My new publication—Grow Your Brain—provides parents and teachers models of simple conversations with 1st and 2nd grade children. It focuses on a few simple tools that should allow children to take off in their understanding and knowledge of essential arithmetic. It gradually builds up a whole library of topics and strategies from which classroom teachers and parents home-schooling or just helping their child can choose and improvise short, daily, Grow Your Brain conversations. It stresses mathematical knowledge as a thinking activity and as a network of connections —as opposed to an accumulation of isolated facts committed to memory. In response to prompts and questions, children respond and answer; they learn to speak about numbers.
The very first tool is a vertical number line to 10 with marks for each number but no numbers except for zero and 10. 5 is shown as a bolder mark. Why vertical? Because on a vertical line higher numbers are higher and lower numbers are lower. It’s a small detail among many small details implemented throughout the book that all aim to make learning facts and concepts just a little bit more intuitive. In fact, far from being a small detail, a complete switch from horizontal to vertical number lines has major beneficial consequences on children’s ability to make sense of essential concepts throughout their mathematical studies and beyond. Why no numbers except 0 and 10? If the marks are numbered, when a child is asked to point to 7, for instance, or to put a token on 7, the child just goes there without any need to think or learn. With no labels, identifying marks without having to count implies some thinking taking place and does not always come easily. A dialog then helps the child formulate that thinking. The parent/teacher’s side of the dialog could be as follows:Show me 7 on the line. Is 7 closer to 5 or to 10? (To 5) How far is 7 from 5? (2 steps) So what’s 5 + 2? (7) How far is 7 from 10? (3) So what’s 2 + 3? (5) And what’s 7 + 3? (10) Show me all the fingers of one hand. How many is that? (5) Show me 2 more fingers of the other hand. How many fingers are you showing now? (7) How many fingers are you hiding? (3) So what’s 7 + 3? (10) Show me all your fingers. Show me 7 again on the line. How far is 7 from 10? (3)
Some children find such questions easy, and indeed they are just the first steps in what we do in Grow your Brain. But for others children, even those first steps will take significantly longer to master. So some improvised version of this conversation will take place as long as needed at the same time as other approaches and other connections are blended in the daily dialogs.
Discussion cards giving different images of the 5+ numbers (5+1, 5+2, 5+3, 5+4) are used to confirm and expand the understanding experienced with the line to 10 (and later the line to 20 where 10 is used as a benchmark.) With 7 by then well known as 5 + 2, a child holding a card with some version of 7 (a 5/2 domino or Roman number VII, for instance) can be asked:
What’s 5 + 5? (10) What’s 2 + 2? (4) So what’s 7 + 7? (14)
The 4 of 7 + 7 = 14 is now connected with the 2 of 7 seen as 5 + 2. Knowledge of essential facts gradually builds up, not as isolated memorized phrases which are nothing more than skeleton knowledge, but as actual experiences which the child acts out and formulates and as meaningful elements in a network of connections.
The powerful engines used to lift a plane from ground level to 40,000 feet are not used to push the plane up but to thrust it forward. It is the forward thrust that is used to lift the plane. In the same way, Grow Your Brain focuses on helping children experience knowledge as connections with other facts and discover knowledge as a thinking activity. Knowledge is a consequence of the connections and the thinking.
For more on Grow your Brain, go to www.GrowYourBrain.education.