Mastery versus Spiral (Part II)

This is a long post, the second part to this, so I'm adding some pictures to easy your way into it ;-) Several points I'd like to make:

1) I want to take you to Heather's post about MEP being Charlotte Mason friendly. She is using level 1 and reception and gives us her reasons to state that MEP supports Charlotte Mason.

2) For the use of MEP in older grades Jeanne has posts about how she does that with her daughter in level 3, if I'm not mistaken. Jeanne also promised us a post on this and HERE YOU CAN READ her extensive and as always delightfully written Aussie post.

3)Kris from the Livingmath yahoo forum (free to join) writes about how US teachers teach 'mastery of a drill or a how to' (such as multiplication),  I believe when we speak about mastery we generally allude to the mastery of ALL what that learning encompasses, for example in multiplication it will be not just to 'know' how to multiply, but to have experienced what multiplication involves, and know why we proceed how we proceed when we multiply two digits, which has the concept of place value at its core and the understanding of what digits are and represent. She added her whole comment in the Part I of Mastery versus Spiral, and has elaborated more on this in HER BLOG.



She refers to Liping Ma's book I've also read and written about, and found exceptional to point these differences between teaching a concept and teaching just a process. Chinese teachers refer to mathematical learning as packages of knowledge, they'll make a graph with a concept in the middle box and other boxes branching out. This will be a MASTERY perception of math, where things are connected and related, versus the floating platforms described by Englemann.

Kris says that teachers in the US would teach for example mastery of an algorithm, while Chinese teach the mastery of a concept. She then says how MEP is aligned with this way of teaching the concept and mastering it, and that will help you through other lessons. For example, if a student learns why you multiply two digits the way we do for all possible cases, the Chinese teachers would tell you the students should and would figure out by themselves how to do it with three, four, and ten digits. If you just learned how to proceed with a two digit multiplication and did drills, a few months after you may very well make the mistake of forgetting how to align the results correctly because you just memorized that 'rule to multiply', without mastering the principles of multiplication, and the mistake won't jump in front of your eyes.

This is why teachers in China stay on the important lessons for a LONG time and devote an entire 45 minute class to ONE PROBLEM. And also why they don't care about the right result as much as the questions, the explanations, the findings, the relationships, the connections, the examples, applications, contradictions, advantages. They write something on the wall and ask the students: is this a sick or a healthy 'patient'? Meaning if they are looking at a right mathematical statement or at something faulty. They play DOCTOR and try to find what's wrong and how to cure it. And as in real life, there are more than one solution, and they have to discuss why some are different and which can be more adequate.

As to what curriculum endorses mastery or spiral I can't say much without getting into trouble, I believe, but we have to remember that it's also the WAY we ultimately use anything we have, if we are catering for MASTERY or just flying over different clusters of knowledge. However it's possible to categorize some programs as more conducive to mastery than others. I'll venture to say that after reading Heather and Kris' comments, MEP is a good starting point. Amy is using it too, and has used Math-U-See which she describes as a mastery program also. I also like Jimmie's talk about how she teaches math in her TRANSITIONING LIVING MATH to Sprite. She had started with Singapore but that alone wasn't working and now she does the Singapore practice after she uses other resources and means to teach the concept.

The following programs I've just heard about being fine and I'm sure that you either use them, have, or are will at some point: Math-U-See, Right Start Math, and I hear great raves about Life of Fred. Mathmammoth is also a hs-ers favorite. Do not forget these are programs, I believe a true MASTERY approach would require more than using one of these.

And Carol (who commented in my first post), don't be afraid to move on to something more than SAXON. There won't be 'gaps or lack of challenge', you don't have to invent the wheel either, just leave what you are doing for a while and experience and experiment with other type of books for math. After all, there aren't that many crucial math concepts, it may look like lots of things to be learned, but they steam from good understanding of some basics and then you take those to different levels of complexity that will not make sense if those basics aren't well absorbed. From LIVINGMATH I took the suggestion of reading books by Marilyn Burns, and she has great books with lesson ideas, and literature based math books. Mathemagic, the 13th volume of Childcraft Encyclopedia is also one of my favorites.

4) Susan Lemons, in her resourceful book Hompreschool and Beyond, has a whole chapter about curriculum and MASTERY and SPIRAL programs. I will not reveal all her ideas, it won't be ethical, but she goes to the heart of the matter when she says that the three R's are mastery subjects, (Reading, Writing, Arithmetic), and they should be taught to mastery and using mastery programs. She then defends the teaching of phonics (in line with Charlotte Mason), but also like her, preferably not earlier than six years of age. As for reading she sees it though as much more than reading and has extensive and helpful lists of books, programs, games, activities, etc, on how to teach specially young children. Her preference for math is Rod and Staff for math. Science, History, and Geography are what she calls exposure subjects, and as Jimmie, she says that a spiral program can be good for these. I believe the classic education method as presented by Susan Wise Bauer advocates the spiral introduction of History in cycles, children then will be learning about Egypt three times in their life, at different depth and levels.



5) Back to Englemann
when he talks about MASTERY and SPIRAL, he is framing this to a classroom environment. He says the dangers of teachers who don't focus on the students mastering, in part because they don't know their subject themselves (which Liping Ma's book spoke about. Statistics say -and I heard that from a mom not long ago- that in the USA parents are most concern with teachers CARING about students, while in CHINA they are concerned about teachers KNOWING, MASTERING what they teach). In return, from Englemann's paper I read that he criticizes how in our US schools teachers give students tests and when the results are poor they always put the blame on the student (try to test for learning disorders, or claim the students did not learn what was taught), he sharply points to this fact that I find provocative and true, if students 'fail', teachers 'fail'. He talks about how a teacher was presenting a lesson about Sweden and while he states that it would have been easy to teach it to MASTERY, the teacher was just wasting the opportunities of delivering a great lesson by asking these questions:

“It says that Sweden is a country in Europe. Do you live in that country? . . . What’s the name of the country you live in? . . . Can you find Sweden on the globe? . . . Can you show me where Europeis on the globe? . . .
Instead, Englemann proposes this: 
It would not have taken the teacher more than five minutes to teach students to mastery on all the information they would have needed to fit the worksheet material into the framework of knowledge they already possessed. They had read about Herman the fly, who flew around the world, landing in Italy. Students were able to locate Italy on the map. This is a good reference point for going north to Sweden. Once they saw Sweden on the globe and saw its distance from Italy and from the US, they would have had a good schema of its size and its relation to places they already knew.
Englemann has a couple of beautiful quotes, what mastery (and direct instruction -which I contend it's what many homeschooling parents do, that others call the tutorial method), achieves in the child.

Teaching to mastery also instills self-confidence in students because they learn they are capable of learning whatever new skills or material the teacher presents.
Students who are immersed in mastery, in all subjects for at least three years, will become much smarter than comparable students taught in a traditional manner. Mastery-taught students will not only know more—these students will be far more proficient and faster at learning new academic material of any kind
Nancy pointed as prevalent in some schools and even homes and what Englemann denounces? It may not be like this everywhere, but I was THAT teacher, I made THOSE questions, and I believe I was part of a system where spiral and traditional (described by Englemann as non teaching to mastery) teaching prevailed. As for my homeschool, I'd try to do things differently.

7) Curriculum. What programs are mastery oriented, which are the bad ones.ones who don't focus that much on mastery. (One moment, where is number six? There is no number six, I just wanted to see you were 'paying attention' ;-) That's your assignment Blossom (not really, Blossom is a fantastic mom and she is back to school, so she has enough on her plate, so I'll pass this to any who wants to tell us what they think. But remember that curriculum or programs are your slaves, and not your MASTER (no pun intended).

* All the quotes by Englemann are from his writings.

* NOTE: If I have misrepresented your thoughts, or made inaccurate conclusions, please let me know and I'll make the needed corrections.



21 opinion(s):

Silvia said...

Sorry guys, Amy posted this before I replied and I deleted by accident:
"but we have to remember that it's also the WAY we ultimately use anything we have, if we are catering for MASTERY or just flying over different clusters of knowledge"...

I very much agree :)

*just a note, I think that MUS claims to be a mastery program... since they spend an entire year with a focus and emphasize the fact that you should let your child master the concept before moving on, regardless of the pace.
Alpha: single digit addition/subtraction
Beta: multiple digit addition/subtraction
Gamma: Single and multiple-digit multiplication
Delta: Single and multiple-digit division
etc.
However, even though these subjects are taken over an entire school year, there are other subjects sprinkled throughout so that some might call it spiral as well...

oh boy. confusion sets in.

MEP though it is more obviously spiral as it does move from concept to concept throughout the lesson (we are using y4, y1 and I've looked over Reception as well), it turns out students who really have a more thorough mastery of concepts than any other math I've ever used. LOVE it. :) I think the higher mastery levels may have to do with the fact that it is rather teacher intensive and that the teacher works VERY closely with the student going over as much practice as necessary in order for the student to grasp the concept. I highly recommended adding more manipulatives even in the upper years. The younger levels recommend using LOTS of manipulatives. It is a very positive approach as well encouraging the teacher to lather on the praise.

SO... all that to say, I only have experience in using these curriculums, but am rather confused/ignorant as far as classifying everything spiral/mastery...

I very much agree with you that you can adjust a good curriculum to suit your own needs and that they exist to serve us :) you've written a great summary!

amy in peru

April 9, 2010 4:46 PM

Amy is in my blogroll, http://fisheracademy.blogspot.com/

Silvia said...

Thanks Amy for your well written and thought comment. I'm with you, it starts getting confusing and I believe this happens because of different programs and things we are applying the word mastery or the concept of spiral. I think it could be a college paper truly, to define and study these terms across different subjects and what it means for different individuals.
Not everything that moves you across different lessons is a spiral, not everything that just presents a few concepts at a time is necessarily mastery oriented, and then US, right? the crucial part of the equation.
However, when many coincide in the completeness and thoroughness of a program, such as MEP, I believe we have good reasons to believe it's good. I also see that what you describe for MPE and the use of manipulatives is in agreement with what I read the Chinese do for math. We LOVE the word manipulatives, but teachers kind of throw them to the kids like peanuts to the elephants, pick them up, move on with the spiral, while in China the manipulatives are the CONCRETE and support for the students to present their explanations, they are ASKED why they used that in that way etc, and not just use unifix cubes and tiles to make a pretty picture (which by the way, it's what my 5 year old did yesterday ;-). That to me, the way you see or use manipulatives is another example of that thought of using something to achieve mastery or in a superficial way to just cover the objectives and look good when the principal comes to observe your class and you can make the claim you are teaching with manipulatives (and Susan Lemons eloquently says in her book that smart moms call them 'stuff' LOL)

Jeanne said...

My post is up here, Silvia.

http://ohpeacefulday.blogspot.com/2010/04/are-mep-and-cm-compatible.html

You have done an incredible amount of work in preparing these two posts, thank you.

Jeanne said...

Thanks for the link!

PS Don't you think our posts are beginning to spiral as we achieve mastery over this topic?!! Hee Hee

wonderinthewoods said...

Very interesting, Silvia. I have these posts book-marked for future reference. In math we are still working on learning to count to 100 (without missing a number) and skip counting. We play math war and bingo and use simple math in real life but my focus with math instruction has been with counting. I want him to have this down before we move on to other things. Perhaps this is mastery? I have Math-u-See Alpha and need to pull that out soon. I so appreciate posts like this, so I can learn too. =)

Pam said...

Very thorough post. Thank you so much Silvia.

I think our learning styles as home school teachers play into this also. I teach in a large group of diverse learners, at all levels. I know where I lack: namely details, follow through, and inability to cover as much one on one as I should. That means I will likely miss "complete mastery" in many areas. Knowing this, I must be ever watchful, always keying into what the kids are telling me, and how they are grasping/or not grasping a thing. *Narration. What they do not grasp reflects poorly on me. That place is my opportunity and responsibility to clarify things for them. It's important to know where we lack, and try to give ourselves accountability. Hey, thanks again.

Jimmie said...

Our math instruction is continually morphing. We've started using Life of Fred books now. We had done quite a bit of work with fractions, and I felt that Sprite had mastered it fairly well (on her level at least, surely not for all time). I was ready to move onto decimals. But instead I gave her the option to begin the LOF fractions. She chose to use LOF. So we're digging in to fractions MORE rather than bouncing off onto another topic.

I admit, I was feeling the pressure to "do decimals" because that's even in the 4th grade Singapore text! (She's 5th grade.) We've NEVER studied them. But who cares? My bottom line belief is that as long as instruction is continually moving forward, it will all work out in the end. So what if we spend an entire year on fractions? We'll get to the decimals eventually.

Silvia said...

Jimmie, you are right, but I UNDERSTAND that pressure. If she has a true and deep understanding of fractions (a big concept that merits lots of time and practice) she'll do good with decimals, not just "covered, check". Let us know ho your math continues looking.

Kris said...

I have so many things to address in this wonderful post I don't know if I should put it all into one long one or several topic-based shorter ones. I think the latter because I may not get to write it all now :o)

My fist comment is regarding whether MEP is a spiral program or a mastery program. My answer is: both! I currently have children in Y1, Y3, and Y4; the older two have each completed the entire previous year. What MEP does is it introduce a big subject, like multiplication and division (started together) at its simplest level, 1 digit numbers. Once that is shown and kids are doing it, MEP then presents a wide variety of different types of math concepts to which multiplication and division is used--tables, graphs, word problems, etc. In doing so, the program also slowly advances the difficulty of the multiplication and division, but never beyond small, easy-to-grasp problems.

I blogged about this on my homeschool blog
here.

By this construct, MEP is teaching mastery of multiplication and division--from early in Y3 to where we are so far in year 4. It did the same thing with addition and subtraction in Y1 through the early part of Y2. Yet MEP has a spiral construct in all the other concepts to which basic arithmetic is applied. Geometry, measurement, and fractions are all scattered throughout Y2 through Y4 (in fact during the same weeks in Y3 and Y4 as I am learning with my two older sons!)

How this will work with fractions I am unsure. I know Ds#1 is ready to learn more about fractions than he does, yet because we switched to MEP later in his school years, he is only on Y4 though in 5th grade while the younger two are in MEP years that match their grades (Ds#2 being about a half year "behind" in that sense.)

I am not one to worry about keeping up with the public school kids (more about that in a subsequent post about Englemann.) I think it will be revealing as to the real strength of MEP as we do learn more about fractions, decimals, and percentages.

Blessings, Kris
At Home Science
A Private Eye Nature
Science of Relations

Amy in Peru said...

yay. I love the discussion...

@Silvia, I just came over to make sure you got Jeanne's link...

@Jimmie, I think Sprite's learning decimals by doing fractions!! I bet that it will transfer over naturally! Afterall decimals are just fractions of 10s...

@Kris, THANK YOU!!

I'm so glad I came back by...
amy in peru

Kris said...

And now about Englemann...

Again, I can't disagree with him in terms of mastery programs compared to spiral programs, but I disagree with him on the bigger picture.

For example, he states when a child does not learn, that reflects badly on a teacher. Yes, but then, in his Sweden example, he basically is saying, "If you just teach it like this then they will understand." Well, maybe not, and that's the point. A teacher has to know a topic well enough to explain it in a variety of ways to reach a variety of students. I think that is what US parents mean by having a teacher care about their children, while Englemann advocate a master teacher.

Yet central to Charlotte Mason's homeschool philosophy is that education is the science of relations. Every child must form a relationship with some knowledge in order to learn it. Compare that with cramming for a test, or even memorizing a method. Englemann's "mastery of the method" is his definition of learning.

He is right, though, especially in math, that by knowing these methods hard a fast a child will do better in school--but that's because school assessments are designed to reward those who do! And that is why, according to Marilyn Burns, vast numbers of people have Math Phobia. For some people memorizing the algorithm simply does not work, and Englemann's answer seems to be, "Well, they just didn't master it."

This leads to the phenomenon of what I very irreverently call Stupid Human Tricks. This type of mastery creates intelligent-appearing children because they can perform all the wonderful 4 and 5 digit multiplications and divisions, spit back science and history facts, and diagram sentences with ease. They even get rewarded on tests for doing so. Yet they do not forge any relationship with the material, and so they will retain but a fraction of what they so brilliantly regurgitate. Ultimately, is that really education? Do they understand mathematical concepts? (Most do not.) Can they discuss the impact of historical event? (Most cannot.) Can they write well? (Not generally.)

Yet the science of relations should not be an excuse to avoid challenging our children, as if our child's enjoyment is the sole indicator of them forming relations. On the one hand, Miss Mason states that we should not interfere with a child's narration for they will glean what is important to them on their own terms (which is why choosing content reflecting my values is the first priority in selecting books and media.) On the other hand, she wrote the facts of a passage, say from a history text, on the board to which the children could refer while narrating. I guess there's no getting over mastering those facts, yet that should be the servant of forging a greater understanding of knowledge and not its master. In short, teach our children to think and not just to know.

Blessings, Kris
At Home Science
A Private Eye Nature
Science of Relations

Silvia said...

Kris, I'm going to definitely cut and paste your last comment, and use it to initiate another post. What a wonderful debate! Thanks for bringing up so many interesting thoughts.

Suzuki Mom said...

Hey, thanks for the link to my blog. I think I understand the concepts of Spiral and Mastery better than I had.... and I like the comment that MEP is a program of concept mastery - and that it is a Spiral-Mastery program. It really doesn't fit that much into either catagory.

Heather

Silvia said...

Your welcome Heather, thanks to you for writing so well about MEP in your homeschool.

Ellen said...

Silvia, another great discussion! I particularly loved reading everyone's comments. Jimmie's about "instruction continually moving forward" hit home for me. And I also have enjoyed the postings regarding MEP, since that's what we use but I've been questioning it for one of my dc. I'm not a math person, although I understand more than I did when I was their age! However, my dad (math and physics double major, physics master's, career as a nuclear health physicist) firmly maintains that my problem was the way schools taught math when I was growing up. So when I see dd10 struggling, I automatically panic. I SO want her experience to be better than mine!

Silvia said...

Ellen, I strongly advice you to look for other programs (remember, your slaves) and ways to teach math to your dd who isn't doing that great with MEP. At the end of the day, mastery or spiral are secondary to 'what is your girl benefiting from', and anything that causes bad experiences in an otherwise willing to learn child has to be changed.
Jimmie says how that was happening to Sprite with Singapore math, and she changed to a different approach, where they do other things (read books on math, make visuals cutting, etc. for fractions) and then, when that is understood, the Singapore practice this time as a success and drill of something already learned.
And as for you, the teacher, don't be anxious about stepping out of a program and leaving the list of lessons to be completed for a while. Anything understood well will transfer (mastery) so something hard in math is worth learning in different ways and depth and it'll make other lessons down the path.

Books that I find very valuable: Family Math and Family Math for young children (this is for 4-8) the other for older, by Grace Davila and Jean Kerr. And A collection of Math lessons from grades 1 through 3 by Marilyn Burns and Bonnie Tank, she, M.Burns, may have other books for older. And from the livingmath.net (which Jeanne says it's an unorthodox approach, which I agree it is and I don't know how many go full way), I take the idea of incorporating literature, many mix that approach with a 'program' of their choice and it seems to work great.

Good luck and let us know what you'll be doing and how it's going.

marion said...

I recently came across your blog and have been reading along. I thought I would leave my first comment. I dont know what to say except that I have enjoyed reading. Nice blog. I will keep visiting this blog very often.

Lucy

http://businesseshome.net

Suzuki Mom said...

Hey, thanks for the link to my blog. I think I understand the concepts of Spiral and Mastery better than I had.... and I like the comment that MEP is a program of concept mastery - and that it is a Spiral-Mastery program. It really doesn't fit that much into either catagory.

Heather

Pam said...

Very thorough post. Thank you so much Silvia.

I think our learning styles as home school teachers play into this also. I teach in a large group of diverse learners, at all levels. I know where I lack: namely details, follow through, and inability to cover as much one on one as I should. That means I will likely miss "complete mastery" in many areas. Knowing this, I must be ever watchful, always keying into what the kids are telling me, and how they are grasping/or not grasping a thing. *Narration. What they do not grasp reflects poorly on me. That place is my opportunity and responsibility to clarify things for them. It's important to know where we lack, and try to give ourselves accountability. Hey, thanks again.

wonderinthewoods said...

Very interesting, Silvia. I have these posts book-marked for future reference. In math we are still working on learning to count to 100 (without missing a number) and skip counting. We play math war and bingo and use simple math in real life but my focus with math instruction has been with counting. I want him to have this down before we move on to other things. Perhaps this is mastery? I have Math-u-See Alpha and need to pull that out soon. I so appreciate posts like this, so I can learn too. =)

Silvia said...

Sorry guys, Amy posted this before I replied and I deleted by accident:
"but we have to remember that it's also the WAY we ultimately use anything we have, if we are catering for MASTERY or just flying over different clusters of knowledge"...

I very much agree :)

*just a note, I think that MUS claims to be a mastery program... since they spend an entire year with a focus and emphasize the fact that you should let your child master the concept before moving on, regardless of the pace.
Alpha: single digit addition/subtraction
Beta: multiple digit addition/subtraction
Gamma: Single and multiple-digit multiplication
Delta: Single and multiple-digit division
etc.
However, even though these subjects are taken over an entire school year, there are other subjects sprinkled throughout so that some might call it spiral as well...

oh boy. confusion sets in.

MEP though it is more obviously spiral as it does move from concept to concept throughout the lesson (we are using y4, y1 and I've looked over Reception as well), it turns out students who really have a more thorough mastery of concepts than any other math I've ever used. LOVE it. :) I think the higher mastery levels may have to do with the fact that it is rather teacher intensive and that the teacher works VERY closely with the student going over as much practice as necessary in order for the student to grasp the concept. I highly recommended adding more manipulatives even in the upper years. The younger levels recommend using LOTS of manipulatives. It is a very positive approach as well encouraging the teacher to lather on the praise.

SO... all that to say, I only have experience in using these curriculums, but am rather confused/ignorant as far as classifying everything spiral/mastery...

I very much agree with you that you can adjust a good curriculum to suit your own needs and that they exist to serve us :) you've written a great summary!

amy in peru

April 9, 2010 4:46 PM

Amy is in my blogroll, http://fisheracademy.blogspot.com/

 

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