The K-1 Math Conundrum
May 22, 2020
A kindergarten parent email reads, Sabrina needs more challenging work. She can already count to 100 and you’re still teaching the number line to 10.
A first grader, who hasn’t yet mastered adding within 10, stares perplexedly at the problem 8 + 6 = with no strategy to solve it.
In a class of 25 kindergarteners, a teacher sits at a horseshoe table, helping five confused children. Meanwhile, 11 students are raising their hand and nine of their classmates - having already completed the math assignment - are reading a book.
Ten minutes after the start of class, 24 first graders are evenly divided into four math pods. Two children at the Computer station are still having login trouble. Instead of solving embedded number problems, three students at the Game station are stacking counters to see who can create the highest tower. One child has left the Skill Work station and is wandering around the room, and another is leaning back in their chair whistling.
These scenarios are common in kindergarten and first grade math classes, but they don’t need to be. To avoid them and in turn optimize lower school math learning, educators must customize their teaching and curricular expectations to emergent early elementary school realities.
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As a teacher trainer, I’ve observed hundreds of math lessons spanning all elementary and middle school levels. Regardless of the topic, a rhythmic flow in which 90% of the class meets the objective almost never occurs. Instead, lessons usually possess an unpleasant friction, resulting in a cohort of frustrated children; a few bored, fast processors; and a teacher futilely scrambling to try and meet every students’ needs.
Mathematics’ cumulative nature lends itself to this grind. Learning math is like climbing a ladder. Advancing to higher levels requires a student to establish sure footing on one rung before moving on to the next, and so on for the duration of their schooling. Skipping steps becomes more precarious and inevitable the higher one climbs. Students with strong foundations comprehend new topics quickly and become bored as they sit through lengthy reviews aimed at catching up their less skilled classmates. Although the dynamic is common in all math classrooms, it is most conspicuous in kindergarten and first grade. This is due to several factors, some of which are obvious and others that are more subtle.
In all grades, classes consist of children with a ten to 12-month age gap. The lower the grade, the greater challenges the younger edge of this range faces. A kindergartener born in September has lived 16% percent longer than a July-born classmate. This alone is a developmental chasm for five, six, and seven year-olds, but is amplified when parents hold their children back a year to start school and/or teach them math concepts prior to entering kindergarten. As a result, lower grade teachers are tasked with delivering instruction to an expansive range of skill and school readiness levels.
Many early elementary students struggle to sit still and listen, while others can maintain focus for extended time periods. The latter is at an enormous mathematical advantage, as are those with advanced written dexterity. Being able to draw clear diagrams keeps children’s work and thought processes organized while they attempt to learn and practice new topics. Others take so long to craft their numbers, symbols, drawings, etc. that their practice time becomes frustrating and inefficient.
Teachers, of course, play a large role in curtailing these obstacles. Tight structures, choral responses, and carefully selected graphic organizers can minimize the range of behavioral and fine motor skills. But educators who master these managerial tasks still have a tendency to accidentally widen the math gap.
Kindergarten and first grade mathematical competencies are often very pronounced, but for a few obvious reasons go ignored. It’s easy for kindergarten teachers to introduce numbers six through ten while a few children haven't mastered one-to-one correspondence within five. Because many of their students are ready for more complex work, instructors rationalize that it’s unfair to hold them back.
First grade teachers can be tempted to move on from Adding and Subtracting within 20 even when half of their students haven’t mastered the content. Having already taught the curriculum, they know that spending several more weeks on the topics isn’t likely to increase proficiency and failing to master the content won’t create huge learning problems later in the school year.
Content depth compounds lower grade teaching challenges. As a class’ collective understanding stretches and splinters, it’s difficult for instructors to facilitate independent work for early finishers. Math teaching requires frequent informal assessments to keep students progressing. Without them, a strong student can quickly become a weak student if they lack necessary foundations to access the new content they’re learning. Upper elementary students are more capable of moving between tasks without an adult’s help. They can also check their work and self-correct when they get answers wrong. Kindergarteners and first graders are not as autonomous and thus more reliant on their instructors.
The end result is widespread dysfunctionality and few K/1 students reaching their mathematical potential. Teachers routinely feel torn between moving forward in the curriculum and reteaching topics that will bore advanced students. Faced with this regrettable decision, it’s understandable that they often choose the former, passing burdens along to the next teacher without affecting their own work.
Because early elementary concepts are often children’s first exposure to them, moving on from unmastered units has a way of resetting all students at a common entry point. Introductory lessons on Ordinal Numbers, Length, and Shapes can neutralize a class’ pre-existing math gap, reducing the need for differentiation. This creates a mirage of functionality, but is actually a cheap patch for foundational fractures. Passing over concepts too quickly seems like a minor decision to the teacher, but choosing to do so holds enormous implications for students’ future learning and colleagues’ teaching.
Minor surface fissures quickly expand and many times this gap never narrows.
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Elementary math achievement gap discussions often center on how best to serve upper elementary and middle school students. Fourth and fifth grade classes commonly contain 30-40% of children functioning a grade level or two behind, and many middle schoolers begin pre-Algebra having not mastered basic second and third grade concepts. This puts enormous pressure on their teachers to differentiate instruction without alienating grade level, proficient, and advanced students with whole-class reviews.
Concurrently, learning support teachers, interventionists, and after school programs are devoted to catching up those below grade level. Although well intentioned - and sometimes heroic - these efforts are usually misaimed and futile, because they fail to address the problem’s source. The best method for plugging foundational math holes is to prevent them from ever forming. Doing so requires extraordinary attention to lower grade learning, specifically in kindergarten and first grades.
The math gap is so prevalent in elementary schools that administrators usually accept it as an uncontrollable reality that they are powerless to change. This is a false assumption. Dynamic K/1 classes filled with students maximizing their mathematical potential is possible, but demands restructuring traditional teaching practices and student support systems.
Below are measures that successful schools implement:
Pacing Graphic
Institutionalize intergrade meetings. Before beginning the school year, administrators facilitate cross-grade level meetings to coordinate the essential topics that need to be taught. By prioritizing concepts, teachers can map out plans to pace the school year and maximize the content each child learns (see Pacing Graphic on right). Philosophically, the school believes that not all students need to learn the exact same content, but there are a few topics that all children should master before moving on to the next grade. Proficient first graders can learn Pictograph and Money concepts, while their classmates spend more time practicing Numbers to 20 and Place Value to 100.
Frequent informal assessments. Once the school year begins, teachers identify misconceptions early and immediately address them. This is done by performing frequent informal assessments followed by small group pullouts, while those who master the topic work on projects, enrichment activities, or units that were deemed less important during the intergrade level meetings. This, of course, is managerially challenging for K/1 teachers who don’t have in-class assistants. Schools that don’t staff them streamline help to those classrooms. Paraprofessionals and other staff members support teachers with classroom management and modest instructional assistance, allowing the primary instructor to channel their energies towards students who need the most help.
Cumulative reviews are embedded into daily instruction. Constantly mending foundations eliminates the need for large-scale reviews and, because students never go long without practicing past concepts, they rarely feel overwhelmed when starting new units. An Adding crossing over the ten within 100 unit feels less overwhelming when children have consistently practiced Adding crossing ten within 20 during the weeks leading up to the new topic. When educators don’t deliver consistent cumulative reviews, students who struggle at the beginning of the school year tend to fall further and further behind as the year wears on. By plugging foundational holes early and often, future topics aren’t as challenging and the math gap narrows.
Cumulative assessments. Three or four times each year, teachers deliver cumulative assessments, followed by targeted review days. After the data is collected and analyzed, each instructor provides whole-class and/or small-group reviews, addressing topics of collective weakness. Throughout the remainder of the year, they intermittently assess their students on individual skills and concepts. Each time the class’ proficiency dips below 90%, they perform short whole-class interventions until returning to the desired mark.
Universal math times across all grade levels. For one period each day, advanced students can have class with the grade level above them and – if children, parents, teachers, etc. don’t mind – struggling students can work at a lower grade level. This streamlines interventions which, especially in the early grades, prevents the math gap from ever growing.
Fluid longitudinal learning threads across grade levels. If children demonstrate proficiency with all the grade level content before the school year ends, they begin working on the next grade’s content. This is often dynamic in the moment, but can lead to boredom the following year if their next teacher doesn’t keep them on a mastery before moving on track. To prevent this, administrators set up fluid schoolwide systems in which students can progress at their own rate, working their way further up or down a metaphorical math ladder.