Promoting Life-Long Learning in Mathematics

Learning mathematics requires deep-rooted intrinsic motivation, motivation to learn, to problem solve, and to discover the best methods for solving those problems.  When we, as educators, attempt to offer doing well on assessments or being prepared for college algebra as intrinsic motivators, we often find that the results are marginal and superficial.

The role of mathematics educators is to promote reflective practices that promote connections within the realm of mathematics, as well as prepare students for the mathematical elements that are the foundation of so many aspects of the daily lives of citizens, consumers, and workers in their communities.

Mathematics teachers need as much training as possible promote discussion and reflection in their math lessons. Some considerations for best practices include the following:

  • Rather than expecting the teacher be the source of knowledge, a mathematics classroom should offer opportunities for the students to explore, collaborate, and make decisions on methods to solve problems. Such guided interaction among the students will add excitement to the development of student problem-solvers.
  • Instructional feedback needs to be more than whether the answer is right or wrong. Students need guidance on which elements of the process were misguided, help with identifying the flaws in judgment, and what adjustments need to be made. In solving most puzzles, we need to step back and determine where we are missing some information or going in a wrong direction. Working in the mathematics classroom can offer the similar intangible gratification when the problem is solved.
  • Problems can be solved using different approaches. Allow time for students to discuss in whole group activities or in small groups to share the different methods and styles of thinking. In the social studies or science classrooms, many discussions lead to the phrase “I never thought of it like that.” Sharing tactics in the mathematics classroom can certainly lead to such discoveries, also.

In order to develop mathematics classrooms that foster reflection, discussion, engagement, and discovery, math educators should be trained at every level. Teachers without strong insights about the reasons for the various algorithms in mathematics will not have the confidence to promote dialogue that might go in unexpected directions. Even the teachers in the lower grades need to understand how topics in mathematics are interwoven so that “math talk” promotes that connectivity. Students who become engaged in learning become life-long learners. This should be the case in all content areas, including mathematics.

The Importance of Leisure Reading – Both at School and at Home

Leisure reading—independent, self-selected reading of a continuous text for a wide range of personal and social purposes—is a critical habit to develop in students as they strive to become college and career ready. And all of us—educators, parents, and communities—have a role in fostering a love of reading and encouraging a lifelong habit of reading for pleasure among our youth.

We also know that leisure reading is on the decline, not only in the United States but around the world and not only at home but at school as well. As the content of a student’s coursework continues to expand each and every year, setting time aside to encourage and promote leisure reading grows more and more difficult. Even the connotations of the term leisure reading may lead many educators to erroneously believe this time is not a wise investment. Research, however, suggests otherwise.

The benefits of leisure reading are well-documented: improved comprehension, language, vocabulary development, general knowledge, empathy, as well as self-confidence in reading, motivation to read throughout one’s life, and positive attitudes toward reading. Additionally, high school students who regularly engage in the practice score significantly higher in reading than do their peers who do not read for pleasure as often.

To address the decline of leisure reading in our classrooms, the joint position statement of the International Reading Association (IRA), the National Council of teachers of English (NCTE) and the Canadian Children’s Book Centre, promotes several recommendations for providing students with leisure reading opportunities during the school day, among them encouraging student selection of reading materials and opportunities to reflect upon, respond to, and share the materials that have been read.

For the benefit of our students we must begin to think of instructional reading and leisure reading not as an either/or initiative but a both/and endeavor. In fact, if we think about the progression of students’ reading skills, we might imagine something similar to what is seen in the visual below. This graphic shows a progression from modeling for students the reading of challenging texts (Read Aloud), to a collaborative, instructional approach to reading challenging texts (Read Along), to finally the independent reading of challenging texts (Read Alone).

Reading

 

(Click image to enlarge)

Just as parents should read with their children and encourage independent reading for pleasure at home, educators should make time for students’ leisure reading at school as well.

As the position statement suggests, “To ensure that students experience the benefits of leisure reading, teachers and families should support students’ reading choices by making available a wide range of print, digital, and multimodal texts that align with and expand on students’ interests and that students are able to read without great struggle”(p. 2).

As we gear up another school year, it is important to consider the advantages of leisure reading and also of utilities such as Find a Book that allow educators and parents to help pair students with books that both capture their interests and also match their level of reading readiness.

Inside the classroom, the Find a Book utility can help students create personalized book lists for leisure reading that are tailored to their interesting and readiness and can also help teachers explore a variety of texts that can complement, support, and enrich the content of their curricula while keeping an eye on the staircase of text complexity each student needs for instructional reading to reach college and career readiness.

Outside the classroom, students can continue to access their personalized book lists for leisure reading through Find a Book and parents and communities can support their students’ growth by using the Find It! feature to locate the nearest public libraries and booksellers that have the chosen books available.

With tools such as these in hand, we can match the right student with the right book at the right time and have a profound impact on the reading habits and learning of our children for a lifetime to come.

A Response to Simon: College & Career Readiness for All Students

Not long ago, Stephanie Simon reported for Politico.com on what she called a “standards rebellion” in America.  According to Simon, “The backlash stems, in part, from anger over the Common Core … But it’s more than that. It’s pushback against the idea that all students must be ready for college — even if they have no interest in going.”  From Simon’s discussion, it appears that on the one hand, some policy-makers want to empower all students for college and/or rewarding careers; yet, other policy-makers call this elitist and say that many students need vocational rather than academic preparation.  Ironically, American College Testing found that high school students “need comparable levels of reading and mathematics, regardless of their post–high school plans.” (ACT, 2013, p. 6).

Recent research on the difficulty of reading materials associated with access to individual careers sheds some light on the issue. Williamson and Baker (2013) examined a randomly chosen linear systematic sample of 1/6 of the Bright Outlook Occupations identified by the National Center for O*NET Development using data collected by the U.S. Bureau of Labor Statistics.  Fully 28.8% of the occupations in the study required only a high school diploma for access.  However, all of the rest required additional education beyond high school.  Using the Lexile® Framework for Reading to measure text complexity, the study examined the difficulty of reading materials associated with individual occupations and found that the reading levels associated with different careers varied widely.  However, while typical high school texts have text complexity at around 1130L (i.e., 1130 Lexiles), almost 70% of the Bright Outlook Occupations had median text complexities above 1200L.  Nearly 29% of the occupations had text complexity above 1400L.  Perhaps the truth is that the postsecondary world offers something for an extremely diverse population of high school graduates.  There are indeed a few occupations that may be accessible with only a high school diploma and typical high school reading ability.  However, the large majority of occupations require substantially more reading ability than is represented by the texts that high school graduates were required to read as they were nearing the end of high school.

The Lexile® Framework for Reading evaluates reading ability and text complexity on the same developmental scale. Unlike other measurement systems, the Lexile Framework determines reading ability based on actual assessments, rather than generalized age or grade levels. Recognized as the standard for matching readers with texts, tens of millions of students worldwide receive a Lexile measure that helps them find targeted readings from the more than 100 million articles, books and websites that have been measured. Lexile measures connect learners of all ages with resources at the right level of challenge and monitors their progress toward state and national proficiency standards. More information about the Lexile® Framework can be found at www.Lexile.com.

Adjusting Math Terms for the Common Core World

Valerie Faulkner of North Carolina State University argues for a shift in the mathematical language we use.  The Common Core should give us pause and force us to reconsider the terminology and vocabulary we employ in describing certain skills and concepts.  Here are a few examples:

Old Habit (eliminate)                                     New Habit (adopt)

Defining equality as “ same as”                   Defining equality as “same value as”

Calling digits numbers                                    Clearly distinguishing between digits, numbers and numerals

Addition makes things bigger                      Addition is about combining

Subtraction makes things get smaller      Subtraction is about difference

Let’s borrow from the tens place               Use regrouping, trading, decomposing

Multiplication makes things bigger          Teach 3 structures of multiplication

Divison makes things smaller                     Teach the different structures of divisions,

Doesn’t go into                                                 Prepare students for later learning by using accurate language

Saying “and” means decimal point         Don’t create false rules for language using and

Canceling out                                                   Explicitly use and discuss the idea behind simplifying

Referring to “the answer”                           Use the model or the relationships to justify your answer

Guess-and-check as a strategy                 Teach systematic math representations

                                                                                                                             

Old habits die hard, but this is food for thought as many districts get farther into implementing the Common Core.

Identity Confusion: The Problem with the Equal Sign

Henry Borenson explains how we use the = sign in two very different ways.  The first way is operational, for example 10 + 15 = ______.  The second way we use the sign is relational, indicating equivalence between two sets of expressions, each of which includes one or more operations, for example 8 + 4 =_____ + 5.  But in 1999 a study of hundreds of first through six graders only 5%  solved problems like this correctly.

Borenson believes that because of this study we can conclude that the relational meaning  of the equal sign is not something that students find intuitive or self evident. When asked to fill in the answer to the problems above most students said that 12 belongs in the space because the answer follows the equal sign.  The equal sign seems to trigger the operational definition in most students’ minds. Some students thought the  + 5 was just there to confuse them.

Borenson recommends  introducing students in the second or third grade to the idea of balanced equations using concrete objects rather than numbers and the equal sign. Once students get the idea the equal sign can be introduced with the balancing explanation. Studies have shown that if the relational meaning of the equal sign is introduced in this way students are much more likely to grasp both ways.

How to Encourage a Love of Mathematics

Here’s Lisa Medoff from Stanford University suggesting eight helpful ways that educators can build students’ tenacity with a subject that frustrates many of them: math!

  • Empathize. It helps to imagine a situation where you are out of your confort zone and feeling frustrated and agry.

 

  • Know your stuff. Be sure to spend time mastering the topic and walking students through their own self-doubts and frustrations.

 

  •  Use a variety of activities and supports. Get students working in groups with structured, hands-on, real world activities with the teacher circulating to troubleshoot and provide one-on-one support.

 

  • Convey the “growth” mindset. Let the students know that some may have to work at it harder and they will approach the problem differently, but they can all master math.

 

  •  Answer all questions respectfully. Even if the question has been asked before, you might say, “ I am glad you asked me again to make sure you understood.

 

  • Be intentional about homework. Think about how many problems students need to practice, which problems will be most helpful, will help be needed, etc.

 

  • Reframe the purpose of quizzes and test. Make it clear to the students that the test are not to determine how smart the student is but to show how well the teacher taught the information.

 

  • Praise effort and reinterpret mistakes. Students should learn to see success as the result of effective effort and mistakes as a sign that more work is needed.

 

Developing Non-Cognitive Skills

Here’s an interesting perspective: Kentucky math teacher, Alison Wright, described how two students in her Algebra II class reacted to a quiz that was returned. One student looked at the test, rolled her eyes, threw the paper on the floor, and complained the test was not fair and should not count. The second student read the comments, reworked the problems to find her mistakes and stayed after class to discuss the test.

After some research Wright came up that a new approach that she is going to implement in her class this year.

·        Teach students that wrong answers are a helpful part of the learning process.  Many students shut down because they are afraid of having the wrong answer and failing.

·        Use cooperative group work as often as possible. By doing this students develop social skills necessary for teamwork while constructing arguments and providing valuable feedback to each other in a nonthreatening environment.

·        Use “A” and “Not Yet” as the only two possible grades. Wright believes this will help students that have bad reactions to failing grades.

 

Jumpstart’s Read for the Record: Otis

Join MetaMetrics®, We Give Books, Jumpstart, and record-breakers everywhere on October 3 as together we read Otis by Loren Long for Jumpstart’s Read for the Record®.

By taking part, you will not only help to set a new world record for the Largest Shared Reading Experience, but also help to model and encourage reading aloud to our children. Within The Lexile® Framework for Reading, Otis is an AD840L book. The AD code designates Otis as an “adult directed” book because such picture books are typically read to a child rather than a child reading them independently. Although seemingly easy reading, many picture books often make for a challenging independent reading experience to an age-appropriate reader for reasons of text complexity and book layout or design. Otis, then, presents a wonderful opportunity to share a great story with children and, at the same time, model for them how good readers navigate complex vocabulary and sentence structure.

The story of a fun-loving tractor and his unlikely friendship with a frightened young calf, Otis explores the themes of courage, determination, nostalgia, and usefulness. The story and its satisfying conclusion likely will appeal to readers both young and old. More information about Otis can be found by visiting the “Find a Book” feature on our website.   To read the book online, click here.

The beautiful illustrations and wonderful story provide readers the opportunity to engage the language, vocabulary, and close reading skills necessary for building success in early education and literacy. 

For example, younger readers might examine the use of prepositions as relationship words as the characters travel over the farm’s rolling hills, or through its haystacks, or even around Mud Pond. Older readers might consider the use of vivid, active verbs as the characters “leapfrog” rather than jump, or “explode” rather than run, or even “skirt” rather than avoid. And all readers can be more active in their close reading and re-reading of the story as they might:

  • keep track of and look up any vocabulary words they do not know;
  • note or mark key phrases or anything that strikes them as confusing or important;
  • keep track of the story as it unfolds;
  • note the repetition of words, phrases, ideas, images, events, etc.; and
  • write down questions they have about the text.

Read for the Record is a campaign that brings together millions of Americans to celebrate literacy by breaking the world record for reading the same book on the same day. This year, October 3 is the official Read for the Record day. More information about Read for the Record can be found here.

Recognizing the Value of Math

In a recent Gallup poll Americans were asked “Thinking about all the subjects you studied in school, which one, if any, has been the most valuable to you in your life?” The top three subjects were Math (34%), English/Literature/Reading (21%) and Science/Physics/Biology (12%). This is similar to the results from the August 2002 results where 34% of the respondents listed mathematics as the most valuable subject. 

With the emphasis now in school curricular standards on critical thinking, innovative problem-solving and effective communication skills, these results should be no surprise.  Many schools are emphasizing the importance of STEM (science, technology, engineering, and mathematics),  education and attitudes towards these once-dreaded subjects are changing.  Many now recognize the importance of mathematics and science in the preparation for post-secondary studies and career training.

While many Americans believe in the importance of the “three Rs” (reading, writing, and arithmetic) in public schools, the demands of curriculum should be fashioned  to promote student focus on logic,  reasoning skills, and the ability to report and justify their conclusions. As a mathematics educator, it is good to see mathematics listed as a top priority.   With new frontiers in science, technology, and engineering opening up, it is imperative that mathematics and language arts go hand-in-hand as the classroom subjects that need the most emphasis.  But overlooking creativity, innovative and logical thinking must also be included in the daily expectations of student inquiry.

One Less Excuse

Those of us who like to believe we are artistic and creative because we are left-brained or that we are analytical and that we reason logically because we are right-brained may be disappointed to learn that these explanations may be more myth than fact.

With the ability to collect data from neuroimaging in brain scans, scientists have observed that the functional connectivity and networking within brain functions are not concentrated in a specific hemisphere of the brain based upon the type of activity a person is performing. In a study out of the University of Utah, researchers found that there is little evidence that one side of the brain has a stronger influence upon our personalities or interests than another.

In other words, the functional network system of the brain seems to be so interconnected that many personality traits, strategies for thinking or creating, or personal areas of interest cannot be attributed to the stronger lateral side of our “gray matter.”  It’s not uncommon to hear individuals use the mythic right-brain left-brain theory to support certain abilities or account for specific deficiencies, e.g. reading or mathematics.  This study, however, casts doubt on such claims.  In fact, both hemispheres of the brain are tightly connected and are necessary for proper functioning.  Remember that next time you hear someone say they ‘don’t do math’ because they’re a right brained person.  That reasoning appears to be a fancier, dressed-up version of the idea that they lack the ‘math gene’ or they lack mathematical reasoning as an innate ability.  Math and reading, like any skill, can be learned and improve with practice.

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.