Doing Math without a Curriculum

[Hunter]

A few of us were talking about using just the math scope and sequence in the What Your Grader Needs to Know series, and then using nothing but library books, hands on, and handwritten worksheets. I guess that is kinda sorta doing math without a curriculum.

 

Does anyone do math without a curriculum?

[Farrar]

We did that for kindergarten.  And then when one of my ds was having a lot of school anxiety, we did that for a little while.

 

But I don't think it's the best option.  Math needs practice.  It takes extra time to print or create practice paper.  Or it takes sitting there creating it with the kid all the time.  When ds was winging it for math, he made close to zero progress.  Some of that was anxiety related, but some of it was just that it was so much more effort for me to be organized with math.  It took a huge amount of time.  And I didn't do it as well as a curricula did.  And when there are so, so many good curricula out there for elementary math, with so many varied presentation styles...  Well, why do it?

 

Since there are so many good resources available, many of them pretty inexpensive or even free like MEP, I don't really get why anyone should sink the amount of time into this that it would take.  I see a payoff when I sink more time planning and customizing into subjects like writing, history, science, geography, etc. because then I have greater ownership of the subject material and the lesson.  But with math, a huge percentage of the work is not setting up the lesson - that I do with a curriculum anyway, and tweak and find games and so forth - but it's setting up the practice pages.  I don't want to demean math to being out of a workbook only, but I find it's more key with math than any other subject to have such a workbook and for that paper work to have a useful scope and sequence and a good presentation.  Math is just more slow and steady and needs to be more constant.

[justasque]

I've done something similar.  

I looked at scope and sequence lists and a few state standards.  Then I assessed where my child was in each general topic (fractions, etc.), then made a list of goals for the coming year.  For each topic, I listed skills I wanted to teach, and for each skill I planned the appropriate level /depth of the lesson.  The idea is that each skill will be foreshadowed the first year (generally at the end of that topic's unit), introduced the second year (worked on some, but without expectation of complete mastery-for-life), worked on more deeply the third year (the "meat" of the unit), and reviewed the fourth and perhaps fifth year.  

Once I knew the goals, I pulled resources from all different places.  Hands-on activities, games, worksheets from the web, vintage math texts, etc. etc. etc.

 

Then I made a rough schedule.  To keep it interesting, we often had two topics going at once - doing some fractions and some geometry, for example.

 

Doing it this way meant I could keep the child's interest by including lots of multi-step word problems, problems that involved more than one skill (like a perimeter problem that used decimals or fractions), interesting activities, assorted manipulatives, and so on, rather than rote learning.  It also meant that the day's lesson's objective was to learn the skill, not to complete the worksheet. 

I found that by mid-year, I usually needed to reassess and adjust the curriculum, as often I needed to bump things up a notch by then.

It went very, very well.  I've used the same approach with many of my tutoring students.

ETA:  I did it this way for several reasons.  It worked well- 

~For students who were working at a higher level in math than in their ability to use traditional textbooks.  For example, a young child who could do complex math but had a age-appoppriate hard time physically writing numbers, or a student who needed fourth grade math but could only read at a first grade level.)  

~For students who were at very different levels in different areas of math (so needed, say, more complex addition but less complex fractions).

~For students who wanted or needed to progress at a much faster rate than a typical textbook.  
This included students who were gifted at math and simply didn't need all the repetition built into most texts.  (And who needed/wanted to be challenged with a bunch of more complex problems instead of the less complex ones that are the bulk of many texts.  We could spend the same amount of time for the lesson but go *much* deeper, content-wise.  For these kids, I pulled "challenge" pages/problems from many different sources and put them together to create a unit.)  
And it included unschool-y students who had little to no formal math work/learning in the past, yet at age twelve or so wanted to "catch up" with their peers.  (Otherwise known as "first to fourth/fifth grade in a year", for which a custom curriculum based on an assessment of the child's existing skills and updated frequently is ideal.)

[OneStepAtATime]

If you don't want the extreme structure and step-by-step limitations of many standard math curricula and if you are looking for something more practical application and outside the box, but still providing practice I would recommend Math on the Level.  It is an excellent program for math through 8th grade.  You don't use workbooks.  There are teacher manuals and they have an amazing tracking system for all math concepts that you should know through 8th grade.  You go at the speed and developmental level of your child and interweave the concepts in with each other, using lots of practical application along with practice in daily problems.  It is something a parent might create that did not want a workbook approach and had lots of time to do the research and lay out the lessons and find the resources, but here it is already done for you.  The system shows you how to weave in math into your daily life and still get all the necessary concepts mastered.

There is an overall big picture, along with the checklist and suggestions for when to approach different topics, and several spiral bound teacher manuals to help you through each set of math concepts, along with math games, etc..  Also, so that no concept is lost, there is daily review but it is only 5 problems a day and they show you how to do the review pages.  As long as your child isn't struggling in math because of learning issues prep time for lessons should be about 15 minutes once you get the hang of the system.  It is really unusual but a great program.  Also, it is a one-time purchase and covers all the math you will need from pre-K through 8th grade.  Obviously, if your child is in 7th grade, it probably wouldn't be worth the money, but if they are still in elementary, I think it would be well worth it for anyone seeking something other than the standard workbook approach.

[dbmamaz]

I havent done what you said, looked at scope and sequence and made my own program.  But i've flitted around a lot using supplements instead of curriculum, because my kid hated all curriculum.  Its not been a clear linear progress.  

 

approximate order:

Singapore 2A (we started homeschooling in 1st grade and he was advanced)

Time4Learning

living math books (whatever the library had)

timez attacks and math ninja for facts

Murderous Maths

Random multiplication and long division worksheets from the web

Primary Challenge math

a few math worksheets from Scholastic Teachers Express

Life of Fred Fractions and Decimals/percents

Elements of Mathematics, foundations

 

Then we tried a few gifted middle school programs (Challenge Math and Descartes Cove) and both were too much for him, so we are reviewing with Spectrum 6th grade math, which he is happy with.  I suspect we'll move on to LOF pre-algebra, which i already own, having used them with my teen.

[Hunter]

When I afterschooled/homeschooled my younger son in the 90s, I didn't even have a scope and sequence. It was a different time, I was a different person, he was not like my current students. But I never questioned myself. How did I know what to do? I just did. Or I was too stupid to know I didn't know what I was doing. Testing showed it worked, though, and he went on to Saxon algebra well prepared.

 

I have Math on the Level, but it's a big pile of books, that are all supposed to be used at once. It's not like I can grab one book and head out the door with it.

 

I just can't seem to recreate the ease of teaching math, that I had back in the 90s. It was just no big deal. Now it seems so complicated. How did things get so complicated?

[regentrude]

I have tried that when I started homeschooling, because I thought with my background (I have a PhD in theoretical physics) it would be easy. Far from it!

Creating good problems that illustrate precisely the concept that they are supposed to teach, creating problems that progress in exactly the right pedagogical order, and creating enough of those problems that the student receives adequate practice is an insane amount of work and requires subject expertise far beyond the concept that is being taught.

 

If you use worksheet generators or books or programs, that is all "curriculum".

I doubt anybody is able to teach math beyond the earliest elementary without resorting to such resources. It would be a full-time job just to write one's own math problems.

 

[mathmarm]

A few of us were talking about using just the math scope and sequence in the What Your Grader Needs to Know series, and then using nothing but library books, hands on, and handwritten worksheets. I guess that is kinda sorta doing math without a curriculum.

 

Does anyone do math without a curriculum?

I am not familiar with the What Your Nth Grader Needs to Know series, but it is easy enough to make your own Scope and Sequence of math, Counting - Calculus, if you know what your doing. Even if you don't, you could just look some up online.

 

My mom is a master teacher and an expert tutor. She did something like this, she had a big list of Concepts and Skills for mathematics that went from sorting and counting to writing proofs and calculus. She created, by hand, several note-pages and worksheets that spanned 3 basic 'levels' for almost every topic and she personally diagnosed/placed each student in her own scope and sequence. Being a teacher in the local schools, she was very good at helping kids to catch up, keep-up or prepare them for where they needed to be next year.

 

She taught me, not just how to do mathematics, but how to learn and teach mathematics. I feel perfectly capable of teaching a kid math without any outside resources, just time to create and prepare my own materials ahead of time.

Whether or not I'd want to do that long-term, like my mom, remains to be seen, but I have done it before and I could do it again.

 

Now, however, I'd probably be more likely to use a worksheet generator, rather than hand-write the pages. But I feel that I definitely could do it and seeing as how I haven't found a math program that I like, just yet, I may wind up in the position of creating my own 'foundations' course based on how I like math to be taught and how my child will learn. It'd save me the trouble of spending $$$ and time on math programs that don't work...

[mama27]

For my current 7 and 8 yos I grab random worksheets from MUS, Some workbook I can't think of the name of, games from Rightstart, activities from Queens living math (I'm skipping the stories, they make me want to stab myself in the eye!), and anything else I think they need or that comes up, or that I see they need to work on. Once they are reading on their own, they will have a curric, either MUS (most likely), or Saxon. It will depend on the child.

I've done this before with my 14 dd and she does not seem to have the math issues my 18 yo does.

[mom2bee]

I could easily cover K-3 math with a scope & sequence, games, books and pre-made worksheets. Maybe even up to 5th, but I think it gets tougher and tougher each year from there. I'd be willing to try up to 6th grade though, so long as I have an SnS, worksheets and books on hand if I needed to.

[dbmamaz]

I do think a lot of things have been added in that arent actually necessary.  They want to introduce higher topics by dumbing them down in to rote memorization - which imo adds NOTHING for most kids.  Like khan academy, which i liked for a while, added some probility to elementary age - but only 1 kind of problem, where they clearly expected the kid to memorize the method.  I think it makes more sense to wait until they understand the CONCEPT of probability, when they can understand why you would use one method in one instance and another method in another instance.  I mean, memorizing a method for one specific type of problem (probably if there are 10 blue and 7 black, what is the chance of choosing 1 blue) - how is this helping them be math literate, OR preparing them for higher education?  It makes math meaningless.  IMO

 

Anyways, when I was first looking at curriculum, someone recommended Math Mammoth as the easiest to use.  However, a friend recently said she's hating it - too boring lol

 

IDk - i try not to worry too much.  Mmm, try.

[OneStepAtATime]

I do think a lot of things have been added in that arent actually necessary.  They want to introduce higher topics by dumbing them down in to rote memorization - which imo adds NOTHING for most kids.  Like khan academy, which i liked for a while, added some probility to elementary age - but only 1 kind of problem, where they clearly expected the kid to memorize the method.  I think it makes more sense to wait until they understand the CONCEPT of probability, when they can understand why you would use one method in one instance and another method in another instance.  I mean, memorizing a method for one specific type of problem (probably if there are 10 blue and 7 black, what is the chance of choosing 1 blue) - how is this helping them be math literate, OR preparing them for higher education?  It makes math meaningless.  IMO

 

Anyways, when I was first looking at curriculum, someone recommended Math Mammoth as the easiest to use.  However, a friend recently said she's hating it - too boring lol

 

IDk - i try not to worry too much.  Mmm, try.

I agree, teach concepts appropriate to the age and developmental level of the child, not rote memorization of higher concepts that they don't actually understand.  Really poor way to approach anything...

 

Actually, this is one of the reasons I like Math on the Level, even though it is not as well known.  They give you things to look for in your child to determine if they are ready to go to the next level in a concept.  Not based on arbitrarily assigned grade levels but developmental levels for each child.  Created by a homeschooling family, one of whom has an advanced math degree.

[Hunter]

Mathmarm, thank you. I think I need to create my own notebook of lessons. Some of us were talking about creating a Teacher Commonplace Book, and I need to buckle down and start doing that.

 

OneStepAtATime I think I can sometimes photocopy pages right from Math on the Level. I don't think Math on the Level is not well known. I just think the upfront price is high, and that there are reasons why it doesn't work well for many families.

 

I'm just really struck by the fact that the more I learn about teaching math, the less confidence I have to teach it. There is something SERIOUSLY wrong with that. Humbleness is a good character trait, but lack of confidence doesn't make a better teacher. Sometimes youthful and naive arrogance is more effective than wiser humbleness.

 

With my son, school had been so bad, that when I pulled him from school, we were just two kids running through a buffet, grabbing and tasting everything we saw. We didn't worry about "complete" because we were so busy gorging on what was right in front of our face. And somehow it all worked out. Even math was "Yum yum, this is awesome! Oh wait, before you eat more of that, look at that over there. Let's try that! Let's see if we can do it." If we could do it fine. If we couldn't, we backed up and shored up the weak point. It was just so easy and so fun. I miss that.

[Jean in Newcastle]

Right now I'm letting Khan Academy be the curriculum.    I think she needs more practice than they give and it "promotes" dd to a higher level faster than I think is wise.  So - I just click on the subject again and have her practice it longer.  I sometimes get a gently scolding note that pops up to say that the student has practiced this enough and needs to go to the next level, but I ignore it.  The software doesn't know how much I'm helping dd with the problems at first.  

[Hunter]

I'm not feeling good today. Looking through math curricula is just making me dizzier and more tired. I never felt that way when I was winging it back in the 90s. But I had such a different attitude. It was fun, not a checklist. Whatever DID get done was one less thing that didn't need to get done later. Obviously teaching a gifted child who was years ahead took the pressure off, but I can't let go of how fun that was.

 

Okay, yes, skills need to be taught more systematically than content. But have we taken it too far? Have we squeezed the life out of this, to the point that we are doing more harm than good? Does it REALLY need to be THIS systematic for ALL students?

 

I think I need to read some unschooling math articles right now. I know, I know--blasphemy at this board. I'm not talking about ONLY doing unschooling for math. But...I feel like I've gone too far on the other end of the spectrum. I think I've forgotten that a lot of math can be learned a little less rigidly.

 

EDIT: Replaced "painfully" with "rigidly".

[Jean in Newcastle]

Have you looked at the these?  http://simplycharlottemason.com/store/your-business-math/

 

Or these https://www.educents.com/national-deals/deal/daring-math-science-bundle

[Hunter]

Have you looked at the these? http://simplycharlottemason.com/store/your-business-math/

 

Or these https://www.educents.com/national-deals/deal/daring-math-science-bundle

I'm thinking outloud here, more than anything. These resources are not wowing my socks off, but I'm not sure exactly what I want. I'm just kinda...I don't know, looking back over--what--maybe 20 years of hardcore math instruction and much longer of helping younger siblings and neighbors, and just thinking about public school instruction, good and bad.

 

Most kids like all the shiny colored pieces of wood and plastic, and the card and board games, and the pretend scenarios but I'm not so sure that is the most efficient way. I didn't always have those, and I didn't feel like they actually increased learning long term, when we did.

 

I think this might be more about my attitude than anything.

 

I struggled with using Student of the Word curriculum until I came up with the phrase "SOW is not a crossword puzzle." I didn't change the materials, but instead my attitude, expectations and scheduling. SOW uses a lot of generic worksheets and writing templates. When I started using the generic resources as more open ended prompts instead of puzzles to be completed, things went so much smoother. It was ME that needed to change. I didn't need games or more fun, just an attitude adjustment.

[Farrar]

Hunter, I don't find that reading most math curricula gets me all excited for learning math either.  But combining resources the way that some people have described - like what dbmamaz said - can be more invigorating.  That way you can use various resources that teach a progression for a specific skill or approach lots of problems more wholistically or creatively.  And you don't have to write your own problems and structure them in an in depth, building way the way that a really good curricula would.  We use the second resource that Jean linked - the Book of Perfectly Perilous Math - it's cute and funny and all the problems are supposed to be to save your life before you're killed in a hilarious but terrible manner (it's really written with boys who like science fiction in mind).

 

[Hunter]

The Book of Perfectly Perilous Math is kinda cute. Is there an eBook format of any type?

[sunnyday]

Someone recently linked me to a speech given by the guy who spearheaded Common Core. He was talking to an association of data analysts, and describing some of the statistical research that backed up the creation of the standards. He made an interesting point. He stated that there are just three topics of k-2 math that have predictive value for later mathematics success -- and just three topics that the international benchmarks had in common. These are there are only three topics in common in high performing countries—they are "addition and subtraction and the qualities they measure."
 
He also said that, similarly, there are very few topics in grades 3-5 that predict mathematical success, but these few are "multiplication and division and...Fractions." He states that one of the failures of our current American curricula is that there are so many topics covered, that you could completely fail at fractions and still pass elementary math because fractions have become such a small part of the curriculum. Yet fractions comprise the "knowledge and skills earned during those grades [that] most predict your ability to handle the near equations in eighth grade" and are the one thing that most predicts the students' readiness for Algebra.

 

So anyway, although I would still like to find the actual statistics and analysis that lead to these conclusions, I am really kind of heartened by this. If you can hit those important subjects confidently, everything else is gravy.

 

(Oh, right, the transcript I got this from. It was transcribed by an opponent of Common Core who was clearly just picking it apart to find statements to use out of context, so the bolding is distracting, but it's the only version I can find in print. http://missourieducationwatchdog.stopcommoncore.info/?p=164)

[Hunter]

He stated that there are just three topics of k-2 math that have predictive value for later mathematics success -- and just three topics that the international benchmarks had in common. These are there are only three topics[/s] in common in high performing countries—they are "addition and subtraction and the qualities they measure."

 

He also said that, similarly, there are very few topics in grades 3-5 that predict mathematical success, but these few are "multiplication and division and...Fractions."

Thanks for this. I seemed to instinctively know what was more important when I was younger. When my boys were still in PS and in 4th and 5th grade, I used to make them do one long division problem every morning before school. I can still remember them hopping on one foot at the table, squealing, while trying to finish the problem, wanting to be OUT the door with time to play before the bell rang at the school. That was all I stressed over. One long division problem a day.

 

By the end of the year, the youngest was the only child in the 4th grade who could do long division, and the older was the strongest divider in the 5th grade.

 

This morning, I was just sick, flipping through curricula. The number of topics was just dizzying. I like math. I even like textbook style math. It's just the avalanche of topics, and the complicated and lengthy systems of review and scheduling that overwhelm me.

 

And integrated is good, but often I cannot see the main point in all the details and integrations and prerequisites. It's soup with mystery stuff floating in lots of broth.

[regentrude]

Someone recently linked me to a speech given by the guy who spearheaded Common Core. He was talking to an association of data analysts, and describing some of the statistical research that backed up the creation of the standards. He made an interesting point. He stated that there are just three topics of k-2 math that have predictive value for later mathematics success -- and just three topics that the international benchmarks had in common. These are there are only three topics in common in high performing countries—they are "addition and subtraction and the qualities they measure."

 

He also said that, similarly, there are very few topics in grades 3-5 that predict mathematical success, but these few are "multiplication and division and...Fractions." He states that one of the failures of our current American curricula is that there are so many topics covered, that you could completely fail at fractions and still pass elementary math because fractions have become such a small part of the curriculum. Yet fractions comprise the "knowledge and skills earned during those grades [that] most predict your ability to handle the near equations in eighth grade" and are the one thing that most predicts the students' readiness for Algebra.

 

This makes perfect sense. One thing that bugs me about US math instruction is the many different topics that are skimmed, but never studied in depth, and the fact that it seems that every year the same topics are covered again, without making any actual progress.

 

Addition and subtraction in 1th and 2nd grade,  multiplication and division in 3rd and 4th, fractions and decimals in 5th, begin with algebra and geometry in 6th. This would be beautifully simple (and is pretty close to how math is taught in my home country).

[mathmarm]

Mathmarm, thank you. I think I need to create my own notebook of lessons. Some of us were talking about creating a Teacher Commonplace Book, and I need to buckle down and start doing that.

 

It might help to record yourself explaining a concept, then working a sampling of problems in that category. For K-5 math, it can (and in my opinion should) be very simple. You're right, the "scope and sequence" of topics and skills is pretty ridiculous in a lot of math books. The point isn't to trick and disorientate students with a number of riddles, topical spirals and conceptual somersaults. Just teach the darn mathematics which, at the elementary level flows nicely along the basic continuum of counting*, addition and subtraction, multiplication and division, multi-operation problems, fractions.

 

If I were to write a super simplified (but still effective) Scope and Sequence for K-2 math, then it might look something like this. Notice that we are essentially working on only a few skills, but to varying degrees and the steps you use to master the skills will vary drastically, but

 

K--End Goal: Efficient and fluent counting. Sorting and classifying. The Zero property of addition (and subtraction)

Counting

Counting on and back (Foundation of addition and subtraction--i.e starting at 13, counting on 5, starting at 12 counting down 8 )

Skip counting by 2-10 (Foundation of multiplication and division)

Place value through 100

Addition and subtraction of facts that total 5 and 10

Greater than, less than, equal to

Addition and subtraction facts 0-4 and 6-9, this year these are used as exercise with no expectation of mastery.

Regrouping units to tens and tens to hundreds.

Special topics:

like terms. 1 cat + 6 cats = 7 cats, 1-ten + 3-tens = 4-tens, 5Xs + 7Xs = 12Xs. So 9-cats + (3 dogs and 4-cats) = 13-cats & 3-dogs.

Concept of Multiplication and Division to whatever capacity the student can take it.

 

1--End Goal: Efficient and fluent counting. Comfort with place value to thousands. Understand associativity and commutativity. Introduce infinity. Develop and practice standard algorithms for addition and subtraction. Inverse operations (+ and -)

Counting on and back

Exercises in +/- using the ideas of 5 and 50 and 10 and 100 as 'check points'

Using the inverse to check work

Addition and subtraction facts 0-10, this year these should be learned.

Skip counting by 2-10, forward and backward to 100 (This year, move towards automating these)

Complements to 10, 100 and 1000.

Place value through 10,000

Using (and vocalizing) associativity and commutativity.

Addition and Subtraction of quantities to 1,000 with regrouping

 

2--End Goal: Efficient and fluent counting, comfort with place value to 100,000s, introduce multiplication (and division), use the properties learned to date (zero property, associativity, commutativity, inverses) to solve problems. Practice and learn the standard algorithm for addition and subtraction. Infinity. Use place value to explore distribution. Develop and practice the standard algorithm for multiplication and division.

Skip counting on and back from a given point (that follows the pattern)

Exercises in +/- using 5, 50, 500, 5000 and 10, 100, 1000, 10000 as 'check points'

Constant revision of addition and subtraction facts, 0-10

Place value through 100,000

Addition and subtraction of quantities to 100,000 with regrouping

Using complements to 10,000

Multiplication and division facts for 0-10

Using (and vocalizing) associativity, commutativity, zero property of + and *

Develop the algorithm for multiplication and division

 

There might be a couple of gaps there, or the sequencing could be off, but I just jotted something down to illustrate the point. The idea is that by the end of Grade 2 (after essentially 3 years of study) students should have some command over counting, place value, addition and subtraction, and be gradually taught how to justify their answers using legitimate math-lingo that summarizes the properties of real numbers.

 

The idea is to explicitly teach up to a definite point, but give the student enough of a 'broad view' that they can extrapolate their skills to extend at least a bit beyond their teaching. If you build that sort of 'take it to the next level on your own' into

the lessons then students get used to going to the next level on their own, they learn to look for the chance to generalize. Plus, many little kids are show-boats. They love to point out the new things that they figure out and then you just sit back and pretend to be wowed in due measure.

 

Also, it is worth mentioning that a certain amount of attention to detail and neatness should be emphasized every single day from day one. Take the time to work out problems neatly for your students, don't allow kids to be in the habit of scrawling all over the page. Teach them to line up digits by place value, to record their work in the margins, to draw lines that separate problems, box their final answer as soon as their handwriting is up to snuff. There is nothing I hated more than 13 and 14 yo kids who wrote like they had no sense what so ever. Okay, I hate when adults in college do it to...). I'm not talking about simple formation of numerals and letters, I'm talking about they don't write in any discernible sequence, scratch out, draw arrows, leave out steps with no real rhyme or reason etc...

 

Beginning (no later than) 3rd grade it is important that students be directed to do their homework exercises and create notes in such a way that their own work is legible and intelligible to them a week later. Their completed homework should be excellent preparation for a test or review, it should serve as a thorough review in and of itself. If they cant study from their own homework a month later, then it typically isn't well done.

 

 

In my experience, it is better to have kids deliberately practice neatness and order in writing their (math) from get go, even if just for a couple of problems each day then try and change 5+ years of bad habits and instructional neglect during the course of one-year (usually pre-algebra). Even if you have to give them worksheets that have that structure built in, then please do it! Middle school, high-school and college teachers all over the country will thank you for it.

I'm just really struck by the fact that the more I learn about teaching math, the less confidence I have to teach it. There is something SERIOUSLY wrong with that. Humbleness is a good character trait, but lack of confidence doesn't make a better teacher. Sometimes youthful and naive arrogance is more effective than wiser humbleness.

 

Yeah, that is weird. I think you may have reached the point of over-saturation. After a while, you just need to find the approaches that work for you and stick with them. There are many ways to teach math well (and many ways to teach it poorly, but we're just going to focus on the good ways.) and you as a teacher, only need to master a few of them. Teaching isn't about chasing students everywhere, but getting their attention and guiding them until they are capable of exploring on their own, while following a few basics. Despite all of the modern propaganda, you can't teach 'critical thinking', number sense, reasoning or any of that other crap that many modern curricula are supposed to be teaching with their bloated scope and sequences, contrived (and often baffling) exercises. However you can guide students to developing these latent skills and abilities, and that is the goal of early math and basic arithmetic. I think that the humility is in recognizing that you are but a humble guide and not some master navigator here to show students 'the light'. Given time, space and the occassion many students will recognize, internalize and learn the axioms (properties of real numbers) mostly on their own anyway, so you aren't really doing anything special. However, whether or not they'd apply them to math on a wide scale is a lot harder to say so our job as teachers becomes pointing out that "Hey, kids, all those nifty, real life relationships and observations that are 'common sense' look! They are powerful ideas in math also!!! They make our computations easier, our calculations more efficient--come here, check it out!" Recognizing that, and keeping that in your heart is a big part of the humility required to teach well. Also, respecting each students right to the knowledge that you hold and your obligation to see that they get it (even if it means getting out of their way) is required. But yeah...There is only but so much to learn about teaching math before you begin to over-saturate...

With my son, school had been so bad, that when I pulled him from school, we were just two kids running through a buffet, grabbing and tasting everything we saw. We didn't worry about "complete" because we were so busy gorging on what was right in front of our face. And somehow it all worked out. Even math was "Yum yum, this is awesome! Oh wait, before you eat more of that, look at that over there. Let's try that! Let's see if we can do it." If we could do it fine. If we couldn't, we backed up and shored up the weak point. It was just so easy and so fun. I miss that.

 

Math is infinite. There is no real quantifying what constitutes 'complete', even at the most basic level of counting. A  large part of the problem is people have become obsessed with this never wanting students to struggle or fail by their own merit--so they try and prepare them for every.single.thing. Starting in K or 1st grade and that is impossible and unbelievably foolish that so many people are still trying. Kids aren't idiots and their is no rule that says kids should be taken and shot if they fail. Since we aren't in the practice of shooting kids over failing, I don't see what the whole "anti-failure" hoopla is about anyway.

 

We need to simplify the Scope and Sequences and any exposure to extra topics needs to be directed by the students mastery over basics up to that level. Student need to learn to count; properly, fluently, systematically and efficiently. They needed to be guided to take counting "to the next level" to do arithmetic and they need, not cute and colorful cartoon animals all over their school books, but capable teachers and an emphasis on the foundations and a mastery of basic skills that build in a logical fashion.

 

Personally, I'm pretty frightened by many of the Kindergarten, 1st and 2nd grade math books that I see. They have whole chapters on graphing, fractions, money, temperature, measurement, clocks and time, calendars, geometry etc...Yet, kids are arriving on the first day of 3rd, 4th, 5th, 6th, 7th and 8th grade  without having ever known any of their basic math facts and without ever having had a firm understanding or command of place value. Many of the kids will leave on the last day of that grade with no more command over them than the day that they walked in...They arrive at the next grade and can't add or subtract 2- and 3-digit numbers easily using paper and pencil. You can hang it up on students knowing how to multiply and of those who can multiply a 1- or 2-digit number by a 3-digit number can't tell you why the algorithm works...And then we wonder why kids can't distribute in algebra and those who can can't do it without mumbling incomprehensibly about "foil". To hell with "FOIL". It's called distribution, its a property of real numbers and you should have been aware that you were using it since 4th grade! I'd wager that many of my college math students can't do long division. Actually, I don't have to wager you, I used to put 'long' arithmetic problems on my high school algebra, pre-calculus  and calculus exam as bonus and many students would miss them. Even the 'A' students.

:glare:

 

Anyway, this is a topic that tends to get my goat. I just can't get over how widely spread poor math instruction is and how the idea that taking emphasis of elementary mathematics further and further away from the basics is what we need. Of course students fail in the topics of algebra that parallel arithmetic almost exactly. They can't do the problems in base-10, which is our natural everyday experience with numbers. Why would they be able to do them in base-X?

 

I also don't know what the hyper emphasis on small numbers in the lower grades is. Kids are masters of generalization, they can take place value in the thousands, we just have to explain it to them in a way that is appropriate. Most kids I have worked with, loved the fact that they could use numbers in the 1000's. It gave them confidence and made them feel big! I can appreciate the idea that for some kids, keeping the scope low helps them, but there is little reason to aim to keep all kids at numbers below 20 (in K). Or 100 (K and 1st). It's absurd and for many kids, they freak out at bigger numbers because they have been trained that "XYZ" is big, scary, or too much. Its ridiculous!!! Anyway, I've been at this one reply for over 10 minutes now, so I'm going to cut it off now...

 

--mathmarm. Who won't drink modern elementary math teaching philosophy kool-aid.

[Hunter]

This makes perfect sense. One thing that bugs me about US math instruction is the many different topics that are skimmed, but never studied in depth, and the fact that it seems that every year the same topics are covered again, without making any actual progress.

 

Addition and subtraction in 1th and 2nd grade, multiplication and division in 3rd and 4th, fractions and decimals in 5th, begin with algebra and geometry in 6th. This would be beautifully simple (and is pretty close to how math is taught in my home country).

Regentrude, what would you use for K-8 if you had it to do over again?

[regentrude]

Regentrude, what would you use for K-8 if you had it to do over again?

 

Well, I never homeschooled K-5 at all; my kids were in public school until I pulled them out in 5th/6th. So, I am not familiar with any elementary math curricula. We used Saxon 8/7 for a miserable semester when I pulled them out, before AoPS Intro to Algebra in 6th/7th.

 

If I had to do it again, I would use AoPS prealgebra in 5th grade now that it is out.

I do not know what I'd do for 1-4 (I would skip K entirely, as I do not believe in early formal academics; I would probably get math books from back home (Germany) or from Russia.

[Hunter]

Mathmarm, wow! Thanks. I need to read that a couple more times.

 

I'm already starting to work through this. Last night, I was again reading through my What Your _ Grader Needs to Know books (original edition), and it was like I was reading a whole other curriculum.

 

As I looked at each lesson, I decided whether it was a critical lesson or a non critical lesson. I realized most of them were not critical and the ones that were critical are easy supplemented by ANY basic worksheet generator.

 

The noncritical skills lessons can be handled just like I handle the grammar and music lessons. They need to be talked about and journaled about and reviewed and applied, but don't need a problem set. And some of the noncritical lessons can even be taught like the content lesson for history and science, and not a skill.

 

Grade 2 has a couple very advanced geometry lessons on line. But those are meant to be integrated with the art lessons on line. Those need to just be taught as a fun topic, not a problem set. Now those lessons have a whole new tone and goal. They no longer seem any more overwhelming than the long list of history topics. They are just interesting opportunities to learn about interesting topics I might not otherwise have thought to cover.

 

I feel some of my old confidence and enthusiasm coming back.

[mama27]

Thanks for this. I seemed to instinctively know what was more important when I was younger. When my boys were still in PS and in 4th and 5th grade, I used to make them do one long division problem every morning before school. I can still remember them hopping on one foot at the table, squealing, while trying to finish the problem, wanting to be OUT the door with time to play before the bell rang at the school. That was all I stressed over. One long division problem a day.

 

By the end of the year, the youngest was the only child in the 4th grade who could do long division, and the older was the strongest divider in the 5th grade.

 

This morning, I was just sick, flipping through curricula. The number of topics was just dizzying. I like math. I even like textbook style math. It's just the avalanche of topics, and the complicated and lengthy systems of review and scheduling that overwhelm me.

 

And integrated is good, but often I cannot see the main point in all the details and integrations and prerequisites. It's soup with mystery stuff floating in lots of broth.

And this is why I choose random things for my youngest kids. There's just too dang many different topics they are trying to cram into a kids head. I don't see the point in it. Granted I'm pretty relaxed with my younger kids, I don't see any reason to rush them into math or reading. I'm aware I'm way more relaxed about schooling than a lot of people here.

I have had times, phases, where I freak out a little, but thankfully they go away.

 

[Hunter]

Mathmarm, you covered so many topics I need to respond to some of them one at a time.

 

I too focus on neatness. I'm not sure if How to Tutor states the need to practice writing numbers, or if I just inferred that I should, from the way the book is set up, but it's a natural thing to do that when using HTT. I make sure I teach the student a way of writing the numbers that complements their handwriting. And we do some Waldorf mainlesson book style copywork. We copy charts. We address envelopes.

 

I think math copywork is as important as any other type of copywork. Students learn best by first seeing, and then copying, and only then being able to write without a model. We forget how much math is like the other subjects. We tend to segregate it as something that is so alien to the other subjects, we don't cross over any of the techniques that work with those other subjects.

[regentrude]

I think math copywork is as important as any other type of copywork. Students learn best by first seeing, and then copying, and only then being able to write without a model. We forget how much math is like the other subjects. We tend to segregate it as something that is so alien to the other subjects, we don't cross over any of the techniques that work with those other subjects.

 

Sorry, but I have to disagree.

There are aspects of math that make it very different: it requires conceptual understanding of something very abstract. Which reading and, to some degree, simple writing, do not. I see no value in copying math problems- all that does is create the false sense of understanding. (I see that time and time again with my students.)

 

Yes, they should definitely be taught to write neatly, and correct writing of problems should be modeled and then expected from the student. But really, once they can write numerals, there is not much gained from copying problems, because it does not do anything for the understanding. It only creates a replication of algorithmic procedures.

 

ETA: I find worksheets to be detrimental to developing the skill of writing out math problems neatly and correctly. It is far preferable to make students write the entire problem out in their notebook, beginning form the beginning, and showing all work.

[Hunter]

Anyway, this is a topic that tends to get my goat. I just can't get over how widely spread poor math instruction is and how the idea that taking emphasis of elementary mathematics further and further away from the basics is what we need. Of course students fail in the topics of algebra that parallel arithmetic almost exactly. They can't do the problems in base-10, which is our natural everyday experience with numbers. Why would they be able to do them in base-X?

 

Algebra is often called the "gateway" to so many things, but the wider the curriculum gets the less likely the student will be able to do algebra.

[Hunter]

I actually completely disagree with this.

There are some aspects of math that make it very different. I see no value in copying math problems- all that does is create the false sense of understanding. (I see that time and time again with my students.)

 

Yes, they should definitely be taught to write neatly, and correct writing of problems should be modeled and then expected from the student. But really, once they can write numerals, there is not much gained from copying problems, because it does not do anything for the understanding.

I think it's important for a student to copy a few example problems before trying to set up their own problems. I don't they should have to master solving a new type of problem at the same time they are learning to set up a new type of problem. Some students really struggle with where they are supposed to place each number, and earlier they struggle to even write numbers at all.

 

And not all math is problems.

[regentrude]

And not all math is problems.

 

???? What else?

[regentrude]

Some students really struggle with where they are supposed to place each number,

 

But they should not memorize "where to place the number" by copying a procedure. They should be taught to understand WHY the number goes where it goes. Otherwise there will be no long term retention.

 

I have seen that in DD's tutoring student. She simply recalled some algorithms she has seen ("It is a fraction, so I need to flip, right?" without the slightest idea why, and in which situation, that would be appropriate.

 

This leads to the kind of students who think there are ten different kinds of linear equations with one unknown that require ten different algorithms to solve: x on one side, x on both sides, fractional coefficients, integer coefficients, negative coefficients, positive coefficients - when, in reality, they are all the same problem.

[Hunter]

But they should not memorize "where to place the number" by copying a procedure. They should be taught to understand WHY the number goes where it goes. Otherwise there will be no long term retention.

 

I have seen that in DD's tutoring student. She simply recalled some algorithms she has seen ("It is a fraction, so I need to flip, right?" without the slightest idea why, and in which situation, that would be appropriate.

 

This leads to the kind of students who think there are ten different kinds of linear equations with one unknown that require ten different algorithms to solve: x on one side, x on both sides, fractional coefficients, integer coefficients, negative coefficients, positive coefficients - when, in reality, they are all the same problem.

I raised an aspie little boy that could solve problems in his head, but was completely unable to set up a problem on paper, without being shown.