Independent Practice

August 10, 2020

Note to the Reader: This essay references five different lessons. For reading ease, I recommend selecting a grade level prior to beginning the essay, and then toggling back and forth between this page and the appendix you choose.

Rationale

Explicit math teaching introduces students to new concepts, temporarily advancing their understanding.  For this learning to internalize, children need to autonomously engage with the work soon after the lesson.  This is vital to develop student confidence and help teachers gauge what students do and do not understand.  Immediately practicing new topics helps children organize information and solidify their understanding, preparing them to later enter it into long term memory.

When planning the lesson, the teacher establishes intermittent stopping points to ensure student practice time.  This is done through scaffolded sample problems (see Appendix).  One lesson might be broken into four complexity tasks, each of which can stand alone as a mini-lesson.

Always time conscious, the teacher never ends a class period with instruction.  Instead, they aim to finish their teaching with a pre-established amount of practice minutes.  Even if they plan to cover four sample problem complexities, they will choose to stop after teaching three rather than shortchanging independent work time.

Master math instructors deliver informal assessments throughout their lessons, but until they see their class working independently, it can be challenging to know exactly how well individuals have learned.  Therefore, as independent practice begins, they resist an urge to quickly help struggling students.  Allowing everyone to engage in productive struggle empowers each child, while providing instructors immediate feedback on their teaching. This helps guide future interventions.

Unconfident math students have a tendency to wait for help if they feel confused and/or challenged.  When teachers provide it for them, they enable this helplessness.  Doing so also prevents the instructor from gaining insight into their students’ thinking.  The expert practitioner demands that their students begin working before providing help.  Even if a child struggles for 30-45 seconds and writes answers that are wildly incorrect, the teacher might still be able to interpret their mistakes and help correct them.  When they don’t allow students to try and fail, they perpetuate passivity without diagnosing what children do and do not understand.

Preparing the Assignment

After planning the lesson, the teacher examines the curriculum’s corresponding practice problems.  Through the lens of their weakest student, they study the problem sequence, trying to imagine their comfort level and emotions following a hypothetically smooth lesson.  They ask themselves:

Will the child feel overwhelmed by the first problem?

If the answer is Yes, they find or create a few simpler problems on topic so the child will feel confidence and momentum leading into the curriculum’s problem set (see Appendix).  By creating their own problem subsets, teachers have the power to address individual needs.  They’re able to write sequences that illuminate challenging concepts and, in the process, improve students’ skills and number sense.

Next, they reexamine the practice set from the opposite perspective:  Will the strongest student race through the problems, feeling under-challenged?  If the answer is yes, they find or create extension work (see Appendix).

With remediation, grade level, and extension work established, they examine the ladder of work they’ve created and ask themselves two more questions:

1. Will the independent practice assignments give every student an opportunity to solidify their understanding and stretch their thinking within the allotted work time?

2. Will the assignment flow in such a way that there will never be a rut in which more students need help than can be accommodated?

If the answer to either question is No, they add problems in their laddered work assignment.  These additional rungs provide more leveled practice, bolstering all students’ confidence.  In the end, the assignment is a summation of the differentiated mathematical journey students experienced during the lesson.

Using this approach, the teacher acknowledges their classroom reality.  The larger the class’ achievement range, the longer the assignment and independent practice*.  The combination of the former and the latter ensures that all students engage with meaningful work.  Number one in the curriculum’s problem set could be remediation for the strongest students but extension work for the weakest.  With the ladder of work established, they might choose to start every child on the simplest problems and have them work linearly until independent practice time ends.  They also have the option to start students at different entry points on the ladder.  Either way, they’re providing each student an opportunity to optimize their learning.

When creating a laddered assignment, the teacher is challenged to determine how much practice to provide.  Depending on their comfort with the content, some students need more repetition than others.  Ideally, every child would practice as much as they need to master content, but no more.  At the same time, it’s unrealistic to always find this balance.  By consistently trying to meet each student’s needs, teachers occasionally provide too much or too little practice.  When in doubt, they always aim for the former rather than the latter.  It’s far easier to tell students to skip problems and move on to more challenging ones than it is to supply additional impromptu practice.

Finding the correct balance of how much practice to provide requires a counterintuitive approach.  Educators often think that mastering skills and concepts comes from burning it into memory through repetition.  Although this intuitively makes sense, new learning is retained better when practice is interleaved.  A young pianist benefits more by practicing a challenging piece in two spaced out one-hour sessions than a single two-hour session.  Likewise, solving crossword or Sodoku puzzles is more efficient in short focused spurts than in one sitting.

* As a general rule, the older the child, the longer they can sustain concentration.  Thus, work sessions are usually longer in upper grades than lower.

Extension Work

For students to stay interested in mathematics, they need incremental challenges while they practice. Whenever possible, the teacher selects or creates extension work that aligns with the lesson.  Immersing children in a singular topic makes the work more interesting and the practice more meaningful; studying something in a concentrated manner always leads to better memory than disjointed learning sequences.  The most skilled teachers commonly enrich practice by synthesizing previous topics and/or providing the next logical step in its strand.  In upper elementary grades, geometry connections are frequently the greatest outlet for extension topics. This is because geometric concepts are often overlooked, forgotten, and only practiced in specific weeks of the school year (see 4th & 5th Grade Appendices).

During computational units, word problems create natural extensions for early finishers because they apply recently learned skills in an authentic context.  This makes it easy for teachers to crutch on them as default extension work.  Doing so, however, creates a different set of challenges.  If the word problems are too easy for the early finishers then the teacher has simply given them more busy work.  Regardless, practicing word problems often becomes tiring for children.  They might be interested in working on one or two, but will invariably lose interest if the practice starts seeming monotonous.

Other Extensions

Deco Trees

Extension work is often considered advancing mathematical understanding and/or learning new topics, but it can also be used as an outlet for retrieving and integrating previously learned concepts.  One tool to achieve this is Deco Trees, a decomposition activity that synthesizes fluency and problem solving skills.  Their design and simple to complex sequences stimulate creativity by guiding students to combine and decompose like and unlike units.

Far too many children see numbers as static rather than flexible figures.  This not only encumbers their ability to understand harder concepts, but it also stifles their creativity.  Many third graders know that 8 x 7 is the same as 8 sevens, but not 5 sevens and 3 sevens.  Fourth graders always seem to know that 2 1/3 is two wholes and one-third, but not one whole and 4-thirds. For students to excel in elementary mathematics, they need to recognize numbers and units in varied forms.  Deco Trees guide this process.

Each Deco Tree consists of a series of missing parts that center around a singular whole (see Appendix).  The first six problems consist of a given part that prompts students to fill in a missing number, which combines to make the whole.  The combinations begin simple and then gradually become more challenging, exposing students to multiple formats so that they are prepared for deeper complexities in the Make up your own section.  Some students need stimuli to induce creativity, while others might feel restrained by the prompts.

After completing the first side of the Deco Tree, students exchange papers and correct each other’s work.  They check all of their partner’s combinations but the last three provide the best springboard to stimulate creativity.  When students are finished discussing their answers, they turn to the back and decompose the same, or an analogous, whole.  In either case, the formats from stage one can be used to make combinations in stage two.

Some students will opt to treat the second stage as a race, trying to complete as many decompositions as possible, but most children gravitate to higher order thinking and, without consciously trying to do so, generate more complex, creative answers without being prompted.  This reality speaks to an intrinsic tendency of our species.  When a task or puzzle is within one’s skill set, people tend to embrace the challenge, stretching the boundaries of their creativity and understanding.

Games

Rather than sacrificing dynamic practice, teachers might direct early finishers to work on projects or logic games.  Making the former meaningful is far more challenging than the latter.  Some of the best mathematical games include:

• Parcheesi

• Battleship

• Mastermind

• Sodoku

Parcheesi is a dice game that naturally differentiates to student skill levels.  Beginners can move their pieces by counting all, while stronger students can advance by grouping numbers.  The board consists of symmetrical seven-, five-, and ten-jump benchmarks.  Some students will organically figure out that they can move their pieces by pivoting on these facts.  They, or the classroom instructor, can share counting strategies with those who don’t.  Subitizing dice dots and adding single-digit numbers together helps K-2 students with addition practice.  Upper elementary students benefit by using probability to guide their decisions.

Battleship provides upper elementary students with intensive practice using coordinate grids while simultaneously weaving in strategy and logic. Neither Mastermind nor Soduku provides children with direct mathematical practice or application, but both strengthen mathematical thinking through reasoning, and logic.

When to Practice

Independent practice normally takes place in the last quarter of a class period, immediately following explicit instruction.  However, some teachers opt to bridge their math lessons over a break in the school day, e.g. delivering fluency, problem solving, and the lesson before lunch, and then providing independent practice time when students return to the classroom.  Although this practice normally arises out of scheduling necessity, it holds distinct advantages over the traditional, condensed approach.

Segmented class periods help teachers make adjustments in the same way that halftime allows football coaches to change their game plan.  When instructors perform dozens of informal assessments throughout a lesson, they are alert to unforeseen misconceptions and challenges they hadn’t accounted for during their planning.  By providing a 45-minute to one-hour break in the middle of their math lesson, teachers can use their prep time to alter plans, problems, etc. before students return.

The first three stages of the four-part lesson can take anywhere from 30-45 instructional minutes.  At this point in the class period, students are often tired and struggle to concentrate while practicing independently.  Children who return to work after lunch, recess, or a specialist class, do so with renewed energy and enthusiasm.  Their mind has rested and they are ready to refocus themselves on mathematics.

Bridged class periods also complement the nature of mathematical learning.  An interactive lesson immediately followed by independent practice condenses new content into a compressed time period.  This can exhaust any child, but it takes an especially hard toll on struggling students who are straining to concentrate.  Mental fatigue is the mind’s way of telling them self:  I’m trying to concentrate and learn, but I’ve been working hard and need to rest.  Scheduling breaks between lessons and independent practice provides brains with the rest they’re craving.  Meanwhile, as children shift their focus to physical endeavors, social interaction, or even a different academic subject, their subconscious continues processing the lesson they just departed.  When they return to independent practice, concepts that they had been struggling to understand are often crystalized.

Internalizing math concepts works best when students are exposed to a new topic, struggle to understand it, practice it for a short time, and then - following another minor retention drop - return to practice it more.  Knowing that learning is strongest when practice is interleaved, the best math teachers prompt their students to work during lessons, soon after instruction is over, and later at home.  If they’re forced to break up their math block, they use the disjointed schedule to their advantage.  If they’re locked into isolated class periods, then they begin building their students’ mental and working stamina starting on the first day of school.

Optimizing Practice Time

During the early days of school, the instructor explicitly teaches their class how to practice and stay organized.  Thus, after the first week of lessons, whenever they direct their students to begin working, there are no lengthy delays in getting started.  Although the organizing mechanisms vary from teacher to teacher and grade to grade, each child has easy access to a sharpened pencil, paper, and workbook (see Table 1).

Independent practice routines also differ from classroom to classroom, but the overarching philosophy is the same.  Students understand that independent work is an integral part of daily math lessons.  Unless otherwise told, they are responsible for trying to solve problems on their own before asking their neighbor.  If their math partner can’t help them, then they know who to ask next.  This might be a specific student, a classroom assistant, or a student teacher.  If all of these options have been exhausted, they quietly raise their hand and wait for the teacher to help them.

An order of assignments that grows from simple to complex is always posted in the same place, e.g. a table folder or a corner of the classroom board.  At all times, children are working, moving on to a new assignment, or waiting for teacher help, but never asking, What am I supposed to do?  Logic puzzles, ancillary texts, and other resources that might be used as tasks are neatly organized on a shelf, where students can quickly retrieve them without disturbing their classmates.  All children are intimately familiar with this section of the classroom, so they never have to ask where to look for an assignment.

The master instructor understands that 90% of their attention must be directed towards the weakest ten percent of the class. To ensure this, middle and upper elementary teachers post a system of answer keys around the room.  The students’ desk position dictates which answer key they use (see image on right).  After completing an assignment, children check their answers.  They never move on to the next task until their work is completely correct.  Trying harder problems without first correcting and understanding mistakes is not in the child’s best interest, because the ensuing work becomes more challenging.  If students answer a problem incorrectly, they do the following:

1. Try to understand the mistake they made.

2. If they don’t understand their mistake, discuss the problem with their partner.

3. If they’re still confused, they ask a different classmate or raise their hand and wait for teacher help.

The teacher emphasizes that students should never expect to finish all of the listed tasks. The entirety of independent practice time is spent engaging with incremental challenges, so there will always be extensions to work on.  When they project a problem set, students know to record their answers in personal notebooks.  While practicing, children are held highly accountable.  No one works long without self-correcting or having their answers checked by a teacher.  Errors are quickly recognized, corrected, and oftentimes practiced further before moving on to harder work.  This ensures that students don’t continue practicing under misconceptions.

Malpractice

There are two pitfalls that math teachers typically encounter when preparing independent work.  One is a common practice and the other is a relatively new trend.

Many educators structure their lessons by teaching several example problems and then directing their class to corresponding textbook or workbook practice.  This works better with some curricula than others.  Traditional textbooks often provide dozens of problems that allow some students to get the necessary practice they need, while simultaneously keeping more advanced students busy.  Although they provide adequate practice quantities for most (if not all) students, it allows teachers to be complacent in their preparation.  Instead of building intricate practice sequences that target specific children’s needs, they direct their students to bland problem sets that rarely interest or appropriately challenge them.

The Eureka Math and Singapore Math programs are exceptions to this rule but present teachers with an alternative problem.  In both curricula, lesson-aligned problem sets scaffold, starting with basic problems that quickly escalate in difficulty.  The laddered sequences are often too steep to hold meaningful, attentive practice for entire classes.  As a result, a few students are frustrated before or after finishing the first problem, while others blaze through the entire practice set and become bored with nothing to do.  The end product is a dysfunctional dynamic in which 60-70% of the class waits for help as the instructor buzzes around the room trying to stamp out small fires.

In recent years, more and more teachers have allowed their students to choose independent work from a collection of assignments that span a large complexity range.  The theory goes that children will naturally gravitate to assignments that stretch, but don’t overstretch, their understanding.  This implies that each student is motivated to reach their personal best, and always possesses the discretion to choose appropriate assignments.  Recognizing themself as the classroom’s math expert, the wise educator understands the absurdity of this rationale, and sees flaws in the theory’s logic even if all children comply with the utopic reality.  Seeing no point in wasting instructional minutes with students looking over and selecting from an assignment buffet, they choose the problems and activities that best lead their students to mastery.