From Counting to Calculus: Weaving Mathematical Connections

Many curricular frameworks for teaching mathematics tend to be only a list of mathematics topics to be learned, with no clear elaboration of key ideas or organizing principles.  Because of this, students may not be taught to integrate mathematical ideas, which causes gaps in their knowledge and limits their understanding.  The Quantile® Framework for Mathematics, used in conjunction with the Common Core State Standards for Mathematics (CCSSM), helps teachers identify key connections and provide ways to ensure that students gain a comprehensive understanding of mathematics.

Mathematics is hierarchical and lends itself to learning progressions.  Development of mathematical concepts depends on a student’s understanding of prerequisite concepts.  Learning progressions are curricular frameworks that provide sequencing and guide teachers on proportional use of instructional time.  The CCSSM lend themselves to the development of learning progressions because they provide critical areas for instruction.  The CCSSM aligns content across K-12 so new material clearly builds upon concepts learned previously.

The Quantile Framework for Mathematics and its taxonomy provides a unique way to support the implementation of the CCSSM and address individual student needs by reporting both student ability and difficulty of concepts on the same scale—the Quantile scale.  The taxonomy of the Quantile Framework comprises approximately five hundred skills and concepts called QTaxons.  Each QTaxon is linked to related QTaxons, and these groupings form a knowledge cluster.  Knowledge clusters form a tightly woven web that encompasses the mathematics learned from kindergarten through high school.  By using information about student mathematical ability, the difficulty of the mathematical concepts, and the relationship among mathematical concepts, teachers can effectively target instruction for their students.

For a more detailed description, be sure to check out our latest white paper: Weaving Mathematical Connections from Counting to Calculus: Knowledge Clusters and The Quantile® Framework for Mathematics

Math Differentiation in a Common Core World

Just last week, I was invited to speak at the CCSSO Rural Chiefs Conference in Kansas City on the topic of “Supporting Math Differentiation in a Common Core World”.  While there is much written and discussed on the idea of differentiated instruction, in practice there are limited tools and resources to support math differentiation, a deficiency well-documented in this recent Ed Week article, ‘’Educators in Search of Common Core Resources”.

A theme permeating much of my presentation was the seemingly benign but pernicious neglect of math in our country.  By almost any measure, e.g. instructional time, professional development, number of assessments ,instructional programs, etc… math runs a distant second to reading in the amount of instructional attention given.  At least part of the challenge we face in addressing our math crisis in k-12 education will require that we remedy this neglect.

In my suggestions for addressing this imbalance I focused on four critical strategies. While the adoption of the CCSS is a huge first step in the right direction, its real success will rest upon how effectively we implement these standards.  Along with the implementation of these standards, it is critical that we recognize that math – like any other skill – can be learned.  Too often we subscribe, consciously and unconsciously, to the notion that math achievement is an inherent ability, as if math achievement was based on a “math gene”.  If we take more of a Carolyn Dwek growth perspective, as opposed to a fixed mind set, we will go a long way toward promoting the idea that math achievement is possible for all of our students.

Secondly, we need to build math tools and resources that support differentiated instruction.  Once, when leading a math workshop for a school district, the head of the math department informed me, tongue in cheek, that all math teachers know how to differentiate instruction:  “We say it louder and we repeat it”. This RV (repetition and volume) model is likely to only work if the student is hearing impaired.  Yet I suspect we have all seen variations of this model, this when we continue to drill a student on a math problem or concept to no avail.  Meaningful differentiated instruction is really only possible when we are able to measure a student’s math level and the difficulty of the math concepts and skills on a common scale.  This possibility is now a reality with the Quantile Framework for Mathematics.  Once you know a student’s Quantile measure you know what math skills they are ready to learn.  And just as importantly, one can make sure that the learner has acquired the necessary pre-requisite skills.  Unfortunately, we often continue to employ the “RV” model and fail to drill down and provide differentiated content and instruction to meet the unique needs of the learner. (more…)

Resources to Implement the Common Core

As this recent Education Week article, Educators in Search of Common-Core Resources, makes clear, educators are clamoring for resources aligned to the Common Core State Standards for Mathematics (CCSSM).  Many states have already adopted these standards and are currently developing assessments tailored to them.  

Shifting from previous standards to implement the CCSSM in its intended manner is no easy task, particularly at the elementary grades.  The Common Core was developed to move our country’s mathematics curriculum away from breadth and rather, vertically articulate the curriculum from kindergarten through high school to develop depth of understanding.    Developing depth of understanding however, requires an emphasis on the connections between concepts.

The Quanitle® Framework for Mathematics has information to help educators better understand these connections and is also aligned with the CCSSM through its freely accessible website.  Using the Quantile website, www.Quantiles.com and its tools, educators can find thousands of free, web-based resources aligned to the CCSSM.  

The Quantile website tools, the Math Skills Database, and the Quantile Teacher Assistant have a two-fold purpose: 1)  These tools leverage the Quantile Framework’s interconnected web of almost six hundred skills and concepts and align them with the CCSSM.  This taxonomy is also aligned to states’ previous standards, thus helping educators in their transition from one set of standards to the next; and 2)  Each of the skills and concepts delineated by the Quantile Framework is linked to freely-available web resources, providing educators with the much needed resources to implement the Common Core State Standards for Mathematics.

If you haven’t already tried it, be sure to take a look.

Khan Academy in the Classroom

We’ve written before on the efforts of Khan Academy – a free, online classroom that is available to anyone with an Internet connection.  Khan Academy offers thousands of video lessons on everything from specific mathematical concepts to explanations of the mortgage loan crisis.  Because Khan’s videos are easily accessible, students (and parents) are able to take advantage of its ‘always-on’ access to review videos in their own time. 

Khan’s work has gotten the notice of educators across the US and a number of Foundations and educational organizations, including the Bill & Melinda Gates Foundation, are looking to capitalize on Khan’s approach to individualizing education:

Each student’s math journey shows up instantly on the laptop Mr. Roe carries as he wanders the room. He stops at each desk, cajoles, offers tips, reassures. For an hour, this crowded, dimly lighted classroom in the hardscrabble shadow of Silicon Valley hums with the sound of fingers clicking on keyboards, pencils scratching on paper and an occasional whoop when a student scores a streak of right answers.

The software program unleashed in this classroom is the brainchild of Salman Khan, an Ivy League-trained math whiz and the son of an immigrant single mother. Mr. Khan, 35, has become something of an online sensation with his Khan Academy math and science lessons on YouTube, which has attracted up to 3.5 million viewers a month.

Now he wants to weave those digital lessons into the fabric of the school curriculum — a more ambitious and as yet untested proposition.

This semester, at least 36 schools nationwide are trying out Mr. Khan’s experiment: splitting up the work of teaching between man and machine, and combining teacher-led lessons with computer-based lectures and exercises

The most promising aspect of Khan’s work is that it harnesses technology to promote individualized instruction.  We’ve written before on the importance of blending technology platforms with instructor interaction to promote differentiation, or even individualization for each student.  By providing a platform that monitors each student’s progress and then responds with more instruction for students who are not ready to move on or with new concepts for those that are, Khan is upending the more traditional assembly-line model of the classroom, allowing teachers to monitor student progress and respond to struggling learners, while allowing proficient students to move forward.

Not it appears that Khan is getting the chance to put his model to the test at larger sites and with more classrooms:

In the past, math class at the Summit schools was always hands-on: the class worked on a problem, usually in small groups, sometimes for days at a time. But getting an entire class of ninth graders to master the fundamentals of math was never easy. Without those, the higher-level conceptual exercises were impossible.

That is where the machine came in handy. The Khan software offered students a new, engaging way to learn the basics.

Ms. Tavenner says she believes that computers cannot replace teachers. But the computer, she recognizes, can do some things a teacher cannot. It can offer personal feedback to a whole room of students as they work. And it can give the teacher additional class time to do more creative and customized teaching.

“Combining Khan with that kind of teaching will produce the best kind of math,” she argued. “Teachers are more effective because they have a window into the student’s mind.”

Khan’s efforts are worth noting.  Khan’s work is inspiring and is likely just the beginning of the work that can be done with virtual classrooms.  We’ve incorporated Khan’s work into our own tools on the Quantile Framework for Mathematics website.  In Math at Home, for example, students can select textbook chapters and lessons and search for supplemental material by which to review their primary lessons.  In many cases, they will find a variety of Khan videos available to help review core skills and concepts.  If you haven’t already, be sure to take a look.

Digital Promise: Math for Every Student

Tip of the hat to Scholastic’s Math Hub for posting this piece on the state of technology in math education.  Though many math educators report still relying on a basal textbook, many more are employing a variety of digital resources to help reach struggling math students:

On average, math teachers reported spending more than one full class period per week using digital tools or content, and many spent significantly more time utilizing technology. Specifically, among teachers who report using digital content or tools during more then 26% of class time (high digital use), the highest percentages are remedial math teachers and grades 6-8 math teachers. The most commonly used digital tool is interactive whiteboards. Teachers considered interactive whiteboards to be the most important supplemental material in addition to textbooks. This demand for whiteboards is a change from 2008 when interactive whiteboards were not even part of the survey. Math teachers and educators value the “faster reporting” and “detailed student/class information” generated by computer-based programs, features that traditional textbooks and workbooks cannot provide.

What many math educators have discovered is that moving from whole-class instruction to differentiating for struggling students requires going beyond the textbook to solutions that harness technology to adapt and respond to a student’s learning trajectory.  Technology of that sort can take multiple forms, but some important features include the ability to individualize for a student’s needs, provide supplemental resources, and multiple explanations for math skills and concepts.  As many educators now understand, one size does not fit all when it comes to math instruction; and ensuring that students graduate ready for the mathematical demands of the post-secondary world entails matching student math ability to the level of the lesson. 

At MetaMetrics, we’ve attempted to harness technology to supplement and strengthen student math ability through Math at Home.  Math at Home serves as a portal for matching students to targeted math resources across a variety of mediums.  Because each student has a different preferred learning modality, Math at Home offers online resources, video tutorials, skill practice sites, literature guides, games, and hand’s-on activities – a wide variety of resources to keep students engaged in math activity.  But Math at Home is more than a mere portal.  There are plenty of activity portals widely available.  What distinguishes Math at Home from other student portals is the Quantile Framework.  Math at Home uses the student’s Quantile measure to establish the student’s math level.  The list of available resources differs for each student and is based on their Quantile measure, or math level.  Additionally, Math at Home utilizes a large database of textbooks to match students with resources of their choice based on their current textbook lesson, but at their own math level.  If you haven’t already tried it, be sure to take a look.

More on Khan Academy

Here’s Wired on the innovative work of Salman Khan:

Khan Academy is an educational website that, as its tagline puts it, aims to let anyone “learn almost anything—for free.” Students, or anyone interested enough to surf by, can watch some 2,400 videos in which the site’s founder, Salman Khan, chattily discusses principles of math, science, and economics (with a smattering of social science topics thrown in). The videos are decidedly lo-fi, even crude: Generally seven to 14 minutes long, they consist of a voice-over by Khan describing a mathematical concept or explaining how to solve a problem while his hand-scribbled formulas and diagrams appear onscreen. Like the Wizard of Oz, Khan never steps from behind the curtain to appear in a video himself; it’s just Khan’s voice and some scrawly equations. In addition to these videos, the website offers software that generates practice problems and rewards good performance with videogame-like badges—for answering a “streak” of questions correctly, say, or mastering a series of algebra levels.

We’ve written before on Khan’s work.  By offering a free, virtual classroom available to anyone with a few minutes and an Internet connection, Khan Academy provides students with easy access to information on their own terms.  And because the lessons are videos, students are free to review again and again, allowing them access to the content as often as needed.  This means that students can move at their own pace, moving ahead when ready or reviewing material where necessary.  Khan Academy stands in stark contrast to the assembly line model of traditional classrooms and represents individualized instruction where students are free to move ahead as they master prerequisite material.  And it appears to be paying off:

Initially, Thordarson thought Khan Academy would merely be a helpful supplement to her normal instruction. But it quickly become far more than that. She’s now on her way to “flipping” the way her class works. This involves replacing some of her lectures with Khan’s videos, which students can watch at home. Then, in class, they focus on working problem sets. The idea is to invert the normal rhythms of school, so that lectures are viewed on the kids’ own time and homework is done at school. It sounds weird, Thordarson admits, but this flipping makes sense when you think about it. It’s when they’re doing homework that students are really grappling with a subject and are most likely to need someone to talk to. And now Thordarson can tell just when this grappling occurs: Khan Academy provides teachers with a dashboard application that lets her see the instant a student gets stuck.

…The result is that Thordarson’s students move at their own pace. Those who are struggling get surgically targeted guidance, while advanced kids like Carpenter rocket far ahead; once they’re answering questions without making mistakes, Khan’s site automatically recommends new topics to move on to. Over half the class is now tackling subjects like algebra and geometric formulas. And even the less precocious kids are improving: Only 3 percent of her students were classified as average or lower in end-of-year tests, down from 13 percent at midyear.

Those results are worth noting.  Khan’s work is inspiring and is likely just the beginning of the work that can be done with virtual classrooms.  We’ve incorporated Khan’s work into our own tools on the Quantile Framework for Mathematics website.  In Math at Home, for example, students can select textbook chapters and lessons and search for supplemental material by which to review their primary lessons.  In many cases, they will find a variety of Khan videos available to help review core skills and concepts.  If you haven’t already, be sure to take a look.

Using Social Media to Support Instruction

Earlier this year we mentioned how educators are using social networking sites including Facebook, Twitter and others to extend instruction outside of the classroom, even in cases of inclement weather.  As EducationNews.org reported this week, teachers in California are continuing to incorporate social media into their English classes.  “Rather than just the teacher reading student work, an entire class can read, review and give feedback on other students’ writing.”  Teachers claim that using social media as an educational conduit has also encouraged participation from students that are often too shy to raise their hand in a more traditional classroom setting – giving a voice and outlet to all students. 

And the utilization of social media sites as a tool to supplement instruction is not limited to the English classroom.  At MetaMetrics we’re trying to incorporate popular social media outlets into our own tools.  Our Math at Home utility, which allows students to locate targeted math resources at the right level of difficulty, now includes social networking features.  Based on the Quantile Framework for Mathematics, “Math at Home recommends various resources that students can add to their list of favorites.  Students can then print, email or share their list on social media sites like Facebook and Twitter.

In addition to encouraging participation in the virtual classroom during the school year, social media sites also support year-round instruction efforts by making resources available to students from any place at any time.  This is especially critical during the summer months when research shows so many students are susceptible to summer learning loss. As social media sites near ubiquity we hope that students will take advantage of these tools to review and solidify their skills.

Planning Math Lessons with the Quantile Teacher Assistant

Scholastic’s Math Hub has posted a thorough entry on our own Quantile Teacher Assistant (QTA) utility.  QTA allows math educators to differentiate math instruction by quickly identifying prerequisite skills for each state mathematics standard.  Furthermore, once identified, QTA helps math educators locate free resources that have been calibrated to that particular skill level – allowing a targeted match between a math skill or concept and the ability level of the math student.

If you haven’t used the Quantile Teacher Assistant yet, do yourself a favor and check it out.

Rethinking Math Education: An Experiment with Video

We’ve written before on how organizations like Virtual Nerds or individuals like Salman Khan are capitalizing on easy access to video and then harnessing the Internets capability to rapidly disseminate short chunks of information through multiple channels.  The ability to access specific, targeted material – and repeatedly view on-demand content – means that many students are able to engage with the content in a setting of their choice, a setting hopefully free of other distractions.  Plus, accessing information on-demand means that students are able to free up valuable mental real estate; students don’t have to be distracted by monitoring social cues or focusing on facial expressions, or even worrying about the speaker’s perception.  Instead, the student is free to focus almost exclusively on the actual content.

Here’s Khan at a recent TED talk explaining what he thinks may account for the appeal of his math videos.

Khan is not alone.  Vi Hart has garnered recent acclaim for her ability to take high-level, abstract, mathematical concepts and render them both accessible and fun.  Hart offers her visual work and explanations through both YouTube and her own site.

Video formats – like the Khan Academy YouTube videos and those created by Virtual Nerds – offer math students a valuable way to reinforce their current lesson or access more in-depth explanations in a setting of their choice.  That sort of accessibility and ease of use were a part of our thinking in making both Virtual Nerds and Khan Academy videos available through our own Math at Home utility.   Math at Home allows parents and educators to link students to resources at a targeted level based on the Quantile Framework for Mathematics.

If you haven’t already checked out this valuable new resource, be sure to take a look.

Video Tutorials Now Available in Math@Home

Here’s some good news for students who are struggling with mathematics: our free Math@Home web utility now includes Virtual Nerd video tutorials. The instructional videos provide students with the extra help they need to understand the textbook lessons being taught in school. It’s like having unlimited access to a personal tutor in the comfort of their own homes!

Plus, each Virtual Nerd tutorial has a Quantile® measure to describe the difficulty of the math lesson. This ensures students are matched with videos at the right readiness level for their unique abilities.

Access to the Virtual Nerd video tutorials in Math@Home is free.  However, subscriptions are available for students and parents who want to access more interactive tutorials on the Virtual Nerd website (access is free for registered educators). Each of these tutorials features an expert tutor and diagrams to explain all of the steps necessary to solve math problems spanning Pre-Algebra through Algebra II. They also allow students to determine the level of support they need. By clicking on a diagram or a specific step, term or symbol, students can launch other tutorials on those topics that require more instruction, and then continue with the main video. Check out this YouTube video for more details.

Math@Home uses Quantile measures to connect students with a variety of family-friendly math resources, like books, worksheets and websites, that best match their ability level and the textbook lesson they’re studying in school. In addition to the Virtual Nerd tutorials, the utility also provides access to instructional videos from the Khan Academy.

Why not get started now? Visit Math@Home and simply enter some basic information—the student’s state, grade, and Quantile measure or comfort with grade-level mathematics—to search the growing library of video tutorials and other resources that can help students practice mathematics at the right grade and ability levels.

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.