Developing attention, focus, and working memory
These are not part of mathematics itself, but part of the necessary "infrastructure" for learning and doing the mathematics.
- being able to picture (or otherwise represent) things and manipulate the pictures/representations,
- being able to hold two or more ideas, numbers, or pictures in mind at once, and
- being able to keep one's place in a process, keep track of what step or idea one is working with.Example 1: A young child who has not yet memorized 7 + 5 as a basic fact might think: 7 + 5 is greater than 7 + 3, which I know; 5 is exactly 2 greater than 3; so 7 + 5 is the same as 10 + 2. This mental process requires thinking of a known fact (7 + 3), using that fact (10), remembering the unknown one (7 + 5), comparing the right part of it (5 to 3) to notice what correction is needed (adding 2), and recalling the first result (10) to make that correction (12). And, of course, it requires a kind of mental "overseeing," to keep track of where, in this chain of thinking, one is. (See Grade 2, Ch 2, Lesson 7.)
Example 2: We never expect "half of 48" to be a memorized fact, but we certainly do expect students to be able to do it mentally. This process requires lots of keeping track. We must hold 48, 40, half of 40, 8, half of 8, and the goal in mind, and then choose the right two of those numbers to recombine. When we're "fluent" with this kind of mental arithmetic, we hardly notice the process, but with beginners one can literally see the effort it takes to build the focus, working memory, and metacognitive skill to keep track of the process.
Most children by age 7 can quickly develop these skills. They do not always think all the steps explicitly, and some "steps" may happen simultaneously (or differently from this description), but focus, concentration, imagery, memory, and the ability to "watch one's own thinking" and keep track of one's place (and what the question is) are all required.
Think Math! starts in the early grades and systematically builds this "mathematical infrastructure" of focus, attention, holding two things in mind at once, and keeping one's place in a process. Think Math! students become very agile at mental computation, continually facing new challenges to keep stretching that ability. This set of skills is never listed in the table of contents of a math book, but it is at the root of success in all those other mathematical competencies that are listed in the table of contents.
These essential mathematical skills are typically so totally ignored that people tend not even to notice they're needed. Think Math! carefully develops them. These skills also underly the development of problem solving stamina: one cannot persevere without the ability to keep one's place in mind.
Making and keeping mental pictures
Show how many
- Hold your two hands up showing no fingers at all, then flash a number of fingers (like 7) for about one second -- long enough to see, but not long enough to count -- then go back to closed fists.
- Then ask children to use their hands to show how many fingers you had up. (7, in this example)
What's My Number?
Logic games often develop the same skills. We play this one (see more detail at What's My Number?), early in grade 2. It is essentially a “20 questions” game in which the children, in only 4 yes-no questions, must guess which number (from 1 through 8) the teacher has in mind. The teacher is silent (not even repeating the questions) and only nods yes or no to answer. The numbers are arranged in a pattern making it easy to ask questions like “is it in the circle” (equivalent to “is it odd”) or “is it below the line” (equivalent to “is it less than 5”).
If their questions are perfect, they need only 3 questions to figure out the number, but second graders typically first ask “is it 2? Is it 5?” and then someone who wasn’t listening, or forgot what was asked, repeats “is it 2?” again. The most important learning, at this age, is not the logic of the search strategy (which they do develop over the course of some weeks), but learning
- to listen to other students’ questions,
- to watch for the answer,
- to remember what was asked and answered,
- to combine that information with prior knowledge to reduce the number of possibilities,
- to think of a new question…
The game is about the logic of the search (asking “is it greater than 4” is better than asking “is it 4”), but the purpose of the activity is about learning to listen, focus, and remember.
Thunk! Swoosh! Pop!
Thunk! ————— Swoosh! ————— Pop!The hand signals are designed as in-the-air drawings of the base ten blocks. The 100 roughly outlines the shape of the "flat" block, the 10 carves out the long shape of the "rod," and the 1 points to the tiny spot that the smallest block takes up. Each sound is invented to go with the blocks, too: pop is the smallest, swoosh describes the swooshing movement of the finger, and thunk is, well, the greater weight of the 100 block.
The thunk-swoosh-pop hand gestures and sounds have a serious purpose as well as being playful and fun for children. (The playfulness is an important purpose, but not the only one.) The idea of using gestures and sounds is designed specifically to help children internalize the ideas -- not leaving them concretely on the table -- and to increase their attention, focus, and working memory.
Base ten blocks concretely represent the kind of grouping (place value) on which addition and subtraction processes (algorithms) are based. The concrete step is important, but we also want children to bring the ideas into their heads. Drawing the blocks is one step in that direction: moving from objects to images is one level of abstraction. But drawings, like objects, are permanent.
The next step is to use language. Unlike the objects and drawings, language is ephemeral; we hear/see it but then the sound/sight is gone, and only the ideas remain. We keep those ideas in our minds! (Either signed or spoken language -- gestures or sounds -- would do but, together, they are easier to "grasp," especially as they are introduced, because the gesture suggests the shape of the object or picture.)
Decoding a simple thunk thunk swoosh pop pop pop into a number takes only a bit of memory: in order, that's two hundreds, one ten and three ones, or 213.
But decoding thunk swoosh pop thunk pop pop requires mental "sorting." It is addition, and follows exactly the logic we want children to apply when they add numbers. To decode this one, a child needs to keep a mental space for “thunks” and add to that space (and no other) each time a thunk goes by. You could think of thunk swish pop, thunk pop pop as 111 + 102 because that is the order in which the sounds came. Children probably don’t think of it that way, but do have to add thunks only to thunks, pops only to pops, and so on, and they do this naturally without even thinking about it. When they later learn a paper-and-pencil notation, the idea of "lining up the columns" becomes not just an arbitrary rule, but another way of showing that we add hundreds only to hundreds, the same idea they've already used spontaneously. That, and the mental focus and memory this approach develops (and requires) is its goal.
Boom! Teacher Michele Tonge wrote: "There are sounds and motions in second grade for place value: the ones (pop), tens (swoosh), and hundreds (thunk). We have 2nd grade students who want to know if there is a sound for thousands."
We had not invented a sign or sound for 1000, so we invited the class -- Mrs. Walsh's 2nd grade at Francis A. Desmares School in Flemington, NJ -- to invent its own and tell us about it. They pictured the movement of setting a large heavy box on a table, and imagined the "boom!" that you'd hear as it landed. They sent us two photos, the first showing their initial stance -- their hand position holding the heavy box -- and the second showing their position after they've moved their arms downward to set the box on the table, and said "boom!"