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The nitty-gritty of mathematics education.

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Anyone want to do some detailed discussion of early math education? I'm a bit tired of threads that only want recommendations of curricula and don't want to engage in how, exactly, we're teaching. I'd like to talk how to communicate certain ideas carefully and clearly. 

I think of mathematics education as largely conceptual. I also think there are only a few concepts you're really supposed to learn in elementary school, and a lot of the other stuff is a) icing on the cake and b) stuff that's easy to pick up through real life. However, there are some concepts that are actually quite non-trivial, and that are often very poorly communicated. 

I also think that there are difficult conceptual leaps at a variety of stages, and that it's really interesting to think about how to get past them. 

Here are some early things I think of as concepts that are important to communicate. 

-- The meaning of symbols as connected to language (What are we doing to two numbers we're "adding"? What are we doing to two numbers we're "subtracting"? What does an equals sign mean?) 

-- Place value and how to use it to add and subtract

-- The dual meaning of subtraction (it's both "taking away" and "what's left over if we have so many") and the connection with addition. 

-- The commutativity of multiplication

-- The distributive and associative properties of multiplication. 

There are lots more, but as a start, this is a good list, I think. I have lots of pedagogical theories about how to make concepts stick, and I would love to hear from other people who do as well. What I'm not interested in are lists of programs. This thread is, of course, open to anyone to using a program, but what I'm asking about are actual teaching approaches. 

 

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I think I'm going to love this thread! I have no awesome pedagogy as far as math (or probably anything else!), that is unique but I'll be tuning in when I can. I've gotten some good ideas here over the years and from the many math publishers I've used LOL. Sometimes I do hate how much we've spent on math programs- it is by FAR the most experimented upon subject in this house. But I will say having now used a lot of the math publishers- I feel each one has made me a better teacher. 

Your thread makes me think of one of the things the KONOS founder mentioned in the videos I watched. I'll butcher it, but she said something to the effect of "your child can figure out fractions while baking in the kitchen but that doesn't mean they've MASTERED it. Discovering something doesn't mean mastery," and I thought of that in relation to a lot of "child led" homeschooling stuff I've read over the years. So I think the concepts are important, but then I think most kids need a decent amount of practice to internalize something. Not just fall across it once and then "check" it's done. Drill and kill has gotten a bad name, but it seems for a lot of children, there is a fair amount of practice required to latch on and keep a concept and getting it to stick long term. The bigger question for me has been at what age/level does that drill become beneficial. We hear so much about "little kids can memorize things so early, they're like sponges! Or empty buckets just waiting to be filled!" No matter what the ability of a 5 year old, I am not sure that is the time to drill them on their multiplication facts. But on the other side, I wonder at what point would it have been easier to learn these sooner? That's when a kid manual would be nice. Teach D at X years, Y months. 

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I've been thinking about this topic and look forward to reading the discussion! 

My foremost thought right now is that cuisinaire rods are the ultimate teaching tool, ime. We have started with Miquon and moved to Beast twice now,  and the rods have been foundational. Ds uses them for new Beast concepts and anytime he feels overwhelmed (which is every day... but he loves Beast anyway).  

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💖 CSMP and the way they use visuals to teach, but I particularly love the minicomputers. They are great for place value, and after working with them, dd had absolutely no problem with the concepts of regrouping or fractions, despite her dyscalculia.

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The way I do place value, starting in about 2nd grade (but it does depend on the kid), is to teach the concept of binary.  It sounds silly, but that's how I was taught in 2nd grade and it made a lot of sense to me later and also sort of manages to separate the base-10 system from the idea of place value, so that kids (at least me and mine) aren't just memorizing that two spots left is 10, three spots is 100, etc., but see that the concept of 10 is arbitrary and the way we use numerals is just a language.  After I do binary, before I do base 10, I do a brief bit on base 12 (or base 15, or whatever I'm feeling like).  I go through the whole thing: "how would you express this many (hold up 7 fingers) in base 12?  How would you express this many? (hold up 3 fingers)" etc. , just like I do for binary.   Then I say, okay, how would you express this many (hold up all 10 fingers and wiggle 4 toes)?"  They are generally baffled for a while and I let them be baffled.  It's baffling!  

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I spend very little time if any on a lot of concepts that are standard in most curriculum: basic geometry, money, applied math (eg rates) and instead focus on the basic operations. In our home we start with practice with numerals using a hundreds chart. I find that this lays a very solid foundation for place value as the child discovers relationships while practicing writing numerals. I also introduce multiplication alongside addition and spend only a small amount of time discussing “taking away” and “less than” before jumping into division. And soon after jump into fractions. We then go deeper and deeper with all four operations simultaneously.

 I have found this works infinitely better than trying to follow any single curriculum. I do own tons of resources but I mainly just pull what I need to save myself time from creating my own worksheets. I strive for as much discovery based learning as I can, but one child is slow at making leaps so I find it far more efficient to spend only a set amount of time on exploration and then teach the concept directly followed by lots and lots of practice. I have another child who simply thrives on discovery but who wilts at challenging problems.

The DC who are old enough to do math are very tactile learners. It took me a long time to realize this because I never needed manipulatives. I like having both crods and an abacus as well as little cut up “pies” for fractions. We’ve recently added games too. I am not a “fun” mom so this is quite a leap for me, lol.

great topic, square!

Edited by mms
Clarifying.
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These 2 concepts have been helpful for me as I teach my kids:

Subtraction isn't just taking some away from a group, it's also finding out how many jumps on a number line are between two given numbers. Otherwise they don't know what to do when a problem asks them to find the difference.

Reading numbers such as 502 as "five hundred two" instead of "five oh two", whether it's in a math problem or finding a page in a book. The kids I've been picky about this with have a better concept of place value, I think.

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My youngest really benefited from the hands-on style of Montessori math. She is a perfectionist and would get frustrated not knowing all of her math facts. I printed up the Montessori math charts and problem strips. Then she’d have time set aside where she would pick a problem strip out of the cup, look up the answer and then say it out loud. Having the answers right there at that age helped her get past shutting down or completely melting down. That’s just one example of how they show kids the answers are right in front of them, they just have to go through the steps to find them. 
I found Montessori math to be really good at helping kids visualize the math and discover the connections on their own. We used C Rods and Miquon which worked for my middle but for dd, I used a lot of Montessori methods. If I could do it over, I’d probably go all Montessori in the early years. 
I can’t find the original website I used. That one would explain the steps and had free printables. This just has printables. 
http://wikisori.org/index.php/Worksheets/Printouts:Math

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This YouTube channel has some good step by step guides on how to use Montessori. The Stamp Game works with +-x/. It teaches place value, regrouping in a visual,  hands-on, self-learning way. It’s worth adding into any math program to solidify abstract concepts. 
You can see how it’s different from C-rods but they can be used in the same way  

 

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Place value understanding is a hill I will die on. After reading Liping Ma’s book, I felt validated in this. Here is how I teach it (don’t know how this formatting will work).

For multiplication of larger numbers, say 234x43, I first draw a rectangle and split the sides into 200 + 30 +4, and then 40 + 3. We see that we need to multiply all the combinations together. We never, ever drop zeros. I then have them write out:

                              200      30      4

           x                               40     3

________________________________

                                         1            2               (3x4)

                                         9          0               (3x30)

                              6         0       0                  (3x200)

                              1            6      0                (40x4)

                 1          2            0        0              (40x30)

   +             8           0           0         0           (40x200)

______________________________________________

    1             0            0             6           2

 

My middle child could not get his brain around adding and subtracting right to left (least to greatest). I searched around and discovered that subtracting and adding (and in the multiplication example above) left to right is perfectly valid and it really helps with understanding place value. It is also how we do mental math. We are adding greatest to smallest in our brains. Check out how to do this on youtube.

I also make very certain that they fully understand multiplying and dividing by 1 in all its forms. For instance, fractions are put in simplest form because we are finding where we have multiplied by 1 (but it could be as 3/3 or 5/5). That we do not just “cross out numbers”. We simplify equations because we are either multiplying and dividing by one or adding and subtracting zero (I don’t think I am explaining this well) and doing the same to both sides.

So for the equation 3B+5 = 20

                         3B +5 -5      = 20 – 5   

                         3B + 0         = 15

                         3B                =15

                         3B/3           =15/3       (and here I make them tell me that 3/3 is multiplying by 1)

                          B                 = 5

Edited by annegables
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I am convinced that many algorithms were developed in the days of slate and chalk, when space and paper were precious. I cannot fathom "dropping zeros" for any other reason. It takes an additional 10 seconds to write the zeros and almost completely prevents so many mistakes and faulty thinking. Kids, we are not being timed and paper is abundant. Spend ten extra seconds and another 2 inches of paper to do it correctly.

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1 minute ago, annegables said:

I am convinced that many algorithms were developed in the days of slate and chalk, when space and paper were precious. I cannot fathom "dropping zeros" for any other reason. It takes an additional 10 seconds to write the zeros and almost completely prevents so many mistakes and faulty thinking. Kids, we are not being timed and paper is abundant. Spend ten extra seconds and another 2 inches of paper to do it correctly.

Yes, yes! Or that silliness that we used to do in vertical multiplication where we'd write an x in the ones place rather than just saying we're multiplying by 10, not 1! I prefer the explicit. 

I didn't understand what I was doing as a kid. I could multiply in my head, or do the thing on paper,  but didn't understand why the paper version "worked."

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40               2 x 2 x 2 x 5           2        x          2      x     2      x      5        

_____   = _____________  =   __                __          ___          ___       = 1 x 1 x1 x5/6 = 5/6       

48              2 x 2 x 2 x 2 x 3      2                  2            2               2 x3        

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4 minutes ago, elroisees said:

Yes, yes! Or that silliness that we used to do in vertical multiplication where we'd write an x in the ones place rather than just saying we're multiplying by 10, not 1! I prefer the explicit. 

I didn't understand what I was doing as a kid. I could multiply in my head, or do the thing on paper,  but didn't understand why the paper version "worked."

According to Liping Ma's research, this is mainly because the teachers do not understand what they are doing or why they are doing it. 

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Count me as someone who disagrees. I will not teach multiplication like that.

My kids learn that multiplication distributes over addition, so 25 ×6= 6 x (5+20) = (6×5)+(6×20), but it doesn't take long to master the concept and then do simple straight algorithmic approach to solving. 

As long as they understand conceptually what they are doing, I dont see the big deal. Not a biggie to me. My kids have obviously not been negatively impacted conceptually.

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3 minutes ago, 8FillTheHeart said:

Count me as someone who disagrees. I will not teach multiplication like that.

My kids learn that multiplication distributes over addition, so 25 ×6= 6 x (5+20) = (6×5)+(6×20), but it doesn't take long to master the concept and then do simple straight algorithmic approach to solving. 

As long as they understand conceptually what they are doing, I dont see the big deal. Not a biggie to me. My kids have obviously not been negatively impacted conceptually.

My oldest and youngest have been able to see math the way you are describing and move quickly on to the algorithm, but my middle has never been able to after two years of trying. He is a kid where the algorithm would have meant never understanding what is happening. But he is really good with math, he just needs to see the place value very explicitly. 

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place value -- used SM trading tokens and just explained that's the convention, demo'd a few different abacus styles. did math in other bases

properties -- SM does this so well I didn't need to add

convention -- b&m school did a unit on measurement (pre-full inclusion); measure length with paperclips, shoes, etc and discussion on how standards are chosen; we extended to include other things and viewed museum displays

measurement of time -- used cub scout resources - sun dial, walking stick, water drip, sand drip and showed how geared clock is measuring; viewed museum exhibits

precise language is important...I would never say 'what are we doing  to a symbol'...we are expressing relationships when we use symbols. We should be able to visualize that at this level, and lots of lego play as preschoolers is how my dc gained familiarity.  I had a visitor point out to me that my son, at age 4, was finding midpoints, but had no language for it...that made me realize I needed to step up my part in giving the children vocab...after all school doesn't do that topic until grade 5...even when paper folding in kindy, the vocab is not appropriate to what they are actually seeing.

Edited by HeighHo
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@8FillTheHeart brings up a good point. I teach those concepts that way, but my kids are allowed to move on to the algorithm whenever they are comfortable. I dont force extra steps for those who understand what they are doing AND can do it without mistakes. 

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1 minute ago, HeighHo said:

precise language is important...I would never say 'what are we doing  to a symbol'...we are expressing relationships when we use symbols.

+1

This has been *the* biggest thing in our house. We just draw to illustrate points ("show me what this means") after some abacus use in the very beginning of school, unlike your extended use of manipulatives, but the idea is the same, and it's made all the difference for my kids and the kids I have taught. 

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7 minutes ago, HeighHo said:

place value -- used SM trading tokens and just explained that's the convention, demo'd a few different abacus styles. did math in other bases

properties -- SM does this so well I didn't need to add

convention -- b&m school did a unit on measurement (pre-full inclusion); measure length with paperclips, shoes, etc and discussion on how standards are chosen; we extended to include other things and viewed museum displays

measurement of time -- used cub scout resources - sun dial, walking stick, water drip, sand drip and showed how geared clock is measuring; viewed museum exhibits

precise language is important...I would never say 'what are we doing  to a symbol'...we are expressing relationships when we use symbols. We should be able to visualize that at this level, and lots of lego play as preschoolers is how my dc gained familiarity.  I had a visitor point out to me that my son, at age 4, was finding midpoints, but had no language for it...that made me realize I needed to step up my part in giving the children vocab...after all school doesn't do that topic until grade 5...even when paper folding in kindy, the vocab is not appropriate to what they are actually seeing.


I don’t think a plus sign represents a relationship. It represents an operation. How would you describe that operation yourself? I describe it as putting numbers together, and that has been clear to my older girl, but perhaps there’s a better way.

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13 hours ago, square_25 said:

I'm a bit tired of threads that only want recommendations of curricula and don't want to engage in how, exactly, we're teaching

 

To be fair, this is the curriculum board* 😊 And lots of us really like talking about curriculum!

 

 

*just teasing you... obviously the boards are largely used interchangeably in practice. 

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I have tons more to write, but my daughter has hijacked my laptop for her Russian cartoons, so I gotta be brief while using my phone.

One thing I’ll explain in more detail later is that I’m EXTREMELY picky about definitions ;-). I give precisely one ease to verbalize definition for each operation, so that we can reason about it whether we understand all of its ramifications or not. For instance, a - b is always “we take b away from a” for us, even though we learn that it is the same as “what number do we add to b to get a” later. But there has to be a primary and internalized definition to enable reasoning about something, in my experience. 

As an illustration, what do we mean when we write a * b * c? Is that even well-defined? If so, how exactly is it defined? 

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Just now, OKBud said:

 

To be fair, this is the curriculum board* 😊 And lots of us really like talking about curriculum!

 

 

*just teasing you... obviously the boards are largely used interchangeably in practice. 

Hahaha, you’re so right!!!! I should have posted this on the General Education board!!

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3 hours ago, square_25 said:


I don’t think a plus sign represents a relationship. It represents an operation. How would you describe that operation yourself? I describe it as putting numbers together, and that has been clear to my older girl, but perhaps there’s a better way.

 

group or regroup would be my euphemism

plus means more and minus means less, in Latin.  we use a symbol alone to stand for an idea because its hard to have a relationship with oneself..but the relationship is that the symbol represents the idea.. We use symbols together to express a relationship.  4-3 =1 is a relationship.  We need to start with precise definitions before going to the sloppy everyday language that requires experience with cases ( or a kindly grandfather with a porch rocker who is willing to chat with the young 'un).

1 put together with 1 is eleven unless 'put together' has been described more specifically...more precise language is needed.  Or there will be 'mistakes' that cause students to hate ed, because they didn't get the teaching to figure out that they have stumbled on to a case not obvious from the language used in instruction, and have erred, making them 'bad at math' in their young minds.

 Being a classroom volunteer and a scouting volunteer also taught me to use precise language, because many youngsters understand cases and trivial solutions and often present them in amusing ways in order to get the discussion going.

Again, I encourage you to contact the gifted ed program at the Renzulli Center at UConn. You will find the people you are looking for, from what my friends who studied there have shown me.

Edited by HeighHo
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7 hours ago, moonflower said:

The way I do place value, starting in about 2nd grade (but it does depend on the kid), is to teach the concept of binary.  It sounds silly, but that's how I was taught in 2nd grade and it made a lot of sense to me later and also sort of manages to separate the base-10 system from the idea of place value, so that kids (at least me and mine) aren't just memorizing that two spots left is 10, three spots is 100, etc., but see that the concept of 10 is arbitrary and the way we use numerals is just a language.  After I do binary, before I do base 10, I do a brief bit on base 12 (or base 15, or whatever I'm feeling like).  I go through the whole thing: "how would you express this many (hold up 7 fingers) in base 12?  How would you express this many? (hold up 3 fingers)" etc. , just like I do for binary.   Then I say, okay, how would you express this many (hold up all 10 fingers and wiggle 4 toes)?"  They are generally baffled for a while and I let them be baffled.  It's baffling!  

 

We did binary, too! We didn't do it before base 10, but I thought it really solidified place value for us. 

The way we did place value is that we used manipulatives or visuals to do lots of two digit addition and subtraction. There's nothing like actually using the grouping visually or physically to get a feel for how place value works. The way I did it with my daughter is that we actually drew 10s and 1s (boxes for 10s, dots for 1s). The way I've been doing it with my homeschooling class is that we use poker chips (which is really just like using Montessori manipulatives, I think!) I actually don't think it makes much of a difference for most kids whether the "manipulatives" are visual or tactile, as long as they feel like they can arrange them in a way that makes sense. At least, I'm not seeing much difference in terms of how kids are engaging with the poker chips versus how my daughter engaged with the pictures.

The most useful thing I've done, I think, is to give hands-on definitions... then step aside. I think it's hard for people who already understand a concept to really estimate how long it takes for a kid to fully internalize an idea. Even when the idea is something that seems really obvious to us, like "we take b away from a." When I tell the kids in my classes to take b away from a, half the time, they make two piles, one of a and one of b, then they don't know what to do! That's because thinking about "taking away" is actually surprisingly hard! It's considerably more abstract than putting things together. 

I actually had a very instructive experience with this with my daughter... I taught her subtraction, figured she must have already internalized it, and started teaching her "different methods." It was annoying me that she couldn't immediately see that 51 - 49 is 2, so I started teaching her that yes, subtraction is taking away, but also that we can calculate it in these other ways. And then I noticed that the spark of understanding slowly leaving her eyes. She started thinking "what method should I use NOW?" instead of thinking "What does this mean? How can I visualize what this symbol is describing?" So.... I backed off. I went back to "51 - 49 is what we get when we take 49 away from 51," and started trying to be patient as she slowly and laboriously took 40 away from 51, then another 9 away from the 11 she had left. But she understood what she was doing again, and she could again verbalize what things meant, and that made a huge difference for us. (And yes, she now knows that 51 - 49 also means "what do we add to 49 to make 51?" and can do this quickly. But it was important for her to really internalize subtraction in one way before she could branch out.)

By the way, we've always done mental math from left to right! I think that's how people naturally reason almost uniformly. It hasn't in any way prevented us from learning the algorithms later.

 

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I would suggest that a teacher or home educator should understand the concepts at three levels:

1. What exactly is going on, when you do XYX (distribute over multiplication or addition, or factor, or compare fractions, or figure percentages, or whatever), and

2. Multiple ways to convey that information, and

3. How to recognize where the student's thinking goes wrong, in any part of the processes or algorithms. This would include knowing what prior knowledge/skills are essential to what you're trying to accomplish, and how to back up to address those gaps.

@square_25, I'm going to disobey your request and discuss curriculum, but only because I'm recommending programs as a means for the *teacher* to learn, and not just suggesting what to buy to hand over to your kid, or to try to march him through without either teacher OR student understanding it.

If you're new to teaching math, this is where curriculum with explicit teacher's guides can be invaluable. I am NOT talking about programs that drill and kill an algorithm; I'm talking about programs that clearly teach the "why" and help you know what to watch for. I always recommend Rod and Staff, because there is a strong assumption that the young Mennonite teacher is not a mathematician, so the TG explains everything very thoroughly.

This program is not a favorite of the pro teachers. I am not recommending it for them - I'm recommending it for those new teachers who need to understand traditional math and how to teach it, as a bridge to learning more conceptual processes later. Work through the lessons. Teach them to yourself. Work every problem and don't move on until you understand WHY you made a mistake. The next stop, in my "learn to teach" program, is Math Mammoth. Maria Miller makes a conceptual approach very accessible - the visual aids and the progression of skills in the exercises will teach you. You'll get a lot more depth of understanding for what you've already learned with R&S's traditional math, which will make you a better teacher.

One more program in my mix is Ray's Arithmetic, especially for the basic four operations. If you want to learn the field axioms and properties through a conversational and interactive approach, while moving around endless piles of beans in a muffin tin, this WORKS. The fractions and decimals lessons are very, very useful, too - the exercises will lead your understanding. I have personally found no benefit in using Ray's beyond the Primary and Intellectual books. But I think every one of my students spent a little time in each of these books, with the beans and muffin tins, to really "see" the operations. It's also beneficial for students who are bogged down by any physical writing in math, in the early elementary years. My alternative recommendation to this is Rosie's Cuisenaire Rods - sorry, I don't know the current name or website.

Another resource would be books that are specifically about tutoring in math. I like Sam Blumenfeld's How to Tutor, Ruth Beechick's The Three R's and How to Teach Your Child Successfully, and (the very vintage and esoteric) Eclectic Manual of Methods for the Assistance of Teachers (written in 1885). Read also Charlotte Mason, especially volume 6, and John Holt (How Children Learn, How Children Fail, and Teach Your Own). Read Liping Ma, specifically for math, and then you may feel VERY smart because you learned traditional math first (R&S) and then learned a more conceptual approach (Math Mammoth and some of Ray's). You need to understand the learning process and how to recognize when to push, when to back off, when to wait for a developmental leap...math seems to be the most fraught subject for homeschool families. You need to be a dedicated and compassionate teacher. 

Having pretty much conquered the teaching of arithmetic and elementary math, if you want to go on to Algebra, I would recommend Foerster's. Maria Miller recommends it, too, and also recommends David Callahan's videos. Same method, teach yourself, and then teach your child.

Lastly, I've often considered offering classes for home *educators* - not students - similar to those that Charlotte Mason, herself, would teach. I think there's a need for this, even though the percentage of homeschoolers who want to personally teach is diminished from the past. I am starting to see glimmers of a return, as the promises of virtual, hands-off programs have failed so many families. Just this year, just within a few months, I've been seeing more posts from home educators saying, "No computer programs, please, that didn't work for my child. I need to learn how to teach this myself."

If we are getting back to that, I am ALL in. I would happily spend my homeschool retirement helping home educators learn to teach. ❤️ 

Edited by Lang Syne Boardie
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34 minutes ago, square_25 said:

I actually had a very instructive experience with this with my daughter... I taught her subtraction, figured she must have already internalized it, and started teaching her "different methods." It was annoying me that she couldn't immediately see that 51 - 49 is 2, so I started teaching her that yes, subtraction is taking away, but also that we can calculate it in these other ways. And then I noticed that the spark of understanding slowly leaving her eyes. She started thinking "what method should I use NOW?" instead of thinking "What does this mean? How can I visualize what this symbol is describing?" So.... I backed off. I went back to "51 - 49 is what we get when we take 49 away from 51," and started trying to be patient as she slowly and laboriously took 40 away from 51, then another 9 away from the 11 she had left. But she understood what she was doing again, and she could again verbalize what things meant, and that made a huge difference for us. (And yes, she now knows that 51 - 49 also means "what do we add to 49 to make 51?" and can do this quickly. But it was important for her to really internalize subtraction in one way before she could branch out.)

 

Not for nothing, but when people talk about waiting until kids are older to work on math, this is what they mean. I literally laughed out loud when I read this, because I had to deal  with that exact situation and it stands out in my mind because I handled it differently with my two older kids and worried about it for years (until, spoiler alert, it became clear that all was well). 

With my oldest I did what you did. I sat there and did everything the long way around. 

When we butted up against it with his little brother and Medium found it enragingly frustrating (because he didn't want to be there doing that in the first place and he's full of feelings), I said OK let's just drop it. And drop it, we did! For....like three years.

Cut to, he's nine and we get back into the swing of things. A problem like this comes up, he reads it, he says, "oh," and proceeds to do it in his school work. I boorishly drag him from his legos later in the day and tell him to show me how that works. He grudgingly does so. He is correct. He has instantly seen the relevance of working some subtraction problems in that manner, and working others as "take-away." ... by the time he's ten he's doing all the same things his big brother did at 10 for official school work-- and to be real, he does it better than Big . Big likes to play around with written math more than medium, but that's innate.  I'd say they use math in the wild about evenly, though to different ends. 

This is totally an aside. 

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Looking at your list, I think I used food to introduce my kids to a lot of concepts : ) 

Addition -- you have two cookies now. If I give you two more, you'll have four.

Subtraction -- taking away those cookies.

Division -- sharing cookies equally

Multiplication -- you have two cookies; if each of your friends have two cookies, how many do we have altogether? Also rows of cookies on a baking sheet / muffins in a muffin tin. 

We did a lot of looking at calendars / talking about how often things happen in a day, in a week, in a month. Calculating how many meals we'd have over the course of a few days. 

My husband also watches a ton of sports with both kids, which lends itself to a lot of math talk. How much is x team leading by? How many points does x team need to make in order to catch up? etc etc 

My son jumped quickly from this kind of thing to doing calculations on paper. If anything, he resented my efforts to use manipulatives and number lines with him. My daughter is different -- for her, Cuisenaire rods and concrete things like clapping and jumping and moving objects along number lines were really helpful. 

Edited by Little Green Leaves
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2 hours ago, Lang Syne Boardie said:

I would suggest that a teacher or home educator should understand the concepts at three levels:

1. What exactly is going on, when you do XYX (distribute over multiplication or addition, or factor, or compare fractions, or figure percentages, or whatever), and

2. Multiple ways to convey that information, and

3. How to recognize where the student's thinking goes wrong, in any part of the processes or algorithms. This would include knowing what prior knowledge/skills are essential to what you're trying to accomplish, and how to back up to address those gaps.

@square_25, I'm going to disobey your request and discuss curriculum, but only because I'm recommending programs as a means for the *teacher* to learn, and not just suggesting what to buy to hand over to your kid, or to try to march him through without either teacher OR student understanding it.

If you're new to teaching math, this is where curriculum with explicit teacher's guides can be invaluable. I am NOT talking about programs that drill and kill an algorithm; I'm talking about programs that clearly teach the "why" and help you know what to watch for. I always recommend Rod and Staff, because there is a strong assumption that the young Mennonite teacher is not a mathematician, so the TG explains everything very thoroughly.

This program is not a favorite of the pro teachers. I am not recommending it for them - I'm recommending it for those new teachers who need to understand traditional math and how to teach it, as a bridge to learning more conceptual processes later. Work through the lessons. Teach them to yourself. Work every problem and don't move on until you understand WHY you made a mistake. The next stop, in my "learn to teach" program, is Math Mammoth. Maria Miller makes a conceptual approach very accessible - the visual aids and the progression of skills in the exercises will teach you. You'll get a lot more depth of understanding for what you've already learned with R&S's traditional math, which will make you a better teacher.

One more program in my mix is Ray's Arithmetic, especially for the basic four operations. If you want to learn the field axioms and properties through a conversational and interactive approach, while moving around endless piles of beans in a muffin tin, this WORKS. The fractions and decimals lessons are very, very useful, too - the exercises will lead your understanding. I have personally found no benefit in using Ray's beyond the Primary and Intellectual books. But I think every one of my students spent a little time in each of these books, with the beans and muffin tins, to really "see" the operations. It's also beneficial for students who are bogged down by any physical writing in math, in the early elementary years. My alternative recommendation to this is Rosie's Cuisenaire Rods - sorry, I don't know the current name or website.

Another resource would be books that are specifically about tutoring in math. I like Sam Blumenfeld's How to Tutor, Ruth Beechick's The Three R's and How to Teach Your Child Successfully, and (the very vintage and esoteric) Eclectic Manual of Methods for the Assistance of Teachers (written in 1885). Read also Charlotte Mason, especially volume 6, and John Holt (How Children Learn, How Children Fail, and Teach Your Own). Read Liping Ma, specifically for math, and then you may feel VERY smart because you learned traditional math first (R&S) and then learned a more conceptual approach (Math Mammoth and some of Ray's). You need to understand the learning process and how to recognize when to push, when to back off, when to wait for a developmental leap...math seems to be the most fraught subject for homeschool families. You need to be a dedicated and compassionate teacher. 

Having pretty much conquered the teaching of arithmetic and elementary math, if you want to go on to Algebra, I would recommend Foerster's. Maria Miller recommends it, too, and also recommends David Callahan's videos. Same method, teach yourself, and then teach your child.

Lastly, I've often considered offering classes for home *educators* - not students - similar to those that Charlotte Mason, herself, would teach. I think there's a need for this, even though the percentage of homeschoolers who want to personally teach is diminished from the past. I am starting to see glimmers of a return, as the promises of virtual, hands-off programs have failed so many families. Just this year, just within a few months, I've been seeing more posts from home educators saying, "No computer programs, please, that didn't work for my child. I need to learn how to teach this myself."

If we are getting back to that, I am ALL in. I would happily spend my homeschool retirement helping home educators learn to teach. ❤️ 

Love your post. It sums up what is necessary for teaching in a practical way. I personally dislike bar models and the entire SM model. I dont use the program bc it makes me a poorer teacher bc I have such a dislike for it. Now, HOE, itoh, is exactly how I think.

I can take ideas gleaned from various sources and teach my kids how to think about how to solve problems. I dont have to be a slave to any one approach. I can adjust what we do according to their individual understanding. My own kids all tend to have similar thought processes. My granddaughter comes from left field and makes me teach everything from a different perspective. My grandson is somewhere in between.

My granddaughter needs to perform operations and master them before the explanation makes sense. She has to be able to do before she comprehends why. My kids easily grasp why and then just do. My grandson needs both simultaneously.

As much as I believe understanding the concept matters, I know after working with my granddaughter who could not intuitively understand 1+2=2+1 even when using manipulatives that there is no single way that works "best."

Knowing how to teach is the key piece to the puzzle for the child.

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46 minutes ago, Little Green Leaves said:

Looking at your list, I think I used food to introduce my kids to a lot of concepts : ) 

Addition -- you have two cookies now. If I give you two more, you'll have four.

Subtraction -- taking away those cookies.

Division -- sharing cookies equally

Multiplication -- you have two cookies; if each of your friends have two cookies, how many do we have altogether? Also rows of cookies on a baking sheet / muffins in a muffin tin. 

We did a lot of looking at calendars / talking about how often things happen in a day, in a week, in a month. Calculating how many meals we'd have over the course of a few days. 

My husband also watches a ton of sports with both kids, which lends itself to a lot of math talk. How much is x team leading by? How many points does x team need to make in order to catch up? etc etc 

My son jumped quickly from this kind of thing to doing calculations on paper. If anything, he resented my efforts to use manipulatives and number lines with him. My daughter is different -- for her, Cuisenaire rods and concrete things like clapping and jumping and moving objects along number lines were really helpful. 

I use food and siblings names in word problems from Singapore or Beast Academy to make the problem make sense for my one kid. So if the problem says, "Mariella, Petra, Hoosieface, and Barfbag need to split 63 thing-a-ma-jigs evenly, how many does each person get?" My kid used to look at me blankly. I would then say, "you and your siblings need to split 63 cookies. How many does each person get?" No more confusion. He can do it almost instantly. He can explain how he did it. I refer to the original problem. LOTS of thinking time later, and he gets it. And then sees the connection. But he needs to personalize it first. 

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More than specific techniques, I think five things are critical to ensuring that a student gets a good math education:

  1. The teacher must have a "profound understanding" (to use Liping Ma's words) of the math themselves.  This is the foundation upon which everything else is built.
  2. The teacher must have an interest in communicating this understanding to the student.  This is different from simply presenting lessons in a math program.
  3. The teacher must be able to use their understanding of math and their interest in communicating it to come up with flexible teaching strategies.  In other words, if the student doesn't get whatever it is using one method, the teacher has several more up their sleeve and is able to invent new ones beyond those.  This is not the same as switching math programs when the going gets tough.
  4. The teacher must have the patience and skill to uncover what is hanging a student up.
  5. The teacher must believe that a solid conceptual foundation in arithmetic is critically important and be willing to move mountains to facilitate it.

Generally when there are problems with a student's math education (unless there is a learning disability involved), it is one or more of the above that is lacking--in other words, it is a teaching problem and not a problem with the student or math program.

As for specific techniques, here are a few that I found to be particularly helpful.

  • Mental math techniques to reinforce place value
  • Studying different bases--for example, my older son and I made "heximal" blocks and then used them to learn how to do calculations in base 6
  • Using manipulatives to introduce certain concepts, but not requiring their use if the kid balks.  Also, using a variety of manipulatives to demonstrate a concept (over time, not necessarily all in the same lesson!) rather than rely on just one for everything
  • Bar diagrams for word problems, but again, not requiring their use

 

Edited by EKS
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I am pretty sure I went to K'er knowing place value. If not by then, soon after. All I can think of that would have specifically taught me that is finding the right hymns in church.

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6 hours ago, square_25 said:


I don’t think a plus sign represents a relationship. It represents an operation. How would you describe that operation yourself? I describe it as putting numbers together, and that has been clear to my older girl, but perhaps there’s a better way.

 

CSMP's dot and string pictures make a good foundation for operations as relationships.

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6 hours ago, square_25 said:

 

 

I actually had a very instructive experience with this with my daughter... I taught her subtraction, figured she must have already internalized it, and started teaching her "different methods." It was annoying me that she couldn't immediately see that 51 - 49 is 2, so I started teaching her that yes, subtraction is taking away, but also that we can calculate it in these other ways. And then I noticed that the spark of understanding slowly leaving her eyes. She started thinking "what method should I use NOW?" instead of thinking "What does this mean? How can I visualize what this symbol is describing?" So.... I backed off. I went back to "51 - 49 is what we get when we take 49 away from 51," and started trying to be patient as she slowly and laboriously took 40 away from 51, then another 9 away from the 11 she had left. But she understood what she was doing again, and she could again verbalize what things meant, and that made a huge difference for us. (And yes, she now knows that 51 - 49 also means "what do we add to 49 to make 51?" and can do this quickly. But it was important for her to really internalize subtraction in one way before she could branch out.)

By the way, we've always done mental math from left to right! I think that's how people naturally reason almost uniformly. It hasn't in any way prevented us from learning the algorithms later.

 

I needed to read this exact thing today, because I came perusing this thread to ask for concrete ideas to help teach subtraction.  My oldest is balking at subtraction, and I'm realizing I just need to let him internalize it much more.  I am completely confident that he has internalized place value and addition, but I'm realizing that he needs to spend more time with the relationship between subtraction and addition and just plain more time with subtraction.

Any ideas to help him practice subtraction?  I am wildly fascinated by math because my own education was so lacking, but it makes me insecure in teaching things I probably know better than I feel.

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33 minutes ago, medawyn said:

I needed to read this exact thing today, because I came perusing this thread to ask for concrete ideas to help teach subtraction.  My oldest is balking at subtraction, and I'm realizing I just need to let him internalize it much more.  I am completely confident that he has internalized place value and addition, but I'm realizing that he needs to spend more time with the relationship between subtraction and addition and just plain more time with subtraction.

Any ideas to help him practice subtraction?  I am wildly fascinated by math because my own education was so lacking, but it makes me insecure in teaching things I probably know better than I feel.

 

I usually just let them practice it with visuals or manipulatives until it really clicks! And I do stick to one definition to start with -- we've done "taking away" so far, and that has worked fine (although I'm sure I'd tweak it if there was confusion.) 

By the way, I've been teaching homeschooling classes, and almost uniformly, my students find addition easier and more intuitive than subtraction, even they are already solid on the relationship between place value and addition. For one thing, to subtract you don't necessarily need to make both numbers -- instead, you make a single number, then you do something to it! It's much less hands-on.  

I've found that in general, "inverse operations" (operations that reverse simple, commutative operations like addition and multiplication) come harder than the initial operation, even at much higher levels of math. In the same way, my AoPS kids found vector subtraction much less intuitive than vector addition when done visually...  

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6 hours ago, OKBud said:

 

Not for nothing, but when people talk about waiting until kids are older to work on math, this is what they mean. I literally laughed out loud when I read this, because I had to deal  with that exact situation and it stands out in my mind because I handled it differently with my two older kids and worried about it for years (until, spoiler alert, it became clear that all was well). 

With my oldest I did what you did. I sat there and did everything the long way around. 

When we butted up against it with his little brother and Medium found it enragingly frustrating (because he didn't want to be there doing that in the first place and he's full of feelings), I said OK let's just drop it. And drop it, we did! For....like three years.

Cut to, he's nine and we get back into the swing of things. A problem like this comes up, he reads it, he says, "oh," and proceeds to do it in his school work. I boorishly drag him from his legos later in the day and tell him to show me how that works. He grudgingly does so. He is correct. He has instantly seen the relevance of working some subtraction problems in that manner, and working others as "take-away." ... by the time he's ten he's doing all the same things his big brother did at 10 for official school work-- and to be real, he does it better than Big . Big likes to play around with written math more than medium, but that's innate.  I'd say they use math in the wild about evenly, though to different ends. 

This is totally an aside. 


Yes, I totally understand why people wait until kids are older! You're describing a real phenomenon. I actually have some theories related to this :-). 

I think that mathematical understanding comes both from having the correct developmental level and from what I think of as "rich conceptual experiences" with the operations. By the time kids are older, they've had to take things away lots and lots of times, and if they don't have a mathematical LD, they've probably observed along the way that you can also figure out what's left over by seeing how many more they need to add. Subtraction and addition are operations we are exposed to a lot in day-to-day life. As a result, we can wait for kids to get the necessary rich conceptual experience just from exposure, and it makes things come easier. And of course, it also ensures that the developmental level is there as well. 

Here's the reason I didn't take that path myself, though. As I said, I had to back off on this with my daughter when she was around 5. But instead of waiting for her to have the rich conceptual experiences outside of schoolwork, I tried to create them in her schoolwork. She was very good at connecting symbols to visuals and manipulatives, so what we did is to write lots of different equations about the same picture. These are often called "number bonds," I think, but I don't like that name: I just think of them as "How can we describe this picture symbolically?" We'd have picture where one box had, say, 13 things, and another had 18 things, and we would calculate that the total number was 31, and then we'd write the equations 

13 + 18 = 31, 31 - 18 = 13, 31 - 13 = 18 

and we'd explain how the picture gave us each equation. And after that, we talked about how to check whether we calculated a subtraction correctly by doing addition. All of these were things she could actually grasp and that didn't stretch her too far. And after a while of doing questions like this intermittently, she moved towards using the fact that subtraction is the inverse of addition more and more. She's now 7.5, and this fact is blindingly obvious to her. I think she's been able to use it since 6 or so.

To me, the advantage of thinking about how to create rich conceptual experiences within the context of schoolwork is that a) it leaves somewhat less to chance and b) you won't have a choice for higher math. As I said, addition and subtraction are things we simply see day to day. However, how often do we have to use variables day to day? Even if you live an algebraically sophisticated life, you probably don't spend a lot of time solving quadratic equations or calculating derivatives. When I was teaching AoPS kids what vectors are in precalculus, I definitely had to think about how to make sure they got sufficient conceptually rich exposure to the very idea of vectors, because I knew that they wouldn't get enough exposure through life. And I found that working on that, which basically meant consistently making them visualize vectors and use the definition of a vector, really helped later understanding of the material. 

Of course, one problem with this approach is that it's quite easy to bump against developmental readiness. You probably won't be able to teach your 1 year old to read, no matter how hard you try! So I do think the "pick when green" approach works best if you're willing to back off if you can't make progress... which isn't easy for everyone. 

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11 hours ago, mms said:

I also introduce multiplication alongside addition and spend only a small amount of time discussing “taking away” and “less than” before jumping into division. And soon after jump into fractions. We then go deeper and deeper with all four operations simultaneously.

 

Absolutely. I think this is incredibly valuable, because sometimes just introducing the operation and allowing kids to calculate it for small numbers with it allows the kid to have more conceptually rich experiences. And it often helps a child recognize the same operation in real life. My daughter definitely uses multiplication when she need to. She multiplies if she knows she has to buy 5 toys, each of which costs $4; she multiplies if she needs to calculate the number of windows in a building that has 7 windows horizontally and 9 vertically; she multiplies if she needs to check whether she split 324 correctly into 18 parts. I don't think she'd be able to get all of those "real-life" experiences if she hadn't been exposed to what multiplication is!  

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20 hours ago, Æthelthryth the Texan said:

I'll butcher it, but she said something to the effect of "your child can figure out fractions while baking in the kitchen but that doesn't mean they've MASTERED it. Discovering something doesn't mean mastery," and I thought of that in relation to a lot of "child led" homeschooling stuff I've read over the years. So I think the concepts are important, but then I think most kids need a decent amount of practice to internalize something. Not just fall across it once and then "check" it's done. Drill and kill has gotten a bad name, but it seems for a lot of children, there is a fair amount of practice required to latch on and keep a concept and getting it to stick long term.

 

Oh, absolutely. I have a very mathy kid -- math comes easily to her. She's working on algebra at 7.5 right now (mostly because she pestered me to start it this year -- I was going to wait until we did long division and some more combinatorics) and so far, it's really smooth. She can solve any linear equation I can throw at her and she didn't need to be taught how to expand expressions like (a+b)^2 and (a+1)(a+2) at all -- it just came naturally after all the multiplication practice she had gotten in arithmetic. 

And yet... on one of her recent worksheets, I asked her whether it's true that 1/a*1/b = 1/ab in general. Now, I'll note that this is a child who CAN calculate 1/a*1/b for whatever positive integers a and b you pick. She has a very solid grasp of fraction addition, subtraction and multiplication. And she can also explain

(a+b)c = ac + bc

as "well, you are supposed to take (a+b) copies of c, which is really just a copies of c and another b copies of c, which is what's on the right." So she can talk about variables very fluently! However... the question of whether 

1/a*1/b  = 1/ab

for general a and b reduced us both to tears. She can do the calculation for any specific numbers. She can reason with variables. And yet her understanding is not quite deep enough to actually generalize from her experience! I thought it was fascinating. 

We're continuing to calculate fraction products using her intuition about what fraction multiplication means. And I'm sure that she'll eventually get enough exposure (or drill, if you like) to be able to generalize her fraction experience, in the same way as she can generalize her multiplication experience to allow her to very quickly conclude that 

(a+b)^2 = a^2 + 2ab + b^2,

but we're not there yet. 

This is a long-winded way of saying that I absolutely agree with you. Exposure is not mastery. Exposure is sometimes about 1/500th of the way to mastery. There are so many possible levels of comfort with operations, with "being able to generalize using variables" at the very top, and with a LOT of intermediate stages in between. And I respect how much practice even very mathy kids need to really internalize things. To me, the interesting question is how to give the kids the KIND of practice that helps the student actually internalize the ideas (instead of merely internalizing the procedures.) 

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7 hours ago, Lang Syne Boardie said:

How to recognize where the student's thinking goes wrong, in any part of the processes or algorithms. This would include knowing what prior knowledge/skills are essential to what you're trying to accomplish, and how to back up to address those gaps.

Yes! I emphasize to DS’ the ability to explain his reasoning over his ability to “solve things”. I want to see his method, so that I can identify misunderstandings right away. Whenever he hits a snag we back up & proceed from concrete to symbolic to abstract. 

I remember hating “showing my work,” largely because it took so long for teachers to expect that skill from me. I had been doing things so automatically for so long that I couldn’t slow down my thinking that much! Hopefully starting now with having DS clearly write out his work or teach concepts back to me will mean it’s more automatic once he gets to more complex work.

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3 hours ago, EKS said:

More than specific techniques, I think five things are critical to ensuring that a student gets a good math education:

  1. The teacher must have a "profound understanding" (to use Liping Ma's words) of the math themselves.  This is the foundation upon which everything else is built.
  2. The teacher must have an interest in communicating this understanding to the student.  This is different from simply presenting lessons in a math program.
  3. The teacher must be able to use their understanding of math and their interest in communicating it to come up with flexible teaching strategies.  In other words, if the student doesn't get whatever it is using one method, the teacher has several more up their sleeve and is able to invent new ones beyond those.  This is not the same as switching math programs when the going gets tough.
  4. The teacher must have the patience and skill to uncover what is hanging a student up.
  5. The teacher must believe that a solid conceptual foundation in arithmetic is critically important and be willing to move mountains to facilitate it.

Beautifully stated, & I wholeheartedly agree. I am terribly envious that my son is getting to learn the way he is at the age he is... his work is engaging, challenging, & FUN!

I love that even when he’s pushed to his limit, his idea of “taking a break” is to pull out a math game... 😂

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2 hours ago, medawyn said:

Any ideas to help him practice subtraction?  I am wildly fascinated by math because my own education was so lacking, but it makes me insecure in teaching things I probably know better than I feel.

IME, Part-Whole Circles make the relationship between addition / subtraction & multiplication / division more clear. They are especially useful when first introducing word problems. 

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13 hours ago, Shoes+Ships+SealingWax said:

IME, Part-Whole Circles make the relationship between addition / subtraction & multiplication / division more clear. They are especially useful when first introducing word problems. 

What's a part-whole circle? 

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2 minutes ago, EKS said:

I think it's also called "number bonds."  Here's an example: https://mathsnoproblem.com/en/mastery/number-bonds/

 

Oh, I see! I don't love having the same visual representation for multiplication and addition... but yes, the pictures we drew of boxes "containing" 13 and 18 and the total number in both boxes was basically the same idea. 

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17 hours ago, EKS said:

The teacher must have a "profound understanding" (to use Liping Ma's words) of the math themselves.  This is the foundation upon which everything else is built.

 

Yep. All the rest has to be built on this understanding. I've come to this topic from a somewhat privileged position -- what I've been observing at both the university level and at AoPS for more than a decade now is that you absolutely can have instructors with a "profound understanding" of the mathematics they are teaching, and yet still have kids who come out without anything of the sort. But I do think that without this understanding, there's nothing to talk about. 

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16 hours ago, square_25 said:

You probably won't be able to teach your 1 year old to read, no matter how hard you try!

 

I love what you bring to this board.  I love your questions and what you share about math and teaching math and your bright daughter.  She is lucky to have you. I’m so glad you are here.

It is a very good point that some of “developmental readiness” is prior exposure dependent. I just wanted to point out that a lot of people do teach one year olds how to read. 

It isn't a welcome topic on this board. People here throw vegetables at me when I bring it up. Haha. I thought it might interest you though.

OK. Now I’m going to duck.

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20 minutes ago, drjuliadc said:

 

I love what you bring to this board.  I love your questions and what you share about math and teaching math and your bright daughter.  She is lucky to have you. I’m so glad you are here.

It is a very good point that some of “developmental readiness” is prior exposure dependent. I just wanted to point out that a lot of people do teach one year olds how to read. 

It isn't a welcome topic on this board. People here throw vegetables at me when I bring it up. Haha. I thought it might interest you though.

OK. Now I’m going to duck.

 

Interesting. How so? Do they wind up reading as a result? 

My older girl could have probably learned her letters at that age (she learned at 1.5 or so), but it didn't occur to me to teach her to read, so I can't say I've done the experiment! How does it work? 

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4 hours ago, drjuliadc said:

 

I love what you bring to this board.  I love your questions and what you share about math and teaching math and your bright daughter.  She is lucky to have you. I’m so glad you are here.

It is a very good point that some of “developmental readiness” is prior exposure dependent. I just wanted to point out that a lot of people do teach one year olds how to read. 

It isn't a welcome topic on this board. People here throw vegetables at me when I bring it up. Haha. I thought it might interest you though.

OK. Now I’m going to duck.

My eldest son learned to read CVC words and some words with consonant blends, before his third birthday. He also said his first words at 4.5 months and was talking by six months, and walking by eight months. In college, he carried a 3.9 GPA for a religion degree, minoring in philosophy and Greek, while working literally 40 hours per week and learning a second career on his own time. (He's chosen to work in that second career after graduation, and is climbing the ladder at a shockingly fast rate.)

He's not normal. He's the reason we started homeschooling.

But I'm never, ever going to believe that "many" children are taught to read at age 1, or that there will ever, EVER be a good reason for literally anyone, anywhere, to try to teach a child to read at that age and stage.

I'm aware that you might consider this to be the first vegetable thrown, but I'm posting it anyway. I'm fine with this concept of "Teach your 1yo to read" never feeling truly at home on these boards. That doesn't mean that I hope you will feel unwelcome; the opposite is true. I'm just saying that I hope that evidence based opinions on education and science will remain the dominant perspective here.

 

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8 minutes ago, square_25 said:

 

Interesting. How so? Do they wind up reading as a result? 

My older girl could have probably learned her letters at that age (she learned at 1.5 or so), but it didn't occur to me to teach her to read, so I can't say I've done the experiment! How does it work? 

Not the person you were asking but I do have a child who learned to read as a toddler. I didn't teach her though, her sister did.

My now-18 year old daughter saw the "teach your baby to read" infomercials for the first time when we returned to the states when she was about 6yo. She was just learning to read herself at the time and made it her mission in life at that time to teach her baby sister, then only about 1.5yo or so. So every day she had her phonics lesson and then she would play school with her baby sister and teach her the same concepts she had learned. Great practice for older dd to solidify her phonics skills by teaching another person. She's also very much the mother hen and loves little kids (and still does). I wasn't really concerned with whether or not younger dd actually learned anything from these phonics lessons her sister gave her at the time. I also never encouraged or discouraged these lessons. I just kept an ear on making sure there weren't any bad habits being taught. Amazingly, older dd was a wonderful and patient reading tutor. Even at 6yo and only teaching concepts she had barely mastered herself.

But learn to read younger dd did. By age two she could read on a kindergarten level (cvc words and such mostly). By the time she was 5 years old and starting kindergarten herself, she could read on a second grade level. By second grade, she maxed out the test at a 7th grade level. Could she have read on a higher level earlier? Maybe but I wasn't interested in pushing her at a young age. I let her go completely at her own pace. If she wanted help sounding something out, I helped her. Other than those "lessons" with her sister as a toddler and preschooler, she never had phonics or reading lessons. She picked up reading on her own after those first lessons from her sister.

Just some observations and other things, younger dd was a very verbal child. By 18 months, she regularly stunned me with her vocabulary and sense of language. Her older brothers and sister were saying things like "want more" and "all gone" and "play outside", 2 and 3 word phrases really, at that age. Ydd was not only speaking in complete sentences but using rather abstract vocabulary correctly. I was holding her in front of a mirror one day, practicing pointing out body parts, and she out of the blue says, " My shirt is very beautiful".  Her verbal prowess only got better as she got older. I think she was predisposed to having no issues learning how to read. Would she have read as early as she did if her sister did not set out to teach her? I cannot say because I don't have another copy of ydd to try out that theory. If I had to guess, based on my experiences with teaching 6 very different kids to read who were all raised in the same environment, my answer would be yes, she probably would have picked up reading on her own, with or without lessons, before kindergarten. She would have been my second child to do so. The lessons from her sister absolutely helped her learn more quickly, but I think she would have picked it up regardless.

Now, the question of early exposure and the results. I think it still comes down to every child is wired a little bit differently. All 6 of my kids have, more or less, been raised in the exact same literature rich environment. They all saw their parents read for information and pleasure regularly. They were all read to aloud from birth. They were all taught in, more or less, the same manner the basic phonics skills at some point before kindergarten. However, when they could actually read on their own from that exposure and instruction varied widely with no real rhyme or reason that I can discern other than we all have different strengths and weaknesses.

oldest ds was 9yo before he could read on his own

second oldest ds was 4yo when he could read on his own

older dd was 6.5yo when she could read on her own

younger dd 2yo when she could read on her own

younger ds was nearly 7yo when he could read on his own

youngest ds just turned 7yo this month and is teetering on the edge of reading on his own but still not quite there, he's been stuck at this point developmentally for about a year now

My definition of "reading on their own" is the ability to sound out most words on their own and only needing help with phonetic constructs that they have not encountered before. They can fluently blend sounds and decode words but may not know what every word they can read means. However, decoding words is no longer such a strenuous act for them that they cannot derive meaning or think about what they are reading as they read. 

So, all this to say, I think exposure and environment are important. It is important enough that I go out of my way to provide exposure to reading, literature and phonics from infancy. However, in my experience, it is not a determining factor for all children when it comes to how easy or difficult it will be for them to learn how to read. They will still read when they are developmentally ready to understand all the underlying skills for reading. Parroting rules and sounds and the like is not understanding. The light bulb has to click on in their head for true understanding to move them forward. Exposure and environment may help a child who is developmentally ready, but it will not push ahead a child who is just not wired to be developmentally ready to read.

As always, this is just my experience, YMMV.

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