Memory, Math facts, Flashmaster

Addition facts were moving very slowly for my daughter. Flash cards weren't helpful, writing the fact out wasn't that helpful, worksheets were helpful but slow.

On the advice of several people at the Well-trained Mind Forum, I got a Flashmaster. It has worked really well!

After you miss a fact 3 times in a row, they tell you the answer. Any fact you miss at least one time, they give you another problem, then the problem after that is the problem you just missed. That seems to be the perfect timing, it's been very effective. It also has several different modes. One mode gives you the last 15 problems you have missed. This is very nice, it's hard to keep up with normally. Doing all the ones she didn't know at once was frustrating to my daughter, so we do them offline. I've found that having her repeat the correct fact 3 times orally is the best way to get her to remember them.

The Flashmaster is also much faster than a worksheet for a young child who writes slowly, they can push a button much quicker than they can write out a number, so you can do a lot more math facts in the same amount of time. (It also does subtraction, multiplication, and division.)

Mathway

From Massachussetts high-school math teacher Mr. D:

Mathway is a website where you can type in problems ranging from basic math to Calculus and not only get the answer, but a step-by-step explanation. All of the key vocabulary words in each explanation are linked to an online glossary. You can graph answers on the coordinate plane when needed as well.

This should be on every students' list of sites to use for homework help, as well as a great self-directed way for students to check their own work. You could do anything from a set of problems, project or test and then have students correct everything themselves and get detailed explanations all from one website. It's fantastic.

You might check out the rest of Mr. D's blog as well.

Sum Swamp--fun adding and subtracting

We recently ordered "Sum Swamp," an addition and subtraction game, from Amazon. It is helping with basic math facts, and also covers odd and even. It's a fun game, and very cute. It only has two 6-sided dice, however. I'm planning on buying two 10-sided dice to make it cover all the basic addition and subtraction facts.

There is an "endless loop" where you have to land on the exit to get out. For a shorter game, we don't make the endless loop endless. You could play with multiplication as well, and then either add the digits or for any answers 10 or over, divide by 10 and round to the nearest number.

We lost a bit of math knowledge this summer, this game will be a fun way to keep up basic math facts during holidays.

Now, if I could only find a fun DVD! Something along the lines of Leapfrog's "Talking Letter Factory" for letter names and sounds would be great. Unfortunately, leapfrog's math circus doesn't teach much, although it is cute.

Comments on Chisanbop (Chisenbop) or Finger Math, Anyone?

Chisenbop is a method of computation using the fingers like an abacus.

Here's a tutorial.

www.cs.iupui.edu/~aharris/chis/chis.html

I think it was popular in the early 1980s, but fell out of favor.

There were a couple of books published:

The Complete Book of Chisanbop : Original Finger Calculation Method (ISBN: 0-442-27568-4)

Complete Book of Fingermath (ISBN-10: 0070376808)

Comments on the value or lack thereof?

Dyscalculia: What Is It? International Dyscalculia Awareness Day

Dyslexia is a nickname for "Specific Learning Disability--Reading", listed as 315.0 in the Diagnostic and Statistical Manual-IV (DSM-IV). Most people understand that dyslexia is a persistent difficulty with reading, despite good instruction and at least average intelligence.

There is also a Specific Learning Disability -- Mathematics (315.1, DSM-IV), known as dyscalculia.

The Dyscalculia Forum chose today as the day to raise awareness.

According to the National Center on Learning Disabilities (NCLD):

Dyscalculia is a term referring to a wide range of life-long learning disabilities involving math. There is no single form of math disability, and difficulties vary from person to person and affect people differently in school and throughout life.

Here's an index page on math and LDs from NCLD. Here's LDOnline's index page on math disabilities. Here's an overview of math disabilities in children from SchwabLearning. Anna J. Wilson's Dyscalculia Primer and Resource Guide.

There are also a couple of good videos on Youtube: start with Dyscalculia: It Is Not Only Trouble With Math.

Math Program Reviews

Yes, occasionally I have some non-phonics thoughts.

The Well-Trained Mind has an excellent review of several math programs, here's a short excerpt:

Which method is better? In my opinion, the one which the student understands most clearly. In both cases, it is possible for the student to learn the mental trick without thoroughly understanding why it works, although the sheer amount of repetition in the Saxon method makes it easier for this lack of understanding to escape detection. But the strongest mathematical training of all would come from a combination of programs – in which the student is taught to do a mathematical process using several different methods and mental procedures.

Currently, Singapore and Math-U-See are “thought-oriented” math programs available to home school parents; Saxon and A Beka are “skill-oriented” programs. A combination of Saxon + Singapore, or Saxon + Math-U-See, or Singapore + A Beka, or A Beka + Math-U-See, may come closest to fulfilling the goals of classical education. Math-U-See + Singapore would also be an excellent combination, as long as you use MUS’s supplementary drill sheets. Treat one program as primary and the other as secondary; when you cover a concept in the primary program, look it up in the secondary program and see whether it is explained and illustrated differently.

I like her idea of using 2 programs. I do that all the time with phonics--I know what is best about each program and pick and choose accordingly, I have dozens of phonics programs from my tutoring. Also, if a student is struggling with a certain area, it's good to have a few books that explain it in slightly different ways. I hadn't thought to do the same with math.

There is also an interesting review at Sonlight, a basic review of their math choices from grade school through calculus, and then, they explain their choice for algebra and geometry--teaching textbooks:

• CD-ROM-based "whiteboard" lectures and step-by-step explanation of how to solve every practice problem taught by a tutor who has been teaching homeschoolers for years...since the days he tutored probability and statistics at Harvard . . . and . . .
• CD-ROM-based "whiteboard" solutions guide that works every step in every homework problem, so you see exactly how to solve each problem . . . and why you want to use the methods the instructor (or, more accurately, personal tutor) uses.

Their calculus choice, thinkwell math, also includes CD-ROM video tutorials.

We're currently using Math-U-See with our daughter for Kindergarten math. I personally like Singapore or Saxon better, but this is what is working for her. The good thing about Math-U-See is that they have a DVD with each lesson so that if you get that "blank stare," (yes, it is even possible with kindergarten level math!), you can just plop in the DVD and have an actual math teacher explain it in field tested verbage.

Math-U-See's approach to fractions is also interesting, I'm not sure what I think about it. The rectangles do make more sense to me than the traditional circle/pie method. From their downloads page, in sample lesson pages, click on Epsilon.

For an even more interesting and thought-provoking review, read Wild About Math!'s Calculus in 4th grade? I'm really not sure what I think about that one. Luckily, we're still doing kindergarten math so I have time to figure it all out.

What are the underlying problems?

For each academic area, what do you think the underlying main problem or problems?

For example, for reading, I would say lack of teaching basics (phonics) first, and lack of teacher understanding of the phonetic structure of English, which leads them to such things as teaching 150 completely phonetic "sight words" as wholes, and 65 more which can be taught with about a dozen rules or patterns (example pattern: to, do, into, today, together.)

I'l be especially interested to see what people think about history. I know I didn't learn much history in school, and the little I did learn was so dry and boring it made me stay away from reading anything history related for years after finishing my education. (I'm now 1/3 of the way through Albion's Seed and am greatly enjoying it!)

A military friend of mine whose children have gone to many different Catholic schools in several different states said that while all his schools have used good math textbooks and methods, there was a wide variability in teacher knowledge, and that his children learned the most from teachers who truly loved and understood math. He also said someone who truly loved math generally wasn't the type of person who loved working with small children, so that while his children often had great math teachers at the upper levels, they seldom had great math teachers in the lower grades. (I'm not saying this is the problem with math, but it seems like it could be a strong underlying contributor at the elementary level.)

"Orton-Gillingham for Math": "Making Math Real" -- Introductory Post

Making Math Real

http://www.makingmathreal.org/

I just finished the 2-day, 14-hour introduction:

http://www.makingmathreal.org/index.php?option=com_content&task=view&id=38&Itemid=57

My brain is either fried or overloaded. Lots more sympathy for kids whose learning styles aren't supported in the classroom.

[Aside: hush!--already about the "learning styles". I agree that lots of the edu-babble about learning styles is content-free. However -- I sat, listened, and wrote for 120-minute blocks, about a subject that I am deeply committed to. I'm an adult with a well-honed capacity for taking in new information by listening, and retaining new information by writing. By minute 90, I was overloaded. You go shadow your kid throughout her school day and see how well you could keep up. /Aside]

Making Math Real (MMR) isn't a math curriculum, it is a specific, structured approach to math instruction.

A good analogy is:

Orton-Gillingham approaches to reading = MMR approach to arithmetic and mathematics.

The training is expensive.

Question: Is it worth it?
Answer: If your district is using Everyday Math (or other purely constructivist math curricula), it is definitely worth it to offset the fog of confusion EDM engenders in many kids, with or without LDs.

It may also be less expensive in the long run than putting your kids in Kumon, Sylvan, or other after-school remediation/tutoring programs.

Here's a description from the MMR website:

Multisensory structured methodologies deliver all instruction via the three processing modalities: visual, auditory and kinesthetic-motoric. Students who are struggling experience processing difficulties in either one or more of these processing modalities. Best instructional practices require linking all incoming information across the three channels to maximize successful processing.

Structured curriculum means starting with the simplest elemental foundation and building developmentally in an incremental and systematic progression from the concrete to the abstract. The most powerful aspect of a multisensory structured program is that each current activity and lesson builds the essential developmental tools for success at the next level thereby reaching the full diversity of learning styles and educational needs in all classrooms.

Schwablearning.com was a great source of discussion and information -sharing for non-standard kids. Here are links to previous discussions of Making Math Real on the parent-to-parent board (listed in chronological order):

• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=3601
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=4982
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=15320
• http://www.schwablearning.org/message_boards/view_discussion.aspx?thread=18313
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=19498
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=24091
  
• http://www.schwablearning.org/message_boards/view_d iscussion.aspx?thread=25871

BTW #1: , as part of the class, I am now "bombproof" on my 13 multiplication facts, thanks to the "nine lines". I'd lay it all out for you now...but I am totally used up, cognitively.

And actually, I'm kind of appreciative of that experience -- because, for some of our kids, they are "totally used up, cognitively" before the end of the school day. That doesn't happen so often for us as adults.

BTW #2 -- being "cognitively tapped-out" is why I think homework in k-3 is stupid a waste of the (a) teacher's, (b) student's (c) parent's time and energy.

Two Resources for Math Facts Mastery (1) Video Game, Timez Attack, (2) Re-introduction to Dave Marain's MathNotations Blog

Review from Math Notations


Here's the heart. of his review.

Having seen dozens of computer learning software products both online and on CD-ROM, I've reached the point where I can rapidly identify quality and Timez Attack is quality in every sense. I decided to email Ben Harrison, the president of the company, and share my thoughts about his product, the only item his company sells at this point. I explained that I would like to write an objective review of Timez Attack for MathNotations after I had an opportunity to try it myself and to observe a child using it. I asked him if he would be willing to give me access to the full version and he agreed. I haven't had time to thoroughly play around with this new version, but my son has been on it for about a week and he likes it. To derive the maximum benefit of this software, I feel I need to establish consistency for him to be on it for at least 15-30 minutes a day for about a month but that has not yet happened. I already see better retention but he is still only on a beginner level.

Here's the BigBrainz homepage:

http://www.bigbrainz.com/index.php

Repetition is the key to mastery, so this does look really promising. The "starter version" is free.

MathNotation is maintained by Dave Marain

Look for fully developed math investigations that are more than one inch deep, math challenges, Problems of the Day and standardized test practice. The emphasis will always be on developing conceptual understanding in mathematics. There will also be dialogue on issues in mathematics education with a focus on standards, assessment, and pedagogy primarily at the 7-12 level through AP Calculus.

Previous mentions of Dave Marain & MathNotations at KTM:

Interview with Lynn Arthur Steen one, two and follow up commentary: one two

Sigh. Misunderstanding Scripted Learning [whiny voice] AGAIN!

One of the most important things I gathered from the Direct Instruction literature is that is isn't scripted lessons, it is a script for lessons that have been tried, tested, debugged, tested again, further debugged, [repeat ad nauseum until you have a program that has a X% rate of success upon delivery] plus effective training of teachers in script delivery plus constant, rapid testing and feedback to students that make a scripted program effectove

[sidebar: Ken deRosa had an excellent summary, from Engelmann's The Outrage of Project Follow Through: how this works in practice"]

These quality-control features are what is missing from the conversation about "scripted lessons", which are often depicted as "teacher proof".

Sheesh. Would any theater-goer think of a play (which after all, is a script) as "actor-proof"? Haven't they ever been to a wooden production?

It is the "phonics vs. whole language" argument, all over again. Frankly, if I had a child in k-3, I'd prefer a semi-competent whole-language teacher, who was at least reading good stories, over a lousy phonics means workbooks and illogical teaching and drill teacher. At least I could supplement my kid's reading mastery with good phonemic extra-curricular teaching, rather than trying to erase the wrong approach and start over.

But wait. I got away from the point of this post.

The point of Direct Instruction and Precision Teaching is that both approaches are continually refined, based upon accurate* assessment. I can sit at home all I want, and come up with a clever, creative lesson plan (which might well be wonderful based on thought experiments) but until I use the lesson plan, and then test what of the concepts in the plan the students have mastered, on actual students...it is just pie in the sky.

NYC Educator has some doozies from classroom teachers on script-hating.


=====
*another time we will talk about how malformed multiple choice questions have nothing very little to do with getting an accurate picture of what your students have or have not mastered.

Connecticut's Curriculum Standards Revision

The State Dept of Ed website has draft curriculum standards posted. If you are a Conn parent or educator, you might want to take a look at what they are doing. I haven't looked through these things carefully, but I want to get the links up here. The letter accompanying the draft standards invites feedback. I don't see why only educators should have fun with this, so please do take a look. The standards will be in draft form until December 2007.

Draft Connecticut Pre-K through 8 Math Standards

Draft Connecticut Pre-K through 8 English Language Standards

There is a Feed Back Form for both draft standards here.

The biggest change to the standards is the inclusion of "grade level expectations." These are fairly specific and are a big improvement over the old standards. For example, in 2nd grade, students should be able to order simple fractions, tell time to the 1/2 hour, and know the calendar months in order.

The standards are clear. They may not be very high, but they are at least clear.

But there are still too many standards. The mile wide inch deep criticism is even more apparent when you look at each of the tasks to be mastered along the way. Still too much pattern recognition, probability, and graphing in the earliest grades and not enough emphasis on automaticity with basic math facts and fluency with fractions, decimals, and percents in later elementary grades. It's there, but there's no focus and no sense of what is most critical. Because of the sheer number of standards and expectations, how is a teacher or school to wade through them all? If they give equal emphasis to everything, they will not master anything.

So a mixed bag, but a step in the right direction. If you have the time to look at these things, you might consider downloading the feedback form and e-mailing it in to the State Dept of Ed.

After all, how often does anybody in education ask for your input?

Think Math!

Are you familiar with the Think Math! textbooks?

Apparently, this is a new K-5 series published by Harcourt. My school will be piloting Think Math! next month. (Yes, the exclamation point is part of the name.)

At first glance, it appears that the developers have tried to address many of the objections to NSF texts while maintaining some constructivist components.

Here is some info from their website http://www2.edc.org/thinkmath/index.htm:

. . . developed by Education Development Center, Inc. (EDC) in Newton, MA with support from the National Science Foundation.

Think Math! does not pit skill against problem solving. Rather, it builds computational fluency through plentiful practice in basic skills as students investigate new ideas and solve meaningful problems. Lessons provide glimpses of ideas to come, letting students build familiarity and develop conceptual understanding as they apply, sharpen, and maintain skills they already have.

What makes Think Math! unique (not just another NSF-supported program)? Think Math! provides focused practice, which enhances conceptual understanding as it increases computational fluency. The curriculum allows students to get involved in solving real problems, figuring out what to do without first being told. Instruction is then used to provide good explanations of reliable techniques. The materials, and some teaching techniques recommended in the lesson plans, also reduce the number of words, using visual models to convey information instead. By puzzling out what’s missing, students can “read” the mathematics and figure out what to do without written directions.
Perhaps most unique, the program features embedded professional development for teachers both in understanding the mathematics content at a deeper level and in suggested teaching techniques. Professional development is located in a feature of many lessons titled About the Math, and in thoughtful explanations throughout the teacher’s guide.

There’s much more information on their site that I’m just starting to read. The first question that comes to mind is why adopt a text that has no proven track record when there are others (Singapore, Saxon) that have demonstrated success? This sounds like another experiment with our children being used as guinea pigs. Also, will our school receive NSF funding if we implement this?

Background:
My school piloted Growing with Math last fall and had planned to pilot TERC Investigations this spring. I have been vocal in expressing my concerns, and then last month a few days after I circulated an email to a small group of parents the school informed us that they had decided not to pilot TERC. My email, which included the Inconvenient Truth video and other choice references, was apparently forwarded to many other parents and made it to the BOE and to the school administration. At one point a PTA officer asked me to please email my “readers” to let them know that the TERC pilot had been cancelled.

I would appreciate any comments and advice.

teaching problem solving to second graders, part 2

(Part 1 is here.)

Now the students are ready to start solving simple comparison problems with variables.

First the students are taught how to translate a phrase like "T is less than H" or "R is more than W" onto a number family.

The lesson might be taught like this:

Sometimes we refer to a number without telling which number it is. We can call that number J or B or any other letter. Here is a sentence that tells about two numbers: J is less than M.

We don't know which numbers J and M are, but we can put those numbers in a number family. J is less than M. So J is the small number. M is the bug number.

The big number goes at the end of the arrow. The small number goes close to the big number. Here's how you write it.


Then an example using "more," like "R is more than W," is taught.

After the students are firm on this skill, they are ready to tackle a translation like "J is 18 more/larger than K."

This must be a difficult skill because in CMC they scaffold the instruction by circling the number which tells how much more. Like this:


Eventually the scaffolding is faded.

To translate the problem, the students are instructed to ignore the circled number. This makes the problem identical to one they know how to translate -- "J is larger than K." They know that this can be translated into:


Then they are taught to place the circled number, 18, into the only available spot in the number family. Like this:


Once the students are firm on this skill, they can be given a problem that they know how to solve like "F is 12 more than 56." Now they should be able to translate this to a number family and solve for F.

Then the problem can be made more difficult by specifying two variables and the value of one of the variables, such as "R is 250 more than P. R is 881. What number is P?"

Finally, the students are ready to start solving real word problems like "Fran was 14 years older than Ann. Ann was 13 years old. How many years old was Fran?"

Here's how they are taught how to solve these kinds of problems:


This is a good stopping point. This represents a month worth of instructional time for the lessons and the practice. That's for lower performers, higher performers can probably learn this in about a week. Bear in mind that the students are learning and practicing about 10 other strands of material while all this is going on.

The value of this problem solving technique (like Singapore Math's bar graphs) is twofold. First, it reduces solving math problems to a systematic process; this will clarify the student's thought process. Second, the use of the written number families frees up the novice student's working memory which is taxed heavily in solving word problems. Given enough practice, these skills will become automatic for the student and lodged in long term memory. When this occurs, the burden on the student's working memory is becomes much less and the need for the number family prompt is diminished.

Teaser for next lesson: The students learn how to solve word problems like "Jerry weighed 72 pounds. Terry weighed 94 pounds. How much heavier is Terry than Jerry?" To solve problems like this, the students are taught the concept of moving forward and backward along the number family line.

teaching problem solving to second graders

Jill ran 2/3 of a mile farther than Steve. If Steve ran 7/3 miles, how far did Jill run?

If the NAEP is any indication, this is a simple problem that many students can't reliably solve by the 11th grade. Which is a real shame because if a student can't solve a simple problem like this, he can't do basic algebra. The student's math education has effectively come to an end.

The biggest stumbling block is translating the word problem into a mathematical expression. (Calculators are no assistance here.) This kind of mathematical reasoning eludes many students. Fortunately, it can be systematically taught. For example, in Singapore Math this skill is taught using bar graphs starting in third grade. A fair amount of digital ink has been spilled on bar graphs on KTM, so I'm going to show you a diferent way of teaching problem solving.

I'm going to show you how the technique is taught in Connecting Math Concepts (CMC) beginning in the second grade. By the end of the second grade, students should be able to solve a problem, like the one above, correctly at a high rate. Problem solving is taught the entire 2nd grade year in CMC, so it's going to take quite a few posts to cover it all. So let's intoduce the technique in this post and I'll periodically write new posts until we've covered it all.

In CMC, simple problem solving is taught via the concept of number families. Here's a number family:



Beginning in this first grade, the student is taught that number families show three numbers that always go together in addition and subtraction facts. In the example, the three numbers in the family are 2, 3 and 5. You can derive four problems from each number family, two addition and one subtraction:

• 2 + 3 = 5
• 3 + 2 = 5
• 5 - 3=2
• 5 - 2 = 3
  

The students are taught that the "big number" always goes at the end of the arrow and the "small numbers" always go above the line. (In DI courses skills are taught using terminology that the students are familiar with. Proper math terminology like subtrahend, minuend, and addend aren't taught, if at all, until after the students are firm on the skill. This reduces the liklihood of student confusion.)

Next the student is taught how to derive the addition and subtraction problems from the number families. Here's an example of each:



An addition problem can be written for each family that has a missing big number, like the bottom family in the picture. Students are taught that if the big number is missing, they are to write an addition problem that ends with the "how many" box (4 + 19 = []). For subtraction problems, students are taught that if one of the small numbers is missing, they are to write the big number first and subtract the small number from it to find the missing number (57 - 12 = []).

Once the students are firm on this skill, they are given some math puzzles to solve. For example, the students are directed to complete the number family, write the addition or subtraction problem, and the answer to the following set of facts: The big number is a box, the first small number is 38, and the second small number is 39.

the student should be able to derive the problem: 38 + 39 = 77.

Now the student is ready to learn about the concept of variables.

The student is told that sometimes a "letter" is used instead of a box in a number family. The letter works just like a box. It's the missing number.

Here's a problem:

The first small number is 14. The second small number is 56. The big number is P.

The student should be able to construct the proper number family using the skills he's been taught so far.


The student should also know that in order to solve for P, he has to add. The student should also be able to write the correct addition problem 14 + 56 = P and determine that P = 70.

The student is then instructed to cross out the P in the number family and write 70 like this:



This seems like a good enough place to stop for this post. Don't want to overload your second grade heads. This sequence takes about five weeks to go through--the first five weeks of second grade, including practice. I'd estimate that this sequence represents about an hour or two of instruction time and another few hours of guided and independent practice.

In the next post we start to get out of the puzzles and into the good stuff -- real problem solving.

Many of you can probably see where we're going with this already.

Here's a teaser.

A student should be able to set up simple comparison like "A is less than B" or "G is more than H" just by using the number families and rules for placing the "big number" and the "small number." Once that skill is firm it's just a hop, skip, and a jump away from setting up a problem like: J is 5 less than K. Solve for K if J = 3.

(Go to Part 2)

learning math is hard, part deux

(Part one of this post can be found here.)

In the first part of this post on the math program, Connecting Math Concepts, we were discussing about how the program is field tested and how error diagnosing and correction is built into the program. I needed to describe those two aspects of the program briefly to get to the aspect of the program that I intended to discuss -- practice.

Student practice is built right into CMC. That's one of the reasons why the program is field tested beforehand; to determine how much practice students need to retain the material taught. Unlike in most math programs, in CMC material is not just taught, tested and then permitted to lay fallow whereupon it is quickly forgotten by the student. Do you remember the threat of the dreaded end of year cumulative test back in K-12? You dreaded it because you knew that you had forgotten most of the material presented in the first half of the year. You don't have such a luxury in CMC, all tests are cumulative. The only time a skill isn't practiced or tested is because it's been incorporated into a more difficult skill. At the end of the year, students are expected to have retained all the material presented during the year. This is exactly what is needed when learning math.

Since CMC has been field tested with lower performing students and since CMC is designed to accelerate student learning as quickly as possible, you can get an idea for how much practice is needed for a lower performing student to retain the material. The program is designed to provide sufficient practice with a little bit extra to account for things like student absences, but not too much since that would hinder the acceleration. So, the practice provided in the program should turn out to be about what is necessary for a lower performing student to master the material at about the fastest rate he can handle. Cutting to the chase, the amount of practice that a lower performing student requires t o learn math is simply enormous if CMC is an accurate guide.

There is way too much practice for my son. I routinely cut out about every other practice lesson for each topic because I don't want him to get bored and our time for lessons is limited afterschool. Plus, I want to keep the ball rolling and stay far ahead of the wildly inappropriate nonsense that gets taught in his Everyday Math class.

So, you might be thinking that I'm only cutting out about half the material. Nope. I'm cutting out far more than that. I'm cutting out all the "extra practice" lessons that are scheduled for students after they fail a proficiency test. Since he's never failed any portion of any test so far, I haven't had to go back and reteach any lesson. At most he'll get a few problems wrong due to his desire not to being math work at night when he could be playing Lego Star Wars II on his PS2, but so far he's always stayed in the proficiency range no matter how fast I go.

In addition, I've never given him any worksheets from the extra practice workbooks or the blackline master worksheets. And, i skip all the games that sneak in more extra practice since there's no one to play against since he's the only student. Occasionaly, I'll play against him to give him an idea how fast I can work the problems so he has any idea how fast he's going to be expected to work the problems. He's not as fast as I am yet, but he routinely does his problems in half the time allotted in the timed exercises. So, he's starting to approach automaticity on some of the stuff he's learned so far.

Lastly, I've been known to skip the last 30 or lessons at the end of the year since most of this material will be quickly reviewed at the beginning at the next level.

I'd estimate that I cut out about 2/3 to 3/4 of the total practice provided in CMC which accords pretty closely with Engelmann's estimation that higher performers can be accelerated at about 3-4 times the rate of lower performers. And, it doesn't surprise me at all that lower performers need every last bit of all that practice I'm cutting out. Math is all about learning abstract concepts and our brains are not wired to learn abstract concepts easily. It also doesn't surprise me that in most math program, with the exception of Saxon, lower performing kids aren't getting close to the amount of practice they need to retain the math they've been taught.

Hence the widespread failure we see in math education.

And, this assessment doesn't even get into the messy area of the initial presentation of the material enabling the student to understand the concepts in the first place. I'll cover that aspect of CMC in future posts since we're now just starting to get into the interesting areas of math instruction. I'll leave you with this. CMC presents the material so clearly and concisely that I only have to "teach" for about five minutes each lesson. The rest of the time he's working problems using the skills I just taught or practicing previously taught skills. Ironically, that's probably far less teaching that goes on in your typical discovery learning/constructivist heavy math class. I'll show you who's the real guide on the side.

learning math is hard

Update: For more one error detection and correction, take a look at this video (quicktime) starting at about 5:30. He's talks about error correction with reference to reading instruction. It continues on into this clip up until about 7:00 and math gets discussed for the last 3 minutes or so.

That's my current position based on teaching my six year old son math for the past year and a half.

Actually, that observation isn't based on my son having difficulty learning math. So far he hasn't. It's based on the the material we've skipped. It is that differential that separates the higher preforming math students from the lower performing math students. That differential represents an enormous amount of practice.

Unlike most parents who use Saxon to teach math, I'm using Connecting Math Concepts. Both programs are scripted, both use a mastery learning "basic skills" approach, and both have lots of practice built into the program. Both are complete programs which don't require parents to know how to teach math; knowing elementary math is sufficient. For most kids there is not much difference between the two. Contrast this with Singapore Math which does require some teaching skill to present and requires practice to be supplemented. That's not meant to be a knock against Singapore Math, each program has its strengths and weaknesses. I actually think that the ideal K-6 elementary math curriculum would be some combination of all three programs, capitalizing on the strengths of each.

For the purposes of this post, however, I want to focus on the practice aspect of learning math. To master elementary math a student needs to practice what's been learned until it is automatic. Unfortunately, most math programs do not provide sufficient practice to safeguard against the ravages of forgetfulness.

Most parents do not take control of the educational process until there the need to remediate becomes evident. At this point, there is a tension between the need to devote time for practice and the need to reteach the child to get him back on track as quickly as possible. Practice tends to get the short end of the stick at this point. It shouldn't.

One aspect I like about CMC is that it's been field tested so you can be certain that if the student has the math skills to enter a level of the program, the program will teach clearly enough and provide enough practice for the student to reliably master all the material presented in this level within one school year, about 120 lessons.

The most important aspect of CMC, however, is that error diagnosing and correcting are built right into the program, unlike almost every other math program. Let's face it, if students didn't make any errors while learning math, a trained monkey could teach math using almost any commercially available math program. It is in the diagnosing and correcting of student errors where most math programs fail. When students derail, many teachers are unable to get them back on the track. Math, being brutally cumulative is not forgiving at all when students derail.

This is CMC's greatest strength.

CMC is designed to minimize students errors in the first place by providing clear instruction in small instructional steps. Students are then tested frequently (workbooks are checked after every lesson and tests are given every two weeks) to check student errors. based on the ten unit tests, student errors are evaluated and a built-in remedy is provided to the student based on the errors the student made. The student is then retested to see if the remedy worked before the student is permitted to advance. If the student were permitted to advance without mastering the material, then the diagnosing and correction of errors would be become much more difficult come the next ten unit test because now the teacher doesn't know where the student went astray. Was it one of the new skills taught in the past ten lessons of was it one of the previously taught skills? Now extrapolate out 80 more lessons and try to figure out where the problem is for a newly taught skill that the student can't do. Forget about it.

Contrary to popular belief, the greatest shortcoming of the "constructivist" math programs is not the less than clear presentation of new skills, though this is certainly a problem; it is that error detection becomes virtually impossible. This is not so much a problem in a class full of higher performers, but it is deadly in a class where students make errors.

I see this post is getting a bit longish and I still haven't touched on the main point -- practice. So, I'm going to break it up into two posts since there's already much to chew on in this post. More to come.

Part two here.

Christmas day 2006

Christopher, who is 12, wanted to get me a Christmas present of his own choosing this year. We both agreed that the best bet would be a book, so Ed took him to Barnes and Noble, where he found just the thing.

"It will make you laugh," he kept saying. "It's funny. It will be funny to you."

He was right.