OVERVIEW, Part One:
- Why De-Track?
- How to De-Track
- Change Unit Planning, Instruction, Homework
OVERVIEW, Part Two (tomorrow):
- How to Change Assessment and Grading for Wider Success
WHY De-Track Math Classes?
First, three articles on letting go of tracking and grades – from Singapore, Finland and San Francisco.
Apparently, math education CAN work without tracking and overemphasis on grades. We decided to de-track middle school math, replacing tracking with leveling math instruction. It’s a work in progress, but here’s our rationale:
- Tracking damages students who are tracked into the low track. We call tracking the “walk of shame” – students who were in the same class in elementary school are suddenly divided into remedial, regular or honors tracks. The “non-honors” students begin quickly to label their class as “the stupid class” and to call themselves failures at math.
- Tracking damages schools and society Too many adults in the US state with a chuckle “I’m just not a math person.” It is too large a waste for our society, especially in the 21st century, to label people who simply need more time as “weak”, “struggling” or “failing”. Some of the best mathematicians in history were not recognized as “fast” in elementary school.
- Tracking is based on speed, not other measures of aptitude Many students tracked into low tracks share two characteristics: they tend to need to process new concepts visually in order to make sense of them, and therefore they need more time to process these new concepts. In our experience, this learning profile is characteristic of about a third of our students. These are the students who give up on math, or just go through the motions of short-term memorization.If they were able to use their strengths, these students would make progress at or above grade level, rather than failing gradually and cumulatively. Indeed, this is the source of success from countries like Singapore and Finland. They have raised the bottom third of student scores, leading to higher average scores. See this article: https://hechingerreport.org/memorizers-are-the-lowest-achievers-and-other-common-core-math-surprises/In addition, many career mathematicians tell of a dislike of math in elementary school, when speed was the only skill that mattered. Mathematicians are not quick thinkers; they are deep thinkers.
De-tracking can build a positive attitude about mathematics; it can build students’ self-knowledge about their own learning. Our goal is that they all leave our middle school at 8th grade knowing how they learn best.
HOW to Teach for Understanding, not Speed
1. Change Unit Planning
- We changed our curriculum to use longer units, moving Concrete -> Pictorial -> Abstract as slowly as needed, with truly challenging work for the fastest workers. Challenge work is not more computation or the presentation of next year’s algorithms. Instead, it is increased complexity of thinking and problem solving, within the current unit.
- We chose to focus on seven or eight units for a whole fifth grade year.
(a)Intro to Fractions I (b)Patterns (c)Multiplication and Order of Operations (d) Fraction Addition and Subtraction II (e)Division (f)Angles (g)Decimals (h)(if time) Triangle Area
We feel these units are the most important, and anything we miss can be worked into those long units, or picked up in 6th grade. We find that this emphasis on mastery learning (plus challenge) correlates with a rise in standardized test scores, because students are learning to think through problems, familiar or not.
- “Sandwich Units” – Some units are so important (multiplication in 4th grade, fractions in 5th, negative numbers in 6th) that they should receive extra attention in the curriculum. In 5th grade, we approach fractions (and word problems with fractions) immediately in September, but we keep them as concrete/visual as possible. Then we run those fraction concepts ‘in the background’ on homework as we go through the next 2 units. There’s at least one “Draw this fraction addition” problem on each homework for weeks. This gives everyone time to master the concept before we move on to the fraction algorithm in December. Many students who thought they could add fractions (algorithmically) find they cannot actually draw them – they haven’t really internalized what the fraction concept truly means. WHAT IS A SANDWICH UNIT?
2. Change Instruction
It is our belief that all children can learn math, if they are allowed to use their own strengths, at their own speed. Our math program therefore accommodates for these multiple learning styles and speeds.
- We often use low-floor, high-ceiling investigations that give students a sense of satisfaction for the progress they are showing at any level. When given a pattern problem, for example, students can build the next 3 figures with blocks, or draw the next figures, or use a computational list, depending on their own needs and comfort level. Investigations Examples
- On other days, students get “Classwork” (a problem worksheet), and they work at their own speed. The problems move from Concrete -> Pictorial -> Abstract, and get more complex toward the end. We ask students to set their own goals, reflect on their own progress and we reward the whole class for good progress. (“4 marbles in the recess jar if we see everyone making good progress on today’s work.”) Sometimes we collect the classwork – it gives us feedback and puts more accountability on the students, but they do not have to have finished everything, since the last challenge problems can be outside some students learning zone at present. Students see quickly that their classwork closely matches the frequent quizzes, too.
- Games provide a further enrichment of the math curriculum. Games should be pictorial-based (Fraction bars: |—-|—-|—-|, not fraction symbols like 1/3 ) Games should build students’ strategy skills, and games should NOT be speed-based. Example: Capture the Circle Game
3. Change Homework
- Homework is limited by time, not quantity. Some students might need the full 25 minutes to complete Level One (Concrete) and parts of Level Two (pictorial/abstract). This is fine. Other students might move through the first two levels quickly, and spend more time on Level Three (challenge). This corresponds to their developmental level only, not their long-run math ability. Their level can change any time; it is completely up to the student. Often we see a sudden growth spurt in abstract ability around 7th
- We know our students’ working speed pretty well (from classwork) and can tell when they’ve spent substantially less or more time on homework. In this case, we have a talk with the child, and if needed, with the parents. It helps if we keep homework short, avoid repetition, and encourage thinking, which can be fun. See earlier blogs for examples of homework. What About Homework? and More HW examples