Math-U-See has recently revised and renamed their arithmetic program. The new titles are Primer, Alpha, Beta, Gamma, Delta, Epsilon, and Zeta. They also now offer math curriculum from pre-algebra through pre-calculus. (Math-U-See still sells its classical curriculum; for ordering information, visit www.mathusee.com.)

Math-U-See believes students need to memorize facts and formulas, but they also need to understand concepts and how to apply them. With this goal in mind, those who teach children must first understand math concepts themselves. If you are math-phobic, then the cure for this common ailment, according to Ruth Beechick, is "to learn arithmetic yourself. You're an adult after all, and elementary school children are expected to learn arithmetic. So it can't be that hard. Get a good book and learn right along with your child...A teacher who loves learning earns the right and the ability to help others learn." (An Easy Start in Arithmetic, Arrow Press)

The essential components of any Math-U-See level are the DVD, the teacher's manual, the student workbook, the student test booklet, and any necessary manipulatives. The DVD is essential for teaching the program; the teacher's manual is not available separately nor does it contain all the material on the DVD's. Each level of the Math-U-See elementary curriculum has thirty DVD lessons, and each lesson has three practice pages, three review pages, and one test. The workbooks also contain four unit tests and one final exam for each level. The teacher's manual contains answers and/or solutions to all the worksheets and tests.

The four steps of Math-U-See are Prepare, Present, Practice, and Proceed. First, the parent prepares by watching the video to understand the new concept himself and learn how to present it to his child using the appropriate manipulatives. Most users (probably about 99%, according to Mr. Demme) prefer to watch the video lessons with their children, a practice Mr. Demme endorses. He encourages parents to learn along with their child and not to be intimidated if they do not already understand math. I would not recommend it without some caveats, which I will explain in detail later. While he directs his teaching toward the children in the front row, it appears to me that he is really modeling a teaching method for the parents in the audience. As he interacts with the live audience, the at-home viewer sees only Mr. Demme at the whiteboard. He writes and speaks very clearly and has a warm demeanor. He also frequently jokes with the audience, which may irritate some viewers, although it did not bother me.

After watching the video, it is time to work examples from the teacher's manual. Parent and child work problems together with manipulatives and on paper until the child understands the new concept. This may take more than one day, and you may need to watch the lesson again until you thoroughly understand the concept and can explain it with manipulatives and on paper. If you are teaching more than one child, you can stagger lesson days so you can work with one child while others are working on practice or review pages. Math-U-See teaches parents and children to "Build, Write, Say" build the problem using manipulatives, write the problem on paper, and say what you are doing.

Once your child understands a new concept, he should do a workbook practice page. There are three practice pages per concept, but your child will probably not need to do every page. Next, there are three cumulative review pages, which include the new concept. Again, your child may not need to do all three pages. I have found one or two practice pages and one or two review pages to be adequate for most of the lessons. Finally, a test shows if your child is ready to proceed to the next lesson. Mr. Demme frequently reminds parents that math is sequential, and children need to master each increment before moving on. A good test of understanding, according to Mr. Demme, is to reverse rolls with your student and see if he can teach you the new concept. Remember, the goal is mastery.

The teacher's manuals, student workbooks, and test books are in black and white with no frills. I prefer this simplicity, but others may find the books too plain. In my early homeschool experience, I used a colorful math text, but my daughter was highly distracted by all the clowns and balloons. I prefer to let the mental challenge of arithmetic stimulate my children's minds, not the pictures on the page. If color is motivating to your student, you can always let him color the pages or decorate them with stickers as a reward for effort or accuracy. Math-U-See student workbooks are not reproducible; you will need to buy a new workbook for each child. Sometimes the problems are crowded, especially the word problems, so your child may need to work some on scrap paper.

Math-U-See produces its own manipulatives. The blocks are terrific. Not only are there units, tens, and hundreds, but also twos, threes, fours, fives, sixes, sevens, eights, and nines. These are similar to Cuisenaire Rods, but I like these blocks better because each one is not only color-coded but also notched to show exactly how many units long it is. In addition, the backs of these hollow blocks are open, and the open side is used for teaching subtraction. If you are teaching one child, the Starter Set of blocks will suffice. However, if you are teaching more than one child, you will probably need the Completer Set. The blocks are used in every level except Epsilon.

Fraction overlays are used in Epsilon. These are not your usual fraction manipulatives that simply show various fractions of a whole unit. Not only do these unique overlays clearly show equivalent fractions, but they can also be layered to show operations, such as multiplication of fractions. These are the best fraction manipulatives I have ever seen.

Decimal inserts are used in Zeta. These attach to the manipulative blocks to show decimals. The decimal inserts snap onto the back of the hundreds and tens blocks to hide their notches. Hundreds blocks now represent units, and the tens become tenths. New red blocks are included with the decimal inserts to represent hundredths. To reduce confusion, Math-U-See color-coded the blocks and inserts. Unit inserts are green like unit blocks; tenth inserts are blue like tens blocks; hundredth inserts are red like hundreds blocks. These inserts are excellent aids for understanding decimal place value and operations with decimal numbers.

Math-U-See follows an unusual, but logical, scope and sequence. For those who highly value standardized test scores, Math-U-See might not be a good fit because your child may not learn certain topics in the usual grade. However, they are likely to know some topics not on the test. Personally, I do not attach much importance to standardized testing; my concern is that my children learn arithmetic well and are prepared for algebra by seventh or eighth grade. Math-U-See's scope and sequence meets my requirements for this goal. For what it's worth, my children had just finished their Singapore Math for the year, and they needed to go back about half a year to transition into Math-U-See because some topics had not been covered in their Singapore books. However, some of the future topics will be review, and we will be able to move through them quickly. (For a detailed scope and sequence of each level, please visit www.mathusee.com/sequence.html. For placement tests, visit http://www.mathusee.com/placement.html.)

The basic sequence is as follows:

• Primer: introduction to math
• Alpha: single-digit addition and subtraction
• Beta: multiple-digit addition and subtraction with regrouping
• Gamma: multiplication
• Delta: division
• Epsilon: fractions
• Zeta: decimals and percents

Money, measurement (both U.S. and metric), time, Roman numerals, charts and graphs, probability, simple geometry, and many other "peripheral" topics are thoroughly covered in Math-U-See. The difference between Math-U-See and other programs is that most of these topics are introduced at the most logical place in the scope and sequence rather than being set apart in a separate unit. For example, measuring the perimeter of a shape is taught in Beta (adding the length of the sides); area of a rectangle is taught in Gamma (multiplying length times width); and average is taught in Delta (addition, then division). In Epsilon, students calculate the area of a circle using 22/7 for pi; in Zeta they use 3.14. As much as possible, Mr. Demme ties in every mathematical application where it is most useful and appropriate.

Primer is the preschool program. This is the only Math-U-See program that does not require mastery. Topics include counting, place value, addition up to ten, beginning subtraction, shapes, telling time, and skip counting by 2's, 5's, and 10's. The suggested activities using manipulative blocks are fun and appropriate for preschoolers. I had fun making "Decimal Street" on a piece of poster board, and my younger children liked exploring place value with the units house, tens house, and hundreds castle. I would skip the student workbook unless I had a child who was begging to "do school" like his older siblings; most of the work at this level can be done orally and with manipulatives. Mr. Demme repeatedly cautions parents to keep the math association positive by not overdoing it at this young age. He also reminds parents that Alpha thoroughly covers everything that Primer introduces. The only prerequisite for Alpha is that your child can count to nine and write the digits (0-9).

Alpha focuses on addition and subtraction of single-digit numbers. A heavy emphasis on place value offers your child an excellent foundation for arithmetic. Again, creating a poster board "Decimal Street" will make this concept come alive for your children. Composing a ten is a focal point of this level, and once you see how it works, you will be amazed at how quickly your children learn all their addition facts. My first-grader went from knowing a few easy facts (with sums less than 9) to knowing all his basic facts (through 9+9) in just one day once he learned to compose a ten. When I taught using Saxon, the fact 8 + 5 = 13 was called "an oddball." Math-U-See teaches children to think of five as 2 + 3 so they can make a ten with the 8 + 2 and then easily add 3 to make 13. This skill is initially practiced using the manipulative blocks, but your child will soon learn to do it quickly in his head. No basic fact is really an "oddball" once you learn to compose a ten. Once the addition facts are mastered, Mr. Demme does an excellent job showing how subtraction relates to addition using the hollow side of the manipulative blocks.

Beta covers multiple-digit addition and subtraction with regrouping. This level builds upon Alpha with multiple-digit adding and subtracting with regrouping. The manipulative blocks make teaching multi-digit addition and subtraction uncomplicated. I appreciate Mr. Demme's continued focus on composing tens.

Gamma concentrates on multiplication. I really like how Mr. Demme links money and measurement to multiplication whenever possible. For example, he introduces pints when students are learning their "times two" facts. Nickels are introduced during the "times fives," and dimes during the "times tens." Quarts and quarters are introduced during the "times fours." This seems completely logical and effective because children meet these coins and measurements one at a time in a meaningful context. Mr. Demme's visual explanation of double-digit multiplication (23 x 12) is the best I've ever seen for teaching this concept. It took me a couple of viewings to understand how to use the blocks to demonstrate that a double-digit multiplication problem is really four separate multiplication problems, but once I figured it out, it made teaching the lesson very straightforward. Not only does Mr. Demme demonstrate multiple-digit multiplication well with manipulatives, he uses expanded notation to ease the transition to figuring on paper and to check the final answer.

Delta focuses on division. The lessons on the meaning of division and the basic division facts are excellent. He also emphasizes the relationship between multiplication and division better than other math programs I've seen. I like how Mr. Demme teaches estimation as a good place to start any large division problem. He also uses expanded notation to show the meaning behind the familiar mnemonic "Divide, multiply, subtract, bring down." Mr. Demme does a good job helping parents and students think through the meaning of division so they can attack word problems with confidence.

Epsilon concentrates on fractions. Mr. Demme drives home the fact that fractions mean division, a simple but important concept. I have never seen a better presentation of certain fraction concepts than with Math-U-See. I actually sat my two older children down and used the fraction overlays to re-teach multiplication of fractions. Both children (aged 11 and 13) had previously been taught this concept, but it never really made sense until I used the overlays. This level's word problems are challenging and show how we use fractions in real life.

Zeta covers decimals and percents. I like how Mr. Demme teaches viewers to think about place value when multiplying a decimal number by another decimal number (for example, 0.2 X 0.4). Rather than lining up the problem like a regular multiplication problem and telling students to solve it like a normal problem first and then to count over the same number of decimal places in the product as in the factors, he works the problem out using place value. What is one tenth of one tenth? One hundredth! Therefore, two tenths times four tenths is eight hundredths; the eight goes in the hundredths place. Using his method, all the decimal points line up. On the video, he gives viewers the option of using the thinking method or the traditional method of counting decimal places. As the parent, I might require my child to practice the thinking method for a couple of days and then teach the formula method once he demonstrated a good understanding of decimal place value. Zeta's word problems are challenging and show how to apply decimal and percent concepts to life.

Math-U-See rightly recognizes that knowing how and when to apply arithmetic skills through word problems is the goal of math instruction. Mr. Demme's word problems are fairly complex, with plenty of multi-step problems that require students to stretch their brains. Mr. Demme sometimes breaks up a multi-step word problem for the student instead of letting him analyze it first. He also gives blank templates for the word problems in Alpha: _____ + _____ = _____. I find that this kind of "help" usually backfires; my children just plug in the numbers in whatever order they are mentioned without really thinking about what the problem means. I cross out these templates with a wide indelible marker before handing my child the worksheet.

I have read that place value is one of the key math concepts lacking in most American textbooks. From Primer to Zeta, understanding place value is a high priority of Math-U-See. Mr. Demme does an excellent job teaching place value using both manipulatives and expanded notation. Many say estimation is another one of the most important math skills anyone can learn, and Math-U-See teaches and practices estimation throughout each level. The worksheets often tell students to estimate first and then find the answer. I suppose you can require estimation with whatever program you use, but it is nice to have it taught and reinforced by the program itself.

After years of struggling with Saxon, with its endless repetition and lack of depth, I am sold on the mastery approach to math. Math-U-See is the most mastery-oriented program I know, with its unique scope and sequence and teaching methods. Another accolade for Math-U-See is the systematic review that it provides through the three cumulative review sheets with every lesson as well as the tests. I felt that Saxon provided too much review in every lesson and Singapore not enough. Math-U-See seems to have struck a good balance between practice of a single new concept and adequate cumulative review so students retain their knowledge.

I do have some concerns about Math-U-See. Most of my concerns involve showing the video lessons to children rather than watching them to learn how to teach math concepts yourself. Mr. Demme frequently speaks indirectly to parents in the audience through the children he is teaching. He might mention why he is teaching a particular method and not another, or he might say that students will not need to perform a certain procedure much in real life because of (gasp!) calculators. In the Epsilon video, he says, "Reducing fractions needs to be reduced in emphasis." Later on, he requires it of students, but this kind of offhand comment could lead some astray. On the Zeta video, he says, "Now some of you may be watching and wondering why, when I multiply, I leave the numbers below the line. That's because instead of carrying them and then multiplying, I just leave them under here. So don't be disconcerted. But you have to watch the multiplication video to learn that." Some might just tune out his comments, but many would be distracted and confused.

Another problem with showing students the video lessons is that some of the more complicated lessons just go too fast for children. Some of the easier ones, like tally marks or finding the perimeter of a shape, would be fine to watch together. I show my children some of these simpler lessons for a fun break from the norm. They like Mr. Demme because he is "funny and nice." However, if you are teaching long division or multiplication by a decimal, you will probably need to watch the video yourself and work with your child at whatever pace works for him. In general, the more complicated the lesson, the less he actually teaches the children in his audience and the more he races through simply modeling a method of teaching for the parents.

A final concern I have with the video lessons that Mr. Demme is a bit loose with his terms and his presentation. Although he admits they are "lousy" words, he often uses the words "carrying" and "borrowing" on the video rather than the correct term "regrouping." He also likes playing around with math terms, such as calling mixed numbers "fracbers" or "numtions" and the numerator the "numberator." Usually he is just trying to show the meaning behind the correct term. The workbooks use the correct terms, such as "regrouping" and "mixed numbers," so you don't have to use his words as you teach your children. Also, because Mr. Demme does not appear to follow a formal outline or script as he teaches the lessons, his presentation is sometimes a bit disorganized. He often says things like, "Let's back up a minute," or "Hold that thought" as he redirects his lesson. Again, this could confuse children. Mr. Demme's casualness with terms and presentation are yet another reason I would not treat the video lessons as a student tutorial.

In the early levels, Mr. Demme nicknames certain numbers "onety-three, twoty-seven, threety-four, fivety-six" in an attempt to teach the place value meaning of the numbers. I understand the idea behind this counting method, but I would not want to practice this style of unconventional counting with my children. Thankfully, this style of counting is not practiced in the student workbooks, yet another reason not to show your children the video lessons. Mr. Demme also invents names for his shortcuts, such as "Rule of Four" and "Same Difference Theorem." Your child should understand that these are not established math theorems; they are simply Mr. Demme's own little tricks. Your child will need to learn these nicknames because they are used in the student workbooks.

Mr. Demme also occasionally teaches unconventional math procedures. Many of these tricks are clever and teach children to think mathematically rather than just to perform operations by rote, but others seem lazy or overly complicated. An example of a procedure I consider lazy is Mr. Demme's "Rule of Four" from Epsilon. This is a quick method for finding common denominators. In short, by multiplying the denominators, you can easily find a common denominator, such as changing sixths and eighths to forty-eighths. This does work, but I prefer to teach my children how to find the least common denominator, which in this case would be twenty-fourths. With Mr. Demme's method, the final fraction often ends up needing to be simplified.

An example of a procedure I found complicated is dividing by a fraction using the "Rule of Four." Mr. Demme initially teaches this by changing both fractions to a common denominator, and then dividing only the numerators. Students practice this method until thirteen lessons later, where he expands the lesson by showing how to divide fractions using the reciprocal. This seems unnecessarily complicated, especially since children already get confused about which operations need to use common denominators. I prefer to move quickly from dividing easy-to-visualize fractions (3/4 divided by 1/2) to the development of the procedure of multiplying by the reciprocal (3/4 times 4). In fact, the important principle, which can be shown with whole numbers first, is that dividing by any number is the same as multiplying by the reciprocal: 8 divided by 2 is the same as 8 X 1/2.

On the positive side, a procedure I found clever is what Mr. Demme dubs "The Same Difference Theorem." This trick is useful when subtracting mixed numbers that would otherwise need regrouping, such as 5 1/3 minus 2 2/3. If you add the same amount to each number, the difference between them remains the same. So by adding 1/3 to each number, you change the problem to 5 2/3 minus 3, which requires no regrouping. Do remind your children, however, that no one except other Math-U-See users will understand what they are talking about if they mention "The Same Difference Theorem."

Another interesting procedure is taught in Gamma. In multiple-digit multiplication problems (like 34 X 57), Mr. Demme regroups below the equals line rather than above it. Neither my husband nor I had ever seen it done that way. It does make sense, though, when you focus on place value, and you might end up liking it. It definitely helped my son understand how to regroup multiplication problems, so you might consider teaching it before showing your child how to regroup above the problem. Because Mr. Demme presents the traditional procedure along with his tricks, you can choose to teach your children whichever method you prefer.

An optional Skip Count and Addition Facts Songs CD supplement is available for Primer through Gamma. An accompanying booklet contains all the lyrics and music. The songs are all performed by children in a style similar to "Wee Sing" music. Seven addition facts songs teach all the basic facts in categories: adding nine, adding eight, doubles, doubles plus one, making ten, making nine, and extras. These songs use Mr. Demme's special method of calling the teens "onety-one, onety-two, onety-three" and so on, but they also give the correct names. The CD also includes two sets of skip count songs, one with a Bible theme, and one with a science and literature theme. The music for each set is identical; only the lyrics are different. You can choose to sing about forgiving your brother seven times or about Snow White and the Seven Dwarves. After each skip count song, Mr. Demme says the multiples slowly and clearly. The goal of these songs is to know the multiples and addition facts without having to sing them.

I'd like to close by encouraging parents who are looking into various math programs to consider two things. First, if you truly desire to give your children an excellent foundation in math, you are going to have to learn it yourself. (Remember Ruth Beechick's comments above.) Your positive attitude toward math will most likely rub off on your children. Secondly, no math program is perfect. You are not a slave to the curriculum you use; it is simply a tool, and you can supplement where it seems lacking and delete what you don't like. Having used several of the most popular math programs on the homeschool market, I can assure you that there is no magic math bullet. All programs have strengths and weaknesses. Some, however, have more of one than the other, and Math-U-See is loaded with strengths. Mr. Demme provides the tools you need to learn arithmetic yourself so you can give your children an excellent foundation in arithmetic. It is the most sequential, conceptual, understandable, complete, and user-friendly program I've seen thus far.

-Product Review by Heather Jackowitz, Contributing Writer, The Old Schoolhouse® Magazine, LLC