Deep End Math
Accelerating Math Instruction for Gifted and High Achieving Elementary Students:
What are we doing for the kids who DO get it?
Data-driven instruction frequently reveals students who enter a new grade already having mastered the majority of instructional content. This workshop presents pedagogically sound strategies for both in-class differentiation and accelerated replacement instruction, drawing on ten years of successful practice.
Why Differentiate Math Instruction?
1. Students are in school to learn.
As educators, it is our responsibility to actually teach every student. Obvious, right? But there is a common misperception that advanced learners do not need to be taught if they are already performing at or above grade level. Perhaps you have heard something like this:
"Those high kids will be just fine."
"You can just let them go!"
Or even, "I take no responsibility for teaching your child," said with a smile.
Grade level skills are NOT an end-point. Our responsibility to teach students does not go away once they have mastered grade level skills. It is our duty as educators to take our students to their next level, whatever that level might be.
You have a right to...
- know about your giftedness.
- learn something new everyday.
- be passionate about your talent area without apologies.
- have an identity beyond your talent area.
- feel good about your accomplishments.
- make mistakes.
- seek guidance in the development of your talent.
- have multiple peer groups and a variety of friends.
- choose which of your talent areas you wish to pursue.
- not to be gifted at everything.
All children, including gifted children, have the right to learn something new everyday, and we have the responsibility to teach them. So when we have students with different levels of skill and ability, we must differentiate to make sure that every student's educational needs are met.
2. Growth Mindset: Students must learn to strive.
Carol Dweck's work on mindset shows the importance of connecting success with effort:
- If students continually "succeed" without expending true effort, they discount the result because it didn't cost them anything.
- Only when students are challenged can they develop the skills of persistence and effort.
- Teaching children how to strive gives them the skills to work through the frustration of genuine challenge when they are older.
Elementary grades are NOT too early to learn to strive! Not only that, but waiting until middle or upper grades may leave students with a "fixed mindset."
Students with a fixed mindset often give up more easily in the face of challenge, and they may accept failure as a reflection of self-worth or ability instead of connecting it to effort. A student with a fixed mindset might think, "I'm so bad at math; I'm dumb," and stop trying in math class because they feel doomed to failure due to their (perceived) innate incompetence, while a student with a growth mindset might think, "I didn't do so well on this math test, but if I practice division more, I could do better next time," which sets the student up for future success.
For more information on Carol Dweck's work, visit her website, MindSet.
Gifted children are at an especially high risk for developing a fixed mindset when they don't have to "try" in the elementary grades, so differentiation in the classroom is necessary to teach them to strive and to help them develop a growth mindset.
3. Behavior issues are minimized when kids are engaged.
Learning is the opposite of boredom, and learning is the antidote to boredom. Engaged, interested kids are more likely to be focused on their work and less likely to tune out, goof off, or distract other students. So how do we keep kids interested and engaged?
Students learn best in the Zone of Proximal Development, where they are presented with an appropriate level of challenge that piques their interest and encourages them to use what they know to expand their skills. An article worth reading is "Boredom and Its Opposite," from ASCD.
While high-achieving kids are often characterized as "good students" because they perform highly in their academics, it is not uncommon for these students to start pushing limits and acting out when they get bored, bringing about confusion: why are the "good students" suddenly developing behavior problems?
Differentiation in the classroom is key to keeping these students engaged at an appropriate level of challenge to prevent boredom and behavioral issues.
Identifying Students Ready to Move Forward
While every student should be taught "as though they were gifted" in that every child should be given an appropriate level of challenge, that level will vary from student to student.
Preassess for every unit and chapter and analyze students' scores:
- Scores of 60 - 70% indicate a strong foundation, possibly some areas of strength that can be compacted.
- Scores approaching 80% show strength, likelihood of general understanding.
The higher the score, the greater the need for acceleration. Review students' preassessment mistakes. What kinds of errors were made? Careless? Procedural? Conceptual?
Consider use of "Hardest First" (below) and allow fluency in grouping per students' changing needs (see 3-Tier Differentiation), but let's realize it's not always all about the "hard" scores of objective data:
Are we teaching math or are we teaching children?
Don't let the data blind you to the child that is right in front of you.
Given the opportunity, gifted children often identify themselves. Provide challenges and engaging opportunities, and see who pursues these, and always remember:
The overall goal is CONTINUAL learning and growth for ALL students.
Another great resource for differentiation in a mixed classroom is Teaching Gifted Kids in the Regular Classroom by Susan Winebrenner.
Students who are successful in the completion of the most difficult questions or problems and can demonstrate proficiency with this work then have earned the opportunity to use remaining time to explore this or related or alternative content in more depth.
This way, the students "buy back" time to work on more advanced or engaging challenges. This strategy can be offered routinely to all students or reserved for those who have demonstrated adequate proficiency in the unit pretest.
One of the challenges of a integrated classroom is that students will move at different speeds through the material. This model was developed for use in general ed. classrooms with students of varying ability levels.
3- Tier Model for Mathematics Instruction
The Intro/demo at the start of class serves tooutline the day’s content. For most students, this sets the direction for the day’s instruction. For advanced intuitive learners in a gen ed classroom, this outline is often adequate for them to move directly into practice.
Teacher Table is differentiated based on the level of independence of the group:
- The most dependent students receive direct instruction, teacher demonstration of practice work, and guided group practice. Then the teacher observes as they begin to work on their own.
- The needs of the average-level group are usually met through direct instruction and brief demonstration of practice work.
- The most independent group is usually able to complete practice problems with minimal instruction, for which the general introduction has sufficed. Teacher Table for this group focuses on assessment and evaluating appropriateness of additional practice or challenge.
During written practice and problem-solving, students are expected to work independently but occasionally get “stuck.” Supports are available. They should first reread the question and look back at text supports, such as sample problems. If they are still uncertain, they may consult any other person in their group. Students who have completed all of their practice work may compare answers and work together to resolve inconsistencies.
The cooperative and/or manipulative activity may directly reflect the day’s instruction, or it may be a related skills-reinforcement, such as Fact Football or Equate. Technology can also be incorporated through online programs or software. Most of the cooperative/manipulative station work is not scored; assessment is per time management and responsible participation. This time may also be used for work on more involved ongoing projects (see Activities).
The relatively brief time frame for each station draws on brain-based learning research. Students stay on task because of the limited window of opportunity. As the class reconvenes after stations, the teacher briefly reviews a problem or two from each level of difficulty and clarifies HW assignments based on students’ work completion and understanding.
HW is often differentiated; students are directed to complete the assignment most appropriate to their learning needs. Based on their own confidence and understanding, students are welcome to switch groups. It has been my experience that students strive to join higher groups; those who are not ready realize that they benefit more from additional support. I have had advanced students move to dependent groups for specific units in which they struggled (fractions). As all students address similar content and have similar activities, stigma is minimized.
In curriculum compacting, students move through the regular curriculum at an accelerated pace, with checkpoints of accountability to show that they have indeed demonstrated mastery.
Visit Prescription for Gifted Success for more information on curriculum compacting as well as guidance on curriculum compacting strategies.
Typical student profile: In accelerated replacement instruction, students typically enter the grade level demonstrating 75% mastery of the content for the coming year. Students who have typically been highly successful in this setting are those who show a high degree of curiosity and initiative. Often these students are either extremely driven to achieve at high levels or they are exceptionally creative; some are both driven and creative, though this is unusual (See Types of Giftedness). The structure and goals for the class may be adapted to match the strengths of the individuals in the cohort. We have had accelerated replacement mathematics classes from grades three through six, with as few as two students or as many as twelve (from an average grade level cohort of 105).
Selection of students: Typically in selecting students for such a class, we have considered recommendations by the prior math teacher, the classroom teacher, and the GT teacher. We have considered the pretest score on the content for the coming year. We have considered the NJASK Math scores for the previous two years, if available, and how these compare to the local cohort. We have also considered the student's score on the ADAM K-7: Adaptive Diagnostic Assessment of Mathematics, and, in grades 4 and above, the DOMA Pre-Algebra assessment.
Content of Instruction: In all cohorts, the first priority is mastery of the grade level standards. Our recent practice is to use the grade level Go Math! textbook as a base, with students previewing half chapters at a time, with selected "challenge problems" as written HW. Most chapters were covered quite rapidly, with mid-chapter assessments and then chapter tests for accountability. If students within the cohort are moving unevenly toward mastery, those who are ready may go forward with finishing a chapter and working on the project (see Activities) while the student(s) requiring more support receive targeted assistance.
Planning ahead: Acceleration may seem to naturally lend itself to continuation with subsequent materials. However, please be certain that plans are well articulated with subsequent grade levels before taking this approach. There is tremendous breadth possible in mathematical exploration without simply going on to the next year's textbook.
My recent sixth graders (class of '15) included students who could easily have worked on 7th and even 8th grade math content. However, in the absence of such articulation, we instead took on two distinct challenges: a BASF Science Education grant and collaboration with Drexel's Math Forum to take our Year Game wiki to international participants. The students gained skills in planning, technology, critical thinking, and communication, as well as the confidence of knowing their own powers. This year (16-17) I will be work with fourth graders for semi-replacement, using some content acceleration with Real World Algebra and other strategies for computational fluency but also incorporating mini-robotics, empowering these students as turnkey for peers with this new technology.
Raising the Bar
It's not just that students are solving the hard problems; it's how students are solving them.
You can "raise the bar" for students by setting your expectations beyond getting the right answer. Students should be able to demonstrate computational accuracy, to fully understand mathematical concepts, to choose appropriate strategies and procedures, to draw useful diagrams and sketches when relevant, to explain their work, and to organize their work neatly so that it is easy to follow and identify the answer. Continental Mathematics League provides challenging math problems for kids at grade levels 2 and up that can be used to assess whether students meet these high expectations. For example:
A box of 140 marbles contains only red, blue, and green marbles. There are three times as many green marbles as red marbles. There are six times as many blue marbles as red marbles. How many blue marbles are in the box?
A grocer bought 50 dozen grapefruit at $2 per dozen. He found 30 bad grapefruit and threw them away. He sold all the rest of the grapefruit at 3 for $1.00. What was the grocer's profit on the grapefruit?
There are two mugs, one of hot cocoa and one of coffee. One third of the contents of the first mug is poured into the second and then one third of the contents of the second mug is poured back into the first. What is the ratio of cocoa to coffee in the first mug after the mixing?
These problems should mathematically doable for students, but they are also complex enough to expose gaps in understanding or technical skills. There are different tools you can use to analyze students' work for the criteria introduced above:
Consider teaching students how to assess their own work through Error Analysis exercises. When students correct tests, quizzes, or HW, they analyze their mistakes and tally by category:
Each of these types of mistakes carries different implications for responsibility and remediation.
Activities for Zippy Kids
When someone starts zipping ahead, you can give them activities that lead them to investigate further based on the mathematics they already know. For example:
- The Year Game by Drexel Math Forum
- Φ : The Golden Number (phi). Vihart's mini-series on the Golden Ratio is also a great resource.
- Powers of 2: Positive and negative powers of 2 demonstrate the scale of exponential functions and builds familiarity with fractions.
- Edward Zaccaro's books are comprehensive and fun. Titles include:
- Primary Grade Challenge Math
- Becoming a Problem Solving Genius
- Real World Algebra
- The 10 Things All Future Mathematicians and Scientists Must Know (But Are Rarely Taught)
Math is all around us, but students don't know that yet. Some exciting applications for them to learn about the math surrounding them in everyday life are:
- Architecture: See the Architecture page for a full list of activities students can dive into.
- Geometry and Angles: Students learn about tessellations and how kaleidoscopes work.
- The Cost of Life: Students come up with a "dream life" and balance their budget, including living expenses, salary, supporting a family, student loans, vacations, investments, et cetera!
- Road Trip! Whole class end-of-year activity in grade 4: planning a family vacation including selecting routes, calculating mileage, costs of gasoline/food/lodging, travel time, etc.
Professional Standards for NJ Teachers
The professional standards for New Jersey teachers overlap with the practice of math differentiation in the classroom. Here are some specifics:
Standard One: Learner Development
The teacher understands … patterns of learning and development vary individually … and designs and implements developmentally appropriate and challenging learning experiences. ***In mixed level classes, instruction is often targeted to the median, if not to the remediation of those students who are struggling. Students who have already attained grade level competencies may be provided with texts at higher levels of readability, though lacking conceptual extension. We believe that by targeting the higher levels of conceptual development, not only can the advanced learners benefit, but those students at middling or lower levels of readiness can be motivated to strive by the engaging nature of these challenges.
Standard Two: Learning Differences
The teacher uses understanding of individual differences … to ensure inclusive learning environments that enable each learner to meet high standards. ***Targeting student strengths supports continual growth, thus expanding inclusivity to those already capable of grade level objectives, not only meeting but exceeding standards.
Standard Three: Learning Environments
The teacher works with others to create environments that support … active engagement in learning, and self-motivation. ***Research shows that pairing high levels of challenge and stimulation results in strong student engagement (McCoach and Siegle, Gifted Child Quarterly, 2003); when general education targets grade level skills, gifted children typically become disengaged, resulting in underachievement. Providing enriched environments and higher expectations counteracts this pattern. As highly capable students become engaged, they often contribute to instructional planning, which furthers motivation.
Standard Four: Content Knowledge
The teacher understands the central concepts, tools of inquiry, and structures of the discipline(s) he or she teaches, … and creates learning experiences that make these aspects of the discipline accessible and meaningful for learners to assure mastery of the content. ***Extending to interdisciplinary transference and previewing concepts and content from subsequent grade levels to open pathways to continual growth, equipping students for independent inquiry in self-selected areas of interest, as well as deeper engagement with required materials.
Standard Six: Assessment
The teacher understands and uses multiple methods of assessment to engage learners in examining their own growth, to monitor learner progress, and to guide the teacher’s and learner’s decision making. ***Students not only use checklists and rubrics for self-assessment, they frequently participate in construction of such tools. By grade 6, students design their own culminating projects, considering not just their desired mean of expression but also the objectives as specified in the CCSS.
Standard Seven: Planning for Instruction
The teacher plans instruction that supports every student in meeting rigorous learning goals ... ***Instructional planning is articulated across grade levels, with subsequent units building upon skills and foundational concepts from earlier grade levels.
Standard Ten: Leadership and Collaboration
The teacher seeks appropriate leadership roles and opportunities to take responsibility for student learning, to collaborate with … other school professionals, and community members to ensure learner growth, and to advance the profession. ***Including the gifted education teacher as a teaching consultant and in regular GT-cluster team teaching supports classroom teachers in flexibly adjusting for students’ readiness and prior knowledge, by facilitating differentiation, compacting, extension, and targeted remediation on an ongoing small group and individual basis.
Standard Eleven: Ethical Practice
The teacher acts in accordance with legal and ethical responsibilities and uses integrity and fairness to promote the success of all students. ***One common concern with gifted education is exclusivity, that access to advanced opportunities is limited to a predetermined number of students or only to those meeting criteria at preset points. Our identification procedures are ongoing, with annual nomination of new candidates at all grade levels. Beyond formal identification, we informally identify students that may be ready to benefit from GT cluster classroom settings as class rosters are organized for each school year. Classroom teachers at all grade levels are encouraged to consult with the GT teacher for resources and support in high-level differentiation, regardless of whether they are working with a formally identified GT student cluster.