General Resources
Position Papers
Resources for Principals & Educators
The field of mathematics education reflects the ongoing work of many outstanding organizations and advisory groups, including the National Council of Teachers of Mathematics (NCTM); the National Assessment Governing Board (NAGB), which oversees NAEP; and the National Mathematics Advisory Panel. An important aspect of these efforts has been the identification of key mathematical topics and concepts within a grade, as well as the progression of topics and concepts across grades. The resulting publications and summaries of these efforts have been aligned with The Quantile Framework for Mathematics, where appropriate.
Resources for Families & Students
Quantile Codes & Terms
Quantile Codes
Quantile codes are used in conjunction with the Quantile measure to indicate special characteristics of the material.
Emerging Mathematician (EM)
Material designated as “EM” is that which has a Quantile measure of zero or below. The measure is shown only as “EM” without the zero or negative number appearing.
Not Measurable in Quantiles (NMQ)
Material designated as “NMQ” is content that is extensively diverse in QTaxons or strands. Some examples are quizzes, tests, riddles, review sheets/activities, and process skills.
Higher Mathematical Content (HMC)
Material designated as “HMC” is content that is above the precalculus level. The Quantile Framework spans the content typically taught in kindergarten through Algebra II, Geometry, Trigonometry, and precalculus.
Quantile Terms
Alignment
Alignment refers to the similarity or match between and among content standards, performance standards, curriculum, instruction, and assessment in terms of knowledge and skill expectations. The inferences made on the basis of assessment results are valid only to the extent that the system components are aligned.
Calibration
Calibration is the process by which measures are assigned to a mathematics textbook lesson or other resource.
Content Standards
Content standards are statements of knowledge and skills that schools are expected to teach and students are expected to learn. They indicate what students should know and be able to do as a function of ongoing mathematics education.
Differentiated Instruction
Differentiated instruction is defined as the attempt "on the part of classroom teachers to meet students where they are in the learning process and move them along as quickly and as far as possible in the context of a mixed-ability classroom" (Tomlinson, 1999). Differentiated instruction promotes high-level and powerful curriculum for all students, but varies the level of teacher support, task complexity, pacing, and avenues to learning based on student readiness, interest, and learning profile.
Foundational QTaxon
“Foundational QTaxon” describes a skill or concept that only requires readiness to learn. Readiness is based upon the learner’s cognitive experiences rather than knowledge of specific mathematical concepts.
Grade Equivalent
Because the Quantile Framework for Mathematics is a developmental scale, Quantile Measures are NOT grade equivalent. Measures are indicative of developmental progress.
Prerequisite QTaxon
“Prerequisite QTaxon” describes a QTaxon which is a skill or knowledge that provides the foundation necessary for another QTaxon.
Quantile
A Quantile, of the Quantile Framework for Mathematics, is the unit of measurement along a developmental scale that locates an individual's ability to think mathematically in a taxonomy of mathematical skills, concepts, and applications. Currently, the scale’s range is for K-12 education (10Q to 1400Q).
Quantile Framework
The Quantile Framework for Mathematics is a developmental scale for the study of mathematics. The scale’s unit is the Quantile. Both the math ability of students and the solvability of problems can be measured in Quantiles. After test results are converted into Quantile measures, young mathematicians can be matched to instructional materials on their own level, and comprehension can be forecast accurately.
Quantile Map
The Quantile Map provides a graphic representation of the Quantile Framework, showing several example QTaxons identified at a variety of Quantile measures along the five mathematics content strands. It provides a context for understanding how test results relate math ability to problem solvability.
Quantile Measure
A Quantile measure applies to both the math ability of students and the difficulty of a skill/concept in mathematics. The measure is expressed by a numerical score followed by the letter Q (e.g., 650Q). A 650Q student is ready for instruction on a 650Q QTaxon. Students receive measures through formal methods such as linking studies wherein the reporting scale of a norm-referenced or criterion-referenced assessment is linked with the Quantile scale.
QTaxon
A QTaxon is a mathematics skill or concept that has a specific level of difficulty along the Quantile scale. The term QTaxon encompasses the varying terminology used by different states to describe what each student must learn. Some states call these student expectations, objectives and goals, grade level expectations, benchmarks, focus areas, etc.
Quantile Range
The Quantile range is the suggested range of Quantiles at which the mathematician should be receiving instruction. This learning frontier is fifty Quantiles above the student measure and fifty Quantiles below the student measure.
Supplemental QTaxon
“Supplemental QTaxon” describes a QTaxon which provides supplementary skills or knowledge that assists and enriches the understanding of another QTaxon.
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Typical Grade Ranges
Based on studies, we have determined the following approximate Quantile ranges for each grade level. In understanding student measures, these represent about the middle 50% of the students.
To interpret what a Quantile measure means for a specific student, two pieces of information are needed: (1) the Quantile measure, and (2) the grade level during which the student received his or her Quantile measure.
For example, a higher Quantile measure within a specific grade range indicates that a student probably has very few problems with grade-level material (textbooks and assignments) in school. A lower Quantile measure indicates that a student most likely struggles to understand and be successful with grade-level material.
| K-1 | EM | EM to 30Q |
| 2 | 100Q to 480Q | 85Q to 250Q |
| 3 | 340Q to 660Q | 250Q to 470Q |
| 4 | 495Q to 815Q | 470Q to 620Q |
| 5 | 635Q to 955Q | 640Q to 840Q |
| 6 | 700Q to 1020Q | 720Q to 900Q |
| 7 | 750Q to 1070Q | 790Q to 990Q |
| 8 | 820Q to 1140Q | 840Q to 1050Q |
| Algebra 1* | 870Q to 1190Q | 990Q to1140Q |
| Geometry* | 940Q to 1260Q | 1100Q to 1220Q |
| Algebra II* | 1000Q to 1320Q | 1100Q to 1270Q |
*Algebra I - typically taught at Grade 9
*Geometry - typically taught at Grade 10
*Algebra II - typically taught at Grade 11
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Expectations in Mathematics
A child’s Quantile measure provides you with additional information about his or her progress in mathematics. Because mathematical skills build upon one another, continuous practice is key to your child’s academic success. Children should be encouraged to think mathematically in school and at home. Challenging him or her to engage in activities which require mathematical thinking and providing help when necessary will support a child’s computation and problem-solving skills.
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What can teachers and parents do with the measure?
Teachers and parents can search the QTaxon database for a list of specific skills and concepts the child is ready to learn by following the instructions below and going to the Tools page.
By clicking on each of these skills/concepts, one can see the knowledge cluster for that QTaxon. The knowledge cluster is a group of related skills to the QTaxon selected. It provides information about what skills come before that skill and what skills would provide support or enrichment.
Further, after searching for skills the student is ready to learn, teachers and parents can click on the links provided leading to resources such as worksheets, games, websites, mathematics textbook lessons, and more.
Parents can also send the measure and child’s grade/state to our research associate who can provide a more detailed explanation of what the measure means.
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Mathematics Activities
Parents and educators can find worksheets, textbook lessons, games, and websites that align with the specific skills and concepts on the Quantile scale. You can help to advance your child’s mathematical development by challenging his or her level of understanding of numeration, measurement, algebra, geometry, and data analysis by doing the following mathematics activities in everyday situations:
Increase Your Child’s Exposure to Music
Research shows a positive effect between early and continuous exposure to musical development and mathematical achievement. Music fosters flexible ways of thinking, which is a powerful problem-solving tool.
Use Math Vocabulary in Everyday Events
Point out mathematical concepts in everyday life and help your child to use vocabulary that reinforces those concepts. Start with simple examples, such as using a cereal box to demonstrate units of measurement and volume. At the grocery store, discuss the graphs and charts found there, or the measurement abbreviations on packages (i.e., lb. for pounds). Open a bank account for your child to help teach him or her how to make connections to positive and negative numbers, as well as basic skills such as counting, sorting, and classifying money.
Use the Car as a Journey in Mental Math
Car time can be learning time. For example, call out coin values for your child to add in his or her head; count, sort, and classify objects; or work on the basic facts that will strengthen his or her computation skills. Both of you will have a wonderful time exploring fun, mathematical activities that you can do together, without relying on formal mathematics instruction.
Cooking is All About Math
You and your child can bake or prepare dinner—all with mathematics. He or she can help read the recipe, which connects to mathematics vocabulary. Or you can divide your recipe in half and ask your child to figure out the correct proportions. You can also talk about conversions; for example, how many cups are in a pint? You can have your child set the table and use multiplication to figure out how many utensils are needed in all for each place setting. Mathematics mastery requires your child to apply skills to everyday life. Cooking is an ideal way to do this!
Form a Close Relationship with the Teacher
In addition to having a positive attitude about mathematics at home, let your child’s teachers know that you support them. Ask what specific areas of the mathematics curriculum your child can focus on at home and what vocabulary would help to reinforce these concepts. You can use this information to connect mathematics to everyday situations and to discuss with your child what he or she is learning in mathematics.
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Six Jobs for Parents
Discuss your child’s mathematics homework with him/her.
Parents and home environment play a crucial role in encouraging a child’s academic success and motivation.
Visit your child’s math teacher to find out what he/she is learning and how you can help.
Only 27 percent of children whose teachers said their parents were not involved took geometry by the tenth grade; 63 percent whose teachers said their parents were very involved took geometry by tenth grade.
Ensure that your child is prepared to enroll in Algebra I or a course that demands algebraic concepts in middle school grades.
The demands of mathematics in a child’s classroom should reflect mathematical thinking, applications from the real world, and an expansion from the world of arithmetic to algebra, geometry, data analysis, and measurement.
Ensure that your child is developing through a rigorous curriculum in grades K-7 that prepares him/her for the transition to Algebra and Geometry.
Parents who said they rarely talked to their child about high school plans showed that 37 percent of the students took geometry by tenth grade. Parents who said they regularly spoke to their child showed that 48 percent of the students took geometry by tenth grade.
Encourage your child to take challenging mathematics courses for their future.
Only eight percent of students who said they did not discuss programs at school with their parents took Algebra I by eighth grade; 17 percent who discussed school programs three or more times during the previous semester took Algebra I by eighth grade.
Show the importance of mathematics for career choices, educational advancement, and life skills.
High school graduates who had not furthered their education but scored in the top quartile on the math portion of the Armed Services Vocational Aptitude Battery (ASVAB) earned an average of 38 percent more per hour than those who scored in the bottom quartile of the math portion of the test.
Source: Mathematics Equals Opportunity, U.S. Department of Education
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How can I help my child in math?
Consider these ideas:
How do I feel about math?
- Your attitude about math is easily conveyed to your child without saying a word. Try to avoid excusing your child’s difficulty with topics in mathematics because you were never good in math. Usually, just a little extra work and encouragement will pay off with a successful and confident young mathematician.
- Be more conscious of where you use mathematics in everyday life. At the grocery store, you compare prices or determine unit prices. In the car, you estimate the time it might take to arrive at your destination. In your daily routine, you estimate the elapsed time you have before a next appointment. Cooking and using recipes, you use thermometers, read fractions, and even divide or multiply fractions when the measuring tools are not exact. (With a ¼ cup, how will you pour in ¾ cups of sugar?)
- Be aware of the ways you use math on the job, such as determining your net pay from your gross pay using percents, writing formulas into spreadsheets, budgeting your time for meetings and projects, or projecting a budget for the coming term. What math do you use that your child could actually see?
Positive attitudes about mathematics are important to display in orderforyour child to appreciate its importance.
How do I react when my child gets a wrong answer?
- Most importantly, be patient. Your child wants to succeed so that you will be proud. Often you can determine your child’s mistake by letting him or her explain how the conclusion was reached. Allowing your child to explain the problem-solving process may help him or her to recognize the error, which can be enlightening to you both. If you and your child appreciate what was learned from the mistake, your child will become more confident and be more likely to experiment with mathematical reasoning powers.
- Sometimes the wrong answer means your child did not understand your question. When he or she explains his answer, you will know if that is the case.
- By asking your child to explain how the problem was solved, you will find out if the problem is with basic facts or operations.
- Your child’s explanation might also offer you information of his or her misunderstanding of a topic that might alert the teacher. Stay in touch with your child’s teacher so that you will have an active part in your child’s mathematical development.
Remember to have fun with your child as he or she grows along the mathematics continuum. Once in a while you will be wrong and your child will be correct. Enjoy the moment!
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