Velocity Norms for Academic Growth

Shuttleworth (1934) suggested that growth standards for stature should be expressed in terms of progress rather than status. Tanner (1952) provided a theoretical framework for the development of clinical standards for growth and advocated velocity standards. Bayley (1956) made the first effort to produce standards for height that took account of tempo. Her paper foreshadowed the landmark paper by Tanner, Whitehouse and Takaishi (1966) on longitudinal standards for height velocity and weight velocity. Incremental growth charts for height and weight have since been produced for use in the United States (Baumgartner, Roche & Himes, 1986; Roche & Himes, 1980).

Have you ever heard of growth velocity norms for academic growth—i.e., the growth rate of reading ability or mathematical understanding? There are three reasons you haven’t, which persisted for most of the 20th century: (a) the absence of sufficient longitudinal data on which to base investigations of academic growth; (b) the analytical methods available to educational researchers who wished to study growth; and, (c) challenges of educational measurement (e.g., dimensionality, lack of scale comparability and common units across instruments). Yet, I submit at the dawn of the 21st century, these obstacles have been overcome.

The most recent two reauthorizations of the Elementary and Secondary Education Act (ESEA) required states to assess reading and mathematics in multiple grades. States have been accumulating data for more than a decade. So, longitudinal data are now feasible for reading and mathematics.

Rogosa, Brandt and Zimowski (1982) advocated the use of longitudinal data collection designs gathering more than two waves of serial measures on the same individuals, accompanied by an analytical methodology focused on the individual growth curve. In their landmark book, Raudenbush and Bryk (2002) included a chapter on formulating models for individual change. Singer and Willett (2003) gave book-length treatment to the modeling of individual change. Perhaps the most enabling resource for the educational research community was Singer’s (1998) article demonstrating how to implement multilevel (including growth) models using one of the most widely available general-purpose statistical packages.

Finally, near the end of the 20th century, a new scale was developed for measuring reading ability. Its significant advantage over previous scales was a new kind of general objectivity, attained by calibrating the scale to an external text-complexity continuum and double-anchoring the scale at two substantively important points, much as temperature scales are anchored at the freezing and boiling points of water (Williamson, 2015).

Combining longitudinal data, multilevel modeling and state-of-the-art measurement scales from The Lexile® Framework for Reading and The Quantile® Framework for Mathematics, Williamson (2016) premiered incremental velocity norms for average reading growth and average mathematics growth. Based on an individual growth model, the incremental velocities reflect the long-term developmental growth of students in a well-established reference population (n > 100,000). Now, it is possible to refer the reading or mathematics growth rates of students observed during schooling to a clearly defined population of growth curves derived from serial measures of students whose reading ability and mathematical understanding were systematically assessed over time.

References
Baumgartner, F. N., Roche, A. G., & Himes, J. H. (1986). Incremental growth tables: Supplementary to previously published charts. The American Journal of Clinical Nutrition, 43, 711-722.
Bayley, N. (1956). Growth curves of height and weight by age for boys and girls, scaled according to physical maturity. Journal of Pediatrics, 48, 187-194.
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd  ed.). Thousand Oaks, CA: Sage Publications.
Roche, A. F., & Himes, J. H. (1980). Incremental growth charts. The American Journal of Clinical Nutrition, 33, 2041-2052.
Rogosa, D. R., Brandt, D., & Zimowski, M. (1982). A growth curve approach to the measurement of change. Psychological Bulletin, 92, 726-748.
Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 24(4), 323-355.
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.
Shuttleworth, F. K. (1934). Standards of development in terms of increments. Child Development, 5, 89-91.
Tanner, J. M. (1952). The assessment of growth and development in children. Archives of Disease in Childhood, 27, 10-33.

Summer Math Challenge Kicks Off on June 19th!

On average, all students, regardless of socio-economic status, lose approximately 2.6 months of grade level equivalency in their mathematical repertoire over the summer months each year. This means students can enter a new school year in August or September having lost about a third of the ground they covered the year before. Fight the summer slide, and keep math skills sharp with the Quantile Summer Math Challenge, a FREE math skills maintenance program based on grade-level standards that help prepare students for college and careers.

For the past several years, MetaMetrics has tried to help stave off the erosion in learning that can occur during the summer months with the Summer Math Challenge. Last year, almost 20 State Department of Educations and over 26,000 students across all 50 states signed up to take the challenge. This year, we have expanded our successful program to include those students who have just finished 8th grade. Now, the program is targeted to students who have just completed grades 1 through 8 and is designed to help kids retain math skills learned during their previous school year.

The Summer Math Challenge lasts for six weeks. The challenge focuses on one math concept per week, so as to help the student remain sharp but not feel overburdened during the summer. From June 19th through July 28th parents will receive daily emails with fun activities and links to educational resources. Activities will be grounded in everyday life and be engaging for both parents and children. This program also helps parents to understand that they do not need to be math experts to talk about math with their kids! When the program ends parents can print an award certificate to celebrate their child’s summer math accomplishment! To learn more, visit www.quantiles.com/summer-math.

Summer Learning Tools From MetaMetrics

Fight summer slide with free tools from MetaMetrics!

Visit Lexile “Find a Book” to submit your Summer Reading Pledge and download our Summer Reading Log. Use the log to track a child’s reading throughout the summer break. Search our database of over 270,000 titles for books within a child’s Lexile range. Enter the child’s Lexile measure, and then narrow the search by selecting topics of interest. You can also use “Find a Book” to check the availability of books at local libraries or purchase titles from major booksellers. When school starts again, share the reading log with the child’s teacher to show his or her dedication to reading.

Keep math skills sharp with the Quantile Summer Math Challenge, a FREE math skills maintenance program based on grade-level standards that help prepare students for college and careers. The program is targeted to students who have just completed grades 1 through 8 and is designed to help kids retain math skills learned during the previous school year. The Summer Math Challenge lasts for six weeks and focuses on one math concept per week. From June 19th through July 28th parents will receive daily emails with fun activities and links to educational resources. When the program ends parents can print an award certificate to celebrate their child’s summer math accomplishment! To learn more, visit www.quantiles.com/summer-math.

See the Quantile Framework in Action at NCTM

Are you attending the National Council of Teacher of Mathematics Annual Meeting and Exposition this week in San Antonio? Take this opportunity to learn more about the benefits of the Quantile Framework for Mathematics. We’ll be visiting the conference and meeting with some of our partners whose products utilize the power of the Quantile Framework.

Quantile partners exhibiting at the conference include: Big Ideas Learning, The College Board, Curriculum Associates, Houghton Mifflin Harcourt, Imagine Learning, McGraw-Hill Education, Mentoring Minds and Origo. We hope to see you there!

New Partnerships Increase Use of Quantile Measures

A series of new partnerships have greatly expanded the reach of the Quantile Framework for Mathematics. In the last year, Quantile measures have been added to Pearson’s aimswebPlus and Istation’s ISIP Math. Quantile measures have also become available through state assessments in Kansas and to the 15 member states of the Smarter Balanced Assessment Consortium. These new partnerships are in addition to the numerous state level assessments and assessment products that already report student mathematical ability in Quantile measures, including Kentucky’s K-PREP, North Carolina’s NC READY, Curriculum Associates i-Ready, HMH Math Inventory and Imagine Math. Also this year, new materials and textbooks from Pearson, McGraw-Hill, Origo Publishing and Big Ideas Learning have been calibrated to the Quantile scale, adding to the dozens already featuring Quantile skill or concept measures.

Quantile measures accurately match students with instructional materials by measuring both mathematical capability and the complexity of mathematical skills and concepts on the same developmental scale. There are two types of Quantile measures: a measure for students and a measure for mathematical skills and concepts. The Quantile student measure describes what the student is prepared to learn next. The Quantile skill or concept measure describes the difficulty, or demand, in learning that skill or concept. Both measures are represented as a number followed by the letter Q (e.g., 640Q) on the Quantile scale. Quantile measures can improve mathematics teaching and learning by helping educators target instruction and determine if students are on track to pass year-end assessments and succeed in college and careers. Visit www.Quantiles.com for more information about the Quantile Framework.

Celebrate Pi Day with Rich Math Tasks!

Every year Pi Day is celebrated in classrooms across the nation. This year we are offering examples of rich math tasks on the topic of pi that have been calibrated to the Quantile Framework. Use these tasks in your Pi Day celebrations to help students make meaningful connections to the concept of pi. Visit quantiles.com/pi-day to access our Pi Day resources!

And with the roll-out of this new resource, we’re giving away a pie! What do you need to do to be eligible to win a free pie? Simply “Like” our Quantile Framework Facebook page by April 14th and you’ll be entered to win*.

*Pie giveaway for residents of the United States only. Winner will be contacted via Facebook.

Celebrate Storytelling in Mathematics

September 25th is Math Storytelling Day! MetaMetrics® offers Lexile “Find a Book” so that educators and families can use student Lexile® measures to make informed decisions about reading materials that both interest children and are at reading levels appropriate for them to understand the material. Likewise, on quantiles.com MetaMetrics offers Math Literature Guides that accompany children’s books so that teachers and parents can use student Quantile® measures to engage children with appropriate topics in mathematics. Because this is the week of storytelling in mathematics, we want to share some Math Literature Guides that serve as samples for a variety of mathematics skills and concepts.

A simple topic for the early mathematics learner is working with ordinal numbers, such as first, second, third, and fourth. A fun book to read is Trouble on the T-Ball Team by Eve Bunting. The Math Literature Guide for this book offers ideas for using ordinal numbers at sports events (third quarter of a football game), in routine events of the day, or when losing the first tooth. In the Math Literature Guide, notice the blue title of the book. This means that when you click the title, you will be taken to the Lexile “Find a Book” page for that book.  On that page in the right-hand column is a drop-down menu called “Find This Book.” Select “World Cat” in that drop-down list to see libraries near your home where you might find the book. You can also select “Barnes & Noble” or “Amazon” to purchase the book.

For children a little older, another more challenging math topic is counting to a million, the subject of the book, A Million Dots by Andrew Clements.  Dots are everywhere in the book and the challenge is to find the dot that is indicated by the number on the page. The Math Literature Guide for this book offers ideas for questions that will encourage children to think critically when comparing such large numbers.

Geometry topics are the subject of many children’s books. One example is Sir Cumference and the Great Knight of Angleland by Cindy Neuschwander. This book includes some plays on words about angles and circles. By fifth grade, students are learning about right angles, acute angles, and obtuse angles. So the Math Literature Guide for the book offers ideas for finding different types of angles in the book and for thinking about where angles are found in architecture, design, and nature.

Mathematics is everywhere in our world but can seem to be invisible unless we take the time to point it out. Reading children’s books that accentuate the uses of mathematics offers insight and appreciation for the role of mathematics in our lives. Talking about the books we read makes reading more fun as well.  Sunday, September 25, is Math Storytelling Day, so take a trip to the library where you will find books that reveal mathematics in unexpected places to kick off a week of pleasure in reading books about mathematics. Read and enjoy!

Story Telling in Mathematics

Many people have fond memories of mom or dad putting them to bed with favorite bedtime stories. I recall the precious memory of a parent reading a fun story just before I snuggled into bed for the night. I also have a special memory of the bedtime hour when my dad was home to tuck me in. He did not read stories to me. He gave me “algebra” problems. From the time I was in third grade and up, he would ask, “A number plus 8 is 15. What is the number?” Now as a grandmother, I play “Mystery Number” with my grandchildren at bedtime. They never fail to follow their answer with “Give me another one, Grandma!”

My bedtime memories were a good combination of mathematics and literacy. The benefits of reading to our children are enduring. Through books, children appreciate experiences of others that often relate to their own experiences, expectations, and dreams. Children’s books can also help to teach the concepts and uses of mathematics.

Math Storytelling Day is coming up! According to their website, September 25 is set aside every year to celebrate the many ways that mathematics is used in our daily lives. “Math Storytelling Day is a great opportunity to get children excited about math through stories and games. Math stories can include logic, patterns, puzzles and numbers.”

A great place to find mathematical resources, such as games, activities, websites, tutorials, and videos is quantiles.com. Parents and teachers can access resources that are targeted to a child’s mathematical ability level based on The Quantile® Framework for Mathematics.  This learning community for teachers, parents, and students also includes helpful articles such as “The Quantile Framework for Mathematics in the Home.” As well as those ideas listed in the article, another simple convenient way to encourage a strong appreciation for the mathematics people use every day is to read books about math topics that inspire children to appreciate and enjoy mathematics.

The website quantiles.com lists children’s books that teach mathematics and offers Math Literature Guides that detail specifics about how to connect mathematics and literature. Each Math Literature Guide includes a series of questions and activities that promote ways for parent to talk about mathematics with their children. One example is the Math Literature Guide for the book Keep Your Distance, a fun and humorous book about measuring lengths with inches, feet, and miles by Gail Herman.

To celebrate Math Storytelling Day this year, the Quantile Team at MetaMetrics will share more book titles and Math Literature Guides about various topics in mathematics. Math storytelling can be an enjoyable experience for parents, teachers, and students as they focus on the ways mathematics is used daily. Stay tuned!

Gold in Math Olympiad

The 2016 Rio Olympic Games finished almost a month ago with the U.S. winning a total of 121 medals, which is almost double that of Britain in second place.  While we are still celebrating our nation’s athletic prowess, we should also be touting our academic success — the U.S recently won its second consecutive gold medal at the International Mathematical Olympiad (IMO). The IMO is an annual competition for high school students held each year in a different country. The two-day competition requires students to complete 3 problems each day in 4 ½ hours. Here is an example problem:

“In Lineland there are n ≥ 1 towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the 2n bulldozers are distinct. Every time when a right and a left bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, the bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes. Let A and B be two towns, with B being to the right of A. We say that town A can sweep town B away if the right bulldozer of A can move over to B pushing off all bulldozers it meets. Similarly, B can sweep A away if the left bulldozer of B can move to A pushing off all bulldozers of all towns on its way. Prove that there is exactly one town which cannot be swept away by any other one. (Estonia)”

In the podcast “Count One More Gold for The U.S. — In MathJody Avirgan interviews Po-Shen Loh, the U.S. team coach. Loh, once a team member and now a professor at Carnegie Mellon University, thinks that just as the U.S. Olympics inspires young athletes, the IMO can likewise encourage students to be interested in mathematics. “I think that mathematics is similar to sports in the sense that if you do more practice, you actually get significantly better,” states Loh. “And the big question then becomes, ‘How do we convince people that doing that practice in mathematics is a fun, fruitful, exciting thing to do?’”

Good question, Professor Loh. One way we can encourage fun mathematics practice is through the use of math games. A quick internet search yields hundreds of math game sites. For a more precise approach to finding math games, teachers and parents can use the free resources available on quantiles.com.  At the site, click “Use the Quantile Framework” at the top of home page. Then select “Math Skills Database” and “Keyword Search.” Enter a keyword such as “Operations.”  A list of Quantile Skill and Concepts (QSCs) targeted to the topic will appear. Suppose you want to practice “Order of Operations.” Click the QSC “Use order of operations including parentheses and other grouping symbols to simplify numerical expressions.” Then click” Show 18 Resources” to view the free resources calibrated to that skill. There are several math games listed to help students practice math AND have fun.  

Fighting Summer Learning Loss

In partnership with the Council of Chief State School Officers (CCSSO), MetaMetrics freely provides the annual “Chief’s Summer Learning Challenge” to state education agencies to develop and sustain their summer reading programs. Led by CEO and President Malbert Smith, Ph.D., who serves on the National Summer Learning Association’s Board of Directors, the summer learning challenge is a favorite project among the MetaMetrics staff. It is fulfilling work that relies on the collaboration and dedication across the MetaMetrics team—from engineering to government relations to the marketing department. This year MetaMetrics celebrates its 5th year of leading the fight against summer learning loss.

This summer, MetaMetrics collaborated with 21 state departments of education to combat the negative effects of summer reading loss. Among the 21 states participating in the Chief’s Summer Learning Challenge, the Lexile “Find a Book” search tool, Summer Reading Log and Summer Reading Pledge were promoted to encourage targeted reading practice. These promotional efforts have not gone unnoticed! For example in Kentucky, with many thanks to the tireless efforts of Kathy Mansfield at the KY Department of Education, over 7,200 summer reading pledges have been submitted. The Summer Reading Pledge is available until August 31st, and so far more than 120,000 books have been pledged to read nationwide.

MetaMetrics also leads a charge against summer math loss. The Summer Math Challenge launched in 2013, and has gained great momentum and popularity over the years. Participants have reported:

“I think that this is a great COST-EFFECTIVE activity for ALL parents. I look forward to each activity so that I have “scheduled learning” time for the summer with my daughter.”

“The summer math challenge was great…the exercises were right on target and fun.

This summer, 19 state education agencies have promoted the Summer Math Challenge through press releases, listserv emails and social media outreach. To see which states particpated this year, and past years, visit: https://www.quantiles.com/content/summer-math-challenge/state-participants/. Like the resources for the Chief’s Summer Learning Challenge, the Summer Math Challenge is made freely available to all participants. For more information about the summer math and reading initiatives, visit: https://www.quantiles.com/content/summer-math-challenge/ and https://lexile.com/chiefs-challenge/.

MetaMetrics is an educational measurement organization. Our renowned psychometric team develops scientific measures of student achievement that link assessment with targeted instruction to improve learning.