People with opinions on math education!

[lewelma]

1 minute ago, LMD said:

What do you guys think about the dragonbox apps?

Just asked my younger if it helped his algebra at all.  He said it provided a bit of a framework to slot stuff into a bit easier, but it didn't help a lot. Just gave him a bit more clarity on solving equations.

[lewelma]

1 minute ago, square_25 said:

 

To be fair, I'm not sure we DO all that much with the math we learn, either! I mean, it comes up in daily life, so in that sense we do? But even in a non-computer-based math classroom, what exactly are kids doing with their math? If you're saying we need more applications, I feel like that could happen with or without a computer. 

 

Well, in NZ in high school the internal assessments are mostly applications.  So for trig. One school I have worked with have the students go outside and measure the space for a non-right angled triangular shade sail over a sitting area (they do this collaboratively). They then come inside and design a 3d model for constructing it and do all the trig calculations and write a brief report. It is a 2-3 hour assessment for 11th grade. 

[wendyroo]

24 minutes 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.

I tend to think it is a bit more nuanced than this.  I agree that it is best to learn straight arithmetic at a pace suited to your capabilities.  If a child has not mastered two digit addition without regrouping, then it is completely counterproductive to move them on to three digit addition with regrouping.

With conceptual learning, OTOH, I think it is stifling to only be presented with things you are ready to fully master.  I would much rather teachers present all sorts of conceptual problems that have something to offer kids at all levels.  For example, I work through an SAT prep problem during dinner each night.  Last night's showed a graphed circle and asked the student to identify which equation would generate that circle.  None of my kids knew the "right" way to do the problem - that is exactly why I chose it!  But as I conceptually talked about circles and graphing and coordinate planes and approaching problems that you feel have insufficient information, I made sure that each of the kids (ages 4, 6, 8 and 10) had to grapple with a challenging idea.  My four year old learned vocabulary and discovered that circles are as tall as they are wide and was introduced to the idea of a coordinate plane.  My 6 year old is teetering on the edge of understanding negative numbers, so this was just one more exposure in a different context to help him broaden his perspective.  My 8 year old had previously been introduced to graphing lines, but this was his first time encountering the idea that other shapes could be represented by equations; it was also a review for him about conceptually and procedurally what it means to square a number.  My 10 year old immediately sensed that there was an equation, an algorithm, a method that he was missing that would simplify the problem.  I had no interest in enlightening him - he'll learn the formula for the equation of a circle eventually, but in the greater scheme of things that was the least interesting part about the problem.  My main goal for him was a conceptual understanding that the equation maps to the circle because every point on that circle (and no other point in the plane) makes the equation true.  I used Socratic questions to lead him to the idea that he could identify the correct equation by testing them with coordinates from the circle.  Then, for kicks, we graphed the other "wrong" equations...both to emphasize that you can graph an unknown equation by simply finding coordinate pairs that make it true, and as a gentle introduction to translating graphs.

When I write it out, it all sounds neat and clean and like each child learned exactly what was in their zone of proximal development, but in my experience it is much richer than that.  The other day when we were hanging Christmas lights around a window, the 4 year old suddenly spouted out the word perimeter to accurately describe the path the lights would take.  That might have been the only thing she picked up during an area and perimeter problem we worked at the board...in fact, I clearly remember that I tried for several minutes to get her to accurately distinguish between a square and a rectangle during that problem and it was a complete failure.  But it is all good, because she will be exposed to those same concepts over and over and each time she will be actively building her mental model...maybe not in the "developmentally appropriate" order, but we're in no hurry.

So, all that to say that I think computers are a wonderful tool to nudge a child along the path to procedural competency.  My kids use Xtramath and Prodigy and Alcumus and Dragonbox and Hands on Equations apps to practice and review at and below their zone of proximal development.  But for all the interesting bits, I think a living, breathing, mathematically adept teacher is vital - but NOT necessarily in a one-on-one situation or highly "tracked" group of students all at the same level.  The most interesting problems have something to offer everyone, and as long as each student is building their conceptual understanding in some way, then I wouldn't really worry if they are presented with some problems that are "too easy" and other than are "too hard".

Wendy

[lewelma]

I should also add that I'm not say don't use computers for modelling!!! That is the purpose of computers - use them to solving problems!  I'm saying don't use computers as the sole or even majority of the instructional approach. 

[lewelma]

3 minutes ago, wendyroo said:

With conceptual learning, OTOH, I think it is stifling to only be presented with things you are ready to fully master.  I would much rather teachers present all sorts of conceptual problems that have something to offer kids at all levels.  For example, I work through an SAT prep problem during dinner each night.  Last night's showed a graphed circle and asked the student to identify which equation would generate that circle.  None of my kids knew the "right" way to do the problem - that is exactly why I chose it!  But as I conceptually talked about circles and graphing and coordinate planes and approaching problems that you feel have insufficient information, I made sure that each of the kids (ages 4, 6, 8 and 10) had to grapple with a challenging idea.  My four year old learned vocabulary and discovered that circles are as tall as they are wide and was introduced to the idea of a coordinate plane.  My 6 year old is teetering on the edge of understanding negative numbers, so this was just one more exposure in a different context to help him broaden his perspective.  My 8 year old had previously been introduced to graphing lines, but this was his first time encountering the idea that other shapes could be represented by equations; it was also a review for him about conceptually and procedurally what it means to square a number.  My 10 year old immediately sensed that there was an equation, an algorithm, a method that he was missing that would simplify the problem.  I had no interest in enlightening him - he'll learn the formula for the equation of a circle eventually, but in the greater scheme of things that was the least interesting part about the problem.  My main goal for him was a conceptual understanding that the equation maps to the circle because every point on that circle (and no other point in the plane) makes the equation true.  I used Socratic questions to lead him to the idea that he could identify the correct equation by testing them with coordinates from the circle.  Then, for kicks, we graphed the other "wrong" equations...both to emphasize that you can graph an unknown equation by simply finding coordinate pairs that make it true, and as a gentle introduction to translating graphs.....

So, all that to say that I think computers are a wonderful tool to nudge a child along the path to procedural competency.  My kids use Xtramath and Prodigy and Alcumus and Dragonbox and Hands on Equations apps to practice and review at and below their zone of proximal development.  But for all the interesting bits, I think a living, breathing, mathematically adept teacher is vital - but NOT necessarily in a one-on-one situation or highly "tracked" group of students all at the same level.  The most interesting problems have something to offer everyone, and as long as each student is building their conceptual understanding in some way, then I wouldn't really worry if they are presented with some problems that are "too easy" and other than are "too hard".

Wendy

You are my  hero! That description was absolutely wonderful! 

[wendyroo]

25 minutes ago, square_25 said:

I actually think AoPS goes far too fast for its current audience. From what I understand, your boy did hours and hours of math every day. I think the only way AoPS classes currently provide you with fluency is if you are willing to invest that amount of time. Since most people are not, I think the classes often don't serve them well. 

I agree, which is why DS is working through the books independently and not taking the classes.  He is currently working through the algebra book and it is taking t.i.m.e!!  It is not the sort of learning that we could speed up and cram into one semester or school year.  He needs time to think and consolidate his learning.  He needs time to go off on mathematical rabbit trails and learn about the things he is interested in.

We LOVE AOPS, but that this point the classes would not be a good fit.

[daijobu]

29 minutes ago, square_25 said:

I actually think AoPS goes far too fast for its current audience. From what I understand, your boy did hours and hours of math every day. I think the only way AoPS classes currently provide you with fluency is if you are willing to invest that amount of time. Since most people are not, I think the classes often don't serve them well. 

 

 

Do you have any inkling of the demographics of your students?  Are many of them radically accelerated in math?  Are they taking precalc for the first time, or again after having studied it at BM school?  Are many of them homeschoolers?  

[lewelma]

If a child has not mastered two digit addition without regrouping, then it is completely counterproductive to move them on to three digit addition with regrouping.

You might find it interesting that NZ does not teach any algorithms for math.  They only do mental math. This is both good and bad. 

[LMD]

8 minutes ago, lewelma said:

Well, in NZ in high school the internal assessments are mostly applications.  So for trig. One school I have worked with have the students go outside and measure the space for a non-right angled triangular shade sail over a sitting area (they do this collaboratively). They then come inside and design a 3d model for constructing it and do all the trig calculations and write a brief report. It is a 2-3 hour assessment for 11th grade. 

Can you recommend any resources lewel? Do you like any nz textbooks? I'm in Aus and most of the stuff I see here is fairly ho-hum. My kids' music teacher is a year 12 maths teacher too so I often ask her advice but she admits it's very procedural and fast paced. She had nothing when I asked how she teaches negative numbers conceptually, for example. She's a great teacher, there's just no time to explore maths like that.

Tangentially, dh has our kids doing lots of woodworking type stuff as part of their school work to support their maths. Building shelves and boxes etc. Ds12 spent hours last week measuring and cutting plaster (we're currently owner-building our house). He and I had a maths education that came from different angles, I did the procedural, higher level stuff but avoided the icky physics (because I was intimidated), he did lots of the hands-on physics stuff (especially electrical engineering) but his maths wasn't good enough to continue due to conceptual gaps. He uses maths up to and including trig regularly, but it is jury-rigged all the way down and quite frustrating for him!

[LMD]

43 minutes ago, square_25 said:

 

I actually think AoPS goes far too fast for its current audience. From what I understand, your boy did hours and hours of math every day. I think the only way AoPS classes currently provide you with fluency is if you are willing to invest that amount of time. Since most people are not, I think the classes often don't serve them well. 

Mind you, they are still better than what the kids are doing at school, because they are conceptual and they force the kids to make sense of the math they are using. But they often don't produce the level of fluency I would like them to. 

Thank you for saying this! My kids take longer to get through the aops books, some of it is me not wanting to push a math pace that destroys the fun, but some of it is just hard! Is this where I admit that my daughter took over 2 years in the pre-alg book... worth it, and I try reaaaaaaally hard to not care about grade levels and shoulds...

Eta- again, my kids aren't accelerated. They are average bright. Aops pre-alg coincided with emergent teen brattiness so a lot happened in those two years but I have faith in the aops process and she likes it (though finds it frustrating) a lot.

Edited November 30, 2019 by LMD

[lewelma]

34 minutes ago, square_25 said:

 

I actually think AoPS goes far too fast for its current audience. From what I understand, your boy did hours and hours of math every day. I think the only way AoPS classes currently provide you with fluency is if you are willing to invest that amount of time. Since most people are not, I think the classes often don't serve them well. 

Mind you, they are still better than what the kids are doing at school, because they are conceptual and they force the kids to make sense of the math they are using. But they often don't produce the level of fluency I would like them to. 

Yes, my ds definitely thought there were a lot of kids in the AoPS classes that were just pounding through the content without deeply understanding it.  My impression from my ds's experience, is that he mastered the content with little repetition because he had to really struggle with the problems. He felt that some kids asked for so many hints, that they converted the deep problem solving approach of AoPS to more procedural because with a hint, they knew exactly what to do.  Without the struggle, they could not master the content with so few homework problems. 

I also think that it was the 3 years of AoPS intro A that was the making of my ds.  He basically pulled high-end problem-solving into the first high school math class, which meant that from then on he had the problem solving skills to master the higher level content. It was this experience, that has led me to have my younger boy to make this third pass through intro calc before we move on to proper calculus. 

 

[daijobu]

24 minutes ago, wendyroo said:

I agree, which is why DS is working through the books independently and not taking the classes.  He is currently working through the algebra book and it is taking t.i.m.e!!  It is not the sort of learning that we could speed up and cram into one semester or school year.  He needs time to think and consolidate his learning.  He needs time to go off on mathematical rabbit trails and learn about the things he is interested in.

We LOVE AOPS, but that this point the classes would not be a good fit.

 

I also did not have my kids take AoPS online classes, except for the extra test prep classes for AMC and MathCounts.  We spent one day on each chapter section, and then 2-3 days each on end of chapter Review and Challenge Problems.  We also school year round, so we had extra time to make up for our snail's pace.

In the time I've been tutoring, I have had several students who are taking the AoPS online classes, helping them with their homework and reteaching material they should be learning in class.  I wonder if the students aren't supposed to also be doing the exercises in the book in addition to the online homework?  Because I think my kids did many more practice problems from the book than the online homework which only has 10-12 problems per week.  

I spent a few days working through my AoPS online precalc student's homework.  I needed to return to the textbook and review a few problems before I could solve his homework.  

Edited November 30, 2019 by daijobu

[lewelma]

But at some point, it does make more sense to deal with the pure math first and then deal with the applications separately. And you can include theoretical questions like that. 

I thought you might be interested in how the NZ curriculum teaches Calculus. In 11th grade they do a mixed up version of American Algebra 2/ precalc/ Calculus.  

So for example, they teach only how to differentiate/integrate basic polynomials.  So f(x)= 3x^2 + 6x . That is all the procedural knowledge taught for the entire unit, and it is not taught conceptually AT ALL because they don't learn limits until 12th grade. But then they use that one piece to do max/min problems, rates, physics, abstract thinking, etc. So my ds on Friday was working on this problem:

The curve y=ax^2 -bx +c passes through the point (3,-13), and has a stationary point at x=2 and a y intercept of 5. Find the equation of the curve and the coordinates of the other stationary point.  

But keep in mind that he is doing this kind of work with procedural knowledge of ONLY basic polynomial differentiation.

Becoming competent in this kind of thinking has taken my ds 3 passes throughout the year. And I *really* like that we are focusing problem solving rather than procedural skills.  So the order I have used: 1) teach one procedure as magic (no understanding with limits) , 2) deep competency in using this one procedure to solve complex problems (2 months, gap, 6 weeks, gap, 3 weeks. I've intermixed these 3 passes with other math), 3) then next week we are attacking limits and the conceptual understanding of a derivative because he is ready for the theory. 4) A full calculus course. 

Edited November 30, 2019 by lewelma

[lewelma]

9 minutes ago, square_25 said:

It would have worked for me, so I'm not knocking it. But extrapolating from IMO (International Math Olympiad) kids is kind of a losing proposition, I think... with kids who are that motivated, all you really need to do is to not bore them and to get out of their way.  

Haha. I know, right! Did I ever tell you that he thought for YEARS that he was bad at math (age 9-11). The challenge level of the Intro A book done independently was so hard that he struggled every day, which is why he thought he was bad at math. Nothing I said would convince him otherwise. 

Sometimes I think I am very lucky to have taught the full range of students from my older ds to my student who couldn't subtract 9-6 at age 17. I think I have honestly seen the full range and many things in between. 

[daijobu]

15 minutes ago, square_25 said:

Well, it's not really precalc, that's the issue! So AoPS precalc is a really "clean" experiment (unlike a lot of other classes), because most of these kids haven't seen complex number exponential form before and hadn't seen vectors before. That means that just about everything they get out of the class is really from the class and not from other sources. And I've definitely seen rather sobering results (especially before I started mucking with the scripts and homework.) 

 

 

Okay, so what does precalc actually mean?  We didn't have that class at my high school.  Our math sequence was:

9th: geometry

10th: functions + analytic geometry

11th: advanced algebra and trig

12th: calculus

It looks we covered complex numbers in 11th grade and vectors in 10th grade.  

And for fun, here's some samples of my work.  (Yes, that is purple mimeograph ink!)

[daijobu]

7 minutes ago, square_25 said:

On a vaguely related note, have you all seen Math with Bad Drawings? 

I'm a fan.  

[lewelma]

I am very lucky that when I take kids in the beginning of 10th grade, they typically will stay with me for 3 years (about 90% have).  So I have the time to create the conceptual learning over a long period of time. I find it astounding that they are willing to pay me close to $20K for this help. But I guess a single mentor/coach/teacher can make a life long impact. 

[lewelma]

9 minutes ago, square_25 said:

On a vaguely related note, have you all seen Math with Bad Drawings? 

Not me, but off to google.

 

[lewelma]

14 minutes ago, square_25 said:

On a vaguely related note, have you all seen Math with Bad Drawings? 

Well the concept on the front page has me hooked, because my students really really struggle with quadratic inequalities.

[lewelma]

6 minutes ago, square_25 said:

For calculus, for example, what I would do is to get us to the ah-ha moment of "tangents are just like secants between two really close points," do some visual derivatives, figure out a few derivatives at a few specific points for given functions with a formula, use these to figure out some tangent lines, define integrals, calculate a few examples, estimate some functions using derivatives, do some examples of definite integrals of functions with a formula, define accumulation functions, graph some accumulation functions, figure out F(a)  - F(b) for some accumulations functions an connect it to the picture, calculate some average rates of growths for accumulation functions, etc, etc. 

As I said, I do a kind of conceptual spiral. It all sounds very complicated but it really isn't :-). It kind of grows the whole knowledge package at once and tries to keep things interesting and making sense at the same time. 

 

I couldn't pull that off. I'm starting to think that you just have a better conceptual understanding of math than I do, and perhaps I have a better modelling background than you do (my PhD was in mathematical modelling of ecological systems). Maybe we teach with our strengths. 

[daijobu]

...and the hilarity that was my academic life continued into my senior year:

 

[lewelma]

Are these *your* math notes from high school!?!?!?  No way do I have mine. 

[lewelma]

Just now, square_25 said:

I'm rather jealous of your older son's education

He has mentioned many times since going off to Uni that he is so grateful for his education. He had the gift of time which led him to be widely read and a deep thinker. The school descriptions from his friends have caused him a deeper and deeper appreciation of what our educational approach gave him. Also, he has had more than one kid who found out he was homeschooled say "your mom must be so smart!" haha 🙂 

[lewelma]

6 minutes ago, square_25 said:

Oh, and here's one of my favorite Math with Bad Drawings posts. I've actually linked it in AoPS feedback before... 

https://mathwithbaddrawings.com/2017/03/01/the-baby-name-book-for-variables/

AWESOME!  I have had this conversation with students, just not so well!

[lewelma]

1 minute ago, square_25 said:

 

Right, time is exactly what I did not have. I do way better with deep learning than with superficial learning, so a lot of my high school learning other than mathematics (and writing) has gone in one ear and out the other. And it's SUCH a waste. I no longer have the luxury of time and energy to learn new things. I wish I hadn't wasted all that time. 

My high school was just a desperate attempt to keep my grades up as I didn't read until I was 12, and my parents tracked me into the honors sequence in highschool. My sister was brilliant and my sole goal was to keep up with her, which was a pretty big undertaking. So I didn't exactly waste time, but I definitely focused on doing what I was told in as efficient a way as I could to get the grades I wanted.  In contrast, my old ds learned what he wanted, when he wanted, and how he wanted. I'm jealous too. 

However, I am lucky that I really love to learn. So this summer (that is now for me) I'm going to be focusing on the equivalent of AP Physics and Chem as I'll be tutoring that next year. And have to brush up on my Calc again as I didn't have a calc sudent this year so likely have forgotten it all!!!!  Then, it's planning my young boy's 11th grade year, and trying to figure out the NZ national Geography exams.  Should be a busy 8 weeks. I had a friend who wanted to know what I would do with all my time now that exams were over and I was done with tutoring for the year.  um, hello. Run an entire 10th grade program for my ds who is not done for 3 weeks, and then plan an entire 11th grade year. People have no idea what homeschooling high school is like. Give a kid some workbooks, they'll be fine. 

 

[daijobu]

17 minutes ago, lewelma said:

Are these *your* math notes from high school!?!?!?  No way do I have mine. 

 

(Fun times reliving my glory days...)

Yes, I kept all my math exams and many of my notes:

The AMCs were a big deal at my high school.  (They were called the AHSME back then.)

 

 

[daijobu]

(Take that, Saskatchewan!)

[chocolate-chip chooky]

3 hours ago, LMD said:

What do you guys think about the dragonbox apps?

My daughter enjoyed Dragonbox. We didn't treat it as curriculum. It was a fun supplement that she just enjoyed as a game. I can't say whether it really helped her or not, but I'm sure it did no harm. She was about age 6 or 7 from memory.

[Ausmumof3]

On 11/30/2019 at 2:19 PM, 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. 

To be fair many screen based learning systems do this as well.  Dd got marked wrong for putting 0.20 instead of 0.2.  I explained to her that her answer wasn’t wrong but we don’t need to include any of the zeroes after the last digit after a decimal point.  But this is why I’ve not been able to deal with purely computer based math 

[daijobu]

22 hours ago, square_25 said:

Oh, and here's one of my favorite Math with Bad Drawings posts. I've actually linked it in AoPS feedback before... 

https://mathwithbaddrawings.com/2017/03/01/the-baby-name-book-for-variables/

 

I was hooked after reading this post:  https://mathwithbaddrawings.com/2015/05/20/us-vs-uk-mathematical-terminology/

"You know who uses scientific notation?  Scientists, that's who."  

[daijobu]

22 hours ago, square_25 said:

I don't know if they are supposed to be doing that, but they emphatically aren't. Most of my kids are swamped by the Challenge Problems and Writing Problems as is. They are doing this after school, on top of their schoolwork, and most of them simply don't have the time... 

 

We have a few AoPS Academies opening locally.  Do you know how the experience of those students will differ from students taking online classes?  

[mathnerd]

14 minutes ago, daijobu said:

 

We have a few AoPS Academies opening locally.  Do you know how the experience of those students will differ from students taking online classes?  

AOPS Academies spread the course out over an entire academic year. For e.g. Algebra 1 is for the entire school year. The one close to me (Santa Clara) is oversubscribed for many of the courses - apparently, the students at the academies are able to spend a much longer time on each unit of the curriculum and hence the pace is more manageable for the school-going kid.

[mathnerd]

32 minutes ago, square_25 said:

What are they covering in Algebra 1? Is it correlated to online AoPS classes? 

They seem to be covering a little more than the online classes (I have no direct experience, I attended an open house to see what they offer). They said that they would also teach C&P during the 36 week Alg 1 course.

[drjuliadc]

So precalculus is trig.  I wondered what they did with trig.  My oldest is only 8 but I graduated a long, long time ago. 

I’m so sorry I don’t have anything intelligent to say about this thoughtful thread.

 Lewelma, I would SO pay you $20,000 to tutor my kids.  Is that for the whole three years?

[lewelma]

3 hours ago, drjuliadc said:

 Lewelma, I would SO pay you $20,000 to tutor my kids.  Is that for the whole three years?

Aw, thanks!  Yes, but just for maths unless I really like your kid and he/she is super motivated. 🙂 And if so, then I have also tutored Physics, Chemistry, Biology, English, Media Studies, Geography, and even Academic PE!!!  Haha.  I've got a lot of work to keep up with so many subjects and their national exams!  I have a couple of families where I'm on their third kid.  🙂 

Edited December 2, 2019 by lewelma

[lewelma]

Precalculus is usually trig, coordinate geometry, and complex numbers if I remember the American system right.  Here in NZ, precalculus and calculus are taught simultaneously over 2 years, rather than sequentially.

[lewelma]

Oh, someone asked for some NZ content.  The best, cohesive, deep-thinking work can be found in the example tasks that NCEA (NZ Certification of Educational Achievement) posts on their website. Internals are large single tasks done during the year and created by schools based on how they taught the set content.  Externals are high end exams done on a set day in November that are the same for all kids (check out the 12th grade probability exam and distributions exams -- very impressive!). The links below give multiple examples of each task/assessment that if worked through with care can really increase a student's level. 

Remember that NZ is not a percent correct system; rather it is a levels of thinking system.  So each assessment has 3 levels of work within each task/exam: 1) achieve is regurgitation and understanding of concepts, 2) merit work is relational thinking and applications to real life, and 3) excellence work is abstraction, generalizations, and insight. Work through multiple tasks/exams and you will up your level of thinking. 

Level 1 (10th grade)

Internals (numeracy, measurement, statistics, linear algebra, basic trig, transformation geometry and many others) http://ncea.tki.org.nz/Resources-for-Internally-Assessed-Achievement-Standards/Mathematics-and-statistics/Level-1-Mathematics-and-statistics

externals (algebra, graphing, geometry, probability) https://www.nzqa.govt.nz/ncea/assessment/search.do?query=Mathematics&view=exams&level=01

Level 2 (11th grade)

internals (advanced graphing, intermediate trig, coordinate geometry, bivariate stats, multivariate stats, questionaires, experimentation, network analysis, and many others) http://ncea.tki.org.nz/Resources-for-Internally-Assessed-Achievement-Standards/Mathematics-and-statistics/Level-2-Mathematics-and-statistics  

externals (algebra 2, basic calculus, probability) https://www.nzqa.govt.nz/ncea/assessment/search.do?query=Mathematics&view=exams&level=02

Level 3 (12th grade)

internals (advanced trig, network analysis, conics, time series analysis, multivariate analysis, bivariate analysis, simultaneous equations, linear programming, and others) http://ncea.tki.org.nz/Resources-for-Internally-Assessed-Achievement-Standards/Mathematics-and-statistics/Level-3-Mathematics-and-statistics

externals (differentiation, integration, complex numbers, probability, and distributions)  https://www.nzqa.govt.nz/ncea/assessment/search.do?query=Mathematics&view=exams&level=03

---------------

Hope that is helpful! I love comparing the American to NZ system, because I grew up and trained to be a teacher in the American system, but have taught in the NZ system.  Fascinating the difference. 

Ruth in NZ

Edited December 2, 2019 by lewelma

[drjuliadc]

On 12/2/2019 at 1:43 AM, lewelma said:

Aw, thanks!  Yes, but just for maths unless I really like your kid and he/she is super motivated. 🙂 

Super motivated? You would throw my kids out post haste. 

I assumed it was just math. You know that is only $13,000 US.  Notice how I actually calculated that.

[lewelma]

4 hours ago, drjuliadc said:

Super motivated? You would throw my kids out post haste. 

I assumed it was just math. You know that is only $13,000 US.  Notice how I actually calculated that.

Haha. With a waiting list until 2021 I can pick and choose. 🙂 I have also been known to become 'too busy' when a parent tries to micromanage me or doesn't like my policy of paying if your kid comes or not.  So you as the parent also have to be nice to me!! As for currency exchange, it is wild here.  Since I have lived in NZ (21 years) it has been as low as 0.39 and as high as 0.91.  Currently, as it drops, we are paying more and more for MIT.  Our bill went up 7K last year due to currency translation. sigh.  But as I posted in the tutoring thread, my goal is to make the same salary as a top teacher here with me working only 20 hours a week.  I figure since my hours are restricted to after school, people are paying me to be available. So currency exchange masks cost of living here and I make double the hourly wage of a top teacher (and I do charge for noncontact hours). But I do have friends who say I should raise my rates again.  🙂  

Edited December 3, 2019 by lewelma

[vonfirmath]

On 11/30/2019 at 2:40 PM, 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?

 

We don't have online school, but the schools have gone to online textbooks. After my son did his first math homework out of the online textbook, we went right up to school and checked out a paper textbook!  He can handle the online textbooks for most classes (and in history it is downright useful to have the additional videos, etc.) but its a HUGE hinderance in math to not be able to flip through the previous chapter to see how the problem was done and flip back and forth while working on your problem on a separate sheet of paper.  It just doesn't work well at all.

 

My nephew (12th year homeschooling) is taking an online course where they just started using matrices. My mom graduated with a math education degree but was trying to help him with his homework but the online textbook was giving her fits trying to refresh herself on matrices to teach the issue.

[vonfirmath]

On 11/30/2019 at 4:05 PM, lewelma said:

 

You might find it interesting that NZ does not teach any algorithms for math.  They only do mental math. This is both good and bad. 

 

This sounds like my son. He was a whiz at math -- as long as he could do it all in his head. It started to trip him up in 6th grade. He couldn't keep all the steps for pre-algebra in his head. This year he is taking Algebra I (I wanted him to take a second year of pre-algebra to cement ideas but he passed the year with a mid-80s so they could not hold him back).  And has FINALLY agreed that yes, he needs to write out all the steps  -- and clearly (not trying to squeeze it all into one page). But old habits die hard and sometimes he's lost major points for just writing down an answer on a problem he felt he could hold in his head.

 

[lewelma]

Not showing your workings is actually not the problem. The problem is when students are not *also* taught how to link word problems to the actual operation being performed (add, subtract, multiply, divide) or taught how to write math in a formal way.  Mental maths teaches students how to break numbers up and put them back together in creative ways. This is a great thing. Without good numeracy skills, students simply do things by rote, which is just memory not math skills.  However, when mental math is done to the exclusion of writing out math formally, students become *very* confused in algebraic word problems. 

What happens in primary school programs with a mental math focus is that students see problems as repeated subtraction rather than division, or a series of operations mixed up, or some such confusion in the actual operation.  When you ask them to write their workings, I call it "crap out of your head". It is typically completely unclear what they have done, with equal signs misused, unorganized numbers/equations/numberlines, no logical order, and then the correct answer at the end.  This does not set students up well for high school math.  They fall over completely in algebraic word problems because they cannot recognize the operation required, they have no idea how to do proper workings, and have been trained to work through intuition rather than logical steps. Here in NZ, students struggle through 8th and 9th grade with these issues, and by the end of 9th grade are failing. At this point, they come to me.   

I teach them to separate out computation (by calculator, algorithm, or mental math) in a box to the side.  Then they have to show proper algebraic workings -- working always down.  I show them which steps are required, and which are optional.  NZ actually marks on "mathematical statements", unorganized crap out of your head will be marked wrong. 

So it is not mental math per se. It is a program limited to this that leads to a major issue in the transition to high school math.

 

Edited December 3, 2019 by lewelma

[lewelma]

5 hours ago, square_25 said:

We didn’t do procedures for ages (as I said, we’re starting them now), and my daughter is really strong with word problems. So I don’t think the lack of procedures is the issue. Perhaps an insufficient focus on definitions is.

I find it hard to believe that your dd would ever struggle with the issues most of my students have. She has you!

[lewelma]

23 hours ago, square_25 said:

It’s true that it’s a different set up!! I just know that for the kids I’ve worked with, focusing on what exactly each operation does using language was really helpful, whereas procedures that didn’t enhance understanding weren’t actually useful.

As I’ve probably mentioned upthread, I find purely mental math programs too constructivist: they don’t give the students enough guidance about basic mathematical structures. I can well imagine the issues you’re describing as a result!

I don't know much about how primary school math works here, just the results I see with almost every student who seeks me out.  There is just a huge disconnect between primary and secondary math -- even the curriculum documents are written by two different non-connected teacher groups. And intermediate math is basically a joke. Stick the kids on a computer and let them self teach. 

So I actually don't like teaching algorithmic methods for computation, I much prefer mental math. But this must be connected to some sort of understanding that you are actually doing a multiplication problem even though your mental math calculation is repeated addition or piecemeal multiplication.  If not, algebraic skills are completely foreign.  How can you possibly understand xy let alone set up an algebraic word problem, when you don't know you are multiplying. So for 8*14, students here would do 8*10=80+5*8=40+80=120-8=112  And they would write it that way too. If you try to clarify what multiplication is, they just don't get it -- they don't think that way. So xy is completely meaningless and they can't use algebraic skills to work real life problems. 

Most students I have worked with have no idea that if you have 80 pies split between 8 people, that you are dividing the pies among the people. So when you have x pies split between y people, you are sunk. It only gets worse from there.

Edited December 5, 2019 by lewelma

[daijobu]

21 hours ago, lewelma said:

 

So I actually don't like teaching algorithmic methods for computation, I much prefer mental math. But this must be connected to some sort of understanding that you are actually doing a multiplication problem even though your mental math calculation is repeated addition or piecemeal multiplication.  If not, algebraic skills are completely foreign.  How can you possibly understand xy let alone set up an algebraic word problem, when you don't know you are multiplying. So for 8*14, students here would do 8*10=80+5*8=40+80=120-8=112  And they would right it that way too. If you try to clarify what multiplication is, they just don't get it -- they don't think that way. So xy is completely meaningless and they can't use algebraic skills to work real life problems. 

Most students I have worked with have no idea that if you have 80 pies split between 8 people, that you are dividing the pies among the people. So when you have x pies split between y people, you are sunk. It only gets worse from there.

Is the problem here that these "equalities" are not actually equal to each other?  Is it bad mathematical notation?  

It's a little hard for me to understand how students might not understand multiplication, but I suppose it is because mental math is not taught much in the US.    

[lewelma]

1 hour ago, daijobu said:

Is the problem here that these "equalities" are not actually equal to each other?  Is it bad mathematical notation?  

yes, it shows a complete lack of understanding of what equals means. It is not so bad with computation, but it becomes very very bad with algebra. 

The problem is that no transferable skills are taught in primary school.  There is no way to connect the mess above to algebraic thinking.  

[vonfirmath]

15 hours ago, square_25 said:

Oh, you'd be surprised. 

Ask a random adult WHY a*b = b*a and you'll realize that most people do not have any kind of definition of multiplication in their head. "But... it... just is.... how could it be different?" 

More people can tell than 6*9 is 54 than can tell you what it is you're actually calculating when you do 6*9. And I think a focus on what a symbol means serves as immediate word problem practice. If you know that 6*9 means you add up six 9s, then it becomes pretty clear that if you buy six things and each one is worth 9 dollars, then the total is worth 6*9... even if you don't see it right away and do 9 +9 +9 +9+9+9, your teacher can point out to you that we have a different name for that and that maybe we remember it or that we at least have some shortcuts for figuring it out. Eventually, it just makes sense. 

 

I found geometry was really good for getting my kids to see this

Perimeter of a square vs perimeter of a rectangle.

 

When we go to area, we draw the shapes on graph pages so we have blocks to count and understand (Sort of).

Volume takes more trusting and learning equations.  Conceptualizing the area and then having many layers of it. (Cylinder is easier to teach volume than a sphere or a cone for this reason)

 

 

[ElizabethB]

On 11/29/2019 at 8:34 PM, 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. 

There is a lot of math about cars that would make him an even better and more informed auto mechanic.  We have an earlier version of this book, my husband went on and on about torque curves after reading it...

https://www.amazon.com/Auto-Math-Handbook-HP1554-Calculations/dp/1557885540/ref=sr_1_1?crid=MUT2WWBC9SEN&keywords=auto+math+handbook&qid=1575621664&sprefix=auto+math+%2Caps%2C187&sr=8-1

[ElizabethB]

On 11/29/2019 at 7:26 PM, Paradox5 said:

I'm seeing 2 books on Amazon with that title. Which is the correct book?

https://www.amazon.com/Teaching-Elementary-Mathematics-Understanding-Fundamental/dp/0415873843/ref=sr_1_1?crid=1J7P6H4NJYD9F&keywords=liping+ma&qid=1575080687&refinements=p_85%3A2470955011&rnid=2470954011&rps=1&sprefix=liping+%2Caps%2C160&sr=8-1

https://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0805829091/ref=sr_1_2?crid=1J7P6H4NJYD9F&keywords=liping+ma&qid=1575080687&refinements=p_85%3A2470955011&rnid=2470954011&rps=1&sprefix=liping+%2Caps%2C160&sr=8-2

They are the same book, but the new one has an added intro with a bit of reflection.  Not worth the extra $ IMO, the meat of the book is in the main section. The book is worth buying overall, but the new book is maybe only worth $1 more...