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lewelma

People with opinions on math education!

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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.

Edited by lewelma
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And my response from the PMs: 

Yeah, that's a tricky exam! Actually, things like this are why I tend to start teaching conceptual skills early. 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. 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. And that's after I rewrote things to be extremely clear on the distinction; before the rewrite, I'd say 90% of the kids were confused about the difference! 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've thought about writing down a curriculum or possibly an app that explains what I do... we'll see how much energy I have after I'm done homeschooling, lol. 

 

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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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Thanks you guys for sharing your discussion.  I have no input, but it's something I've given a lot of thought to and have asked about how some of dh's former employees were educated overseas. The fact that his company has had to literally import "math wizards" as he calls them, because they could not find enough people qualified in the states to fill these roles has always intrigued me. To find people who had the mathematical and computer skills AND function well in a workplace getting along with people.....well, apparently there is a big need for that here in the US, judging by his companies and others and the lengths they go to to recruit.....I've had some really interesting discussions with some of his employees on their math backgrounds and have definitely altered my approach based on those talks. 

I look forward to eavesdropping on y'all's discussion. 🙂

Edited by Æthelthryth the Texan
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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.  

Edited by lewelma

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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?

Edited by daijobu
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Wow, that is an impressive exam.  I would love to see American students take this exam for college admissions.  It's so much more interesting and way more challenging than anything you'd see in the States.  

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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?

Edited by maize
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So where does that leave those of us in the US who have only what materials are available here to teach? Few of us learned another way to approach math other than learn the tables and go from there. Is then the whole UK/Asian math the way to go? How do you teach in a way that is soo different from what you learned if it makes zero sense to you?

My husband is from Scotland. We have had many discussions about the differences in his education compared to mine.

Edited by Paradox5
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5 minutes ago, Paradox5 said:

So where does that leave those of us in the US who have only what materials are available here to teach? Few of us learned another way to approach math other than learn the tables and go from there. Is then the whole UK/Asian math the way to go? How do you teach in a way that is soo different from what you learned if it makes zero sense to you?

My husband is from Scotland. We have had many discussions about the differences in his education compared to mine.

 

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... 

 

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3 minutes 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 too late for my olders but I have a little guy coming up in a few years that I would like to do things differently with. I have read Liping's book suggested here many times. Guess I should try to get a copy. 

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1 minute ago, Paradox5 said:

It is too late for my olders but I have a little guy coming up in a few years that I would like to do things differently with. I have read Liping's book suggested here many times. Guess I should try to get a copy. 

I think it's worth a read, for sure! 

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3 hours ago, lewelma said:

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. 

 

I would personally start more abstract stuff earlier. For example, I think there's very little preventing us from teaching multiplication to younger kids and letting them explore the ideas and the potential shortcuts before learning any algorithms. I would also break things up less. I often see learning broken up into simple, digestible chunks that are supposed to help learners by making things easier to absorb linearly. So, for example, for multiplication, we learn how to multiply by 1, and by 0, and by 10, and by 5 first. However, I think a learner really engages with the DEFINITION of multiplication as repeated addition much more if you just throw a bunch of questions at them and let them create a foundation of experience on which to base their later learning. All the shortcuts make much more sense once you've internalized the definition, used it a bunch, and had your ideas about it organized both in your own head and by your teacher. 

It's the same thing with place value. By breaking up the knowledge into bite-sized chunks, we can cheat kids out of grasping the overall picture of place value. Once place value actually makes sense to kids, a lot of the shortcuts are sensible. Before place value makes sense, the shortcuts feel like random memorization. And by not letting kids engage with the definition, we make it harder for kids to see when the definition applies (and therefore, we make word problems and modeling much harder for them.) 

I have an accelerated kiddo, so you can take this with a grain of salt, but it took us literally one day to learn the standard addition algorithm for an arbitrary number of digits, and then it took us another two days to learn the standard subtraction algorithm with an arbitrary number of digits. She already had such a deep understanding of place value that she could absorb the ideas much, much quicker than she would have if she had been grappling with place value and the algorithm at the same time. 

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4 minutes ago, square_25 said:

 

I would personally start more abstract stuff earlier. For example, I think there's very little preventing us from teaching multiplication to younger kids and letting them explore the ideas and the potential shortcuts before learning any algorithms. I would also break things up less. I often see learning broken up into simple, digestible chunks that are supposed to help learners by making things easier to absorb linearly. So, for example, for multiplication, we learn how to multiply by 1, and by 0, and by 10, and by 5 first. However, I think a learner really engages with the DEFINITION of multiplication as repeated addition much more if you just throw a bunch of questions at them and let them create a foundation of experience on which to base their later learning. All the shortcuts make much more sense once you've internalized the definition, used it a bunch, and had your ideas about it organized both in your own head and by your teacher. 

It's the same thing with place value. By breaking up the knowledge into bite-sized chunks, we can cheat kids out of grasping the overall picture of place value. Once place value actually makes sense to kids, a lot of the shortcuts are sensible. Before place value makes sense, the shortcuts feel like random memorization. And by not letting kids engage with the definition, we make it harder for kids to see when the definition applies (and therefore, we make word problems and modeling much harder for them.) 

I have an accelerated kiddo, so you can take this with a grain of salt, but it took us literally one day to learn the standard addition algorithm for an arbitrary number of digits, and then it took us another two days to learn the standard subtraction algorithm with an arbitrary number of digits. She already had such a deep understanding of place value that she could absorb the ideas much, much quicker than she would have if she had been grappling with place value and the algorithm at the same time. 

Your dd sounds like my Son 2. 

I know Waldorf introduces all 4 operations at the same time. I do not remember about place value. I used a story and picture to explain it to my kids. I can see your point that knowing the whole picture before the parts is beneficial in math for many kids. I have seen it with several of my kids. I have also had a child who could not understand the big or small picture for years. I used every trick and help I could find with him. Like anything, how we approach a thing cannot be inflexible. I know that is not what you are saying.

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

 

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8 minutes ago, square_25 said:

 

That looks like the same book to me, but maybe different editions? One of the editions is absurdly expensive, so I'd guess it's out of print. 

The first one has a publishing date of 2010, while the second is 1999. Hmm..

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1 minute ago, Paradox5 said:

Your dd sounds like my Son 2. 

I know Waldorf introduces all 4 operations at the same time. I do not remember about place value. I used a story and picture to explain it to my kids. I can see your point that knowing the whole picture before the parts is beneficial in math for many kids. I have seen it with several of my kids. I have also had a child who could not understand the big or small picture for years. I used every trick and help I could find with him. Like anything, how we approach a thing cannot be inflexible. I know that is not what you are saying.

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

 

 

I don't actually introduce all the operations at the same time, because I think that would create soup in kids' heads :-). But I do think that after addition and subtraction are solid, it makes a lot of sense to introduce multiplication. And after multiplication is solid, it makes sense to introduce division. That way, the properties of the operations become apparent as you use them (especially if you structure lessons as to make the properties more apparent.) 

I also do not in any way advocate inflexibility! I think kids need to learn things when they are ready. I've generally nudged my daughter into understanding each new thing, but I have tried not to push her, and I've tried to do the same thing with the kids in my homeschooling classes. Kids understand things when they are ready. I would personally wait until a kid seemed conceptually ready for a topic before I taught it to them. It's possible that teaching it via mnemonic or procedurally first would work well, but I've seen that backfire far too often to be willing to try it myself. Of course, that means that I really can't speak about the efficacy of these methods when used mindfully, so my opinion should probably be taken with a grain of salt! 

I don't think you need to be a mathematician or an engineer to have opinions! So you don't need to bow out :-).  

 

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2 minutes ago, Paradox5 said:

The first one has a publishing date of 2010, while the second is 1999. Hmm..

So the older edition is cheaper! Weird... 

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This article advocates for releasing elementary teachers from teaching math altogether, and bring in math specialists instead.

This idea that elementary math teachers are math-phobic is a big reason I decided to homeschool.  

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17 minutes ago, square_25 said:

 

I don't actually introduce all the operations at the same time, because I think that would create soup in kids' heads :-). But I do think that after addition and subtraction are solid, it makes a lot of sense to introduce multiplication. And after multiplication is solid, it makes sense to introduce division. That way, the properties of the operations become apparent as you use them (especially if you structure lessons as to make the properties more apparent.) 

I also do not in any way advocate inflexibility! I think kids need to learn things when they are ready. I've generally nudged my daughter into understanding each new thing, but I have tried not to push her, and I've tried to do the same thing with the kids in my homeschooling classes. Kids understand things when they are ready. I would personally wait until a kid seemed conceptually ready for a topic before I taught it to them. It's possible that teaching it via mnemonic or procedurally first would work well, but I've seen that backfire far too often to be willing to try it myself. Of course, that means that I really can't speak about the efficacy of these methods when used mindfully, so my opinion should probably be taken with a grain of salt! 

I don't think you need to be a mathematician or an engineer to have opinions! So you don't need to bow out :-).  

 

My thoughts exactly. This goes for everything we teach. But that is another topic. 

Thanks for making me feel welcome to express my thoughts. 

I checked my library system (the largest in the Houston area) and not a single copy of Ma's book is to be had. I know hat I am adding to my Christmas list.

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3 minutes ago, daijobu said:

This article advocates for releasing elementary teachers from teaching math altogether, and bring in math specialists instead.

This idea that elementary math teachers are math-phobic is a big reason I decided to homeschool.  

Which article? 

Yeah, the fact that elementary math teachers are math-phobic was absolutely one of the reasons we wound up homeschooling. When I told my daughter's kindergarten teacher she was bored in math, the only thing she could think to do was to give her bigger numbers to add... Actually, I got the sense she didn't really believe us, because she said that she had offered kids "a bigger challenge" if they wanted one (that is, slightly bigger numbers), and DD had never taken her up on this challenge. (She was at this point good at adding numbers under 10 and found adding any pair of them equally boring. She did like negative numbers a lot, though, and multiplication, and two digit addition and subtraction.)

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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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7 minutes ago, lewelma said:

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. 

 

Oh my. This is the kind of "rock and a hard place" situation you often get when tutoring, I bet. Because who knows what's going on? Maybe he's not clear on variables. Maybe he's not clear on multiplication (that one's totally possible.) Maybe he's not clear on the idea of "solving" something. In an optimal world, we'd have some time to dig in and figure out which one it is. Do we live in an optimal world? No. You just need to get him to do his best on his exam... and that means you probably don't have the time to figure out exactly what the issue is. Ugh. 

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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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Just now, lewelma said:

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! 

 

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...

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2 minutes 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. 

 

It's so hard to motivate kids by some sort of nebulous "in the future, you might need this!" argument. That's why I've tried so hard to actually make the subjects I'm teaching interesting and relevant without reference to some hard-to-model future... but I know that doesn't always work. School can be such drudgery, though :-/. 

Edited by square_25

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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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Just now, lewelma said:

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. 

 

 

Whereas I actually really enjoy working with elementary-aged kids! I've been teaching 5 to 7 year olds and 8 to 10 year olds at our homeschooling center, and I find the conceptual issues they have fascinating and absolutely approachable. We've been playing lots of games and working on puzzles and using poker chips as our base 10 manipulatives and all sorts of really fun stuff. And I've been watching them engage with the material and improve, even though once a week classes really aren't frequent enough. It's been fun and gratifying.

But I don't think it's practical versus theoretical as much as much an interest in mathematical education at all levels... my husband is as theoretical as they come (he's still a practicing research mathematician, unlike myself), and he doesn't really know how to teach young kids math. It's just not as interesting to him. 

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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.

Edited by lewelma
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Just now, lewelma said:

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! 

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

 

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. 

 

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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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8 minutes ago, lewelma said:

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. 

 

And this kind of things is why I don't really know how to tell people how to teach math! I feel like a large part of my math teaching consists of figuring out when my daughter is confused, and then explaining the concept to her so she's less confused. However, this is an absolutely impossible instruction to carry out! First of all, it takes a lot of practice to notice when someone is confused. In fact, I've had the hardest possible time trying to convince a variety of people about exactly how confused their students are... people often take smiling and nodding or 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. 

And mind you, these are the difficulties I've had with people who absolutely understand the material! How do you troubleshoot if you're shaky in the material yourself? It becomes exponentially harder. And I don't say that arrogantly: I'd have exactly the same trouble troubleshooting understanding in, say, chemistry, because I simply don't remember it, and what I do remember is incoherent and piecemeal. So I'd never be able to spot that my student's understanding is missing a large chunk, because chances are that I'm missing that chunk myself... 

 

Edited by square_25
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2 minutes ago, lewelma said:

We need to clone you.  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.

My mom has an opinion that back when women had very few job options, very capable women taught primary school. Now, there are so many other options, that capable women often go into better paid and more respected careers.  I don't think she is far off. 

 

Absolutely. My husband says this all the time, and I'm sure he's right. I wish teaching was a better paid and more respected career choice, because elementary teachers really matter....  

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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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2 hours ago, lewelma said:

me sick.  I rescued him from the 'cabbage' class this year, and now I am regretting my choice. sigh. 

Ahem?  Some sort of gardening course?  

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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.....

 

Edited by lewelma
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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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