The Lesson

February 26, 2020

Mathematics is a sophisticated form of communication that has developed over millennia.  Its unique nature makes it vastly different than other academic subjects because of its heavy reliance on the instructor.  A child who knows basic phonics and sight words can advance their reading levels through practice and exposure to more challenging literature.  They can then teach themselves history and - if they’re also a good mathematician - science.  Independently improving mathematical understanding is much harder.  A first grader, for example, is unlikely to absorb second grade math concepts by independently practicing grade level content and extension problems, but they might increase their reading level using the same method.

Archimedes, who is widely regarded as one of history’s greatest mathematicians, died not understanding the decimal system, not because he was intellectually incapable of learning it, but because its development was in its infancy.  Today, average fifth grade math students comprehend a model that the greatest mind of the 200 BCEs did not, simply because the science has advanced and knowledgeable teachers abound.  Although decimal place value is a relatively easy concept, late elementary school students could never be expected to learn it without explicit instruction.  For the same reason, third graders won’t intuitively figure out the long division algorithm and middle school students won’t discover algebra through exploratory learning.

In delivery, expectation, and student experience, quality math lessons are delivered unlike any other subject.  Veering from traditional definitions of instruction, the teacher gives many directives, but doesn’t demonstrate or tell their students how to do anything.  Extracting understanding without explaining, they transform the subject from a noun to a verb.

Baseball is practiced by hitting, fielding, and throwing, and the guitar is learned by plucking, strumming, and transitioning between chords.  Students learn math, not by following procedures, but instead Mathing.  Throughout lessons, the ratio of teacher instruction to pupil participation is heavily weighted towards the latter.  By providing stimulus without downtime, the educator trains their students to think, reason, discuss, and practice for the duration of the instructional period.  Constantly alert, children improve their mental stamina and become better thinkers.  Ruminating to make sense of the world is the subject’s essence and innately human.  To learn the discipline children simply need a teacher to shine light where there is darkness.

Math offers its teachers more accurate feedback on their instructional quality than any other subject.  During lessons, student comprehension, or lack thereof, is immediate and transparent.  Because learning new mathematical topics is predicated on cumulative foundations, the teacher must constantly know what each of their students does and does not understand.  This allows them to advance or retract complexity, keeping their students in a comfortable learning zone.  Collecting this necessary information is not possible without performing frequent informal assessments.

Methods for delivering quality, dynamic math lessons vary, but their preparation, beginning, middle, and end share a few common qualities.

• The teacher is time conscious.

• Instruction starts with student activity rather than passivity.

• Students never go long without feeling successful or challenged.

• The lesson ends before the class is collectively bored and/or frustrated.

Preparation:  The teacher is time conscious.

Fluency and problem solving has awakened and invigorated students’ minds when the math teacher looks down at their wristwatch and tightly transitions to the lesson.  All learning tools have been distributed and each child has them comfortably organized.  Because the tools are easily accessible but not distracting, students’ physical and mental workspace is tidy.  The class is prepared to learn and the instructor - knowing exactly what they want to say and do – is ready to teach.

The efficient math educator always knows what time it is.  Properly valuing and respecting instructional minutes, they understand, determines student success and failure.  They know that successful lessons are more the product of learning conditions and preparedness than the instructional period itself.  Student learning, therefore, is achieved long before concepts are taught.  The lesson is delivered at a time when students’ energy and focus is optimal, and disruptions are minimal.  Bathroom breaks are not allowed, the classroom phone doesn’t ring, and the PA system is silent.

With these structural necessities in place, the teacher takes full ownership of their students’ learning.  All foundational skills needed to succeed in the lesson have been reviewed and practiced in the previous days or weeks.  First grade students automatize their Make 10 facts before a lesson that requires them to Add single-digit numbers crossing the ten by breaking apart the second addend.  Leading into an Add fractions in which one denominator changes lesson, fourth grade students are proficient with modeling and computing equivalent fractions.  By solidifying foundations prior to teaching a new concept, teachers are only tasked with helping their students make a small incremental gain.  When this doesn’t happen, instructors invariably spend more time reviewing previous topics than they do addressing that lesson’s objective.

The Beginning:  Instruction starts with student activity rather than passivity.

Stating an objective is a conventional approach to starting any lesson.  Many administrators require it, rationalizing that students must see a purpose to their learning before they’ll engage with the content.  This is logical in theory, but foolish in practice.  The philosophy implies: 

1. Students will pay attention while the objective is being stated.

2. Students possess the vocabulary to comprehend the objective.

3. Understanding the lesson’s purpose drives student motivation.

4. The lesson tangibly connects to student interests.

Today we’re going to be learning about… are the seven most misused words in math education, because - regardless of what follows - it’s likely that students aren’t listening to the teacher’s message.  Most American children (and many adults) stop paying attention if someone talks to them for longer than fifteen seconds.  For the few students who continue listening, the words they hear are often confusing.  Write base ten three-digit numbers in unit form and show the value of each digit is abstract wording for second graders.  Likewise, most fourth graders will struggle to make sense of Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.  Instead of generating interest, stilted language more often bores, confuses, or intimidates children.

Even when students understand the objective, they won’t always be captivated by its message.  First graders might not care that Grouping teen numbers into a unit of ten and some ones is a gateway to more advanced mathematical learning; and, many fifth-grade students will feel indifferently about Recognizing volume as an attribute of solid figures.  Connecting math inside and outside of the classroom can be valuable, but continually searching for lesson relevancy is poor use of a math teacher’s time.  The wise educator channels their planning energies into question and problem sequences that allow every child to feel successful, but also challenged.

Answering questions correctly, mastering new skills, and developing self-confidence are inherently exciting to children.  Therefore, they don’t need to see an outside of class purpose to be interested learners.  Children find joy in simply getting answers right and moving on to more challenging problems.  Long before they develop cynicism, young people are internally motivated by feeling smart and showing off what they know.  Educators who understand this are more effective, because there aren’t always real-world applications to the content they’re teaching.

Making change and double batching recipes are often cited as reasons to learn mathematics.  Although both are useful life skills, they don’t justify an hour of learning each day for the entirety of a child’s schooling.  The subject’s purpose is not utilitarian, but instead to develop greater critical thinking skills.  Many people go through life never needing to accurately draw a triangle, or divide a fraction by another fraction, but both are catalysts for intellectual development.

The master educator knows this and is careful not to waste an instructional day or word.  They also know that most students become disinterested learners when class starts with verbose teacher commentaries and/or a classmate being called on.  If they’re required to state an objective before delivering lessons, they do so succinctly, moving past the mandate as quickly as possible.  Instead of previewing content with long-winded narrations, they immediately initiate student learning.  To do this, they always begin lessons with a clear, centralized question or prompt that every student in the class can immediately engage with (see table below).

The simple act of listening, thinking, and responding has a profound, positive impact on concentration.  Because its purpose is to stimulate focus, the first question can’t be too easy.  By starting lessons with student activity rather than passivity, teachers instantly know who is and is not paying attention.  Children, in turn, become more attentive.

The Middle, Students never go long without feeling successful or challenged.

The math teacher plans and delivers lessons with specific instructional aims.  Depending on the day, their objective ranges from acquiring a skill to accurately using a new model to computing a problem using multiple strategies.  Regardless, their goal is to advance their students as far along  the curricular track as possible.

At the same time, they know that this is rarely easy, and not always possible.  The sixth grade teacher who inherits a child functioning on a first grade level is foolish to think that they can make up five years of foundational understanding in a single grade, and no fair educator impedes collective learning momentum to accommodate chronically truant or defiant students.  Determining that these outliers represent a problem greater than they can control in a single lesson, the teacher addresses their needs through standard differentiation but never at the expense of their classmates.

Instead, they focus their energies on the weakest non-outlier, a child who attends school daily, possesses average to above average intelligence, and always puts forth respectable effort.  Except for maybe being a grade level or two behind, this student has no discernible learning obstacles.  Their homelife is stable and they come to school each day rested and fed.  The ambitious educator knows that their teaching success is directly linked to this child independently demonstrating the established instructional goal.  If they teach well, the student will reach comprehension; if they don’t, the child will be confused.

Achieving success often requires a counterintuitive pedagogical approach.  The harder the concept, the greater a teacher’s tendency is to tighten their instructional grip, rigidly plodding children through tiny procedural steps (see example below).

This practice of Over-Teaching can spark temporary feelings of success, but normally ends in frustration, because students are left woefully dependent on their teacher.  As a result, children are ill-equipped to think and reason when tasked with solving problems on their own.

Successful mathematicians are willing to try, fail, and persevere, and the classroom is a laboratory for cultivating these traits.  If students are to perform well independently, their teacher must provide them opportunities to engage in productive struggle.  At the same time, children must routinely feel successful if they’re to excel.  Teachers, therefore, build simple to complex questioning into their lesson plans, ensuring that students correctly answer problems before moving on to harder ones.  As the lesson progresses and problems become more complex, they occasionally loop back to simpler questions, maintaining student engagement and confidence in the process (see fourth grade example below).

T:  (Project 3 - 2/3.)  Say the equation.

S: 3 - 2/3.

T:  (Gives students an opportunity to solve.)

T:  Draw rectangles to represent 3.

S:  (Draw. See Image 7.)

T:  What challenge do we have with our model?  Turn and talk.

S:  We don’t have any thirds.

T:  How can we create thirds?  Turn and talk.

S:  Break one of the rectangles into thirds.

T:  How many thirds are in one whole?

S:  3-thirds.

(T & S Draw. See Image 8.)

T:  (Gives students an opportunity to solve.)

T:  What’s 3 – 1?

S:  2.

T:  What’s 3 - 3/3?

S:  2.

S & T:  (Draw and write. See Image 9.)

T:  What’s 3 cars – 2 cars?

S:  1 car.

T:  What’s 3-thirds – 2-thirds?

S:  One-third.

T:  (Gives students an opportunity to solve.)

T:  Take away 2-thirds in your model and complete the equation.

S:  (Write. See Image 10.)

A swimming instructor’s first concern is keeping their pupils buoyant, and a math teacher’s main priority is preventing their students from drowning in complexity.  When afloat, children feel motivated.  The further they sink, the more discouraged they become.

Although superior to over-teaching, a productive struggle approach can be taken too far, especially when teachers prompt their class to practice problems during instruction.  The freedom that it provides often leads students to become unnecessarily frustrated in their quest for independence.  Understanding the important role of confidence in elementary mathematics, the expert teacher provides scaffolds when they notice students close to melting down or quitting.  Finding this balance for each individual requires them to be constantly aware of each students’ mathematical level and emotional fortitude.  The scaffolds act as rungs in a ladder.  Students climb freely, but when they can’t reach the next level, the teacher interjects support, preventing them from dangling in discouragement and later free falling (see examples below).

As children demonstrate comprehension, the instructor uses fewer questioning prompts, increasing student independence in the process.

The End, The lesson ends before the class is collectively bored and/or frustrated.

While focused on the tasks leading to the lesson objective, the master educator constantly gauges individual and collective student focus.  They always aim to meet the objective but understand the futility of pushing through problems when instruction becomes a grind and/or student stamina is waning.  Aware that their teaching is no longer productive, they segue into independent practice.  By maintaining a willingness to alter their lesson plan, the teacher routinely maximizes student learning.

Consistently recognizing a class’ collective breaking point is the product of detailed planning and deliberate practice.  The more a teacher anticipates students’ academic and emotional responses, the better they become at pacing and delivering instruction.  This helps them become more attuned with students’ educational and social needs, resulting in stronger teacher/student bonds and more academic success.