Summary.Because we cover so much ground in this Phase, most parents appreciate having an up-front understanding of the game plan.Of course, our children do not need such an overview; they are perfectly happy just sailing along performing their usual inductive magic on mountains of challenging new material!But we adults are made of less stern stuff; we feel more comfortable with a road map.So, for the benefit of the feeble oldsters out there (i.e., anyone over the age of fourteen), here is our Phase 4 strategy in a nutshell:
General Approach to Both Math and Reading.Our general approach is to charge forward at breakneck speed into exciting new areas in both math and reading.We establish strong outposts in advanced territory, and then use these outposts as foundations for consolidating our advanced positions.Like Alexander the Great and Genghis Khan, like Microsoft and Intel, we grab territory, then consolidate.
Specific Approach to Reading.We want your child to be reading sentences on her own – reading simple books on her own! – in seven months.Many knowledgeable people will criticize this objective as being ridiculous; for example, we have been told by several elementary school teachers that children simply cannot read sentences on their own until they are five or six years old.They say it cannot be done; we do not have much time to deal with such criticism, because we are too busy doing it!Here is how we establish a beachhead in reading territory:
We work hard on phonetic reading, emphasizing the process of sliding letter sounds together to make words;
We learn two key rules that help us read words with multiple vowels, the Two Vowel Rule and the Silent “e” Rule;
We memorize a number of common words, and encourage reading by word recognition;
We encourage the use of contextual clues in reading; and
We learn about important aspects of sentence syntax.
Specific Approach to Math.Meanwhile, we strike like lightning on three different math fronts:
Addition and Subtraction.We learn the important “call yourself” methods of addition and subtraction, and also establish outposts of known addition and subtraction facts by learning the “Zero Rules.”
Fractions.We learn to read fractions carefully, to understand a fraction’s syntax, and to understand the geometric meaning of a fraction.
Intersections.Through complicated intersection drawings, we learn a great deal about the abstract ideas of boundaries, interiors, etc.
Phase 4 Objectives.Here is the customary complete list of important objectives for this Phase:
Continue to reinforce the letters, shapes, numbers and vowels (and especially vowel sounds) from earlier Phases;
Continue to do Field Work regularly;
Introduce all two-digit numbers and all remaining letters;
Introducethe “call-yourself” method of addition;
Introducethe “call-yourself” method of subtraction;
Read using three methods: phonetics; recognition; and context;
Teach the Two Vowel Rule and the Silent “e” Rule, also known as the Boss “e” Rule;
Teach techniques for reading two consecutive consonants;
Emphasize the most important rule of all for reading English:Every rule has exceptions, including this rule;
Introduce fractions: vocabulary; syntax; and geometric meaning;
Memorize the Zero Rules (adding and subtracting zero; a number minus itself equals zero, etc.);
Learn other reading techniques:“s” for plurals; “th” sounds; “ing” words; “ay” and “oy” sounds; etc.;
Introduce two-digit addition;
Introduce intersection pictures to develop a deeper understanding of inclusion and boundary concepts; and
Work on remembering and understanding written sentences (as opposed to just pronouncing them correctly).
 We are not the only ones to use a “conqueror” analogy.Karl Gauss, perhaps the greatest mathematician of them all, said, “It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again . . . . I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.”K.F. Gauss, Letter to Bolyai, 1808.
 Of course, for the benefit of all those Microsoft and Intel attorneys reading this book, let us emphasize that nothing in this sentence is intended to suggest any parallel between these admirable, consumer-friendly corporations and any legendary rapacious conquerors!
 “What,” you ask, “is the exception to this rule?”The answer is simple; this rule is the exception to itself; it has no exceptions. Think about that one for a minute!We have ventured into Gödel, Escher and Bach self-referential territory.Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979 (Pulitzer Prize Winner, 1980).