Ever wonder why the unit on fractions comes after the telling time unit? Don’t you need to know halves and quarters of circular regions to tell time to the half and quarter hour? Why do we teach measurement to the half and quarter inch before mixed numbers and fractions on a number line? If decimal numbers are fractions with denominators that are powers of 10, why does the decimal unit come before the fraction unit? Is it better to first teach decimals in terms of place value or as special fractions?
Most scope and sequences or pacing guides have a procedural flow. Would we be more successful in developing conceptual depth if the pacing guide had a conceptual flow? Working with a teacher development team from Jefferson County Schools, WV we tried to do just that.
- Invest in sticky notes, chart paper, and painter’s tape.
- Record 1 standard or 1 part of a standard per note.
- Backmap: For each standard ask, “Upon what understandings does the standard build? If students really understood the mathematics described in the standard, what would they understand?”
Note: A common teaching approach is to launch at the standard and then use repeated practice to reach “mastery.” In this approach we determine the prior conceptual and procedural understandings needed to extend to the standard and beyond. We first review and solidify these understandings We then build on those to “reach” the standard and extend beyond. The goal is to develop conceptual depth and build procedural flexibility.
- Keep in mind things that could be handled in daily warm-ups, daily number sense/number talk type activities, etc. Record on different colored and sized sticky notes or identify in some way.
- On a half-sheet of chart paper begin to place the sticky notes (one conceptual idea or standard per sheet). The sticky notes make it easy to change your mind and rearrange until you’re comfortable with the flow.
You will find that the standard is often on the right side of the sheet and the backmapping notes are to the left. You may find that the order of the sub-parts of a standard or the order of the standards in Standards documents will be different from the logical flow of concepts.
- Begin to sequence the half-sheets into a flow that makes sense. Continue to rearrange until you are comfortable with the flow.
- What conceptual investigations can be done to launch the year that supports community buiding and establishing classroom norms related to the Mathematical Practices? We chose geometry and data activities.
- What conceptual investigations can be done before and after holidays so that no days are lost? We chose measurement, intro to fractions with playdoh, and other hands-on rich experiences.
- What to do after the “test.” We chose to review and solidify grade level concepts and procedures instead of previewing next year’s topics.
- Estimate day ranges needed for each “unit.”
- Identify quarter or semester divides.
- Revisit the sequence to ensure that all of the standards have been included. When comfortable, tape together as quarters or semesters.
- Begin to identify daily number sense/warm-up activities that can preview an upcoming idea or review a previous idea.
- Determine the best way to represent and share.
Below is the 3rd grade Scope and Sequence/Pacing Guide Overview. The Overview provides a year at a glance. The year is divided into quarters. Color is used to identify mathematical domains. Estimated day ranges needed for each “unit” are included.
This format is used for the Scope and Sequence Overviews for grades 1 through 5.
The kindergarten team decided that a different structure was needed. The kindergarten scope and sequence/pacing guide, is a continuum.
Investigations related to the domains are introduced at different times during a quarter. They then continue throughout the quarter, and year, using centers, play, routines, stories and songs, etc. It is designed to support best practices in early childhood education and emphasizes the development the mathematical thinking.