Meaningful Math Homework

September 14, 2020

Homework often evokes strong emotions among students, parents, and educators.  Although children tend to dislike it, adults have traditionally agreed that it’s a necessary - albeit unpleasant - part of schooling.  In recent years, the practice has been reexamined, and many studies have emerged, proving no correlation between homework and academic achievement.

Parents and educators figure prominently on both sides of the widening philosophical chasm.  Some advocate for more homework, while others think it impedes a well-rounded childhood consisting of play, exploration, and family time.  Some administrators require daily homework for each subject, while others disallow it on weekends, or ban it altogether.

Understanding that their discipline is challenging to learn outside of class, the math teacher assumes meaningful progress will only occur under their supervision.  Still, in addition to their subject specialty, the instructor also feels a duty to teach their students responsibility.  If they choose to give homework, they use the practice, not as a catalyst to advance learning, but instead to develop discipline and academic readiness.  Taking an assignment home, completing it, and turning it in the next day is a valuable life skill, even if it’s not always fun and doesn’t lead to higher academic achievement.

The teacher also understands that happy children learn well, while frustrated, anxious, and intimidated students do not.  Excessive and/or over-challenging homework often leads to student apprehension, creating a ripple of negativity that carries over to the classroom.  When working on these assignments, children are inclined to seek parental help, which creates a different set of challenges.  At best, the child receives instruction that conflicts with their lessons.  At worst, the assignment leads to a toxic parent/teacher relationship, and the child feels wedged between the most influential adults in their life.

Considering all this, the master teacher never assigns homework that might damage individual morale between the end of one class period and the beginning of the next.  Their assignments are solely used as additional practice that either complements a recent lesson or reviews important skills. To optimize the work, the teacher creates confidence-building tasks that provide adequate, but not excessive practice.  In turn, their students arrive to class each day happy, confident, and motivated to learn more. The teacher has no illusions of the planning time this requires.  Therefore, they choose between assigning thoughtful homework, or not giving it all.  Preparing it, they understand, is not nearly as time consuming, detrimental, or calamitous as working with unhappy students and parents.

If they choose to give homework, their assignments normally fall into one of three categories:

1. Conceptualization

2. Skill Building

3. Cumulative Review

Conceptualization

Checking or reviewing homework assignments provides teachers with very little insight into what their students do and do not understand.  A child might get their assignment perfectly correct, but the instructor can never know for sure if they received help from a parent or copied off a classmate.  It’s also challenging for them to know how long the students spent working on the assignment and, in some cases, their methods and rationale for arriving at specific answers.  When a child comes to class with an incorrect answer, the teacher has no way of knowing if the student worked on the problem for three minutes or three hours.

Assigning homework that addresses conceptual and/or model understanding works differently.  Teachers who use this method provide answers and require students to draw diagrams and/or write explanations to show why the answer is correct (see Conceptualization Examples).

Teachers can use the same method without providing answers, but in either case, demonstrating model comprehension and/or conceptual understanding is the assignment’s purpose.

Conceptualization homework provides a summation of that day’s lesson, prompting students to recall its objective.  Because the answer(s) are provided, parental interference is minimized and usually unhelpful.  Instead, it provides excellent opportunities for a child to share their learning and understanding with their guardian, improving home and school relationships in the process.

Skill Building

Homework is a poor outlet for learning new concepts, but it can be an excellent channel for bolstering basic arithmetic skills.  After learning a concept and strategies to solve problems, students need practice to build fluency and later automaticity.  The marriage of conceptualization and efficient solving strategies leads to meaningful practice.  Through intentional, sequenced repetition, students gradually increase their solving efficiency and enter basic skills into long term memory (see Skill Building Examples).

Exemplary teachers who assign this type of homework are careful to provide ample, but not massed practice.  The former keeps students engaged and focused, while the latter can be exhausting and antagonistic.

Cumulative Reviews

Although homework has traditionally been linked to same day lesson content, the master math teacher does not feel constrained to these parameters.  When lessons don’t go well and/or the content doesn’t lend itself to meaningful independent practice, the instructor might assign skill work or cumulative reviews.  Regardless of what they are learning, there is never a bad day for a first grader to practice number bonds or a fourth grader to simplify fractions.  Likewise, during the winter or spring, students can benefit from revisiting topics that were taught earlier in the school year (see examples below).

The thoughtful practitioner rarely, if ever, assigns item-analysis cumulative reviews.  Jumping from topic to topic, they understand, does not strengthen memory.  Meaningful review requires depth, repetition, and concentration.  During fluency practice, the teacher might bounce from concept to concept informally assessing what students have and have not retained from earlier in the school year.  They then provide cumulative review homework that targets areas of collective weakness.