Planning the Lesson
July 12, 2019
Note to the Reader: This essay references many Appendix subheadings for grades 15. 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.
Appendix
The students have left for the day, and other teachers have gone home or retired to their classrooms when the math teacher begins writing the next day’s lesson. Studying units and chapters is done over the summer and maybe reviewed on weekends, but individual lessons are always prepared the night before delivering them. Only then can the teacher reflect on that day’s outcomes and critically decide how best to move forward. Lesson plans from previous years might be referred to, but will always need to be tweaked. Regardless of how successful the delivery was in the past, the students are new and their needs differ. The great educator  whether it’s their first year of teaching or the last days of a 35year career  constantly selfcritiques and is thus always striving to improve.
Sitting at their desk, a textbook and ancillary resources are stacked neatly to their left. A pen and paper or blank word processing document is positioned in front of them. The teacher’s guide, if they own one, is stowed away in a drawer or closet. This is because the content expert knows that their acquired knowledge is a far better resource than any manual a publishing company can produce. During their apprenticeship, they were trained to plan lessons from scratch, and thousands of hours spent deliberately practicing have rendered the guide’s almost useless. If the instructor needs any clarification, they’ll consult a coworker and then, if they still need resolution, will read the teacher manual’s recommendations. At its most useful, the guide is an adequate colleague. More often, it leads the instructor astray with shallow teaching sequences and flimsy suggestions. Given enough time and impetus, the great math teacher could write a better manual than the publishers.
Possessing deep curricular knowledge, the instructor has a clear picture of their students’ mathematical journey, knowing from where they are coming and where he/she is leading them. To teach well, the instructor needs proficient knowledge of at least three grade levels – the one they are teaching and those that precede and succeed it. They also have an intimate understanding of each student’s basic skill proficiency and concepts they have and have not mastered. A fourth grade teacher, for example, finds out how well their students remember third grade fractions content in the days and maybe weeks leading up to their Fractions unit. They rarely, if ever, enter a lesson blindly. With this in mind, the instructor opens their book and begins planning.
Previewing the Lesson
The teacher begins by studying a short set of problems, examining how the lesson grows in complexity from beginning to end. The resource that they use varies depending on the curriculum. Singapore Math® teachers use the student textbook. Eureka Math instructors use the Problem Set. As they study, the teacher makes special note of the simplest and most challenging tasks that students will be asked to perform. This, along with the intermediary problems, helps them understand the breadth of the lesson they’re designing (see Appendix: Lesson Topic & Problem Set).
Learning new topics requires deep thinking and, oftentimes, creativity. Both are hampered when students’ working memory is inundated with calculations and/or trying to remember previously learned concepts. Knowing this, the instructor isolates foundational practice when they need students to innovate the most.
The teacher writes a Fluency heading into their plan, leans back in their chair, and considers what children should know prior to this lesson.
Fluency
The instructor starts by brainstorming a list of every basic skill and concept students need to succeed in the lesson (see Appendix: Prerequisite Skills & Concepts). The former will address the latter’s understanding gaps through a short fluency drill. When their students lack multiple prerequisite skills, the teacher adheres to a simple philosophy:
Intensity trumps extensity.
This leads them to target fewer skills well, rather than shallowly addressing many. In their lesson plan, they write out a fluency script that is customized to their students’ needs (see Appendix: Fluency Script). When they’re finished, they jot down other fluency topics that they want their students to practice. This includes reviewing basic skills from previous grades and/or preparing their class for upcoming lessons.
With fluency planned, the teacher shifts their focus to critical thinking.
Problem Solving
After reflecting on that day’s math class, the instructor chooses one of the lesson’s skills or concepts that they want their students to practice through a thinking, reasoning context. Most frequently, they select a word problem. The task can come from the textbook or an ancillary resource, but the teacher normally makes up their own. This allows them to not only provide context relevant to their students’ lives and interests, but also helps them customize arithmetic. If a large portion of a fifth grade class hasn’t memorized their times tables, the teacher might write in “friendlier” multiplication facts, so that students will engage more with problem solving and less with computation (see Appendix: Previous Lesson Topic & Word Problem).
The instructor doesn’t always use this time to apply the previous lesson’s content. Their main focus for daily problem solving is to stimulate students’ thinking/reasoning skills, using concepts that they’ve already learned. Sometimes, they decide to review a particular type of word problem, drawing, or diagram that was taught earlier in the school year.
After selecting or writing a problem, the teacher asks themselves two questions:
Will the weakest student meaningfully engage with the task?
Will the task challenge the strongest student?
If the answer to either question is No, they modify the problem. This can be as simple as writing a simpler and/or more complex question for struggling and early finishers, respectively.
With half – or close to half  of the class period designed, the teacher reexamines the problem set that guides their planning.
The Lesson
While planning a lesson, the teacher asks themselves three questions:
By the end of the lesson, what should students be able to do?
What do they already know that will help me guide them there?
To arrive at the desired outcome, how can I help fill in their understanding gaps?
Often, they find that the first task will be challenging for their weakest students to access. This leads them to begin class with two or more simpler questions that every child can easily answer. Confidence, they understand, plays a vital role in learning elementary mathematics. Therefore, their lessons always begin with questions several grade levels below the one they’re teaching. Third grade lessons, for example, begin with a kindergarten question, followed by first and second grade questions. Consistently immersing students in this type of laddered instruction peels away their anxiety, preventing math complexes from developing (see Appendix: Laddered Questions).
Strong students often learn and master skills or concepts after one example problem. Most children, however, need additional practice. Knowing this, the teacher prepares analogous problems for each complexity they’re asked to teach. During the lesson, they’ll gauge student comprehension through informal assessments. When the entire class doesn’t correctly answer a question on the first try, the teacher provides more practice at the same complexity before moving on to an incremental challenge (see Appendix: Analogous Complexity Set).
Carefully selected sample problems mean little without strong instructional delivery. Great math instruction requires teachers to begin lessons knowing exactly what they want to say and stimulating students to immediately engage in the content. Rather than ingesting students with information via a lecture, the teacher uses an interactive approach, taking on a role similar to a trekking guide. They lead students to a destination rather than describing the journey or carrying them to it. In the process, they aim for their students to speak and work as much or more than they do. Achieving this lofty goal stems not from natural ability, but instead a rigorous apprenticeship, followed by thousands of hours spent perfectly practicing.
Overtalking is the most common mistake math educators make. To avoid developing this habit, the teacher spent the early part of their apprenticeship writing lengthy teacher/student scripts, focusing on precise wording, along with anticipated student responses. Speaking with brevity is vital to becoming a master educator, so it’s crucial for the apprentice teacher to develop the skill early in their career. The longer a bad habit is practiced, the more challenging it is to break.
During their apprenticeship, they arrived to school daily with tenpage teacher/student plays, covering the first five minutes of their lesson. While teaching, they’d keep the script in hand and read from it when necessary. Several weeks of this intensive practice built their stamina and transformed their brain neurons, allowing them to teach far more efficiently and effectively than they otherwise would have. Their lesson scripts soon truncated and they were able to maintain the mode of delivery longer and longer. In the end, they realized that maximizing succinctness while delivering instruction is far less tiring than explaining mathematical concepts.
Next to the problems they’ll be using, the teacher jots down a few key questions that require careful wording. They then select images to project and/or white board inserts that students will use. Their goal is to end lessons with at least 90% of their students mastering what was taught. Thus, they always overplan, determining logical stopping points throughout the lesson. They’d rather fall short of their target, ending instruction at a point of mastery, than push to get through content and wind up with shaky student understanding.
Classwork
With their lesson plan almost complete, the math teacher studies the curriculum’s practice problems. As they examine the tasks, they ask themselves two questions:
Are the problems simple enough to engage the weakest students?
Are the problems challenging enough to hold the interests of the strongest students?
If the answer to either question is No, they find or create remedial and/or extension problems. This requires additional work, but the extra five to ten preparatory minutes helps eliminate other difficulties. Darting around the room to help individuals is exhausting and often creates student malaise. This can lead to disciplinary problems, frustrated students, and – in turn – unhappy parents, whose phone calls and email correspondence consume more time than creating leveled practice sets.
What is simple for one child is challenging for another. Number one of a sixquestion problem set, in which the tasks become increasingly harder, could be easy for most students, while challenging for a few. A child on grade level might be underchallenged by numbers one and two, but overchallenged by number six, while an advanced student finds the entire problem set too simple. Two minutes after a teacher distributes such a problem set, a few students are finished and bored, while others haven’t yet completed number one. Seventyfive percent of the class is raising their hand waiting for teacher help, and everyone in the room is feeling restless.
The teacher understands that the greatest resource for dynamic independent practice time is their brain. Every class has a range of math levels, and no curriculum can produce practice problems that provide perfectly appropriate challenges for each child. This is why the teacher consistently generates problems that help every child feel a balance of accomplishment and rigor. When a class’ weakest student experiences success during independent practice, and its strongest student feels challenged, every other child feels both and can thus maximize their learning.
Some educators decry laddered problems, claiming that teachers who give them set low expectations, patronizing children in the process. “Struggling students,” they say, “will meet grade level challenges when their teachers set high expectations.”
The wise teacher understands the absurdity of such statements. If improving students’ academic performances were as simple as providing them with grade level practice problems, the term Achievement Gap would’ve never entered educational lexicon. Critics of remedial work that ladders into grade level problems fail to recognize the role emotions play in mathematical learning. Confidence is vital to succeeding in the subject, and feeling consistent success is the best catalyst for generating it. A struggling student is far more likely to embrace a mathematical challenge after they’ve correctly answered five to ten questions than they would be if it were the first problem they encountered (see Appendix: Laddered Subset).
Once the teacher has chosen independent practice problems, they write it into their Order of Work board and decide whether or not to assign homework.
For more on Order of Work, see The First Day of School.
Homework
Homework’s merit is the subject of visceral competing philosophies. Some believe it should be assigned daily, while others site research “proving” that it impedes educational progress. Great teachers think independently and thus aren’t trapped by dead labels such as Traditional or Progressive, Old or New School. Instead, they remain openminded about past and present philosophies, all the while knowing:
No educational practice works perfectly for every child.
Regardless, of whether or not they choose to assign it, the teacher has no illusions about homework’s role in elementary mathematics. At best, it is a catalyst for developing discipline and responsibility through confidencebuilding practice. At worst, it makes children hate the subject, while driving a wedge between them and the most influential adults in their life. Therefore, if the teacher gives homework, their biggest priority is to make sure that students return to class as happy or happier than when they left.
Knowing that children with a low morale are difficult to educate, the teacher provides assignments that their students can do independently. When students need help with their homework, they are likely to seek parental assistance. With all the best intentions, parents usually “help” their kids using methods that contradict that day’s lesson, further confusing an already puzzled child. In the process, they sometimes subtly (or overtly) undermine the classroom instructor, damaging the student/teacher relationship in the process.
Weighing all of this, the strong teacher adapts a flexible approach to homework, depending on the school, students, and parent body. Their assignments usually fall into one of two categories:
Skill practice. Because it's straightforwardly presented, this type of work reduces parental interference and – when thoughtfully selected  can bolster students’ fluency and number sense.
Conceptualization. Demonstrating conceptual understanding negates parental interference and students aren’t burdened by finding answers (see Appendix: Homework).
Before packing their bags and leaving for the day, the teacher decides whether or not to give homework. If they choose to assign it, they normally write it themselves.

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