In a react script, for every function in both the parent and child components we must have this magic line using .bind(this)in the constructors:

this.nameOfFunction = this.nameOfFunction.bind(this);

For example:

// Parent with a incrementNumberTurns() function:
this.incrementNumberTurns = this.incrementNumberTurns.bind(this);

// Child with a handleClick() function:
this.handleClick = this.handleClick.bind(this)

Here is an example from the parent component constructor of my tic-tac-toe game. Every function name needs to be initialized in the constructor this way:

class Game extends React.Component {
  constructor(props) {
    super(props)
    this.incrementNumberTurns = this.incrementNumberTurns.bind(this);
    this.populateXLocations = this.populateXLocations.bind(this);
    this.cellClicked = this.cellClicked.bind(this);
    this.updateBoard = this.updateBoard.bind(this);
    this.newGame = this.newGame.bind(this);
    this.changeTurns = this.changeTurns.bind(this);
    this.setGameInProgress = this.setGameInProgress.bind(this);
    this.setCatsGame = this.setCatsGame.bind(this);

See:
https://github.com/chris-relaxing/react-tic-tac-toe/blob/master/tic-tac-toe.js

Here is what the React documentation says about .bind(this):

For more information about .bind(this), see:
https://stackoverflow.com/questions/29732015/value-of-this-in-react-event-handler
http://facebook.github.io/react/blog/2015/01/27/react-v0.13.0-beta-1.html