Anne Collins and Linda Dacey. Zeroing In on Number and Operations: Key Ideas and Misconceptions. Portland, ME: Stenhouse Publishers. This is a set of four books (Grades1-2, Grades 3-4, Grades 5-6, Grades 7-8 ) formatted as spiral-bound flip charts with each page longer than the one before. Primary math can and should anticipate algebra. One reason students have trouble with algebra is that teachers typically lead students to believe it to be a new and harder topic, a different math than arithmetic. Algebra should be taught as a natural problem-solving strategy, even in first grade. On the page entitled “Join and Separate” from the book “Grades 1-2” (there are no page numbers), the authors present this story problem: “Jake had 5 knights for his toy castle. His sister, Emma, gave him some more knights for his birthday. Now Jake has 11 knights. How many knights did Emma give Jake? How do we usually teach children to approach these kinds of story problems?
We drill them on math fact families so they will recognize a math fact buried in words. Now I believe children should memorize their math facts, BUT I also think that language can be an ally rather than the enemy it usually becomes. The natural progressive analysis for this problem starts “5 knights (gave more) ? 11 knights,” proceeding to “5 knights plus ? ” (or box) is what algebra calls “a variable” and if replacing the “? ” or box with a lower case letter makes the expression look like algebra. There are six yellow cubes on the other side. A big advantage is the lack of reliance on numerals. We do not give children enough credit for their ability to think. We discourage thinking by presenting mathematics not as something that can be reasoned about, but as something that must be memorized and accurately recalled. If they do not remember, they have no recourse.
If they fail to remember often enough, they soon conclude wrongly that they are bad at math. Math-phobia is just one more short step away. The text’s idea that join and separate problems both share the start-change-end structure is helpful, but the graphic organizer is not obvious or intuitive to children. The teacher would be better off going straight to the algebra gear which requires a lot less instruction and makes more intuitive sense. Then the problems can be “played” like a game. On page A14 of Grades 1-2, there are three examples of sentences where all three components (start-change-end) are left blank. The idea is to play around with providing any two out of three. The authors rightly note, “leaving the initial state blank is the most challenging, as many students are uncertain where to begin.” The algebra approach addresses and eliminates this uncertainty. Students simply use the turquoise cube to stand for the blank and march on.
Teachers need a resource that explicitly addresses the common misconceptions children (and their teachers) hold about math. Sometimes teachers deliberately teach misconceptions because they do not know any better. The set is comprised of four very slim volumes of fifteen informational pages and about fifteen pages of problems and exercises for a total of thirty pages printed on both side of the paper. Thus the entire set is about 120 pages. The list price per single book is fifteen dollars and sixty dollars for the set of four. In a strategic marketing maneuver, by dividing what normally would be one book into four, the publishers may be able to capture more income. Teachers are likely to be most interested in the information pertaining to the particular grades they teach. A teacher might not be inclined to pay sixty dollars for largely “irrelevant” material (although that point could be argued), but may willingly spend fifteen dollars for grade-specific content. The books could also be useful to education students. I intended to read a sampling of pages from each book very carefully and peruse the rest.
I wanted to get a feel for the quality of the information across the scope and sequence. I ended up reading all four books line-by-line, analyzing the references and working the problems. I made copious notes on every page. Grade 1-2, Counting by Tens and Ones: I like the concept of “counting the tens and the leftover ones.” In fact, I like to rename the “ones” place the “leftovers” because they are not in a group. Grade 1-2, Writing Numbers: Children can learn a lot of math without numerals. This page has a good strategy for using cards to illustrate digit positions. Grade 1-2, Equivalent Representations: This is a valuable trading exercise, all the better because it is done on the overhead, avoiding the possible “magic” of computer trading exercises. Computer simulations of physical activities often look like magic to students. They resign themselves to taking the teachers word instead of understanding for themselves. I would only caution the teacher to make sure the students very intentionally see all aspects of the trade. Students need to be certain that the teacher added nothing nor removed anything.