This was prompted by a discussion at the Ambleside Year 0 group, where Phyllis and Kathy were talking about MEP, the free math program, being a spiral, and Charlotte Mason's defense of the use of a mastery program for teaching math. Phyllis suggested she'd like to at one point study CM's sayings on this, and we also have to be cautious because her idea of mastery may or may not be in total harmony with that understood today by the term. She points to MEP being a CM friendly program in the use of manipulatives and the way lessons are presented.

I'm going to write about mastery versus spiral from Engelmann's point of view, and present the argument that a homeschooling setting is ideal to make our teachings guided by mastery of skills, (even if we are using a program originally designed as a spiral). When we use a program and resources as an aid and are not constrained to follow anything by the letter, such as in a classroom setting, and when combined with a precise knowledge of our children skills (if wholesome or still being formed), this will help us avoid skipping too fast from lesson to lesson without the children fully understanding and mastering of the skills previously presented. Knowing this difference between mastery and spiral will also help us when buying programs or when putting together resources.

I don't intend to give a thorough analysis of CM at this time, I may make some conjectures but take them with pins and feel free to rebuke or add information that may prove or disprove my reflections on the matter.

Allow me to start by quoting Engelmann:

When students are taught to mastery, they become smarter, acquire information faster, and develop efficient strategies for learning. Teachers must have an understanding of what mastery is and how to achieve it in their students. However, teachers cannot teach to mastery without referencing the performance of their students. In addition, teachers cannot teach to mastery without a program design that supports the approach.We have to remember he is writing to a teacher/students audience. In a classroom setting, the program the teachers are mandated to follow makes all the impact. We, at home, are free to implement what we have in the way that best fits our family needs and our principles.

Englemann proceeds to give us five reasons for the benefits of a mastery program. Then he exposes the deficiency of the spiral as follows:

Most programs do not require teaching to mastery. Teaching to mastery is a foreign practice to many experienced teachers because most programs do not require mastery. Instead of providing continuous skill development, these programs present topical or thematic units. Students will work on a particular unit for a few days and then it will be replaced by another unit that is not closely related to the first and that does not require application of the same skills and knowledge. This design, referred to as a “spiral curriculum,” is more comfortable for the program designers, teacher, and students; however, it is inferior for teaching skills and knowledge.

It is comfortable for the designers because the design does not have to be careful. The designers do not have to document that everything that is presented is “teachable”; the amount of new learning does not have to be carefully measured. The amount of time required for a “lesson” does not have to correspond precisely to a period, because the design assumes that different teachers will take different amounts of time to get through a particular “lesson” and “unit.” The amount of new material is not controlled. The expectations for student performance is low because teachers understand that students will not actually master the material. They will simply be exposed.

If the systematic program is like a stairway, the spiral curriculum is like a series of random platforms suspended on different levels. Students are mysteriously transported from one platform to another, where they remain for a few days as they are exposed to information that is not greatly prioritized. Mastery is impractical with a spiral curriculum design because many students lack the background knowledge they need to stand on a particular “platform.” The poor design relieves the program designer of assuring that earlier-taught skills and knowledge are mastered and used. The poor design also relieves students of the responsibility of learning to mastery and it relieves the teacher of teaching to mastery. It therefore promotes poor teaching and poor learning.Reading the first paragraph of this quote, one can easily infer that if this is the premise of a spiral, it wouldn't have been Miss Mason's choice. She criticized the practice of some teachers lecturing about inconexed topics, and called that 'the art of fragmentation', and a spiral program can easily lead to lessons that fit that description.

In summary, a program that teaches to mastery is like a stairway. Mastery is the guarantee that students are able to reach each stair without falling.

I believe that homeschooling parents have a way with knowing if the children grasped a concept, skill, or if they haven't mastered it yet. We don't have to move on, we don't have to present results to the principal, district, etc. We don't have to follow the approved program to teach any given subject. We can, in other words, benefit from any materials, free programs, or what we have and base our learning experience in the mastery of skills. We also know when we have to wait for more maturity, and we are constantly revising and being sure the foundation is being built.

That's what I like about living math more and more, we don't need to follow and finish any text book. Many programs repeat the lessons across the different 'grade levels', many don't present them for enough time, and there are many key mathematical points such as place value that need to be mastered to move on to other aspects such as multiplication, etc. If we provide the children with a rich mathematical experience they will be climbing the mastery ladder with confidence.

## 18 opinion(s):

Please, more on mastery. More examples of each method, and specific curriculum which employ them. My curiosity is peaked, but I am lacking in understanding.

Very good post, thankyou for sharing :)

Give me some time, Miss Pam, and I promise more. I'm glad you like this topic, I'm enjoying researching it too.

Fascinating. I'll be back to chat about this, but haven't time this week. Stay tuned!!

From one who actually has quite a strong opinion on this...

I take you on your word, Jeanne.

That was so helpful! I have been using Saxon for the past two years. I like it less everyday. After reading this I understand why it has not "worked" for my oldest daughter. I have to agree with Pam. I am convinced to switch to a mastery based program. Where do I find that? Writing my own sounds so scary. I worry that I will create a "gap" or that I will not make things hard enough. Thanks for writing about this. Really, that was so helpful.

I am waiting patiently for more on this topic. I just went off on a web search to try to distinguish which programs are considered what...but they don't tell you that on many of the programs. They all mention something about 'mastery'.

I will be patient :)

Obviously we want mastery, otherwise what's the point in teaching something? But I do think that if something is not mastered, and it is obvious that it's becoming a problem, then it may be necessary to step back and revisit the concept later when the child is better able to handle it (either emotionally or mentally). So I still feel spiral learning has a big place in instruction. But mastery learning is quite important for math where concepts do build on one another. Other subjects maybe are more suited for jumping around.

Silvia,

Great post! It addresses one of my "pet peaves" (spiral based math and language arts.) A spiral review can be a good thing, though, as can a spiral based history program (these ususally spiral over the course of years, not weeks, and do so to maintain developmental appropriateness. Each new spiral is more in depth than the last.)

I talk about it quite bit in my book.

Blessings,

Susan www.susanlemons.wordpress.com

Hey there Silvia!

Thanks for this thoughtful post. I'm in the middle on this one. I'm definitely for teaching for mastery! How mastery is attained is another matter :) I think sometimes some spiraling is necessary, as some others have already mentioned for a number of reasons. We loved MUS (mastery) for several years and are now LOVING MEP (spiral mastery of concepts). Both have been good. I agree with Kris on the Living Math list who said, "...given the definitions as presented, MEP is a program of mastery yet is not a Mastery Program".

amy in peru

p.s. I LOVE the pictures of your girls in the flowers!! perfectly precious!!

Here is a reprint of what I posted on the Living Math forum in response:

Interesting post, Sylvia. I suggest that given the definitions as presented, MEP is a program of mastery yet is not a Mastery Program.

Reading Liping Ma's book will help people understand what I mean. Ma makes the case, IMHO, that US teachers teach mastery of an algorithm, or the mechanics of doing problems without mastery of the mathematical concepts, while teachers in Asia have a better understanding of mathematical concepts and teach them to their students. IMHO, this was achieved by mastering concepts on a low level (1 or 2 digit numbers, using 10s, etc.) before advancing kids to greater complexity, like multiplying 37946 x 897.

My oldest is in Y4 of MEP; we switched from Math U See last year. He has not done any large or complex multiplications. He continues to master the regrouping and distributive property, though he does not call them that, by breaking larger numbers into smaller numbers and then multiplying or dividing those. He learned division through place value and no longer gets confused by "bringing down the next number." He can multiply and divide large numbers because he figured it out based on his mastery of small numbers. That is the genius of MEP.

Compare this to Math U See. By the end of Gamma my son was multiplying these large numbers because he learned mechanically how to do it, but he didn't really understand what he was doing or why. Englemann calls that Mastery. I call that Mastery of the Algorithms, while I call MEP Mastery of the Concepts. This difference is why Liping Ma got the results she did.

MEP is a true spiral program because it does not put forth a concept to cover until mastery and then move on to the next concept. But there are, what shall I call them, pseudo-spiral programs that are basically mastery programs in that they move from topic-to-topic only do so in shorter bursts. In fact, I think what we call spiral programs are really these mini-mastery programs, and are like Englemann's floating stairs. I can see why Miss Mason did not care for them. MEP then is something different entirely.

Kris, THANKS for adding your whole comment, it'll help others understand more about it. I've commented more in the second part.

Silvia, I've just now jumped into this discussion and am really enjoying it. We've been using MEP, but one child seems to "take to it" better than the other, and I've been wondering about switching to more of a "mastery" program for the latter dd. Since MPE mixes things up, I've found that she will forget, from one day to the next sometimes, how to do something because there were many other concepts taught in the previous day's lesson. Make sense?

Great job!

Ellen

Silvia, I linked your series on my post about Cookies and Fractions which is more about fun (my post) but I thought on a serious note, regarding mastery, your series is very helpful... =) It's late, not sure if this comment makes sense. lol

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