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Edited to add: New topics to munch on:

Page 2, post 12: What are the ramifications of the move to online math programs as a way of individualizing pacing?

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Square25 and I completely overran a LC thread on math this morning, then we started talking by PM.  I think that some people might be interested in our discussion as we both have opinions on math education.  

Basically we are discussing the ramifications of teaching procedural knowledge vs conceptual knowledge first.  Both of us are high end educators with experience with a range of students, and we have each seen a lot and tried to learn from our successes and failures. 

I'm going to just dump you guys in the middle of the discussion with my last post to Square25 by PM. I am making the case for having no choice but to drill in procedures fast with little conceptual understanding to make time for more complex problems.  

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You would find the NZ exams very interesting because the test for insight, generalizations, abstractions etc.  If you can only do what you have been shown, even if you get 100% correct, you will only get a C. So for algebraic graphing, I have to do basic drill on lines, parabolas, exponentials as fast as I can so that we can move onto *using* the math to actually model complex problems as this takes a LOT of time to build up skill in. 

Check out this exam.  1.5 hours for 10th graders. It requires very high level understanding completed in a very short period of time. This exam is typically given back to back with a 1.5 hour geometry exam so there is an endurance component too.

https://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2018/91028-exm-2018.pdf

If you sit down and actually try to do the whole exam in 1.5 hours including graphing  (because to get full marks you need both a graph and an equation) and using a PEN (so no erasing) you will realize just how hard it is.

Conceptual understanding for this exam is way way more than just *why* does this equation create this type of graph.  So I have to budget my time.  What is more important? Algebraic conceptual understanding or insight into modelling complex problems?  This unit represents 20% of a 10th grade integrated math program of algebra, geometry, and statistics, so only about 7 weeks of time to build up this kind of skill.

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I don't think it's quite fair to expect kids to have a high level of conceptual understanding in a short period of time, especially since you can start working on conceptual understanding considerably before you become fluent in calculations.

I completely agree.  For students here who want to work at an excellence level, my best option is to get them a year early (9th grade) and start using their more basic graphing knowledge (lines only) to model complex problems.  Basically, questions like the first one on the test, which only required line knowledge which is 9th grade knowledge.  But when I have less time, my focus is on the modelling of the word problems rather than a deep conceptual understanding of linking the algebra to the graph. For most of my students, using math to model real life problems is more important than procedural or conceptual understanding of algebra on up.  Most people will not use advanced algebra, but they would be well served by the ability to convert real life problems into mathematical language.  And I am not talking about word problems.  Those are pretty useless.  NZ is now focusing on investigations, where often there is not ONE answer.  Notice the name of the exam -- investigating tables equations and graphs.  It is not called a mathematical understanding of algebraic graphing. This different focus requires different priorities from me. 

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I wish we could write classes where the conceptual skills and the computational skills were separate strands that I tackled at a relatively leisurely pace, combining them as appropriate. 

I agree with this too.  For my best students, I ram them through the basic knowledge as fast as possible, and then give them separate more advanced workbooks to do IN CLASS while the teacher is still working on the basics.  Insight, abstraction, and generalizations take a lot of time to master, and here in NZ they never ever leave enough time.

We have a teacher shortage so currently have a number of foreign trained teachers from America, UK, and even Russia.  Most of the programs there are much more driven by a pure-math approach rather than what NZ does which is much more big picture, modelling, use-in-life approach.  What this has meant for my students with these foreign trained teachers is that too much of their class time is focuses on pure math that is of course important but not any more important that using math for complex modelling.  The fact that these kids' class programs are misaligned with the NZ exams, is just one more thing.

So I drill in the procedures fast and furious and then work for remaining 6 weeks to use them in a deep and meaningful way.  In a holistic way.  

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I think the ideal would be to cover less content so that you could do a proper job with 1) building up the conceptual knowledge of pure math first like Square25 does, 2) then drill in the procedures, 3) then use the math to model complex problems.

Personally, I would completely abandon geometry.  I would also abandon the more esoteric topics of Algebra 2 and PreCalc.  Calculus is wonderful for modelling complex problems so I would keep that. But if I also want to pull in Statistics, then I am sunk.  Too much content to do it well and have kids actually understand the content well enough to use it. Most kids do math because they are told they have to, but they have no expectations they will actually use it. sigh

Have to run and tutor some kids in MATH!  Be back in a few hours. 

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@lewelma  Could you provide some more background on this exam you linked to in the first post?  Is it for all 10th graders?  What score is required to advance to 11th grade?  What is a median score?  Is this exam score used for college admissions?  Is the Level 2 exam more challenging?

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2 minutes ago, daijobu said:

@lewelma  Could you provide some more background on this exam you linked to in the first post?  Is it for all 10th graders?  What score is required to advance to 11th grade?  What is a median score?  Is this exam score used for college admissions?  Is the Level 2 exam more challenging?

This exam is for all 10th graders, and I would say about 80% take it. It is curved to create about 15% As, 25% Bs, 40% Cs, and 20% Fs. To earn a C, you need to get about 30% correct (you will notice the first 2 pieces of each question are very easy).  In NZ, all students take the same exam whether regular, honors, or gifted courses, which means all students can be compared to the same benchmark.  

In NZ the exams are not based on percent correct, rather are marked based on the level of thinking demonstrated. So the test is carefully designed to have a balance of questions - regurgitation earns you a C, relational thinking earns you a B, generalization/abstraction/insight earns you an A.  It is also a forgiving test, so that you only have to show generalizations/abstraction/insight on 30% of those advanced pieces to earn an A. This means you have more than one opportunity to show your level of thinking. So on this test you could have gotten an A if you only answered one question completely, and got only the moderately-difficult pieces of the other 2 questions. So you can shine in your strength. 

To get into university here you must pass 10th grade math, but only 3 of the 5 units you take during the year. And those 3 could be numeracy, measurement, and statistics if your school designs a program as such (there are 12 units available a la carte style for teachers to choose). But I will say that those exams are only slightly less hard.  But a key difference to America -- you do not need to pass an algebra exam to get into university here.    

To move up to 11th grade, you typically need to pass 4 out of the 5 exams your school offers (although I have seen exceptions). So basically, they drop your lowest grade. In addition, NZ has a very high end qualitative statistics course for 11th and 12th graders who want to continue in mathematical thinking but have no interest in the algebra/calculus route (or who have failed the algebra unit). 

About as clear as mud.  But I find it a very good system to get kids working on the higher level content. What you test is what kids try to learn.

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Does NZ give some sort of allowances for students with learning differences, like dysgraphia, with extra time, or other accommodations?  

There is much discussion locally about accommodations in US universities for students with disabilities, even those that are learning differences.  I'm on the fence about it.  I hate to see an LD holding back a student who could otherwise do the work.  But then I wonder what is the point of college anyway if not to give the best learners an opportunity to move to higher learning?  Do we grant accommodations to athletes who are too short for basketball?  

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56 minutes ago, daijobu said:

Does NZ give some sort of allowances for students with learning differences, like dysgraphia, with extra time, or other accommodations?  

There is much discussion locally about accommodations in US universities for students with disabilities, even those that are learning differences.  I'm on the fence about it.  I hate to see an LD holding back a student who could otherwise do the work.  But then I wonder what is the point of college anyway if not to give the best learners an opportunity to move to higher learning?  Do we grant accommodations to athletes who are too short for basketball?  

 

University in the US is, for most students, a job qualification program.

It isn't like basketball because no-one needs to be a basketball player, because basketball teams need a only a very limited number of athletes, and because you don't need anything other than the right physical characteristics plus basketball specific skills and fitness to be a basketball player.

I don't need calculus or college algebra to be an elementary school teacher, but I do need to make it through college math because I need a college degree. I don't need great writing skills to be a computer programmer, but again I may need a college degree. I don't need to expound on controversial points in history to be a dentist, but I need a college degree. I don't need to be quick at completing tests to be a business manager, but I may need to get through that MBA.

Do you question whether my blind sister in law should have received accommodations for her disability in college? If not, why would it be less appropriate to accommodate someone with a disability like dysgraphia?

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35 minutes ago, square_25 said:

I think kids need to learn things when they are ready.

So disaster with my tutor kid just now. He is taking an algebra exam on Monday and I didn't see him last week because of our working bee.  So he comes today with the practice tests on quadratics and factoring/expanding.  He can factor/expand, but only if it is exactly one kind (x-3)(x-2) variety. Any deviation from this format led to confusion.  Then quadratic equations - oh my word, he couldn't understand them. I tried explaining them conceptually with these are 2 things multiplied together, if one is zero, they equal zero.  I tried substituting the answers back in to prove to him what works and what doesn't. I tried comparing it to linear equations. I tried just procedural drill.  Nothing. So I said, 'well, we can do this next year.  Let's focus on your strengths.' This kid is NOT ready for this work, but on the school marches. I am SO glad that I had the privilege to homeschool my kids. This kind of thing just makes me sick.  I rescued him from the 'cabbage' class this year, and now I am regretting my choice. sigh. 

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48 minutes ago, Paradox5 said:

Since I am not a mathematician nor an engineer, I shall bow out.

 Aw, I didn't name this thread, "Mathematician and engineer opinions on math education." haha. Please feel free to post. 

If it helps at all, I'm a biologist and science teacher by training. 🙂  

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1 hour ago, square_25 said:

I would personally start more abstract stuff earlier.

I think this would work if we allowed for more differentiated math learning, but a lot of my students developed mathematical maturity very late.  I have one student who at the age of 14, when I asked her "if you have 6 apples and I give you 3 more, how many apples would you have?" - she had NO idea.  She had used MUS and had gotten through the first 4 books (up to 4th grade) and then just stalled.  She could compute, but had NO idea what adding even was - at 14. 

4 years later she is taking calculus.  yes, believe it or not, I have turned this around. But I have come to believe that part of her brain just didn't mature until she was 16. She is *definitely* not my only student like this, just the most extreme case.  I have come to believe that there is no way that you can guess at where a kid will top out, nor how long it will take for them to master content. For this girl, after working with her for a year, I planned to get her into some sort of consumer math, just basic life skills, because she just could not grasp anything I taught her. Wow, what a change! 

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Just now, square_25 said:

Ugh. 

Ugh is right. Then today he tells me he is thinking about dropping out of school because he has a job repairing cars, and LOVES it. And his boss doesn't have 10th grade math qualifications, so why should he?  Well, maybe he doesn't need them. Not here. Don't know. sigh. 

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1 hour ago, square_25 said:

Honestly, my perspective probably isn't going to win me any friends, but I think to teach elementary math effectively you have to have a deep understanding of the connections between the different parts of elementary math.

I think you are right. I'm very good at what I do, but if you give me a younger student or an older student who is still doing primary school math, I'm often at a loss for how to help.  I am practical, not theoretical; but the theory could help me solve some of these problems. 

 

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10 minutes ago, square_25 said:

To be fair, it's possible that if she'd had hands-on math instruction earlier, she'd have done better earlier? I can't be sure, of course, because I've never met her! But I do think a lot of kids' math education are missing the language component, which leads to all sorts of weird stuff. Like, it's possible was adding lots of numbers without realizing she was actually putting collections together? I saw this firsthand in my homeschooling classes: kids could "add," but it literally didn't occur to them to add when playing blackjack. They didn't think they were actually adding when putting together collections...

She was and is fascinating.  My approach with her in our first year together (age 14) was to get her into the kitchen. I gave her a list of things she needed to practice every week in the kitchen to build up conceptual understanding. Then for all fractional work, I had to draw little measuring cups in her notebook because she couldn't understand pies, like couldn't understand the idea of splitting a circle in half. It was crazy. 

 I also had her mom rewrite Every. Single. Word. Problem in MUS books 1-4, and had her do them all again (she had done them all before while getting through the books the first time).  And after she did them this second time, I realized she was still struggling. So I had her mom rewrite Every. Single. Word. Problem in books 1-4 all mixed up, and she did them again! At that point, she had a vague notion which I considered great progress! At this point she was 15. She is now 18 and doing calculus.

The development of her mathematical maturity has been truly (as in literally) exponential. Very eye opening for me.

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1 minute ago, square_25 said:

 

See, the fact that she didn't know how to split a circle in half makes me feel like there was something seriously deficient around their house. I think "half" is way more approachable when it's half of something you can count. And I notice myself talk about halves all the time with my kids. And I would notice quite early if they didn't really know what that meant, and I would explain. And from the sounds of it, none of that kind of catching and troubleshooting was going on with her. It sounds like she was being taught completely open loop. 

Oh, absolutely. Her mom was *clueless* in math. As in would mark all her dd's work, and if dd wrote 0.1 and the answer was .1, the mom would mark it wrong. Absolutely NO math skills what so ever. She has absolutely NO idea the success I have pulled off. None. 

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This is such an interesting discussion...thanks for including us in the discussion.  I am an engineer and have had many discussions about learning concepts and applications along with math facts vs drilling facts for years.  My kids span a wide age range, but my youngest is in first and my second is in algebra this year so I am sort of starting over again looking for the best balance.  Do either of you have experience with right Start?  It seems balanced, but I tend to second guess myself.

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I have been horrified recently that students in 6th and 7th grade are being put onto computers for math, expecting to self teach.  I've now had 3 of them.  One switched schools because of it, and the other boys told me that not only did the the teacher not teach, she didn't even supervise.  So they didn't do any of the work for 2 years and just socialized!  So I just said, "well, let me be clear, not only have you not learned 6th and 7th grade math, but you have forgotten 5th grade math. So that is where we will start even through you are in 8th grade." Kids like to hear it straight, and one of these boys shaped up, and the other is the boy I described above who is likely to fail algebra next year.

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5 minutes ago, Mom2mthj said:

This is such an interesting discussion...thanks for including us in the discussion.  I am an engineer and have had many discussions about learning concepts and applications along with math facts vs drilling facts for years.  My kids span a wide age range, but my youngest is in first and my second is in algebra this year so I am sort of starting over again looking for the best balance.  Do either of you have experience with right Start?  It seems balanced, but I tend to second guess myself.

Glad you are enjoying it. Hope you have opinions to share!

Is Right start the program with all the games and manipulables? If so, I did look at it and was impressed.  I used MEP and Singapore math with my younger, and then we did 3 years of PreAlgebra (crazy I know, but I just wasn't convinced he was grasping stuff, and like Square25, I was not moving on until concepts were rock solid).  My older has been self-taught since the age of 7.5. He considered teaching to be cheating, which included the textbook I might add.  So I'm not even sure how he learned fractions. My best guess is trial and error with the answers, but he doesn't remember. haha.

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6 minutes ago, square_25 said:

 

 procedural competence or the ability to regurgitate what they just said as evidence that someone is no longer confused. It takes a seriously open mind to be willing to be unpleasantly surprised about how little someone really understands. 

I think this is where the American approach to math tests leaves people thinking they know the content because the tests are mostly procedural.  NZ kids take exams like the one posted above, and it is very clear that they are clueless because they fail them or only achieve them.  And I might add 60% achieve or fail, which means only 40% of students have recognizable higher order thinking skills in math. 

What a test tests is what a student learns.  

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3 hours ago, daijobu said:

Does NZ give some sort of allowances for students with learning differences, like dysgraphia, with extra time, or other accommodations?  

I don't know about university, but high school definitely.  I have had students with reader-writers, extra time for slow processing speed, and private rooms for anxiety.  In fact, my ds is just now starting the national exams and I told the chemistry teacher that ds has dysgraphia and struggled to write down all the workings to support his answers. She told me that he was allowed to *record* on audio how he got the answers as evidence of the thinking. I was like, WOW. 

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7 hours ago, square_25 said:

 

For example, for the AoPS precalculus class, the last section is linear algebra (that is, vectors and matrices), and even with an extremely conceptual focus, I still have a third of the class confusing vectors and points by the end of the 9 weeks.

 

 

I'm really surprised here.  I mean, in my own high school, I remember on the first day we were told "A vector is a quantity with magnitude and direction."  I suppose I wasn't the only one because I was pleased to see this in a popular movie:  

 

Interestingly I just took on a new student who is doing the AoPS online precalculus course.  I've been working on his homework on complex numbers and those last 2 were challenging!  I can't image students who made it that far in the course are struggling with vectors.  

 

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Loving this thread! I'm more with you square_25 I adore teaching elementary math and conceptual/hands on is where it's at, in my very amateur opinion. I'm not a teacher or mathematician, I didn't do any formal maths after grade 12. I did quite well in school but maths was very procedural and a couple of truly abysmal teachers meant I tapped out and instead focused on humanities.

I love maths though, and I love it more having now had the chance to get down into the sandbox with my kids and get my hands dirty making it conceptual. I can't say with authority whether it works best though, I've only got one kid through aops pre-alg so far! 

One thing that inspired me (other than miquon, Singapore maths and aops) is math circles and specifically this book https://www.amazon.com/Math-Three-Seven-Mathematical-Preschoolers/dp/082186873X

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26 minutes ago, LMD said:

Loving this thread! I'm more with you square_25 I adore teaching elementary math and conceptual/hands on is where it's at,

...get my hands dirty making it conceptual.

More elementary math people!  Great! Boy do we need them!  But don't get me wrong, I *definitely* teach conceptually. Square25 and I have never disagreed about teaching conceptually, just what comes first (procedural or conceptual) and the ramifications of that choice. I have argued that for some students it is better to teach procedural first, drill in the concept, and then teach them to understand it conceptually. I think that many teachers who take this approach have good intentions but never get around to the conceptual understanding because they run out of time. I do the opposite.  I drill in the procedural work fast and furious and leave 70% of the time to work on conceptual knowledge and generalizations/abstractions/insights. What I find is that many students understand more when they can see the whole. So with a tool chest of tools, we can then work on deep, long, difficult problems that require the use and understanding of many tools all at once. I have found this approach both incredibly efficient, effective, and motivating. Go look at the test I linked to above to see the kind of thinking that I work towards. 

So for me instead of teaching isolated concepts deeply and in full one at a time, I instead often do a quick drill in of all the basic procedures, and spend all my time then using them in a holistic manner to solve and model complex problems. 

What I need to think about next, is what students do I do conceptual first, and why do I make that choice. I definitely do it sometimes, but the decision is made on the fly based on my judgement of the situation.  I'm curious now, what triggers my different approaches.....

 

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1 hour ago, daijobu said:

Ahem?  Some sort of gardening course?  

I know, right. But kids go into that class to die a slow math death. They don't work extra hard to catch them up. Instead they slow down because they have slower kids, so the kids fall further and further behind.  Here, it means that you are routed into 'unit studies' instead of 'achievement standards.' They can get you the university entrance you need, but mean that you can never do anything remotely mathematical at university including all the social sciences because you have only done consumer math. In my ideal world, kids in that class would work to get into the 10th grade class a year or two late, but that is just not how it is done at the 10 high schools that I work with. sigh.

Lock step. We all must go in lock step. The ramifications of this is often kids barely making it through with little understanding and then moving on to the next class. In reality, it would be far far better to teach high level thinking skills on less content, then to cover more content at a surface level. But this is not the way of schools. 

My older boy took 3 years to get through Algebra. Yes he was young, and yes he was doing AoPS independently, but I gave him the gift of time.  And 3 years felt like a very long time to this first time homeschooler. My younger is currently doing his third loop through beginning calculus before I move him into the advanced course. If he can't think abstractly now, why would I expect him to think abstractly with even harder content?  The best approach in my opinion is to work up to the highest level of thinking with all content before moving on. But this takes time that we often refuse to give our kids. 

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My background to bring to this discussion:

I did extremely well at maths at school and went on to do some maths at uni as well. Throughout highschool and uni I tutored highschool maths. My path changed and I ended up as a Montessori early childhood teacher.

I have three daughters. School didn't work for my youngest, who I've now homeschooled for 8 years. 

She's 13 and recently sat the Senior External Exam in Maths B, here in Queensland, Australia. This exam is two 3hr exams that cover all grade 11 and grade 12 Maths B. So that's algebra, trig, geometry, statistics, financial maths, logarithms, calculus etc. 

I thought I was pretty decent at maths until I homeschooled my daughter. I realised that I'd gone quite a way in maths without real conceptual understanding. My daughter however has demanded full conceptual understanding. Even if she got things correct, she'd refuse to move on until she absolutely understood the 'why'.

I have never had such a thorough maths education as I have homeschooling this daughter, and I'm grieving a little now that she's done.

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2 minutes ago, chocolate-chip chooky said:

She's 13 and recently sat the Senior External Exam in Maths B, here in Queensland, Australia. This exam is two 3hr exams that cover all grade 11 and grade 12 Maths B.

Wow! That is impressive! Go you!

 

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I talked to my younger boy today about this conversation when we were out walking. He said the oddest thing. He told me that he never truly understood fractions until we did exponents.  I was like What?!?!? Apparently, the -1 exponent allowed him to make a break through into division with fractions, which filled in a glaring hole in his mind and finally settling fractions deep into his understanding. 

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1 minute ago, chocolate-chip chooky said:

Thanks, Ruth. But really it's all 'go her'. I did a lot of ...um... 'emotional management' more so than maths guidance. I think you understand.

Yes I understand! But I also think you underestimate your role. So much of what I do for my tutor kids and my own kids is 'emotional management.' This is not a small part of the job, it is at least 30%. I think you deserve some kudos too!

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4 minutes ago, lewelma said:

Yes I understand! But I also think you underestimate your role. So much of what I do for my tutor kids and my own kids is 'emotional management.' This is not a small part of the job, it is at least 30%. I think you deserve some kudos too!

I cringe a bit, thinking back to all my angsty posts about 3 years ago, some on the AL board but many in the private group. My daughter refused teaching as such for quite a period of time, thinking it cheating, and the emotional outbursts at times were challenging, to put it mildly.

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56 minutes ago, chocolate-chip chooky said:

I cringe a bit, thinking back to all my angsty posts about 3 years ago, some on the AL board but many in the private group. My daughter refused teaching as such for quite a period of time, thinking it cheating, and the emotional outbursts at times were challenging, to put it mildly.

Oh, so been there. Don't cringe. Parenting is tough, and we are always newbies with every kid. I also think we do a service for other parents when we put it out there and actually describe some of the difficult times and not just sugar coat everything. So in that vein, my older boy used to punish himself when he was 7 if he got something wrong by giving himself 2 more hours math to do, refusing all help, food, and breaks until it was done correctly. And by age 9 I remember him crying and raging over the AoPS but refusing all help.  That is when I hid his book!  I would only give it back when we came to a long-term agreement about math and his mental health. Now, age 19 he is super excited to be writing up a book chapter for a new textbook on recent research in Combinatorics. 🙂

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7 hours ago, square_25 said:

 

Honestly, my perspective probably isn't going to win me any friends, but I think to teach elementary math effectively you have to have a deep understanding of the connections between the different parts of elementary math. I think Liping Ma's book Knowing and Teaching Elementary Mathematics explains that quite well (although for my money, the Chinese curriculum she described still leans far too procedural.) In the same way that I'm a much more effective Russian teacher than French teacher because I speak Russian, I'm a much more effective math teacher than I would be if I didn't have a thorough grasp of the concepts. I can generally spot what it is that's confusing a student, and I imagine that's a lot a harder if you're only operating procedurally. 

I guess what I'd suggest is to learn math conceptually yourself along with your kids and to be open to ways of doing things that you didn't learn at school. But I know that's hard work and takes a lot of time... 

 

It is worth noting that all the kids taking that exam will have their early instruction from generalist teachers who often have quite limited understanding of maths.  The vast majority will have had no specialist maths instruction until high school (grade 8 to 12) and even then it is likely for the first year or two it would have been from a science teacher required to teach junior maths against their inclination (mine straight out said he didn't want to teach maths).  This may be why Ruth has so many confused students.

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5 hours ago, lewelma said:

Ugh is right. Then today he tells me he is thinking about dropping out of school because he has a job repairing cars, and LOVES it. And his boss doesn't have 10th grade math qualifications, so why should he?  Well, maybe he doesn't need them. Not here. Don't know. sigh. 

He should check.  Unless his boss is in his 20"s there will have been major changes in the apprenticeship structure.  As a general rule the amount of bookwork increases due to more technology and safety regulations.  Also whereas apprentices when I was a kid often started at 15 with or without school cert a bit more is expected now.  Some kids do really well finishing NZCEA at a polytech trade course though.  Really any credits he can get will be useful at some point.

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Sorry Lewel, didn't at all mean to imply you didn't teach conceptually. Honestly, I drink in your posts and consider it a great privilege that you share on these boards. I have some of your posts bookmarked (the science ones).

I can see the wisdom in getting the drill out of the way. I tend to be big picture first and stall on procedural stuff that I can't yet understand the meaning of, so I teach from what works for me. And my kids, so far

I just got a little excited about the early conceptual maths talk 😄

My kids are decidedly average scholars, but I love seeing the lights go on and they feel competent, which is a huge priority of homeschooling for me. 

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So new topic to munch on. It is clear that the best way to learn math is to work at a pace suited to your capability. But individualized instruction is very difficult in schools. I have heard some podcasts recently when extol the virtues of online learning in math because the program can be individualized. But it seems to me that only procedural content is learned. Only understanding that can be *measured* will be taught.  Quoting myself:

Quote

I have been horrified recently that students in 6th and 7th grade are being put onto computers for math, expecting to self teach.  I've now had 3 of them.  One switched schools because of it, and the other boys told me that not only did the the teacher not teach, she didn't even supervise.  So they didn't do any of the work for 2 years and just socialized!  So I just said, "well, let me be clear, not only have you not learned 6th and 7th grade math, but you have forgotten 5th grade math. So that is where we will start even through you are in 8th grade." Kids like to hear it straight, and one of these boys shaped up, and the other is the boy I described above who is likely to fail algebra next year.

Since I started tutoring, I have come to believe that personalized human interaction is key to learning. But it is expensive. What are your thoughts on online math? Not specific programs, but more on their general potential. Their pitfalls. How they can be used effectively in a balanced program? How students respond to this kind of learning? And not just students in a general sense, but students as individuals who are human and experience fear, uncertainty, and doubt?

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1 minute ago, lewelma said:

So new topic to munch on. It is clear that the best way to learn math is to work at a pace suited to your capability. But individualized instruction is very difficult in schools. I have heard some podcasts recently when extol the virtues of online learning in math because the program can be individualized. But it seems to me that only procedural content is learned. Only understanding that can be *measured* will be taught.  Quoting myself:

Since I started tutoring, I have come to believe that personalized human interaction is key to learning. But it is expensive. What are your thoughts on online math? Not specific programs, but more on their general potential. Their pitfalls. How they can be used effectively in a balanced program. How students respond to this kind of learning? And not just students in a general sense, but students as individuals who are human and experience fear, uncertainty, and doubt.

 

Maybe it would be possible to achieve a balance in which a class of 25 students spends 80% of their math instruction and practice time on computers, with programs that personalize to their individual levels, and 20% working individually or in small groups at similar achievement levels with an instructor or tutor. 

Classrooms need to take advantage of efficiencies, no school has the resources for students to get one on one individualized instructor attention 100% of the time, but computers can be used for the things they are good at--practice, pre-recorded instruction, gamified motivation--which is probably a more efficient use of student time than group lectures that can only target the learning level and needs of a minority of the students.

One data point: my household was introduced to the Prodigy math game three years ago and since then my kids have spent countless hours of their self-directed time working math problems because they enjoy the game. From my parent account I can review what they are doing and assign problem sets, or I can let the adaptive program choose its own problem sets. This program doesn't attempt to teach, but it has proven excellent for practice of grade school math skills. My 16 year old bemoans the fact that it doesn't extend to high school math 🙂

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6 minutes ago, maize said:

 

Maybe it would be possible to achieve a balance in which a class of 25 students spends 80% of their math instruction and practice time on computers, with programs that personalize to their individual levels, and 20% working individually or in small groups at similar achievement levels with an instructor or tutor.

But here is my problem with that.  It seems that all of the online math programs focus on procedural math because it can be checked by a computer.  It also teaches students that math is about getting the correct answer, rather than investigating ideas, modelling scenarios, attempting abstractions, communicating ideas. It seems to me that it teaches all the wrong lessons about math in an effort to drill in old concepts and push them forward faster into new concepts.  I would rather see brief drill and then the USE of math.  I think computers are the wrong solution to the problem. 

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1 minute ago, lewelma said:

But here is my problem with that.  It seems that all of the online math programs focus on procedural math because it can be checked by a computer.  It also teaches students that math is about getting the correct answer, rather than investigating ideas, modelling scenarios, attempting abstractions, communicating ideas. It seems to me that it teaches all the wrong lessons about math in an effort to drill in old concepts and push them forward faster into new concepts.  I would rather see brief drill and then the USE of math.  I think computers are the wrong solution to the problem. 

There may be value in time spent as well.

When learning a language, our brains need lots of time engaged with that language. It doesn't all need to be time spent constructing beautiful sentences or engaged in deep analysis. Fluency is largely a product of time.

 

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9 minutes ago, square_25 said:


To be fair, I never said I did isolated concepts in full! I don't do mastery teaching, I do a sort of "conceptual spiral." 

So I'm very much a fan of spirals :-). I just don't tend to use shortcuts until I think the kids are solid on the ideas. And I haven't had trouble coming up with things to do as we get fluent. 

Thanks for clarifying. That is much more the way I have taught my younger. He *must* have conceptual understanding first. must must must. But as he learns, he gains insight, and then we loop back around to concepts that he had good conceptual understanding, and build on that with more connections to broader math. 

Like I mentioned above, he told me yesterday that his fraction knowledge was solidified with the learning of the -1 exponent. 

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8 minutes ago, square_25 said:

 

I've actually been planning to work on an app that addresses things conceptually. I'm not sure what percentage of time someone would need to have personalized interaction with this plan, but hopefully I'll get the chance to write it and use it in conjunction with my classes :-). 

Cool! What level?

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17 hours ago, lewelma said:

I have been horrified recently that students in 6th and 7th grade are being put onto computers for math, expecting to self teach.  I've now had 3 of them.  One switched schools because of it, and the other boys told me that not only did the the teacher not teach, she didn't even supervise.  So they didn't do any of the work for 2 years and just socialized!  So I just said, "well, let me be clear, not only have you not learned 6th and 7th grade math, but you have forgotten 5th grade math. So that is where we will start even through you are in 8th grade." Kids like to hear it straight, and one of these boys shaped up, and the other is the boy I described above who is likely to fail algebra next year.

I walked in to my dd’s 1st grade classroom and began volunteering and discovered they are using computers for math, phonics and handwriting instruction. We are planning to return our children to home instruction.

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9 minutes ago, square_25 said:

I actually have some experience with this 🙂

Yes, I know!  Which is why I asked!

Quote

I've had to write short answer questions for AoPS that helped the kids internalize the concepts by next class (the classes are weekly.)  I found that very hands-on homework that had the kids engage with the concepts using a visual model without doing much calculation helped a lot more than anything else I tried. The writing problems helped, too. So from my experience, at least, a mix of very conceptual questions that made the kids engage directly with the ideas and of deeper questions that got human feedback worked really well.

So my older boy LOVED AoPS classes. Took them all and they gave him great joy.  But do you think that we could implement this style of computer based learning for all kids? The computer based programs that the schools use here are all procedural math. Drill and Kill. And like I said upthread, they are creating kids who can *only* compute; they don't even realize that math is actually not about computing, it is about modelling our world and investigating questions. It is really sad, because they are learning content that is useless if you never use it, and they will never use it because they don't know that you can actually use math to answer questions. 

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16 minutes ago, prairiewindmomma said:

I walked in to my dd’s 1st grade classroom and began volunteering and discovered they are using computers for math, phonics and handwriting instruction. We are planning to return our children to home instruction.

But it is so modern to use computers. It will help your kids join the digital economy. 

The girl I taught who switched schools because all her math was completely computer based (with no teacher instructing), told me they were solely computer based because was a decile 10 school (wealthiest 10%).  I was like ?!?!!? Why would the wealthiest school in the area let teachers off the hook for *teaching* math?! This girl believed that it was *her* fault for not being able to learn with the computer. 

Plus, like I said above, all the computer programs are just basic math. They may even teach fractions, percentages, decimals conceptually, but the kids don't do *anything* with the math they are learning. It is just isolated, random, content they are told to learn and master for a test. 

I am starting to believe in completely integrated programs like what High Tech High in CA uses.  I never thought I would go that route, but I see student after student, from high level to low, all missing the point of what math is about. Even today, let alone in the future, computers will do all the computation.  *People* will set up the models of real world problems. So why aren't we focusing on *that*?

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11 minutes ago, prairiewindmomma said:

I walked in to my dd’s 1st grade classroom and began volunteering and discovered they are using computers for math, phonics and handwriting instruction. We are planning to return our children to home instruction.

Yep, one school we were zoned for required iPads from K, to use their 'apps across the curriculum' in community (k-3 & 4-6) classrooms. The parents I spoke to were not impressed. It seemed to be mainly unsupervised iPad time all day, with added inappropriate sharing from older classmates.

So I have a bit of a visceral reaction to math instruction on the computer, even though I quite like Khan academy for older & motivated learners (like myself needing to brush up ahead of my children!) What do you guys think about the dragonbox apps?

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