## Math Instruction

### Background

The McNatt Learning Center, Inc., focuses primarily on the development of foundational skills and secondarily on the application of those foundational skills to curricula. By extension, we also offer help with teaching the "basic skills" of reading, writing, and arithmetic. In each of these subjects, whenever possible, we recommend teaching concepts first or, at least, alongside their applications. Unfortunately, most math programs in the past have emphasized teaching symbolic operations that some students (particularly young or struggling students) haven't connected with mathematical concepts. Equally unfortunately, many contemporary math programs emphasize work with concrete objects (manipulatives) in constant pairing with symbolic operations. Not only does the repeated emphasis on the concrete not necessarily translate to better conceptual understanding, repetitive emphasis on the concrete (by functioning as a mathematical crutch) can even hinder development of the ability to think symbolically.

To teach mathematical concepts, we often recommend Structure of Intellect Systems' Math Prep, which teaches mathematical concepts from addition to algebra. Generally, we also recommend Professor B Math as most students' primary math curriculum before algebra due to the program's wonderful, sequential unfolding of mathematical concepts and good practice with mathematical operations. Professor B Math also lends itself well to supplementation with cognitive training programs and mathematic games to reinforce the concepts it teaches while continuing to build computational skills.

The following is one possible schedule, prepared at the request of our clients, to outline how Professor B Mathematics can be integrated with cognitive training programs and mathematic games. The schedule, like most everything we do, is presented in "levels." Students may progress through levels at their own pace; they may be in widely different levels across subjects. Nevertheless, to provide a sense of pacing, we have broken the schedule into quarters of a typical school year. We welcome feedback about its scope and sequence. It is important to note that the following schedule does not include work on measurement or time, which are best learned with exploration and which, therefore, fit quite well in science.

For families or schools interested in hosting tournament game play of Equations, a logic and math game, the Academic Games Leagues of America may be of assistance. Rule modifications for various ages of students can be found by clicking *Equations,* above.

To practice 24 Game online, visit FirstInMath.com.

### Preparatory Levels

Text: *Mathematics Power Learning for Children Book 1*

Lessons 1–4

In these lessons, to be completed by the end of the Preparatory Levels, students learn number concepts (correspondence, counting, skip counting, more than, less than, and equal to). We do not formally teach mathematical operations (addition, subtraction, multiplication, division) in the Preparatory Levels.

Another good resource for Preparatory Levels, since it allows for young children to explore math concepts using easy-to-handle manipulatives, is Catherine Stern's Structural Arithmetic for Pre-K and Kindergarten. First grade students who struggle in beginning Level 1, below, may benefit from further "discovery" of math concepts using Stern's 1st Grade math program. Generally, once a student can "feel" basic math concepts in his or her body, (s)he is ready to begin Professor B Mathematics' "Mathematics Power Learning for Children."

### Level 1

Text: *Mathematics Power Learning for Children Book 1*

Text: *Math Prep* by Structure of Intellect Systems (SOI)

#### Quarter One

Lessons 1–4 (p. 3–23): Review

Practice may range from one to four weeks, until all students are ready to move on or have been identified as needing extra help. Students also finish the following SOI Modules, the first ones in the MathPrep sequence, during this time and, if necessary, continuing into the next set of lessons:

- CFS-II (First Half)—Understanding Origin, Destination, & Direction
- CSS-P—Discriminating among Abstract Figures
- CSS-I—Understanding Systems of Numerical Symbols
- CFS-I—Matching Figures

Students may want to begin working with CFT Flipcards (available from SOI Systems separately from the MathPrep workbook) and CFT-I (in the MathPrep workbook) in conjunction with HANDLE® or a pre-packaged program like S'cool Moves for Learning, Balametrics Learning Breakthrough Program, and/or Bal-A-Vis-X. Since any sensory-motor activity may be overwhelming to some students, teachers are encouraged to watch for signs of neurological stress, called "State Changes (PDF File)" in a HANDLE® paradigm—and to help students notice their own State Changes. (If your computer can't read PDF files, click here to download a free reader from Adobe Systems.)

Lessons 5–9 (p. 24–33): Linking Number Concept with Addition and Subtraction

Approximately one week is allotted per lesson. In conjunction with practicing quarter turns in HANDLE® or Balametrics, teachers may want to begin using money to help students apply their number concept and addition and subtraction skills.

#### Quarters Two through Four

Lessons 10–20 (p. 33–67): Addition and Subtraction

Lessons 10–12: Approximately one week

Lessons 13–14: Approximately one week

Lessons 15–16: Approximately one week

Lessons 17–20: Approximately two weeks

Interlude: 24 Game: Addition & Subtraction Primer and More SOI Modules

Approximately two weeks' practice with the 24 Game: Addition & Subtraction Primer provides an opportunity to practice and develop arithmetic proficiency in a different context. It also gives students who need extra work an opportunity to catch up. Students also finish the following SOI Modules (the next ones in the MathPrep sequence) during this interlude and, if necessary, continuing into the next set of lessons:

- CFS-II (Second Half)—Understanding sequences of spatial relations
- CSR-II—Recognizing numerical sequences
- NSS—Using symbolic codes for copying
- CFT-I— Spatial Manipulation of Figures (if not already completed)

Lessons 21–43 (p. 68–159): Developing facility with number concepts, addition, & subtraction

Approximately one week is allotted per lesson. Note that sections in the text beginning "For the Appreciation Of…" belong with the lesson in the text immediately following them; both are taught in approximately one week. Students must finish through lesson 40; lessons 41–43 are enrichment that students and parents should finish over the summer if they are not finished during school.

### Level 2

Text: *Mathematics Power Learning for Children Book 2*

Text: *Math Prep* by Structure of Intellect Systems (SOI)

#### Quarter One

Lessons 1–2 (p. 3–21): Review (Lesson 1) & Introduction of Multiplication (Lesson 2)

Approximately six weeks are allotted for these lessons.

Lesson 3 (p. 22–26): Linking Multiplication and Division

Approximately two weeks are allotted for this lesson. During this time, students use one day to learn the 24 Game: Multiplication & Division Primer. In a classroom setting, students can thereafter be allowed to play this game on Wednesdays if they are finished with their work early. Teachers may want to try to conclude Wednesday's instruction 15 to 20 minutes early to allow for game time but also may want to let students know not to expect every Wednesday's instruction to finish early.

Interlude: 24 Game: Single Digits

Students learn 24 Game: Single Digits and take about one week for a 24 Game tournament.

#### Quarter Two

Lessons 4–12 (p. 27–87): Multiplication and Division of Larger Numbers

Approximately one week is allotted for each lesson. During this time, students may play 24 Game: Single Digits on Wednesdays if they are finished with their work early. When some students are proficient with 24 Game: Single Digits, they can begin using 24 Game: Double Digits among themselves.

SOI *Math Prep*

Students complete their *Math Prep* books through the NSI-I Module. While some students may require assistance with catch-up work, most students will only need to complete the following SOI Modules:

- CSS-II—Understanding order (elevator exercises)
- NSI-I—Understanding systems thinking. Many students will require help with this module and may need to work through it several times. The skills developed, however, are helpful in solving word problems and playing Equations, introduced next. The skills developed are also an extremely important foundation to algebraic thinking.

#### Quarters Three and Four

Lesson 13 (p. 88–91) + Interlude: Solving Word Problems (Lesson 13) & Learning "Equations"

Approximately two weeks are allotted for this lesson. Students alternate applying their knowledge to word problems with learning and playing Equations by Wff 'N Proof publishers, first with addition and subtraction but with all four basic operations by the end of this Lesson. The thinking skills developed by Equations are so beneficial that teachers may want to consider implementing a four-day "instructional" schedule from now through the end of Level 3 Mathematics. During five-day weeks, students can play Equations on Fridays. Students who choose to do so can work by themselves or with a peer on Equations training materials (including DIG Math or IMP Kits 1–11) on their own time.

Lessons 14–17 (p. 92–100) + Interlude: Learning Factorization and "Square"

Before proceeding to Lesson 15, students should demonstrate proficiency with multiples and with applying number traits using the first section (4–5 pages) of the SOI Module ESS-II (provided separately, as the ESC-III and ESS-II Modules remaining in *Math Prep* are used for review during Level 3 Mathematics).

Approximately two weeks are allotted for the entirety of Lessons 14–17 and for practice with PrimePak by Conceptual Math Media. Students play each other with PrimePak cards during the last 10 to 15 minutes of each Monday through Thursday math lesson, assuming Equations is played on Fridays. 24 Game is no longer available during this time.

Lessons 18–21 (p. 101–111): Applying Factorization

Approximately two weeks are allotted for the entirety of these lessons, for completion of the SOI Module ESS-II, and for continued practice with PrimePak during the last 15 minutes of each Monday through Thursday math lesson. 24 Game is still not available for play on Wednesdays.

Lessons 22–28 (p. 112–160): Adding and Subtracting Fractions

Approximately one week is allotted for each of these lessons. In addition, "Equations Fridays" are temporarily preempted to explain and play a new edition of 24 Game, 24 Game: Variables. Thereafter during this session, students may also play with 24 Game: Variables on Wednesdays if they are finished with their work early. For the remainder of this level, teachers may want to allow students who grasp math concepts but need extra practice on computation to continue playing 24 Game: Single Digits, 24 Game: Double Digits, or 24 Game: Variables in place of Equations on Fridays.

### Level 3

Text: *Mathematics Power Learning for Children Book 3*

Text: *Math Prep* by Structure of Intellect Systems (SOI)

Text: *Math Magic* (Scott Flansburg, Harper, 1993)

Supplementary Text: *Problem Solving the Systematic Way, Part 1* (SNP Pan Pacific, Singapore). While not essential in the following schedule, this text is nevertheless well-written and may be of great benefit or interest to some students.

#### Quarter One

Review: Factorization and Facility with Basic Skills

Allot approximately two weeks for this section. During the first week, parents should be made aware that all students in Level 3 will need to be up to speed on understanding, adding, and subtracting fractions by the end of the sixth week; students can come in for extra help before or after class. Parents can also come in for training to work with their students at home, so that the whole Level 3 math class starts out strong. During the first week, students learn 24 Game: Fractions Primer and review 24 Game: Single Digits, 24 Game: Double Digits, and 24 Game: Variables. The first two weeks of Level 3 math are spent reviewing basic math skills with these games. The first two Fridays are spent reviewing PrimePak, least common multiple, and highest common factor and playing PrimePak to practice applying these concepts. As necessary, teachers can review Lessons 14–17 (p. 92–100) of *Mathematics Power Learning for Children, Book 2* to help students remember factors and can assign corresponding homework from those units.

Review & Expansion: Number Analysis and Adding & Subtrating Fractions

Allot approximately three weeks for this section. At the beginning, students take a quiz on fractions, but not for a grade. The quiz is used to match students for peer tutoring and review, as necessary, as early as possible.

SOI Module ESC-III (in *Math Prep*)—Checking Number Traits (Week 1)

During the first half of the period, students apply their review of factorization to proving answers, when requested, from this module. During the second half of the period, students review and then play 24 Game: Fractions Primer. Friday may be used to play PrimePak or 24 Game: Fractions Primer. Students who have difficulty with ESC-III may benefit from working through ESC-II (Checking Phonetic Traits, a separate module) to practice focusing on details.

SOI Module ESS-II (in *Math Prep*)—Number Analysis (Weeks 2 & 3)

Students who have progressed through Mathematics Levels from the beginning have completed this module, separately from the *Math Prep* text, on the previous level. Students re-do the module in the text now for review and to ensure continued understanding. The last two days of this section (likely Thursday and Friday) can be used to review and play Equations, so by the end of the first few weeks students have reacclimated to all of the core math games.

Lesson 1 (p. 3–32): Multiplying Fractions

Allot approximately two weeks for this lesson, keeping Fridays set aside during five-day weeks to play Equations.

Lesson 2 (p. 33–41): Dividing Fractions

Allot approximately two weeks for this lesson, keeping Fridays set aside during five-day weeks to play Equations.

#### Quarters Two and Three

Lessons 3–19 (p. 42–165): Understanding and Working with Fractions, Decimals, and Percents

Allot approximately one week for each of the remaining lessons in *Book 3* while continuing to take Fridays to play Equations.

#### Quarter Four

Chapters 1–10 of *Math Magic*

Allot approximately one week for Chapters 1 and 2, then one week per chapter for the remainder of the year, continuing to take Fridays to play Equations. This practice with thinking about math in a different way allows students in Level 3 Mathematics to receive ample practice with the skills they have learned. This extra practice also helps the math skills these students have developed become automatic. Students who so desire may finish this book over the summer.

### Level 4

Text: *Power Algebra I*

Text: *Math Magic, Ages 11+* by Scott Flansburg (The program was formerly called *Mega Math, Ages 11+*. It is essentially a reworking of the *Math Magic* book on VHS or DVD, with accompanying workbooks.)

Supplementary Text (If not used in Level 3): *Problem Solving the Systematic Way, Part 1* (SNP Pan Pacific, Singapore). While not essential in the following schedule, this text is nevertheless well-written and may be of great benefit or interest to some students.

Note: Even as students grow older, HANDLE® Activities and Balametrics equipment can continue to be used, especially when memorizing or practicing mental math. Students should not only be proficient at mental math while sitting at their desks; they should be able to do the operations in time with a metronome while throwing and catching bean bags, or while setting their own rhythm with Superflight Crystal Stix.

#### Quarter One

Review of Mental Math Skills and Basic Math Concepts (Weeks 1–4)

The first four weeks are spent entirely on review of basic math skills using *Math Magic, Ages 11+* by Scott Flansburg. Teachers may review and allow students to play Equations on Fridays during this time, as well. Teachers are encouraged to review 24 Game: Variables and let students play this game quietly if they finish early, especially on Wednesdays.

Continued Review + Introduction of Exponents and Roots (Weeks 5–6)

While review practice continues throughout this quarter, the fifth week begins an introduction of exponents and roots using *Math Magic* and, subsequently, Equations. Students play Equations every Friday, now using all the cubes (including exponents and roots). They should gain familiarity with standard notation and the notation used in Equations and its accompanying computer tutor, DIG Math.

Lessons 1–2: Pre Algebra + Review of Exponents and Roots (Weeks 7–8)

In addition to work in *Power Algebra I*, students complete Patterns 1–8 of *Pattern Blocks: Building Algebraic Thinking with Progressive Patterns* (Bob Willcut, Critical Thinking Books and Software, 1995), one per day, while continuing to play Equations using all cubes every Friday. Students should be made aware that, though they are encouraged to continue playing Equations on their own, this is the last quarter they will have Friday Equations matches as part of class.

Practice with Variables (Week 9)

During the first part of each math period, students complete one pattern from Patterns 9–12 of *Pattern Blocks* (4 patterns total, one per day). During the second part of each math period, after a brief introduction, they play 24 Game: Algebra/Exponents. Every five-day week, students play Equations using all cubes on Friday.

#### Quarter Two

Lessons 3–5: Beginning Algebra (p. 40–88)

Students learn and begin playing 24 Game: Integers beginning with Lesson 3. Through Lesson 5, available Fridays are set aside to play 24 Game: Double Digits, 24 Game: Variables, and 24 Game: Integers instead of Equations. Students will also continue to complete weekly problems from *Pattern Blocks*, as indicated below:

- Lesson 3—Problems from
*Pattern Blocks*: 13–19 (7 total) - Lesson 4—Problems from
*Pattern Blocks*: 20–22 (3 total) - Lesson 5—Problems from
*Pattern Blocks*: 23–25 (3 total)

#### Quarter Three

Lessons 6–8 (p. 89–133)

Beginning with Lesson 8, it is important that students not mark solutions in their *Power Algebra I* books, as they will be using the same texts at the next level. Students will also continue to complete weekly problems from *Pattern Blocks*, as indicated below:

- Lesson 6—Problems from
*Pattern Blocks*: 26–29 (4 total) - Lesson 7—Problems from
*Pattern Blocks*: 30–33 (4 total) - Lesson 8—Problems from
*Pattern Blocks*: 34–37 (4 total)

#### Quarter Four

Lessons 9–10 (p. 134–213)

It is important that students continue to avoid marking solutions in their *Power Algebra I* books, as they will be using the same text next level. With each lesson, students complete the following problems froom *Pattern Blocks*:

- Lesson 9—Problems from
*Pattern Blocks*: 37–40 (4 total) - Lesson 10—Problems from
*Pattern Blocks*: 41–44 required (4 total); 45–50 optional

__Alternative Sequence__

Some families may find the above schedule too arduous, particularly if a parent/teacher does not have a firm background in algebra or needs to attend to other children. These families may want to consider *Video Text Algebra I* (Video Text Algebra), along with *Pattern Blocks: Building Algebraic Thinking with Progressive Patterns* (Bob Willcut, Critical Thinking Books and Software, 1995), if desired.

### Level 5

Text: *Power Algebra I* …Continued (First half of level)

Text: *Dad's Lessons: Trigonometry* (Second half of level)

Supplementary Text: *Problem Solving the Systematic Way, Part 2* (SNP Pan Pacific, Singapore). While not essential in the following schedule, this text is nevertheless well-written and may be of great benefit or interest to some students.

__For Students Needing Extra Assistance Before Continuing__

*Power Algebra I*, Lessons 8–10 (p.121–213): Review

Beginning in Level 5 Mathematics, students may want to try learning the material from *Power Algebra I* primarily on their own. Ideally, a teacher is still readily available to assist individual students or to teach the class as a whole if requested. At this level, students take pre-tests to aid them in focusing their efforts where to study and post-tests for a grade. This process begins by taking a review test over *Power Algebra I* through Lesson 10. Students who have forgotten substantial portions of materials should schedule tutoring before or after a group class; others may re-work the problems in these lessons for extra practice. All will find that subsequent lessons build upon and reinforce what they have learned.

#### Quarter One

*Power Algebra I*, Lessons 11–12 (p. 214–261): Approximately 4 weeks

Interlude: Beginning Logic

Students learn Wff 'N Proof, Game 1, which they practice for one week. For the remainder of Quarter One, students will play Wff 'N Proof, Game 1, every Friday.

*Power Algebra I*, Lessons 13–15 (p. 262–315): Approximately 4 weeks

#### Quarter Two

*Power Algebra I*, Lessons 16–19 (p. 316–448)

Students learn Wff 'N Proof, Game 2. Students continue to play Wff 'N Proof, Games 1 and 2, every Friday while completing *Power Algebra I*.

#### Quarter Three

*Dad's Lessons: Trigonometry*, Lessons 1–7

Students read through and practice Lessons 1–7. Each student is responsible for teaching one class every other week.

Logic: Students continue to play Wff 'N Proof. Students who are ready may move into playing Games 3–9.

#### Quarter Four

*Dad's Lessons: Trigonometry*, Lessons 8–14

Students read through and practice Lessons 8–14. Each student is responsible for teaching one class every other week.

Logic: Students continue to play Wff 'N Proof. Students who are ready may move into playing Games 3–9.

### Level 6

Texts: *Power Algebra II* + *The Geometry of Wholemovement: Folding the Circle for Information* by Bradford Hansen-Smith (Wholemovement Publications, IL)

No detailed plan yet….

### Level 7

Texts: *Power Algebra III* + *The Geometry of Wholemovement*, continued.

No detailed plan yet….

### Level 8+

The McNatt Learning Center, Inc., currently has no recommendations for upper-level math texts. Students completing the primary track of Levels 1 through 7, above, will have completed the equivalent of the following standard courses:

- Arithmetic: Addition, Subtraction, Multiplication, Division, Fractions, Decimals, Percents, and Factorials
- Algebra I,
- Algebra II,
- Trigonometry, and
- Transformational Geometry.