Implementing Think Math! - Support for Students
First year of implementation, student
Three things are potentially new for students: mathematical ideas, solidity of facts, and special formats used in Think Math!
Think Math!, from the very start, treats addition and subtraction as two sides of the same coin, the same idea viewed two different ways. Likewise, division appears as soon as multiplication does (2nd grade, though only as single digit by single digit arrays), and when division becomes the focus of attention, it is introduced as unmultiplication. Fractions, because they have many meanings (number, represented as position on number line, is primary, but all the usual other senses are accounted for as well), appear in many forms: rods (ratio); dots (and sets); number line (number); function machine (multiplication, process/operator). Think Math! uses its own formats, but these are genuinely also ideas that may be new to students (and teachers).
Kids need to be totally comfortable with certain basic facts as stepping-stones for all their work. We don’t need all facts, and Think Math! doesn’t advocate random fact drills, but kids must find the following mental computations easy. This list is not complete (for one thing, it doesn’t include any multiplication), but represents the kinds of basic building blocks that students find relatively easy to acquire, even if they do not already have the skills, and that serve well enough while additional skill is built up. For a more complete list, see basic facts; for strategies for working with students, see addition and subtraction and multiplication.
- pairs that make 10 (and, equivalently, pairs of multiples of 10 to make 100)
- adding 10 (or 100) to anything
- adding multiples of 10, and adding multiples of 10 and then a single digit number to the result
- adding 8 or 12 to anything by adding 10 and then adjusting
- adding 80 to anything by adding 100 and then adjusting; and, later, using the same strategy for adding 28 or 98 to anything mentally.
For some 5th graders, adding 10 to anything is a challenge, but a return to addition would feel so defeating—like back to first grade—that it’s destructive to do. Yet they must learn. The image of the Number Line Hotel (in the K-3 curriculum materials), and some clearly not first-grade-like mental challenges (like add-29-to-anything after the add-10 is mastered, largely as a linguistic, almost rhyming, pattern) can help. It is an effective way of learning some important conceptual ideas about addition. It is good practice of addition facts.
If the Number Line Hotel is left up in the classroom (which we suggest), it is a useful reference when students are doing activities that require two-digit addition. It should not be, as hundreds-charts are sometimes used, for getting the answers—that we want to become internalized through a combination of the mental computations listed above—but for re-viewing how those computations work: adding 28 could be two steps up (+20) and eight more steps, but one is then stuck “counting” (or knowing the fact) 8; it’s easier to add 30 and subtract 2.
The fact-of-the-day concept can minimize distraction from 5th grade content and allows the class to move on to grade level (and, for the kids, less discouraging and more interesting) material without sacrificing essential basics.
Every lesson in every grade includes a Skills Practice and Review (SPR) component. These essential daily exercises are designed to be short drills. Fifth grade begins with the kids working on filling out a Multiplication Table, which has the same three benefits for multiplication that the Number Line Hotel had for addition – understanding multiplication, practicing facts, and a ready resource for the kids as they work on problems using these facts.
These are the least important, being just format and not mathematical idea, skill, or fact, but they do matter to kids and teachers. For this (and occasionally for ideas), we sometimes drew in a limited way on related lessons in earlier grades. Think Math! avoids lots of specialized formats, precisely because they can be offputting to parents, and an extra burden on new students. But certain representations of mathematical ideas are especially powerful -- for example, intersections, arrays, and area to model multiplication, and a format that we call "cross number puzzles" to model the structure of addition and subtraction -- and so these representations are used. It is worth looking at each of the following articles to get familiar with the idea, and to find strategies for dealing with first-year students.
- Cross number puzzles (CNPs) appear in the second lesson of fifth grade. These puzzles are introduced in earlier grades, but in the first implementation year the fifth grade students (and teacher) will find them new.