How to Use the Materials
Guidance for Accelerating Students in Mathematics
We believe that understanding the concepts addressed in K–8 mathematics is extremely important, and students need time to learn this content properly. Schools should provide students who are ready for more of a challenge throughout K–8 with extension and enrichment opportunities, aiming for deeper understanding (which is great for future learning), before acceleration is considered (which can undermine future learning). (Note that classroom teachers need resources, training, funding, and structures that support this.)
There is recent evidence that too many students are being accelerated in middle school mathematics. When more students were placed into a grade 7 or 8 Algebra 1 course, the pass rates of Algebra 1 declined and the students were significantly less likely to pass Geometry and Algebra 2 (Clotfelter, 2012). Students who were de-tracked in middle school performed better in high school, even the higher achievers (Boaler, 2000). 44% of 8th graders who took Algebra 1 had to repeat it with mixed results in improvement among groups (Fong, et al., 2014). 30% of a sample of accelerated students had to retake Algebra 1 between grades 7 and 12, with very little improvement the second time (Finkelstein, et al., 2012).
Are your current acceleration practices good for students?
Considering historical practices and data can help show whether acceleration has been good for students. Here are some questions schools and districts should investigate before making choices about acceleration policies:
- What are the explicit and implicit reasons for accelerating students in math?
- What are the criteria for selecting students for an accelerated pathway, and what are the possible sources of bias in the selection process?
- What opportunities do students have to enter an accelerated mathematics pathway later in their schooling, if they are not selected for early opportunities to join an accelerated cohort?
- What do your current data say? Of the students accelerated to high school work in grade 8 or earlier:
- What proportion repeats one or more high school level courses?
- What proportion enrolls in and completes a calculus course (or other challenging, advanced math course) in high school?
- How does their racial and ethnic composition compare to the student population of the whole district?
- For all of these questions, what proportions are tolerable?
Use multiple metrics
When identifying students who may be good candidates for an accelerated pathway, it is important to use multiple metrics, including student and parent self-assessment, teacher recommendation, and assessment scores to make placement decisions for accelerated coursework. If that decision process results in disparities in the racial and ethnic composition of the group of students placed in accelerated courses, then consider both how to address opportunity gaps in K–5, as well as how to support students with the interest and aptitude but unfinished learning to enter an accelerated pathway. Providing accelerated pathways that start in high school, or offering acceleration through extra math courses or summer options, will increase the likelihood that mathematically interested students with unfinished learning have access to the same opportunities going forward as their peers who have had more opportunities from the beginning.
Additionally, consider building in a checkpoint with families after the first year of acceleration, so that students who might be better served by building a stronger foundation have a well-defined opportunity to not continue on an accelerated path.
Teacher Notes for IM 6–8 Math Accelerated
Since the lessons, activities, and assessment items for IM 6–8 Math Accelerated were rearranged and compacted from IM 6–8 Math, notes are included in the materials to highlight changes for teachers. Teacher Notes for IM 6–8 Math Accelerated are used for many reasons, including
- adjusting the time allocated to an activity
- explaining how reorganization affects references to other parts of the curriculum
- revisions of activity launches or syntheses to include important pieces of activities no longer in the course sequence
- changing an activity’s priority to or from optional
In places where Teacher Notes exist, the instructions supersede other directions for the unit, assessment, lesson, or section where they are attached.
Here is an example. For a lesson in Accelerated Grade 6, Unit 1, the Teacher Note says
“Adjust the timing of this activity to 15 minutes.
Tell students to skip the question asking about the height of Parallelogram B that corresponds to the base that is 10 cm long. If time allows, include this question as part of the activity synthesis.”
The original activity from IM 6–8 Math is meant to take 25 minutes. In order to compress time, students are instructed to skip an additional practice question.
Each Lesson and Unit Tells a Story
The story of each grade is told in nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson in the unit also has a narrative. Lesson Narratives explain:
- The mathematical content of the lesson and its place in the learning sequence.
- The meaning of any new terms introduced in the lesson.
- How the mathematical practices come into play, as appropriate.
Activities within lessons also have narratives, which explain:
- The mathematical purpose of the activity and its place in the learning sequence.
- What students are doing during the activity.
- What teacher needs to look for while students are working on an activity to orchestrate an effective synthesis.
- Connections to the mathematical practices, when appropriate.
Launch - Work - Synthesize
Each classroom activity has three phases.
During the launch, the teacher makes sure that students understand the context (if there is one) and what the problem is asking them to do. This is not the same as making sure the students know how to do the problem—part of the work that students should be doing for themselves is figuring out how to solve the problem.
Student Work Time
The launch for an activity frequently includes suggestions for grouping students. This gives students the opportunity to work individually, with a partner, or in small groups.
During the activity synthesis, the teacher orchestrates some time for students to synthesize what they have learned. This time is used to ensure that all students have an opportunity to understand the mathematical punch line of the activity and situate the new learning within students’ previous understanding.
Each lesson includes an associated set of practice problems. Teachers may decide to assign practice problems for homework or for extra practice in class; they may decide to collect and score it or to provide students with answers ahead of time for self-assessment. It is up to teachers to decide which problems to assign (including assigning none at all).
The practice problem set associated with each lesson includes a few questions about the contents of that lesson, plus additional problems that review material from earlier in the unit and previous units. Distributed practice (revisiting the same content over time) is more effective than massed practice (a large amount of practice on one topic, but all at once).
Are You Ready For More?
Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. We think of them as the “mathematical dessert” to follow the “mathematical entrée” of a classroom activity.
Every extension problem is made available to all students with the heading “Are You Ready for More?” These problems go deeper into grade-level mathematics and often make connections between the topic at hand and other concepts. Some of these problems extend the work of the associated activity, but some of them involve work from prior grades, prior units in the course, or reflect work that is related to the K–12 curriculum but a type of problem not required by the standards. They are not routine or procedural, and they are not just “the same thing again but with harder numbers.”
They are intended to be used on an opt-in basis by students if they finish the main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in “Are You Ready for More?” problems, and it is not expected that any student works on all of them. “Are You Ready for More?” problems may also be good fodder for a Problem of the Week or similar structure. Students who opt into IM 6–8 Math Accelerated are more likely to opt-in to these problems, so teachers should prepare to discuss “Are You Ready for More?” problems as part of lesson planning.
The kind of instruction appropriate in any particular lesson depends on the learning goals of that lesson. Some lessons may be devoted to developing a concept, others to mastering a procedural skill, yet others to applying mathematics to a real-world problem. These aspects of mathematical proficiency are interwoven. These lesson plans include a small set of activity structures and reference a small, high-leverage set of teacher moves that become more and more familiar to teachers and students as the year progresses.
Some of the instructional routines, known as Mathematical Language Routines (MLR), were developed by the Stanford University UL/SCALE team. The purpose of each MLR is described here, but you can read more about supports for students with emerging English language proficiency in the Supporting English Language Learners section.
- Algebra Talk
- Anticipate, Monitor, Select, Sequence, Connect
- Group Presentations
- MLR1: Stronger and Clearer Each Time
- MLR2: Collect and Display
- MLR3: Clarify, Critique, Correct
- MLR4: Information Gap Cards
- MLR5: Co-Craft Questions
- MLR6: Three Reads
- MLR7: Compare and Connect
- MLR8: Discussion Supports
- Notice and Wonder
- Number Talk
- Poll the Class
- Take Turns
- Think Pair Share
- True or False
- Which One Doesn’t Belong?