Math Education: What's most important?

[Chelli]

What do you think is the most important component to being successful at homeschooling math?

Is it curriculum? The teacher's understanding?  A combination? Something else?

How do you take a kid who doesn't like math or doesn't understand math to a place of confidence?

I have really done a disservice to my oldest, the poor guinea pig, with her math education to the point she almost has math PTSD. I'm thinking through all of these issues trying to make sure I don't drop the ball with subsequent kiddos.

[Rosie_0801]

My dd has dyscalculia, which we "stealthed" and that was a hell of a lot of work.

The most valuable thing I did was teach her there was more than one way to skin a cat. Or do the maths. I had her translate maths from one format to another a lot and that provided a foundation I don't even know how to describe.

[Chelli]

15 minutes ago, Ordinary Shoes said:

Wow - this is timely given the thread on the Chat board! I won't dare jump into this one but thanks so asking a question like this. 

Well, I'll jump in a little. I don't know how you can teach math adequately if you don't understand it. 

Right, so at that point would curriculum choice be more important? If so, what curriculums have really good teacher helps?

[EKS]

I think that both are important, but if I had to pick, I'd say teacher understanding of both the math and the student they are teaching is more important especially as the math gets harder.  

[Chelli]

2 minutes ago, EKS said:

I think that both are important, but if I had to pick, I'd say teacher understanding of both the math and the student they are teaching is more important especially as the math gets harder.  

Any recommendations to help a homeschool mom improve either or both of those?

[Rosie_0801]

1 minute ago, Chelli said:

Any recommendations to help a homeschool mom improve either or both of those?

I think you just have to do maths, and do it in a different way to how you learned it originally.

[Chelli]

I guess my problem is that I'm actually really good at math. I don't like it though, but I can solve problems. 

My stumbling block has been I have no clue how to explain it, especially to a kid who doesn't get it or struggles.

[EKS]

10 minutes ago, Chelli said:

Any recommendations to help a homeschool mom improve either or both of those?

If you're teaching elementary age kids, the book Elementary Mathematics for Teachers is excellent especially if you're going to use Singapore (or something like it).  It's also a good idea to work well ahead of your student--so a year (or better yet, two or more years) before your student will move into algebra, actually work through an algebra program yourself.  Same with geometry and the rest.  It will not only allow you to support your student once they get to algebra (or geometry or whatever), it will also make you a better teacher of everything that comes before it.

I used various resources to prepare myself to teach high school math--ALEKS, Jacobs Algebra, and Derek Owens were the most helpful.  I also did tons of problems ahead of my kids.

As for good programs, I've particularly liked RightStart for K-1, Singapore for 2-5, Jacobs Algebra, and Derek Owens for high school math, but there are many others out there. 

[Zoo Keeper]

Well, I just graduated one who probably does have math-induced PTSD, so I may not be the best person to take advice from. 😉

What I've picked up, since I feel I have been attempting to teach high school math F O R E V E R-- to a math struggler/hater, then to a quick to get concepts kid, now back to a math struggler/hater, with a few more unknowns coming behind...

• If you don't know it, you can't really teach it.  Yes, you can facilitate it, look at the teacher book/solutions manual and try to reverse engineer it,  knock around on Khan Academy to see if that helps, etc., but to actually teach it, you have to have some level of understanding yourself.  If the child is stuck, and you have no idea why, well, then you are both stuck.  I can't tell you how many times I have had to hit pause (so to speak), take a few hours or a few days and work the dang problems myself, until I really get it.  Then I would drag the child through the same process until the light dawned.   Sometimes the process of me getting it has to take multiple sources-- looking up say, factoring, in multiple textbooks, reading, working example problems, until I can see the principle behind it.  Unfortunately for me, this takes time.  Waaaay too much time.  Math is not my language, and I have to really work hard until I see the process behind it.   A combination of Keys to Algebra, Jacobs' Elementary Algebra, MUS, Tobey and Slater's Beginning and Intermediate Algebra books, and Saxon Algebra were ones that helped, at various times and in various ways.  Yes, ALL of them. 🙂   And I'm still working through some Algebra 2 topics...  Seeing me work hard on this over and over has helped one of mine understand that it is a process, and it takes time.
• Some kids just need lots more time for math to become internalized.  Lots more time.  Lots more practice.  And we as parents/teachers need to give it to them-- this is hard to do.  Since everybody these days is supposed to make it to Calculus (insert eye roll here, folks), that doesn't leave a struggling student with the time to get it solid.  My experience has shown me that some very bright, hard working kids just need more time to park on a topic, review, rework problems, etc.  It is hard, very hard, as a parent/teacher to be willing to take it slow, not feel pressured, not feel "behind", not feel the SAT's breathing down your neck, etc.  That pressure can make it very difficult to relax and LEARN.  That anxiety DID make it harder for one of mine.  I really wish there was more differentiated instruction for high school math--without academic stigma attached to it.  STEM has become king, and a humanities strong kid really isn't seen as "smart" unless he is rocking it in math/science as well.   I wish I had an easy answer for this, other than to just let other people's expectations go.  Again and again.  Keep working hard, every day, and you will make progress.  It just won't be on someone else's timetable.  Understanding leads to more understanding, and it does all build up.

That's all I've got for now.  🙂 I'm sure others with more knowledge and experience will chime in with actual help instead of ramblings. 😉

[frogger]

1 hour ago, Chelli said:

I guess my problem is that I'm actually really good at math. I don't like it though, but I can solve problems. 

My stumbling block has been I have no clue how to explain it, especially to a kid who doesn't get it or struggles.

I'm not sure the age or what types of concepts your DD is struggling with but I'm a big fan of actually understanding the concepts and why it works. 

Things that help: listen, listen, listen.

Target actual problem.

Find a new way to explain be it pictures, manipulatives, a comparison to something else, or relating it to something they see in real life or have seen in past problems. 

 

For example, when multiplying polynomials you can show how it is just like multiplying tens and ones separately. 

We used the rapidly reproducing cute critters from Star Trek, tribbles,  to explain exponents and why any base to zero is actually one. You can also work your way up and few generations and then backwards in a pattern although a 1/4 of a tribble is gets some eyebrow raises. 🙂 

 

 My son is doing something I really thought he would know by now. We are practicing conversions in Physics. I kid you not I used pattern blocks today to show him why you can flip a conversion unit either way. He is in high school! I don't know why he was so hung up on that. We converted trapezoids to triangles and back using pattern blocks but I wrote it out as ratios rather than what I believe he automatically did which was simply remember to multiply and divide.  So simple but I had to make it click and then he can go back to converting ml/s and using scientific notation. The hardest thing is realizing what they are really hung up on after all with Mammoth Math you would think he would know that already.   That is where listening is important.  I thought it would be the exponents or something more advanced! But nope his brain simply didn't like that I was writing the ratio in whichever direction I needed it to be and he was totally hung up thinking there must be a right way to write it! Which is humorous considering he used to write the subtrahend on the top of a subtraction problem. 

 

 

[Not_a_Number]

9 hours ago, Chelli said:

I guess my problem is that I'm actually really good at math. I don't like it though, but I can solve problems. 

My stumbling block has been I have no clue how to explain it, especially to a kid who doesn't get it or struggles.

When you say you’re good at math, what kinds of problems do you mean? Are you good at word problems?

[Not_a_Number]

And to answer the question, I tend to think the teacher’s understanding matters most. That, and willingness to take as long as it takes to teach concepts. People vastly underestimate how long it takes kids to form mental models.

[LinRTX]

I also think it is the parent's understanding that matters most. And giving the child time to struggle with the problem. And what I mean by that comes from the time I worked at a fine arts school. The children of the teachers would often work math homework in our lunch room. One day a child was struggling with the problems and the mom would just jump in and tell them how to solve it. One day I spoke up and just asked questions. Things like what is the problem asking? What do you already know to do. Will that work here? The child soon knew how to approach the problem herself.

 

And if my child was struggling with something, I would have her teach it to me. That way I could more easily see where the problem was.

[wendyroo]

35 minutes ago, Not_a_Number said:

And to answer the question, I tend to think the teacher’s understanding matters most. That, and willingness to take as long as it takes to teach concepts. People vastly underestimate how long it takes kids to form mental models.

I agree that teacher's understanding matters most, but I think it goes beyond an understanding of just the math itself, and also requires understanding of how to teach math...and beyond that, to understanding multiple different ways and orders to teach concepts so that the teacher can adapt to different learners.

I think, especially for some kids, they require a delicate balance of time, challenge, repetition, introduction, review, spiral, motivation, etc.  We have never used just one math curriculum front to back - I am always tweaking and supplementing based on a multitude of factors such as how the child is handling the current concept, how I anticipate they will handle the next one, how they are feeling about math, other priorities or stresses in their lives, whether they are showing good retention of past topics, what executive function skills I want them working on, what concepts could be lightly introduced now so they have a long time to build intuition before mastery is expected, etc.

With my kids, I could never accurately pinpoint and say that they are at X grade-level in math.  I feel that as their teacher I need to have an understanding of the different facets of math and a nuanced view of my students' skills along each axis in order to nurture and challenge them appropriately.  I also need a deep understanding of how math concepts relate to each other.  And a working knowledge of different curriculum and supplement choices (or the time and knowledge to DIY it).

So, IMO, curriculum choice is very important for a less fluent teacher who is going to have their students work through it start to finish.  And largely unimportant for a teacher who is going to extensively tweak and supplement and go off script as required.

[Not_a_Number]

8 hours ago, frogger said:

My son is doing something I really thought he would know by now. We are practicing conversions in Physics. I kid you not I used pattern blocks today to show him why you can flip a conversion unit either way. He is in high school!

When I was working with my sister one summer, I wanted to get a head start on calculus. So we started out with doing some very simple problems with speed and velocity to get the idea of derivatives as "instantaneous velocity." 

My sister is a smart and relatively mathy kid whose math education at school was really flawed. It turned out that she didn't really understand AVERAGE velocity and speed calculations AT ALL. Like, she had done problems at school by division, like she was told, but it hadn't penetrated that what she was actually calculating "the distance traveled per unit of time." We had to practice that a bunch before we could even talk about instantaneous velocity at all. 

This is all to say that having gaps is utterly normal. Most of my college kids didn't understand fractions. The nice thing about homeschooling is that you can take as long as things take. Don't be worried or surprised if things like this crop up -- it's normal 🙂 . 

[frogger]

I will add, kids who often do have a good conceptual understanding of math sometimes just get hung up on a mistake in their problem. Algebra requires staying focused to detail throughout the problem without dropping negative signs or skipping steps because you don't find it hard and you just want to finish quickly.

Ways to help those children are:

Make a list of the most common mistakes that particular child makes (dropping negatives, forgetting the other side of the equation, multiplying a power and a base together). Sometimes they are in a hurry and if their brain glitches a certain way they need to go back and check for those specific errors.

Have them talk you you through the problem. My oldest did math himself because I was busy with the youngers. I couldn't review the whole curriculum and it had been 20 years since I had seen stuff in Alg. 2 or trig, etc. So when he was stuck I'd have him explain what he was doing to me. Basically, he was teaching me what he was doing and he would often catch his own mistake or I'd see where he didn't do what he said he did. He still asks for me to check out problems now that he is home because of Covid and doesn't have anyone to study with. He is taking 300 and 400 level math classes that look like gibberish to me but it helps him focus to explain and he often catches things and once in a blue moon I can still help. 

[PeterPan]

15 hours ago, Ordinary Shoes said:

Wow - this is timely given the thread on the Chat board! I won't dare jump into this one but thanks so asking a question like this. 

Well, I'll jump in a little. I don't know how you can teach math adequately if you don't understand it. 

Oh dear. I *think* she actually meant math, not "math". 

Now I'm trying to think if I agree with you about your point. I've known people and seen transcripts where the mom *never even finished high school* and did a GREAT job teaching math. The key? She stayed one step ahead of her kids.

I'm not sure what is glitching op, so maybe I can read more of the thread. My dd did not have a disability and just required diligent instruction. She could sorta do self instruction with MUS, but really she was always better with Mom driven instruction. So then, the glitches are consistent curriculum and Mom with time/energy/availability. Glitch those, and the kid struggles. I think my dd would have done better with math the last few years had I been available, but I wasn't. I was driving ds to his therapies and it just wasn't reality.

Ds has SLD math and enough delays to be confounding. He also needs that 1:1 and consistent, consistent work. Problem is, curriculum is less likely to fit him. But it's still about Mom time and consistency.

I doubt well chosen curriculum trumps the need for Mom (structure, supervision, whatever) and consistency. 

14 hours ago, Chelli said:

My stumbling block has been I have no clue how to explain it, especially to a kid who doesn't get it or struggles.

I don't know. I went to a gifted school for math/science/humanities in high school and have a kid who couldn't understand the number FIVE when he was 5/6. Gifted IQ in the kid btw. So it happens, lol.

Do you think the issue is you with language? Like it's hard to get it into words? Do you think they had contributing factors, like ADHD that was making it hard for them to attend and get through lessons? Do they have issues with working memory (holding their thoughts in their head), processing speed (bogging down in long problems, difficulty coming up with answers quickly), or writing? 

Is there a reason you didn't use a fully scripted program such as BJU that would have given you very thorough teaching helps to fit a variety of learning styles? It wouldn't be an ideal match for SLD math students, but for most typical students it would work well. 

I think if you have a hard time getting out what is in your head, then you are looking for more teaching supports. So you're wanting something like MUS which has videos and full scripting, or maybe BJU or CLE.

If you're saying they have ADHD and had some issues with attention, holding their thoughts, keeping things straight, fatiguing with lots of math, retrieval and automaticity, then personally I would get evals and learn about accommodations and self-advocacy. My very ADHD dd did well with 1:1 and learned how to use MUS to good effect. 

If the dc is actually struggling and has an SLD, then I would be looking at Ronit Bird. It might not be the problem is you. It might be they are not making the internal inferences, visualizations, and steps that the math curricula ASSUME the dc will. In fact, one of the most basic tests for dyscalculia is to see how a student counts up a field of objects. There's also subitization (the ability to see smaller quantities within a larger quantity). So due to subitization weaknesses, a dc with dyscalculia might look at a field of objects and literally count them ONE BY ONE. My ds even struggles to deal cards by counting and pausing as he goes around. He literally has to count one by one. There are other forms of SLD math as well, but it's the general idea that it's there, it's not always you, and that connecting with materials MEANT for SLD math can make a huge difference in getting explanations that will click.

So not to be trite, but you sorta have to figure out the problem to decide what solution(s) you want to try. There are probably multiple things going on. When in doubt, eval.

[PeterPan]

14 hours ago, Chelli said:

I'm actually really good at math. I don't like it though, but I can solve problems. 

 

14 hours ago, EKS said:

If you're teaching elementary age kids, the book Elementary Mathematics for Teachers

This. Sometimes what happens is people got good grades at math, did what the teacher said, and it was all *memorized*. So then you can't teach it, because you don't really get the why or the thought process. You aren't really thinking in terms of multiple ways to solve problems or proof. It's more here's the equation, this is how you do it, boom done. 

I was talking with someone using the BJU math (an older edition) and telling her how much I liked it because they included some things from set theory, things you recognize if you did Dolciani, and some other goodies. The person was like oh I JUST SKIP ALL THAT. I'm like WHAT??? You skip the meat that actually teaches them how to think mathematically??? Yup. She was "good at math" and knew how to do math. 

So again, you don't have to become a math EXPERT. You only have to get up to what you need to meet your kids' needs. And not my kid either, lol. You can identify the problem (instruction breakdown, attention breakdown, more consistency, whatever) and then target it. 

I personally am not a fan of almost any math generated by/for the homeschool community. MUS is an exception, but it probably would benefit from additional word problems, for which I like the Evan Moor Daily Word Problems series. 

How is that Math in Focus going? That fits the parameters as something that has been used with a wider audience, has a teacher's manual option, presents mathematical thinking, and can be done consistently. Is that going well at least?

50 minutes ago, Ordinary Shoes said:

My dad says everyone can benefit from going through Algebra again and would have recommended that my nephew skip calculus and re-do Algebra but no one does that today. 

My dd went through algebra multiple times. It's somewhat common, and people just don't talk about it. Some kids stall out developmentally. I was busy and we were trying to do things more independently. Also she needed to learn how to learn on her own, how to monitor her comprehension. She *asked* to repeat algebra, so I let her do that. 

Given that many homeschoolers go year round, it's pretty easy to have time to get some repetition. I think those other skills, like learning how to monitor attention, rewind, go back, try again, ask for help, are just as important. It's a sticky age, because they're pushing away but don't stop needing us.

[Not_a_Number]

2 minutes ago, PeterPan said:

This. Sometimes what happens is people got good grades at math, did what the teacher said, and it was all *memorized*. So then you can't teach it, because you don't really get the why or the thought process. You aren't really thinking in terms of multiple ways to solve problems or proof. It's more here's the equation, this is how you do it, boom done. 

Yep. I see this more often then not.

[EKS]

4 hours ago, PeterPan said:

This. Sometimes what happens is people got good grades at math, did what the teacher said, and it was all *memorized*. So then you can't teach it, because you don't really get the why or the thought process. You aren't really thinking in terms of multiple ways to solve problems or proof. It's more here's the equation, this is how you do it, boom done. 

This is a good point.  When working through the material ahead of your student, it is important to always be thinking about the whys and how you might explain things in different ways.  

I know that my approach to teaching writing (writing has always been intuitive for me--I even worked as a professional scientific writer for a few years) changed drastically after I took several writing intensive graduate courses after having homeschooled for 10 or so years where I forced myself to be stretched to the limit.  The entire time I was also thinking about my students' experience with writing and how to convey what I was struggling with and the various ways I overcame it to them.

Edited October 14, 2020 by EKS

[Momto6inIN]

15 hours ago, Chelli said:

Any recommendations to help a homeschool mom improve either or both of those?

I was "good at math" in school because I could memorize processes quickly and could easily figure out which ones to regurgitate when. When I started teaching math to my kids, I realized how very little actual conceptual understanding I had. Going through Math Mammoth and then Video Text alongside with my kids helped fill in the "why" gaps of my knowledge and I'm much more confident now.

That's not meant as a curriculum suggestion - several other conceptual programs would do the same thing. But going through the programs with my kids and learning the why was crucial.

Sometimes I still don't explain the why to them very well when they get stuck, so I'm far from a perfect math teacher. I will never be Not_a_number. But at least the concepts are straight in my mind and now my problem is how to best convey those concepts to the kids, which I feel is a typical teacher hurdle to overcome, not a failure, which I think I kind of may have been before going through the programs myself. Good thing my first kid is natural math whiz and didn't need me to teach him while I was going through them for the 1st time 🙄

[frogger]

26 minutes ago, Momto6inIN said:

I was "good at math" in school because I could memorize processes quickly and could easily figure out which ones to regurgitate when. 

I remember discussing elementary mathmatics with a very successful adult who by all appearances understood math and could when he stopped to think about it. I complained about our English words 11 and 12 and that they don't show the pattern of math.  My words were something like, "What the heck is 11?"

His response was, "It's two ones." He was implying just memorize it or come on stupid.

I said, "No two ones make two, not eleven." 

He got quiet. Lol

It seems silly, but once learned human brains tend to accept things unquestioningly. He knew what 11 was from experience. He could tell you what added up to 11. He could use 11 for all sorts of complicated operations but he also just accepted it not really thinking about why it was written that way or how to explain it to someone who had never encountered an 11 before.  I'm sure it wouldn't take long on such a simple thing but I find many adults struggle with very simple common core problems because they just memorized facts and steps. That is how my generation was taught math.

I'm not saying you did OP but stopping and recognizing that when seeing something for the first time it is just random facts to memorize until we understand is important.

[Not_a_Number]

45 minutes ago, frogger said:

I remember discussing elementary mathmatics with a very successful adult who by all appearances understood math and could when he stopped to think about it. I complained about our English words 11 and 12 and that they don't show the pattern of math.  My words were something like, "What the heck is 11?"

His response was, "It's two ones." He was implying just memorize it or come on stupid.

I said, "No two ones make two, not eleven." 

He got quiet. Lol

It seems silly, but once learned human brains tend to accept things unquestioningly. He knew what 11 was from experience. He could tell you what added up to 11. He could use 11 for all sorts of complicated operations but he also just accepted it not really thinking about why it was written that way or how to explain it to someone who had never encountered an 11 before.  I'm sure it wouldn't take long on such a simple thing but I find many adults struggle with very simple common core problems because they just memorized facts and steps. That is how my generation was taught math.

I'm not saying you did OP but stopping and recognizing that when seeing something for the first time it is just random facts to memorize until we understand is important.

This is why I don't really "teach" math 😉 . I think of our math learning as making sense of definitions and putting them all together in interesting ways. After we understand the definitions well, we learn shortcuts. It takes a long time to learn the definitions well -- longer than we think. And the shortcuts come much easier after the definitions are internalized. 

We subtracted and added two digit numbers using pictures for a really, really long time. And we talked the whole thing through, and I never even mentioned "carrying" or "borrowing." I need kids' brains to make sense of place value, and that takes months of playing with it. All the algorithms make sense after one's brain has a way of interpreting the conventions. But the hard work is understanding the conventions and definitions. The shortcuts are actually later work. But the way we teach, we assume the definitions are easy (false) and that the shortcuts come first. 

[wendyroo]

49 minutes ago, Momto6inIN said:

I was "good at math" in school because I could memorize processes quickly and could easily figure out which ones to regurgitate when.

I find this to be true with so many adults.  My husband has two engineering degrees.  He is "good" at math, but he was trying to walk my son through a  double digit multiplication problem he had gotten wrong and he just hit a wall.

The problem was something like:

     54
   x37

Life was fine multiplying the 7 by 4.  They "regrouped" the 20 from 28 and noted it up above the 5 (which makes perfect sense because that is the tens column).  DS understood perfectly that the next step was multiplying 7 by 50 and then adding on the 2 extra tens.  But then things went haywire.  DS came up with 37 tens and wanted to write the 7 in the tens place in the answer and note the 3 hundreds as a regrouping note up above the empty hundreds place.  DH said that wouldn't work.  Except clearly it could - the next step would be multiplying 7 by 0 hundreds, adding in those 3 regrouped hundreds, and writing it in the hundreds place of the answer.  But DH would not budge; in his mind that is just not how you did it.

As one could foresee, things got worse when they were multiplying 30 by 4.  DS clearly saw this as 12 tens and wanted to write the 2 in the tens answer place and note the regrouped 1 above the empty hundreds place.  DH said no, you have to note it above the tens place.  Which is very common based on YouTube tutorials I have watched, it is even how they teach it in Math Mammoth much to my displeasure, but conceptually makes so sense.  Then 30 by 50, is 1500, plus the regrouped 1 should be 1600, but DS wrote 1510 because he was convinced that a regrouped 1 noted in the tens place should be worth 10...which it absolutely should, but DH just could not see that.  In his mind, it HAD to be noted above the 5, and it also HAD to be counted as a hundred.  He could not explain why that was, he was just sure those were the facts.

When I am deciding if I am ready to teach a concept, I imagine doing an example and having the student ask why about every single little thing I do.  If I cannot conceptually justify every step of the process, then I am not ready to teach that concept.

[Not_a_Number]

2 minutes ago, wendyroo said:

I find this to be true with so many adults.  My husband has two engineering degrees.  He is "good" at math, but he was trying to walk my son through a  double digit multiplication problem he had gotten wrong and he just hit a wall.

The problem was something like:

     54
   x37

Life was fine multiplying the 7 by 4.  They "regrouped" the 20 from 28 and noted it up above the 5 (which makes perfect sense because that is the tens column).  DS understood perfectly that the next step was multiplying 7 by 50 and then adding on the 2 extra tens.  But then things went haywire.  DS came up with 37 tens and wanted to write the 7 in the tens place in the answer and note the 3 hundreds as a regrouping note up above the empty hundreds place.  DH said that wouldn't work.  Except clearly it could - the next step would be multiplying 7 by 0 hundreds, adding in those 3 regrouped hundreds, and writing it in the hundreds place of the answer.  But DH would not budge; in his mind that is just not how you did it.

As one could foresee, things got worse when they were multiplying 30 by 4.  DS clearly saw this as 12 tens and wanted to write the 2 in the tens answer place and note the regrouped 1 above the empty hundreds place.  DH said no, you have to note it above the tens place.  Which is very common based on YouTube tutorials I have watched, it is even how they teach it in Math Mammoth much to my displeasure, but conceptually makes so sense.  Then 30 by 50, is 1500, plus the regrouped 1 should be 1600, but DS wrote 1510 because he was convinced that a regrouped 1 noted in the tens place should be worth 10...which it absolutely should, but DH just could not see that.  In his mind, it HAD to be noted above the 5, and it also HAD to be counted as a hundred.  He could not explain why that was, he was just sure those were the facts.

When I am deciding if I am ready to teach a concept, I imagine doing an example and having the student ask why about every single little thing I do.  If I cannot conceptually justify every step of the process, then I am not ready to teach that concept.

Huh, I'm having trouble following your example 😄 . I think I might do it how your DH does -- I note however many extra of whatever above the next digit I'm multiplying by. I don't keep real good track of whether they are 100s, 10s, or what. But then we only really teach the algorithm after all those kinks have been worked out, so there are no conceptual issues -- just accounting issues, and then we don't worry about place value all that much, since it's all just a shortcut. 

[Lori D.]

Love the previous poster responses.

I found that having a math struggler made me have to learn how to be a better teacher of math, and I was able to do that (up through pre-algebra) by us having used multiple math programs that all came at math from different angles. That helped ME to see some of those math connections early on, and that there was more than one way to problem-solve or even "see" the math, as Rosie said up-thread.

After pre-algebra, while I understood how to solve the problems myself, or could figure out the lessons, I realized that the math was more advanced than I knew how to teach beyond the math program's explanations if we hit a stumbling block. Like Wendyroo and EKS said upthread, what would have best helped me was to have been trained into how to teach upper level math AND be able to see where a student was struggling AND how to have different tools in the teaching toolbox for helping the student approach the math in different ways.

JMO re: the higher maths: if the child is just hitting a rough patch with a particular topic, I do think that can be addressed in the way Zoo Keeper, LinRTX, and others above suggest. But if it is a child hits the higher maths and has bigger and more long-term math difficulties, and the parent is not able to get that "training to be a math teacher" kind of depth of concept understanding needed to tutor the student through it, that might be the point to outsource to an expert *math teacher* or tutor.


Chelli's exact same questions can be applied to teaching writing: 
"What do you think is the most important component to being successful at homeschooling writing?"
"Is it curriculum? The teacher's understanding?  A combination? Something else?"
"How do you take a kid who doesn't like writing or doesn't understand writing to a place of confidence?"

In some ways, I think teaching writing is even harder than teaching math. At least math has objective concepts; writing, and the idea of what makes "good writing" is much more subjective. Sure, there are some very clear guidelines, such as grammatical structure, what goes into a paragraph, formatting requirements, etc. But writing at heart is expression of an individual thoughts, and it's a tricky balance of guiding students into learning the structural and content requirements of formal writing, while still encouraging individual style and voice... 😉 

Again, like Chelli, I am a natural writer, which made teaching writing to (NON natural writer) DSs very difficult. It's taken me a good 6 years of trying to teach Lit. & Writing at homeschool co-op classes -- it has required a LOT of looking at different writing programs, but especially a lot of reading, research, and learning to get a much better handle on how to *teach writing*. And especially how to come at teaching writing from different angles or with different explanations for students who struggle with writing and grammar.

 

1 hour ago, frogger said:

...  I complained about our English words 11 and 12 and that they don't show the pattern of math.  My words were something like, "What the heck is 11?" ...

Side note about this question -- eleven and twelve do show an implied mathematical pattern, but the words are from a different linguistic source than the "teens":

Eleven is from the Old English endleofan "...which is from Proto-Germanic ainalif (“one left”) (i.e., one left over after having already counted to ten)"

Twelve is from the Old English twelf [side note: which explains the adjective "twelfth" rather than "twelveth"] "...which is from Pro-Germanic twalif, an old compound of  twa- (“two”) and -lif (“left over”) (i.e., two left over after having already counted to ten)" 

Edited October 14, 2020 by Lori D.

[frogger]

37 minutes ago, Not_a_Number said:

This is why I don't really "teach" math 😉 . I think of our math learning as making sense of definitions and putting them all together in interesting ways. After we understand the definitions well, we learn shortcuts. It takes a long time to learn the definitions well -- longer than we think. And the shortcuts come much easier after the definitions are internalized. 

We subtracted and added two digit numbers using pictures for a really, really long time. And we talked the whole thing through, and I never even mentioned "carrying" or "borrowing." I need kids' brains to make sense of place value, and that takes months of playing with it. All the algorithms make sense after one's brain has a way of interpreting the conventions. But the hard work is understanding the conventions and definitions. The shortcuts are actually later work. But the way we teach, we assume the definitions are easy (false) and that the shortcuts come first. 

I did that with my first child Number (do you mind if I type number for short?). We used Miquon, real life, I basically just gave him definitions for what he already knew.  I thought if people just slowed down, everyone could learn that way. His first real math program that he completed rather than just toyed with here and there was videotext Algebra.  He is now a double major in Math and EE. 

 My second child didn't give a rip though She just didn't care to do the hard work.  She was much older when I realized I would have to try something different. We may still be in 3rd grade math if I would have kept it up.  LOL I tried interesting puzzles, card games, blocks, stories that used principles of math, and I did have faith since it worked so well with child 1. Child 2 humbled me. 😛 If I didn't directly teach everything, she probably wouldn't have passed even pre-algebra.   She is finishing up Statistics though and will be done with math unless SHE CHOOSES to get interested as an adult.  I will be glad to let her fly. She is stubborn and will do well at whatever she puts her mind to, including math. But I couldn't let her graduate with no math just because she didn't feel like doing it. 🙂 

I'm very glad you are so enthusiastic about math (I do agree in with many of your principals)  but be careful with others and be kind. Life doesn't always work so simply. 

[frogger]

45 minutes ago, Lori D. said:

Side note about this question -- eleven and twelve do show an implied mathematical pattern, but the words are from a different linguistic source than the "teens":

Eleven is from the Old English endleofan "...which is from Proto-Germanic ainalif (“one left”) (i.e., one left over after having already counted to ten)"

Twelve is from the Old English twelf [side note: which explains the adjective "twelfth] "...which is from Pro-Germanic twalif, an old compound of  twa- (“two”) and -lif (“left over”) (i.e., two left over after having already counted to ten)" 

 I do remember hearing that before but unless people know it, it really is no use to them and it isn't often taught. 🙂 But it is an interesting fact. 

[Not_a_Number]

43 minutes ago, frogger said:

My second child didn't give a rip though She just didn't care to do the hard work.

So, interestingly, I haven't found that it's hard work for the kids to do the work the way I do it 🙂 . At least the ones I've taught so far, but that includes a good number. The tricky thing is to give them questions that are GENUINELY doable with the definitions and don't require banging your head against the wall.

So, if I tell a kid that the number with the digits a b contains a copies of 10 and b copies of 1, and that green poker chips are tens and blue poker chips are 1s, and that a + means "putting together," I'm actually giving them enough tools to work out 

23 + 34,

I really am. It's not actually harder than using an algorithm that doesn't involve poker chips (or visuals, or whatever.) But the point, for ME, is to create problems that actually exercise the conceptual understanding, where by "conceptual understanding" I tend to mean having a reliable mental model for operations and a way to reason about them. 

But I'm not arguing all kids are equally good at this kind of work! I guess I have faith that you can give kids conceptual tools and that they will learn to use them at their own pace. But yes, I also would require the work to actually be done 😉.