste of a successful outcome makes that practice a means to mastery. Practice without the analysis of performance and only the process of practice does not lead to significant progress. Ordinary extra practice will not take students beyond their plateau.

Effective practice results in higher achievement in mathematics. For example, practicing addition and subtraction facts by sequential counting is not going to result in mastery—understanding, fluency, and applicability. On the other hand, practice using decomposition/recomposition strategies will result in mastery of facts, not just whole numbers, but also integers.

A focus on process means that you are not just trying to output work, but you are trying to improve strategies. This is the difference between basketball players practicing their skills through playing games and doing drills. The focus of a game is on winning; the focus of a drill is on process. It is this latter focus, which is crucial in sustained improvement. Practicing a skill alone (drill) and solving problems (game) are the mathematics counterparts.

Finally, practice should be intense, beginning with comfortable to moderately uncomfortable to uncomfortable, and be focused on particular skills. Children should not just practice the skills they are good at. The teacher or the student should identify the trouble spots in the process and drill them intensely until the problems are resolved and the weaker points are remedied.

# Building a Mathematics Vocabulary

We cannot receive, hold and manipulate a concept without having either an image/schema or words for the concept. The image could be a picture, figure, drawing, or symbols. In the case of language it can be a word, expression, or an equation.

For effective communication of mathematics ideas, children need robust and rich images and vocabularies (language containers). Without appropriate language containers, children cannot retain and communicate mathematics ideas. Vocabulary—words, expressions, phrases—are the language containers for mathematics concepts.

Learning mathematics, then, is using, creating, extending, and modifying language containers—the vocabulary of mathematics. Students’ proficiency in mathematics is directly related to the size of the set of their vocabulary. Rote memorization of a collection of words is not enough to master the language of mathematics. Instead, one has to acquire the related schemas with understanding. Language proficiency refers to the degree to which learners exhibit control over their language.

The introduction of mathematics vocabulary and terminology should be contextual, but even direct study of quantitative and spatial vocabulary contributes significantly to improved mathematics conceptualization—learning new concepts, creating deeper and robust conceptual schemas, and more effective communication.

When children create and encounter a language container for a mathematics concept, they also create and invoke the related conceptual model in their minds. Each word and expression such as sum, product, rational number, least common multiple, denominator, rectangular solid, conic section, and asymptotic represents a concept with its related schema. For example, if a person understands the definition of multiplication as ‘repeated addition’ or ‘groups of’, then these expressions invoke the conceptual schema. The expression 43 ´ 3, will invoke: 43 repeated 3 times (43 + 43 + 43) or 3 groups of 43 (43 + 43 + 43). If multiplication is learned as the ‘area of a rectangle’, then 3 ´ 43 will invoke an image of a rectangle with dimensions 3 (vertical side) and 43 (horizontal side).

The development and mastery of mathematical vocabulary are the result of a long and continuous interactive process between native language, mathematics language and symbols, and their quantitative and spatial experiences. This begins with play and concrete experiences in children’s environment. Experiences are represented through pictorial and visual forms and means, which then may result in abstract mathematics formulations and problems that students solve. This mathematics formulation—devising of abstract symbols, formulas, and equations, is then applied to more problems, and the result of this process is communicated. Successful communications demonstrate that the child has mastered a concept. The process can be summarized as:

• Understanding the environment (concrete experiences and use of native language).
• Translation (native language to pictorial and linguistic forms).
• Representation (in the native language).
• Description and verbalization (in the native language).
• Discussion (in the native language).
• Mathematical formulation of the problem (in the mathematical language).
• Manipulation of mathematical language.
• Communication of the outcome of mathematics operations (in mathematics and native languages).

This communication furthers not only children’s mathematics achievement but also their language development.

Building the Vocabulary of Mathematics
Many of children’s mathematics difficulties are due to their limited vocabulary—its size, level, and quality. A child’s size and level of vocabulary is the intersection of three language sets:

• The level and mastery of the native language and background the child brings to the mathematics task.
• The level and sophistication of language that the teacher uses and the questions she asks to teach mathematics.
• The language set of the mathematics textbook being used.

The intersection of these three language sets is the available language the child has to learn mathematics. A small intersection means the child has a limited vocabulary. The objective, then, is to increase the size of this intersection. A child’s limited mathematics vocabulary may be for many reasons.

• The mathematics problems of the child with English as a second language in a classroom where the medium of instruction is other than the child’s native language.
• The child’s and teacher’s economic, cultural, and geographical backgrounds differ. For example, the linguistic problems that many urban black children and immigrant children face are an example of a linguistic/cultural mismatch and the assumptions teachers make in instructing children.
• Textbook language sets differ from the language sets of the children and the teacher.

Whatever the reasons for limited language sets, we need to help children acquire a robust mathematics vocabulary. Properly acquired and used in context, a mathematics vocabulary has a profound effect on children’s mathematics achievement and their thinking. Planned activities for developing, expanding, and using vocabulary contribute significantly to better mathematical word problem-solving ability and support learning new concepts, deeper conceptual understanding, and more effective communication.

Although more textbooks are emphasizing the language of mathematics, there is still little attempt to develop a coherent and comprehensive mathematics vocabulary in school mathematics teaching. In one textbook, the expression “find the sum” is introduced quite early. In another series, the expression is introduced much later, and then the words “find the sum” and “add” are used interchangeably. In another text, the word “sum” is used sparingly. Consequently, a child may face different language sets from grade to grade and from school to school. Although the textbooks have a large number of common language terms and vocabulary, many words are not in common. Further, some textbooks use so much language without properly introducing the terms that many children find textbooks frustrating. Exercises do not provide enough practice in basic skills, which prevents children from automatizing the language or the conceptual skills associated with them.

Strategies for Enhancing the Mathematics Vocabulary
Ways in which children’s failure to develop mathematical vocabulary may manifest as: (1) children have difficulty conceptualizing a mathematics idea; (2) they do not respond to questions in lessons; (3) they cannot perform a task; and/or (4) they do poorly on tests, particularly on word problems.

• Their lack of conceptualization of a mathematical idea may be because they do not have the language for the concept to receive it, comprehend it or express it, such as ‘find the sum of’, ‘union of two rays…,’ ‘evaluate…’
• Their lack of response may be because they do not understand spoken or written instructions such as ‘draw a line between…’, ‘touch the base of the triangle’, ‘place a positive sign next to the numeral,…’ or ‘find two different ways to…’
• They are not familiar with the mathematics vocabulary words such as ‘difference’, ‘subtract’, ‘quotient’, or ‘product.’
• They may be confused about mathematical terms such as ‘odd’ or ‘table’, which have different meanings in everyday English and have more precise meanings in mathematics.
• They may be confused about other words and symbols like ‘area’ and ‘perimeter’, ‘factor and multiply’, ‘and’.

To enhance children’s vocabulary, every school system should have a minimal mathematics vocabulary list at each grade level. Mastery of words from such lists will prepare children to communicate mathematics. This list can also be used to assess students’ grade level language of mathematics. This list should indicate the grade of introduction of words, terms, and definitions and the level where they are mastered. It should be developmentally and linguistically appropriate. The teacher should constantly identify, introduce, develop, and display the words and phrases that children need to understand and use.

The teacher should use the same techniques to introduce mathematics words as she teaches native language. She should have a Math Word Wall for every mathematics concept she teaches. When a new word related to the concept emerges in discussion, it is added to the Word Wall. With the introduction of each word, students are exposed to several words and concepts that contain it. Then students use it in their own words, with as many examples as they can. The teacher selects a word and then asks children to use it in mathematics context. The following exchange illustrates this process.

• Give me a sentence that uses the word ‘add.’”
• You have \$5 and I have \$14. Let us add both amounts.”
• That is great! Now use the word ‘sum’ in a sentence.”
• That is easy. If we add our monies, what is the sum of our monies?”
• That is great! Now I am going to write some words on the board. I want you to first to tell me and then write a sentence or two using each word. If you want, you can use more than one word in a sentence.”

The concepts are then reviewed in circular fashion, built upon, and tied into new ideas. This helps children construct a working vocabulary that is constantly augmented, and they are also learning skills to build it.

Once the key root words have been introduced to children, the teacher can begin to extend the mathematics vocabulary words. Among the easiest sets are the words formed with prefixes, suffixes and derivative words. The process is to introduce the math prefixes and roots casually and then formally. In a casual manner, parents and teachers can remark, “You know a tricycle has 3 wheels. Tri- means 3 and cycle means wheels.”

Teacher: What will be the name of the object that has three angles?
Student:  A triangle.
Teacher: Why?
Student:  A triangle has 3 angles and tri- means 3.
Teacher: Now draw a triangle on your paper.
Children draw triangles on their papers.

Teacher: The word ‘lateral’ means a side. What will you call an object that has three sides?
Student:  A trilateral.
Teacher: Now draw a trilateral on a paper.
Children draw a trilateral on their papers.

Teacher: If the word ‘gon’ means a corner, what will you call an object that has three corners?
Student:  A trigon.

Teacher:  If ‘octo’ means eight, what does ‘octagon’ mean?
Student: A figure with eight corners.

As with all language development, there is a sequence in moving from speech ability to writing ability: the input is auditory in its foundation (the child is immersed in oral linguistic experiences), then followed by speech ability (the child produces language) and later by reading and writing ability. When young children have this kind of foundation, they avoid the anxiety of making sense of key foreign words later on in a formal setting. They will be able to generalize and relate math concepts to their daily experiences.

Instructional Suggestions for Language Proficiency
There are practical reasons children need to acquire rich and appropriate vocabulary for them to participate in classroom life—the learning activities and tests. There is, however, an even more important reason: vocabulary, as part of mathematical language, is crucial to children’s development of thinking not only in mathematics problem solving but in general problem solving. Once children have control over their language usage, they begin to have control over the meta-cognitive skills that produce insights into their learning and their interactions with learning tasks. Language and thinking are interwoven in reasoning, problem solving, and applications of mathematics in multiple forms—intra-mathematical, interdisciplinary, and extracurricular. If children do not have the vocabulary to talk about a concept, they cannot make progress in understanding its applications—therefore solving word problems.

Teachers often use informal, everyday language in mathematics lessons before or alongside technical mathematical vocabulary. This may help children’s initial grasp of the meaning of words; however, a structural approach to the teaching and learning of vocabulary is essential to move to higher mathematics using the correct mathematical terminology. This also applies to proficiency. The teacher needs to determine the extent of children’s informal mathematical vocabulary and the depth of their understanding and then build the formal vocabulary on it.

It is not just younger children who need regular, planned opportunity to develop their mathematical vocabulary. All students and adults returning to education need to experience a cycle of concrete work, oral work, reading, writing, and applications.

The teacher needs to introduce new words through a suitable context, for example, with relevant, real objects, mathematical apparatus, pictures, and/or diagrams. Referring to new words only once will do little to promote the learning of mathematics vocabulary. The teacher should use every opportunity to draw attention to new words or symbols with the whole class, in small groups or with individual students. Finally, the teacher should create opportunities for children to read and write new mathematics vocabulary in diverse circumstances and to use the word in sentences.

• Concrete work: Concrete materials/models develop images and the language for mathematics ideas. The concrete materials/models help children (a) generate the language, (b) understand the concept, and (c) arrive at an efficient procedure. Students should be encouraged to explore and solve problems using manipulative materials and asked to discuss and record the activity using pictures and symbols. The teacher or a student can also act the word out.
• Writing work: The teacher should explain the meanings of words carefully. The teacher should refer to a similar word; give the history and the derivation of the word and write it on the board. Children should copy it in their Math Notebook. The teacher should ask the children to say the word clearly and slowly. They should rehearse the pronunciation of the word. The teacher should ask them to spell the word and ask a child to say the word and spell it with eyes closed.
• Oral work: Students describe the work done at the concrete level, using mathematics words and expressions based on the visual and tactile experience of the meaning of mathematical words in a variety of contexts. This oral work may be facilitated in different contexts by
• listening to the teacher or other students using words correctly
• acquiring confidence and fluency in speaking, using complete sentences that include the new words and phrases, in chorus with others or individually
• discussing ways of solving a problem, collecting data, organizing data and discussing the properties of the data for a variety of reasons: to generate hypotheses, develop conjectures or make predictions about possible results or relationships between different elements and variables involved in the problem
• presenting, explaining, communicating, and justifying methods, results, solutions, or reasoning, to the whole class, a group, or partner
• generalizing or describing examples that match a general statement
• encouraging the use of the word in context and helping sort out any ambiguities or misconceptions students may have through a range of open and closed questions.

Because students cannot learn the meanings of words in isolation, I believe in the centrality of reading and conversation in mathematics lessons. Shared reading is a valuable context for learning and teaching not only mathematics language but also mathematics content. Strategies such as using children’s books, stories, DVDs, and videos as a vehicle for communicating mathematical ideas develops mathematical language. Reading word problems aloud and silently, as a whole class and individually, is equally important. During these readings, the teacher should ask questions involving mathematics concepts. This develops strong mathematics language and understanding. Students can be asked to read and explain:

• numbers, signs and symbols, expressions and equations in blackboard presentations
• instructions and explanations in workbooks, textbooks, and other multi-media presentations
• texts with mathematical references in fiction and non-fiction books, books of rhymes, children’s books during the literacy hour as well as mathematics lessons
• labels and captions on classroom displays, in diagrams, graphs, charts, and tables
• definitions in illustrated dictionaries, including dictionaries that the children have made themselves, in order to discover synonyms, origins of words, words that start with the same group of letters (e.g. triangle, tricycle, triplet, trisect…), words made by coding pre-fixes or suffixes, words derived from other words.

All students from K through 12 and adults returning to education need to work on developing their mathematics vocabulary.