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Putting the sense in Number Sense

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The best teaching happens when you make a lesson visual, spatial and auditory. That’s why I love teaching kids number sense with a math balance.

Utilizing a math balance in a whole-class setting of twenty-seven kids would be tricky, but at home with one child it’s easy. The  balance we own came from Right Start and costs $25.

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5 does not equal 8. It’s so easy to see.

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Just like it’s easy to figure out that there are many number combinations that equal 5.

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In fact, we spent a full ten minutes just figuring out the number 5!

In pedagogy, we call this “Constructivism”.  It means learning a new concept through your own experimentation and discovery. Giving children the full Constructivist experience isn’t always possible, but a math balance makes it a lot easier.

For more posts about our math balance please click here and here.

Three Year Olds Can Do Math

In the above picture you see my daughter Jenna learning the greater than/less than symbol, at age two.  She thought she was playing “Hungry Guy”, but really she was learning a first and second grade skill.

I believe that children as young as two and three can do real math.   The trick is to teach them mathematical concepts in a way that makes sense to them. 

Play-based math will get results!

(Click here for some free activities to try.)

Jenna is three and a half now, and I felt like she was ready for something more formal.  So three weeks ago, we begin using Right Start Mathematics Level A I have no affiliation whatsoever with Right Start.  I’m just a die-hard Joan Cotter fan.

We are on lesson 7 now, and I am thrilled.  All of the activities are easy to set up, play-based, and conceptually very deep.

Jenna does between 5 -10 minutes of math every day, but only if she wants to.

Here is a sample of what we have done so far:

We made triangles and quadrilaterals out of craft sticks.

We learned about comparison words like long and longest.

We did ordering work of longest to shortest. (This is the before picture.)

We began to explore the abacus.

This is exciting.  This is fun.  This is easy.

The teacher in me wishes every child in America was benefiting from Dr. Joan Cotter’s wisdom.  The mom in me wishes I had know about Right Start when Bruce was this age!

Right Start Math Balance Fun

I could have also titled this post “How I Fold Laundry”. 🙂 You see, I have our Right Start Math box underneath my bed. About once a week, Jenna(2.5) likes to get out the box and play with the math manipulatives, while I’m folding laundry.

Pictured here is the math balance. Developmentally, Jenna’s not ready for all the learning skills the math balance can teach. But it is teaching her the concepts of “balanced” and “equals”. I guess one of the benefits to being a little sister is that there are extra cool things floating around the house to play with!

Singapore Math and Constructivism

I am a big believer in teaching mathematics from a Constructivist perspective, which means enabling children to develop their own meaningful strategies for solving problems, instead of just blindly teaching traditional algorithms.  It also means giving children time, space, and materials to explore mathematical concepts and create their own understanding, before you start imposing your own thinking upon them.

One of my favorite at-home curriculums for teaching math is Right Start, but I have always been curious about Singapore Math because it so popular with homeschoolers.  It is also popular with families who are Afterschoolers, even if they have never heard about that term before.

Even just a casual internet search will tell you that many parents who are confused about Constructivism, or unhappy with how it is being implemented in their children’s publics schools, choose Singapore as the at home alternative to help get their children “back on track”. I have even seen a blog that is very much dedicated to how much better Singapore Math is than current Constructivist textbooks.

The problem that I see with all of this public school curriculum bashing is that when I look at the Singapore Primary Mathematics Standards Edition textbooks, I see a lot of Constructivism.  In fact, there is enough of a Constructivist influence in the 4A book, that I have no problem with my son Bruce(6.5) using it on an Afterschooling basis.  Here are the main, Constructivist elements that I like about the 4A book:

  • Metacognition: There is an emphasis on helping students think about thinking.  There is also guidance through the thinking process.  This is very Constructivist!!!
  • Spiraling Curriculum: Most concepts are taught, and then revisited again and again throughout the year.  The Constructivist public school curriculums that I have seen also use this spiraling method.
  • Visualization: There are a lot of pictures and modeling.  A good Constructivist classroom would also encourage pictures, drawing and modeling to help reach visual learners.
  • Multiple Strategies: Again and again I keep seeing examples of more than one way to solve a problem.  Alternatives are shown beyond just traditional algorithms.

This is really surprising to me because some of the most vocal critics of Constructivism I have seen online are also parents who choose Singapore as their children’s math program.  They tend to keelhaul Constructivism and hail Singapore as their mathematical savoir. This is really bizarre, because comparing Singapore with a public school Constructivist curriculum like Dale Seymour Investigations is not like comparing apples and oranges; it’s more like comparing apples and pears.

If you were to think about math as a continuum with Back to Basics “just-teach-her-to-borrow-and-carry” on one hand, and pure Constructivism “she will discover every new bit of knowledge herself” on the other, then in my opinion, Singapore would not be considered a Constructivist program nor would it be considered a Back to Basics program.  It would fall somewhere in the middle, like Houghton Mifflin’s Math Expressions.  Falling closer to the Constructivist end would be programs like Right Start, Dreambox Math, and Hands on EquationsSaxon, Horizon and Life of Fred would be closer to a Back to Basics philosophy.

So what do I think of Singapore Primary Mathematics Standards Edition?  I think it’s pretty good, but for a complete homeschool program I still prefer Right Start.  For Afterschooling though, I can see the benefit of using a cheaper, more colorful program like Singapore, if your child preferred it.  The pages are smaller and easier to complete, so if your kid gets a “buzz” from completing pages, then Singapore would facilitate that.

If you do use Singapore Standards edition for a homeschooling program, then you really would need to buy the complete program, including the word problem book.  As a former public school teacher I have to say that there are not nearly enough word problems in the textbook alone to adequately prepare kids for state standardized tests.

Our own experience with Singapore remains limited.  I brought the 4A textbook home for my son Bruce in the middle of Christmas break last week and he completed 23 out of 161 pages in about three days.  Part of this is because the Singapore 4A geometry section is much easier than the third grade Houghton Mifflin Math Expressions work Bruce has been doing.  The multiplication section was all review for him too, because he has completed Dreambox 3rd grade.  But Bruce liked the Singapore textbook.  It is a lot more “fun” to look at than Right Start Level D, and he willingly plowed through pages in 4A.

For a more experienced opinion of Singapore, I’d like to include my friend Claire’s earlier comments from my K-1 Summer Bridge page:

For parents who have sticker shock at the price of RS, I would recommend Singapore Primary Mathematics. Note that these are *NOT* the “Singapore Math” workbooks sold at Barnes & Noble (those are by a different publisher and are substantially “dumbed-down” from the original program).  I prefer using a “hands-on” program like RS to a workbook-based one like Singapore in the primary grades, but RS is pricey and very parent-intensive. Also, Singapore is easier to accelerate and/or up the challenge level for a bright student.

I would recommend getting the Primary Math textbook and either the workbook or the Intensive Practice book (depending on whether the student is average or advanced).  The Intensive Practice book is only available in the U.S. edition but it is very easy to match up the topics in the Stds. ed. text with the ones in the IP book. The Stds. ed. text is full-color and has more of a “cartoony” look to it. I actually prefer the “cleaner” look of the US ed. books myself, but overall feel the Stds. ed. is better.  The books are available at Singaporemath.com, Christian Book Distributors, and Rainbow Resource Center.

Thank you Claire!  Finally, I would like to add my own thoughts about Singapore Math and gifted children.  Just because my six year old can burn through 23 pages of 4A in a few days doesn’t mean he should. Gifted children deserve instruction too.  The deserve attention. They deserve good teaching, and they deserve experiences. They do not deserve to have their curiosity or love of learning drowned in busywork, endless workbooks, or too much isolation.

I think that parents of gifted children should have a clear and consistent message. Too often society says: “Oh, that child is so smart.  Just send him off in the corner with an advanced book and he will be fine.”  In my opinion, that does a huge disservice to gifted children.  I would hope that nobody reading this would use a curriculum like Singapore that way.  Instead, here are some of my favorite 4th grade level activities that you might consider combining with the 4A Standards edition, to “jazz it up” a bit:

Fractions

Geometry

 More to come!

Sylvan Learning Center Alternatives

What do you do if your child is really struggling at school? A lot of parents end up turning to Sylvan Learning Center…if they can afford it. I knew one family who went to Sylvan for a year and it cost between $2,000-$3,000. Did they think it was worth it? Yes, but the financial cost was a real hardship for that family. As an educator I take issue with Sylvan in that they hire credentialed teachers but then only pay them about $15 an hour. However, the families themselves are paying Sylvan between $45-$50 an hour. That data alone begs for other, cheaper alternatives.

As a former K-4 teacher, here is what I would suggest as a cheaper alternative to Sylvan Learning Center:

  • Work with your school district to determine if your child has a learning disability
  • Assess and continually track your child’s learning needs and progress
  • Use quality, scripted tools to deliver one-on-one instruction in the home

Step 1: Working with your School District

If you live in a well-functioning and accountable school district

If you feel like your child is really struggling in school, the first thing to do is to talk to your child’s teacher immediately. She can probably offer a lot of insight, and give your ideas to try at home. If you think your child might be struggling with a potential learning disability, you and your child’s teacher can make a referral for assessment and evaluation to the special education department in the school district. This can sometimes take a couple of months, so in the meantime continue working with your child at home.

If you live in a school district facing corruption and/or poverty issues

Definitely talk to your child’s teacher and get ideas for what to do. The teacher probably really wants to help your child, but her hands might be tied in terms of getting the special education assessment services your child needs. This happened to me as a teacher when I taught in a low performing school district. I finally ended up having parents write letters requesting special education assessment, and then having them hand deliver those letters to the school district office. If you need to go this route, make sure to have your letter time and date-stamped by the secretary, and then have her make a copy of that letter for you. At one of the districts I once worked in, I also had to send a copy to the lawyers who were suing our school district for failure to administer special education services!

I would not recommend requesting assessment in writing if you live in a normal, functioning school district because it might make you as the parent appear overly aggressive and not willing to go through the proper channels. But requesting assessment in writing is a legitimate, valid thing to do, and is sometimes necessary. More information regarding assessment, special education, and IEPs can be found here.

Step 2: Continual and Ongoing Assessment

One of the things that Sylvan Learning Centers does really well is telling you exactly what grade level your child is at in language arts and math. But you can figure this out for yourself at home, if you have the right tools! Some of the ways to do this cost a little bit of money, but some of them are absolutely free.

Reading Assessments: Guided Reading Level, San Diego Quick Reading Assessment,

Math Assessments: Saxon, Singapore, Horizons

Step 3: Deliver Quality, One-on-One Instruction at Home

First of all, you need to choose a neutral parent or adult to be the Afterschooling instructor. If you are lucky enough to be in a two-parent household, choose the parent who does not have a history of homework battles with your child. For example, if it’s been the mom who has been struggling with the eight year old to learn multiplication, then have the dad take this on, at least for a little while. I’m not blaming the mom, but if there is already some “tense” history here, then choose the neutral parent. If at all possible, give your child a fresh perspective.

Second of all, buy a kitchen timer. Make sure your child knows that your Afterschooling time is going to be fun, effective, and limited. Use your own judgment, but think about creating an hour long schedule that is broken into 15 minute chunks. Set the timer so that your child can see that progress is being made. Afterwards, eat ice cream or bring out the DS as a reward for cooperation.

Thirdly, invest in the right materials. For a child who needs remedial intervention you want to choose instructional materials that incorporate visual, auditory, and kinesthetic learning as much as possible. Hopefully the school district will be assessing your child soon to determine if they have visual or auditory learning disabilities, but in the meantime make sure you are using teaching techniques that encompass all possible learning styles. You also want to choose programs that are scripted, or (although I find this term insulting) “teacher proof”. Here is what I would suggest:

Reading/Spelling/Phonics

I would recommend All About Spelling, Levels 1 and 2 to start with. Full Disclaimer: I am an AAS affiliate, but only because as a teacher I really believe in the quality of the program. It is systematic, sequential, hands-on, fun, fast, and will help you diagnosis where the exact gaps in your child’s phonics and spelling knowledge lie. Here’s my full review with pictures: http://teachingmybabytoread.blog.com/where-to-start/all-about-spelling-level-1/ AAS will give you a scripted program to teach reading and spelling at the same time. It can be used with nuero-typcial children, as well as kids with dyslexia.

You could also try Guided Reading using post-it notes, for about ten minutes each Afterschooling session. Be sure you choose the right level of book for your child to read. That’s why you need to be continually assessing and monitoring their Guided Reading level.

Writing

Some kids have horrible writing blocks and can’t put anything on the paper. Other children write page after page, but their writing is riddled with convention errors. Here’s is a continuum of suggestions for how to help kids write at home:

For kids who won’t write anything at all

Kids who want to write, but get stuck with spelling

Encouraging writing in general

Journaling

Working on Conventions (no link yet, sorry!)

Mathematics

I think that the best, scripted, hands-on math program you could buy to help you teach your child math is Right Start. Full Disclaimer: I have no affiliation with Activities of Learning/Right Start Math whatsoever! Right Start is a bit of an investment, but you can use all of the manipulative materials that come with it to help your child with their regular math homework through the years. Start with the free online placement test to determine which kit to buy. Make sure you buy the teacher’s guide, because this will tell you exactly what to do. For more information on math education in general, please see my post here: http://teachingmybabytoread.blog.com/math/

Final Thoughts

You can do it! You can deliver remedial intervention for your child, and get your son and daughter back on track. Start by working with your school district to make sure the possibility of learning disabilities are assessed and addressed. But don’t wait around for bureaucracy to take its course, start working with your child at home right away! Afterschooling three–five hours a week can really make a difference, especially if you have the right tools.

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Right Start Level D

For the past month and half my six year old son Bruce has been working out of the Right Start Level D workbook, which is at the 3rd-4th grade level. Even though it is summer vacation, he is still doing math each day because the rule in our house is that two pages (or one side front and back) of math earn 30 minutes of screen time. So I’m not making Bruce do math all summer, it is his own personal choice.

We do not own the Right Start Level D Teacher’s Guide, but we do have all of the manipulatives at our disposal. I am also familiar with teaching in the Construcvist method and have read the Level C Teacher’s Guide cover to cover. So we even though we are not following the program exactly, I am still able to deliver a lot of meaningful instruction to Bruce, that is a huge step beyond plunking him down on the dining room table and having him do math worksheets.

I don’t know if you can tell from the blown up picture of what he was working on today, but Bruce has quite a sense of humor! He is also still struggling with number reversals. At this point in his math education, I am having him go back and correct the reversals each time. Today however, I chose to ignore them because Bruce was trying to do extra math to earn watching a movie.

We are about 22 out of 150 pages into the Level D workbook, and so far I’m pretty pleased. I think that the workbook can stand alone as an Afterschooling supplement, but that if you were using this program for Homeschooling, you would definitely want to get the Teachers Edition.

Yellow is the Sun

Today was a big day for us!   Jenna, (23 months), laid out all of the sticks in the correct order 1-5 while we sang “Yellow is the Sun” together.  This is a song from the Right Start Math curriculum.  She even held her hand out when we sang the line “My whole hand makes five”.  I’ve modified the words a bit to make them more meaningful to Jenna.  “Yellow is the sun, Jenna shows you one” etc.

Of course, when we got to six she wiped all of the sticks away!

We shelled out $5 to buy the book, but you could do this activity for free at home because the music is posted on the Activites for Learning website:  http://www.alabacus.com/pageView.cfm?pageID=309  I guess I should think about buying the Level A teacher’s guide in the next six month or so.

The AL Abacus

 

AL Abacus front

AL Abacus, reverse side

In a previous post I talked about the Right Start Level C math program Bruce has been using, and its core component, the AL abacus.  I explained about how even though I think the abacus is totally amazing, Bruce hasn’t been too keen on it yet, which made following the specific lesson plans in the Level C teacher guide difficult.  He still benefited a lot from the other parts of the program however.

For those of you unfamiliar with the abacus, here’s a cool and (free) link that let’s you explore it online: http://www.alabacus.com/pageView.cfm?pageID=321

Here are also some pictures of two ways to represent 57.

                         

Pretty cool, hunh?  Hopefully Jenna will be much more interested in using it when she is ready.

Toddler Math

Day after day of watching her big brother Bruce sit at the dining room table working on Math Expressions and Right Start, Jenna has now started regularly asking to “do math” too.  She is 23 months old now, and I’m still trying to work on visualization with her in addition to counting.  

Jenna can count with correspondence from 1-3 and can also rattle off her numbers to 14.  However, I’m trying to phase that part out and work on saying “ten and one, ten and two, ten and three etc.”  This is what is suggested in the Right Start literature, and is a new idea to me as an educator.  I’m really interested in trying it out, so my poor little girl gets to be my experimental guinea pig!

With Bruce at this age, we worked on counting and that was about it.  All of his early math skills were learned at Montessori, and I didn’t begin any formal instruction with him until he was four. 

Here are some of the difficulties I’ve encountered trying to teach math to an almost two year old.  First of all, Jenna keeps trying to eat the math manipulatives!  They are all choking hazards, so I really have to watch her and put them away up high when we are done.  The other problem is her eternal asking of the question “Why?”  Our most recent math session looked like this.

Me: “Can you give me two?”

Jenna: “Why?”

Me: “Because Mommy wants two squares.”

Jenna: “Why?”

Me: “Umm… because one square is not enough.  Mommy wants two.  Can you count out two?”

Jenna: “One’s nough.  One’s nough Mama.”

Me: “No, one’s not enough.  Mommy wants two.  Please give me two.”

Jenna: “Why?  Why Mama?”

At this point in the lesson I decided to just switch back to counting with correspondence.  I’ve been using the counting song from Sesame Street and Jenna can now sing along.  I’m not sure how much she is learning from all of this, but at least it’s a fun activity to do with Mom.

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Review: Right Start Level C

We have been using Right Start Level C as part of Bruce’s afterschooling for about seven or eighth months now.  We switched over to Right Start, after Bruce completed year of Horizons 1st grade math.  I thought Horizons’ first grade program was okay, but that the second grade curriculum focused too much on algorithms.  For more information on our experience with Horizons, please see here: https://teachingmybabytoread.com/2011/05/01/horizons-math/

I believe in teaching children math through the Constructivist method, which is where they discover mathematical concepts and strategy themselves, sometiems through hands on discovery. I was very impressed by the philosophy of the Right Start program which stresses understanding over rote memorization.  Additionally, I was familiar with Right Start, having used components of the program when I taught 3rd/4th grade at a Charter school.

I decided to shell out out the big bucks and bought the homeschool deluxe set for level C, and I am really glad I did.  It has tons of math manipulatives that are useful for Bruce, Jenna, and other math activities we do as well.  It also has all of the books, including the Teacher’s Guide.  I used the Right Start manipulatives a lot to help explain concepts in Bruce’s school math program, Houghton Mifflin Math Expressions, and we have also used them to help with Life Of Fred Fractions.

The Right Start program has a teacher’s guide that is very creative, detailed, and goes above and beyond what is just in the worksheet book.  If you were using Right Start as a stand-alone curriculum, you would really need the teacher’s addition.  We were using the program as a supplement to Math Expressions, so we didn’t follow the lessons plans exactly, although Bruce did do almost all of the worksheets.

Since I’m a teacher myself, I read the teacher’s guide almost cover to cover, and then just “winged it”, or taught the lessons intuitively according to all the training I’ve received in teaching mathematics from a Constructivist perspective.  But I would definitely recommend following the teacher’s guide, to anyone who was new to the Constructivst approach, or who was using Right Start as their child’s sole mathematics curriculum.

The hallmark of the Right Start program is the abacus, which I was really excited about when the box arrived.  My husband is an engineer, who has had the opportunity to work with a lot of coworkers from China, many of whom are abacus devotes. (But a different type of abacus, I should point out.)  One of these friends told my husband he could “see the abacus in his mind,” and that’s why he had such exceptional mental math skills.  This is the same claim that the author of Right Start makes.

I am still super excited about teaching with an abacus, and intend to do this with Jenna as soon as she is old enough for Level A.  The problem with Bruce and the abacus, was that he rejected it from the get-go.  You know how there are little babies who reject the pacifier or the bottle?  You think, “Did that mother really try hard enough?  I mean, did she really try?”  Well yes in fact, I did!  Bruce would not have anything to do with the abacus at all.  Probably his math skills were developed enough already that imposing the abacus into his thinking was something his brain just did not want.  He was five years old at the time and had already finished the Horizons first grade workbooks, and half of Hougton Mifflin 2nd grade.  This is also party the reason that we could not follow the Right Start lesson plans exactly, because so much of them are abacus based.

Not using the abacus turned out to be okay for our family, because I am not teaching the traditional algorithms of borrowing or carrying to Bruce at this point.  The way the Right Start lessons set things up, they use the abacus to teach borrowing and carrying in a manipulative way.  So I would have skipped over those parts in the Teacher’s Guide anyway.   (For those of you thinking, “What the heck!  This lady doesn’t teach borrowing or carrying?”, please see my previous post on subtraction: https://teachingmybabytoread.com/2011/03/15/subtraction/.)

A final point of note.  Bruce has almost finished all of Level C, and I’m still stumped by what exact grade level it is.  I’ve thought about it a lot, and reread the 2nd, 3rd, and 4th grade standards in our state.  I’ve also taught 3rd grade for two years, and a 3rd/4th combo for two years.  And yet, I’m still a bit perplexed, because it introduces some concepts that aren’t usually taught in public school until later.  In general though, I’d say Level C seems to be in the 2.5 grade – 3.5 grade range.  They also have a good placement test online to help you determine which book to choose for your child.

Horizons Math

(Don’t buy this!)

When Bruce turned four and was still going to Montessori, I decided to begin formal math instruction with him at home. He was (and is) so energetic and inquisitive, that I thought his behavior would improve if some of his energy was channeled into academic pursuits.  I looked around blindly for a curriculum to get him started on, and discovered the Horizons math program through Alpha Omega Publications.  There are two workbooks in each curriculum year, for a total of 180 lessons.  There is also a teachers guide to go with it, which I did not purchase, and I suspect a lot of people do not buy either.  (That may have been a big mistake!)  Bruce took the online placement test which scored him as being ready for the first grade.

The first grade curriculum was pretty good, and Bruce sailed through it in about six months.  It is a spiraling curriculum, so there is a lot of coming back at topics previously covered for review and practice.  I also liked that the workbook pages were very colorful, and had pictures.  The workbooks seemed to be very equation heavy, with not a lot of word problems, which is okay for first grade because that way reading skills do not hamper math progression.  ( I have since found out that the Teacher’s Guide includes a lot more word problems.)

The first grade workbook has a lot of drill-and-kill.  Often times Bruce would get tired of actually writing out the numbers (since he was only four), so I’d be the secretary and he would solve problems in his head and then tell me what to write.  Near the end of each book the lessons were getting too easy and repetitive, so we just crossed off big sections of them and skipped to the next page.

When Bruce started Kindergarten I bought him the second grade program thinking we would continue to chug along.  (He was also doing our school district’s second grade curriculum, Houghton Mifflin Math Expressions.)   I purchased the Horizons math, reading and spelling kits.  This was all a big mistake.  By the second grade book it was clear that the Horizons program was designed for back-to-basics home school families, which certainly doesn’t describe my teaching approach!

The second grade math program has a strong focus on traditional algorithms such as borrowing and carrying, to the point that pages are set up with little “carry the one” signs.  They have kids doing 4 digit addition and subtraction by the second or third month of second grade, which is only possible if you are mindlessly solving equations with algorithms but are not doing the deeper work of creating true number sense.  By contrast, Bruce has finished the Hougton Mifflin Math Expressions second grade program, is now half-way through with Right Start Level C, and is only now capable of solving 4 digit equations in his head, meaning he really understands how to do it.  Teaching kids to crank out algorithms is not teaching higher order mathematical thinking.

The Horizons Second Grade reader was truly bizarre.  It was an adaptation of Robinson Crusoe written at a second grade level and broken into 90 chapters, interspersed with excerpts from a second grade reader from the 1800s.  After about ten chapters, Bruce was bored out of his mind and refused to read any further.  It was the exact opposite of the high interest reading material necessary to inspire young children into becoming self motivated readers.

To be fair, I didn’t have the Horizons Math Teachers Addition, which the website clearly states in an integral part of the program.  But based on the workbooks, which have 2nd graders cranking out 4 digit subtraction with regrouping problems using traditional algorithms, Horizons did not seem to be a program I felt comfortable using for Bruce.  We have switched to Right Start, and have been much happier.

The Horizons reading program seemed to be from the standpoint of “If it was good enough for my great-great-great-grandpa, then it’s good enough for my son.”  Um… no in fact, it is not good enough  at all.  I just wish it hadn’t taken me almost $300 to figure that out.