# Where to discuss math pedagogy and how to teach specific concepts

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I am about to launch into a major algebra and problem solving review with my 16 year old.  We will finish the geometry book this week and she needs the review before going on to Algebra 2 probably in the fall.  I am wondering if you have any recommendations about where (if not here) to post specific pedagogical questions. I don't know if there are enough people here interested in this topic, nor if I will have too many questions to post.  I have taught math; I have a degree in math.  This is my 4th child so I have taught my own kids math.  This is the keep you humble child.  Not because she is trying to, but because math is a struggle for her and what worked with others doesn't.

Here is an example of a question I had today.

The section was on writing the equation of a line using the point slope formula or the y=mx+b formula.  I also showed her how to use y=mx+b instead of the point-slope formula if you are given a point and a slope.  That made sense to her.  I don't see a reason to require that she use the point slope.  The only reason that I can think of is the connection to inside and outside transformations of functions that will be discussed in precalculus.  I can absolutely guarantee you that she won't remember it then even if we do it now. We did go through the point slope formula and where it came from.

I also have questions like how to help a student who does not remember math easily.  I have a few things I have tried, but would love to chat with others about them.  I have plenty of people that I could talk to that would just say just do the best you can and not worry because math is just not her thing.  I refuse to give up.  She doesn't mind math at all which is good.

Kendall

Edited by matermultorum
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You can certainly post here.

For your example, no, there is not a need to know anything other than that one form.  All others can be derived from it.  In fact, you would be better served having her understand and be able to show the derivations.

At some point in the future (e.g., with transformations / translations), there will be an advantage to the point-slope style forms.  She can learn the format at that time, and be fine.

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My ds has some lds and has greatly benefitted from additional work in algebra before moving on to precalculus. I started him in CLE's algebra, which is a spiral program and then we used a very light geometry program (MUS) the following year so that we could continue to spend time on algebra that year as well. I used Lial's Introductory algebra with him the year he did geometry. I didn't have him do the complete program, but he did all the chapter review problems for each section as well as the chapter tests and cumulative review problems that year. Last year, we moved on to Lial's Intermediate algebra for algebra 2. I used the dvd lectures with it and we spent one day doing watching the lectures and doing the margin exercises to make sure he understood the lecture and the next day to do the odd problems for that section. I took the end of chapter cumulative reviews and had him spend 10-15 minutes a day, every day, doing cumulative review problems. If we waited until the end of the chapter, which sometimes took 3 or 4 weeks to complete, he would have forgotten things.

All of this repetition seems to have really benefitted him. Now he is doing Lial's precalculus at home with me, which is going great, and also taking an algebra 2 class at a local coop. He is getting all A's in the algebra 2 class whereas many of the kids in his class are struggling by with 70's, so, imo, all the repetition has paid off for him. Originally, the only reason I was going to have him take this class at the coop was so that he could be with his friends since social opportunities are scarce for him right now, but it's turned out to be a big positive because he is continuing to get the reinforcement that he needs and it has been a real confidence booster for him. I honestly think he understands the concepts as well as, if not better than my dd, who is accelerated in math, at this point.

If I was going to suggest one thing, it would be to get the Lial's books and use the cumulative review exercises on a daily basis alongside whatever other program you decide to do (unless you are already using something that builds in daily review like Saxon) They are very inexpensive. I honestly think that was very important for him retaining the information long term -- never going too long without hitting on a topic.

ETA: The other thing I did that really helped him was have him do all his work on a small whiteboard. He has handwriting issues and this made a huge difference for him. Up until this year, I've always sat with him while he did his math and immediately let him know if there was something that needed to be corrected and made sure he understood what went wrong in his initial calculation.

Edited by OnMyOwn
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You can post here.

As for the specific issue, a student who understands linear functions should be able to derive the point-slope formula when she needs it - it is not something I would memorize at all. In general, with conceptual understanding, there really is not all that much to remember.

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Thanks everyone.  I will continue to post here until I drive everyone crazy.  We will stick with the y=mx+b. Thanks for the input on that, Mike.

OnMyOwn, thanks for the info about Lial's.  Continual repetition of everything is what I suspect she needs.  Trying to do it systematically and regularly on the fly is hard.  I end up reviewing anything that comes up that she has forgotten, but I need something planned out. Are the answers in the back or do I need an answer key? What edition do you have?  Do you know if older editions have the review?

This semester we started by reviewing 30-60-90, 45-45-90, trig functions, and the Pythagorean Theorem (Is it really review if they have seen it before but have no recognition??)   In the Jergensen Geometry book these are used regularly in the 3 chapters we have done this year. At the beginning of the year she had almost no recognition of these things that we had done at the end of last year. She has gone from needing to be retaught every day or two, to not recognizing that she needed to use, to remembering that they existed and going to her notebook to find them and needing help using them, to going to her notebook and mostly being able to use them, to sometimes being able to remember them without looking.  This took at least 12 weeks.  Iâ€™m pleased that she has gotten there, but with my other kids this was a 1-2 week progression at the most with maybe a brief reminder after months of not using them.  So Iâ€™ve got to continue to look for ways to help her better.

Regentrude, I wish I was completely sure of the full relationship between remembering and understanding with this student!  She seems to understand and can answer questions and explain it back, but the next day I have to reteach, though each day it is a shorter reteach unless we go some time without hitting a concept. There are also odd confusions.  I try to anticipate trouble spots, but when talking about the slope of a median of a triangle I never anticipated that she was trying to figure out how to do the statistical median. At least she remembered there was another median!  But I understand what you mean and understanding is my priority vs. memorizing.

Iâ€ll keep plugging away. Relationship first, math second.

Thanks everyone!

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I'm a fan of point-slope form for linear equations.  I don't know, it just seems that knowing a point the line passes through and it's slope is more intuitive and seems to be presented more often than y-intercept.

My kids occasionally forget things though they are strong in math, so ymmv here.  What I like to do is if I detect that they've forgotten something, we re-derive it.  Maybe it's a quickie back of the envelope derivation, but we definitely derive it.  How does that quadratic equation go again?  Derive it, beginning with ax^2 + bx + c = 0.  It's also a good opportunity to review completing the square, a bonus!  I don't view it as drudgery or punishment for forgetting, but as an opportunity to exercise a muscle that maybe hasn't been used for a while.

When you explain things, do you also put pen to paper and draw them out?  Maybe after leading her through the derivation, you can throw out the paper and ask her to derive it for you with pen and paper in hand.  Like if she forgets 30-60-90 triangles, begin with an equilateral triangle and proceed from there.

One tip I learned that mathletes use for training fir math competitions is to keep a notebook of all difficult problems with their own hand-written solutions.  Then after some time has passed, attempt to solve them again.  If they can't, study their solution and rewrite it, and then let some time pass again before attempting it again.  Maybe you can have her keep a notebook of difficult concepts:  linear equations, pythagorean theorem, quadratic equation, rules for exponents.  Pull one out every day and have her derive it from memory, on paper.

HTH.

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I am rereading what you said about Lial's cummulative review.  I could do that with the Foerster algebra books I have.  Has the transition from sitting next to him to not needing to been intentional or did he just not need that any more?

My current plan is to do the chapter reviews of Foerster Algebra I(until I have to start reteaching), continue to work through AoPS PreAlgebra(we've taken a break from that but were doing it along side Geometry, we are only in chapter 3 I think), and to do some ACT problems.

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daijobu,

Thank you!!

Yes, if she has forgotten I go back to the whys underneath. What I have not done well is to have her then redo the derivations on her own.  I do to much of the writing myself, having her tell me steps when she can. I will start having her then do it and explain it.

I love this idea and will be starting it tomorrow. " Maybe you can have her keep a notebook of difficult concepts:  linear equations, pythagorean theorem, quadratic equation, rules for exponents.  Pull one out every day and have her derive it from memory, on paper. "

Though we made a notebook this year as we went along, I haven't had her keep a notebook of selected problems solved.  We will add a section for that.  Great idea.

You said, "I don't view it as drudgery or punishment for forgetting, but as an opportunity to exercise a muscle that maybe hasn't been used for a while. "   This has always been my attitude, but I hadn't thought to describe it as exercising a muscle.  I like that analogy.

I'm not sure I understand about the point slope being more intuitive than the slope intercept if you have a point an a slope, but since math intuition isn't something she is consistent demonstrating it probably doesn't matter though it would be interesting for me to think about.  I spent about three days trying to get across the reality that an equation of a line (or circle) "works" for every point on the line meaning that if you substitute the coordinates of a point on the line into the equation it "works" and if you substitute a point not on the line it doesn't "work".  She seemed to be getting that and so I think that is why substituting the slope and one point into y=mx+b and solving for the y-intercept made sense to her, today that is:).  Tomorrow is another day!

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I am rereading what you said about Lial's cummulative review. I could do that with the Foerster algebra books I have. Has the transition from sitting next to him to not needing to been intentional or did he just not need that any more?

My current plan is to do the chapter reviews of Foerster Algebra I(until I have to start reteaching), continue to work through AoPS PreAlgebra(we've taken a break from that but were doing it along side Geometry, we are only in chapter 3 I think), and to do some ACT problems.

My ds does not want me to sit with him any more while he does his work. He's 17 now and needs to work more independently (and has in all of his other subjects for years). I do still check his math each day and have him sit down and go over anything he got wrong, though.

Sure, you could use Foerster's cumulative review. If you want more variety, here are the Lial's books that I have. https://www.amazon.com/Intermediate-Algebra-Lial-Developmental-Mathematics/dp/0321279204/ref=sr_1_9?ie=UTF8&qid=1480567078&sr=8-9&keywords=Lials and https://www.amazon.com/Introductory-Algebra-Developmental-Mathematics-Paperback/dp/0321279212/ref=sr_1_1?ie=UTF8&qid=1480567161&sr=8-1&keywords=Lials+introductory

The answers for all the odd problems are in the back of the book, but you can also get the solutions guides, which have all the answers, inexpensively.

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Echoing Regentrude, for many students, there is a real danger in expecting memorization.  Other than the basic arithmetic tables, there is no necessity to memorize anything by rote.  Practice will drive the key details in.  You mentioned the 12 week progression; this is rather quick, and natural.

From a teaching perspective, what you might consider employing is the overview-detail-overview method that good coaches use.  That is, go through the material - especially all of the theory - quickly, without attention to grades.  Then, go through in detail normally.  Finally, brush up with a quick overview of each section.  This will allow memorization to become a natural consequence instead of a required effort.

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My ds does not want me to sit with him any more while he does his work. He's 17 now and needs to work more independently (and has in all of his other subjects for years). I do still check his math each day and have him sit down and go over anything he got wrong, though.

Sure, you could use Foerster's cumulative review. If you want more variety, here are the Lial's books that I have. https://www.amazon.com/Intermediate-Algebra-Lial-Developmental-Mathematics/dp/0321279204/ref=sr_1_9?ie=UTF8&qid=1480567078&sr=8-9&keywords=Lials and https://www.amazon.com/Introductory-Algebra-Developmental-Mathematics-Paperback/dp/0321279212/ref=sr_1_1?ie=UTF8&qid=1480567161&sr=8-1&keywords=Lials+introductory

The answers for all the odd problems are in the back of the book, but you can also get the solutions guides, which have all the answers, inexpensively.

Thanks, I went ahead and ordered those because my 12 year old is using the Foerster algebra book and they all like to do math first thing in the morning.

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I'm not sure I understand about the point slope being more intuitive than the slope intercept if you have a point an a slope, but since math intuition isn't something she is consistent demonstrating it probably doesn't matter though it would be interesting for me to think about.  I spent about three days trying to get across the reality that an equation of a line (or circle) "works" for every point on the line meaning that if you substitute the coordinates of a point on the line into the equation it "works" and if you substitute a point not on the line it doesn't "work".  She seemed to be getting that and so I think that is why substituting the slope and one point into y=mx+b and solving for the y-intercept made sense to her, today that is:).  Tomorrow is another day!

One of the things that struck me doing linear graph problems in AOPS was how they were able to take a couple equations and apply them to solve a lot of problems that at first glance didn't look like they had solutions.

While I'm not suggesting that AOPS is a good choice for all students (I have to program in a lot more review that the book expects in order to let concepts solidify), I do think that the mix of learning a concept and applying it to lots of situations is really helpful.  This is the same thing I like about the old (1960s) Dolciani books.  They have some fantastic word problems or problems that use things like cross section area of a figure to practice algebra concepts.  You might try collecting more applications of what she is learning, so she can see that many of the alternate equations and approaches are simply coming at a relationship with different base information.

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• 3 months later...

I donâ€™t know if I can put the exact PSAT problem on a public forum, but this was Question 17 from the calculator portion of the Wednesday test 2016.

I will change it some. The relationship between Q and T is modeled by  Q= 5-.8T  If the equation is graphed with T on the horizontal axes and Q on the vertical axis, what is the slope of the line?

Even though we have been reviewing(see below) semi-regularly how to graph an equation in point slope form and how to identify the slope and y intercept, she did not approach this problem correctly. Instead she plugged one of the slope answers from the multiple choice into y=mx+b in place of m.  Then she set that equal to 5-.8T instead of y.  Then she was (obviously!) stuck. She was planning on doing that with each slope to see which one worked. I was a little frustrated that she couldnâ€™t see what to do.  I am trying to think positively and be happy that she remembered y=mx+b and knew that m was the slope.

I used to be pretty negative about standardized testing, but one nice thing about PSAT/SAT/ACT math tests is the mixed practice and being forced to apply what you know to something that looks different. But I am struggling with how to get her to be able to do that.

Since December when I posted last on this thread we have worked through the chapter tests in the Lial Introductory algebra book. In addition each day I give her 4-5 review problems.  I choose the problems from whatever comes up that I have to stop and (re) teach her. Basic operations with decimals, fractions, percents, as well as algebra topics.  Every day we do this. I know this is what she needed all along, but better late than never I guess. I try to space the appearance of each type and also make sure the review questions are quite mixed. But If I space it too far she often needs some help again remembering.

Now I am alternating between test prep (we are moving to ACT math prep in a few days) and having her work through the cumulative reviews in the Lial book. I have AoPS prealgebra I could use as well and AoPS Introductory algebra.  And I am continuing the 5 problem review each day.

I donâ€™t even know why I am writing. She still doesnâ€™t mind doing math and we still like each otherJ and I think there is small progress being made. Just tell me to keep at it and to try not to show my  frustration.

I guess I do have one question.  Do you think there is value in her work through a mixed practice test like this PSAT regularly so that she will get faster at understanding what to do?  I mean doing the same test over and over again until she is fast and solid and can explain it. Maybe In a way this is kind of what was mentioned earlier about keeping a notebook of difficult problems. Maybe I should wait and do that with the ACT.

Thanks for listening. I was just feeling discouraged after our last session. Math with the other 7 kids is going(or already went) really well.

Kendall

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Don't be discouraged.  The problem you cite is a tough one if you don't have a thoroughly deep understanding of graphing and lines.  Relabeling the axes from x and y to T and Q, while not unheard of, does remind me of this recent Math With Bad Drawings: Baby Name Book of Variables, especially the function x(f) = f - f^3, lol.

As your student collects more and more tools in her problem solving toolbox, it can be tough to remember them all.  I like using old AMCs (maybe start with the AMC 8) to throw different problems at my kids so they are reminded of some long-forgotten techniques and theorems.  Keep mixing it up.  The real world won't be telling her which theorem to use; they just want the right answer.

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To add, you might relabel the axes to more accurately relate what they are measuring.  For example, you might be describing volume as a function of temperature.  Then the y-axis would be labeled V and the x-axis labeled T.

Another example might be population versus time.  The y-axis might be labeled P or N (for number of people?) and the x-axis would be labeled t.

So it's not unheard of to use different labels for the axes, and it's worthwhile to explore this in depth.  V = mT + b?

Edited by daijobu
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I've tutored for a few years, and have taught 1 kid who scored 0.1 percentile in math and two other kids who really struggle to *remember*.  What I have found is that flashcards are your friend.  I make probably 5 flashcards per session of questions that were difficult in the lesson that day.  Then I add them to the pile.  We drill 30 cards per day orally. It is not exactly memory but yet it is.  These are all the questions that were hard for the student.  If she has to re-answer questions, and there are 300 cards to work through over a fortnight, she can't memorize them.  But yet, by the constant review, there is more of a chance that it will go into long term memory.  Some times kids need more of a memory component than a problem solving component to have success.  And success builds on success.  So after a stronger memory year, I've seen students become more capable of problem solving.  The goal is long term.  Each year does not have to be the same.

Ruth in NZ

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I feel so encouraged!  Thank you! Daijobu, we thoroughly enjoyed the Baby Name Book of Variables and other posts on that site.  I'd never heard of it.  I passed it on to my Electrical engineer son and he enjoyed it, too. And thanks for the link to the AMC tests!  Those are going into the mix for at least 2 of my kids tomorrow.

I decided today to pause on test prep and spend some time on application problems with linear equations and graphing and also to extending the problems in various ways such as relabeling.

Ruth, thank you for the reminder that the goal is long term. It was encouraging to hear that you have worked with kids like this and to hear what has helped.  It makes so much sense to me that this kind of "memory work" leads to increased problem solving skills. I love the flashcard idea.  Do you put the problem on one side and the worked solution on the other? When you say orally, do they just tell you what they would do? Or do they go ahead and solve it all in their head? This would be a good way to do problems that require more space.  I've been using post-it notes for the review.  I will start building a stack of flashcards. Your explanation of how they help is what I vaguely had in my mind when I mentioned going over and over the practice test questions.

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• 5 months later...

Thanks again for all of the input. I have a few questions, but thought I would update you. We spent last semester going through the Lial's chapter tests, continually cycling through "flashcards" from ACT/PSAT problems, and doing other review questions daily. I had hoped to continue this at least some during the summer, but we did very little. In spite of that, she is remembering a lot. I am really seeing the fruits of the constant repetition and am thrilled that some of this lasted through the summer. I also used some of what I think would be called right brain methods for remembering a few things, such as the sine, cosine, tangent ratios and what an integer is. I think I need to continue that.

We have started the algebra 2 text by Foerster.  She has so much trouble remembering terms and there are quite a few at first, all of the axioms-distributive, associative, commutative, rational, irrational, natural numbers, degree of polynomial.

I'm wondering if I should spend our review time nailing distributive, commutative, reflexive, symmetric, identity, associative? What are your thoughts on that? The author has review questions at the beginning of each lesson so it is reviewed, but not enough for her to remember what the names mean. She knows how to distribute and commute and associate.

The names of the groups of numbers I'm thinking I should keep working on until she remembers them.

Also, how important is it that they remember that y=4 is called a constant function?

Thanks,

Kendall

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We have started the algebra 2 text by Foerster.  She has so much trouble remembering terms and there are quite a few at first, all of the axioms-distributive, associative, commutative, rational, irrational, natural numbers, degree of polynomial.

I'm wondering if I should spend our review time nailing distributive, commutative, reflexive, symmetric, identity, associative? What are your thoughts on that? The author has review questions at the beginning of each lesson so it is reviewed, but not enough for her to remember what the names mean. She knows how to distribute and commute and associate.

The names of the groups of numbers I'm thinking I should keep working on until she remembers them.

Also, how important is it that they remember that y=4 is called a constant function?

Language is important. We must be clear what we are talking about, so knowing the difference between rational and irrational numbers is vital - they are different things! Knowing what the word "constant" means is also essential: it is something that is not changing. A student who has understood this does not need to "remember" that y=4 is "called" a "constant function"; she will understand that y=4 is a constant since it does not change.  I have a hard time understanding what the issue is here; does she have any language processing problems? Does she understand what "constant" means when the word is used in a context different from math?

I do not find it terribly important to name the different properties - associative, distributive, commutative; it is more important that she can apply the terms. Knowing that the order of terms in a sum can be changed is more important that knowing this is called "commutative", and distributing a factor in an expression more important than knowing the word. In my experience, it is much more efficient to learn the terms through talking about math rather than drilling flash cards. When the student narrates a solution and says "we have to distribute the factor", it will naturally cement the distributive property.

Not "remembering" the degree of a polynomial strikes me as strange. Does she understand what a polynomial is? That there are different powers of the variable? I am truly a bit puzzled by this, since one can see the highest power by one glance at the polynomial, and it is obvious that this is a critical quantity for the polynomial's property. Again, I would not drill this; it comes up every single time you talk about a polynomial and identify its degree.

I also used some of what I think would be called right brain methods for remembering a few things, such as the sine, cosine, tangent ratios and what an integer is.

ETA: Does she know word roots? It is obvious what an "integer" is as soon as one notices that it comes from the word for "whole".

For the trig functions, teach her the mnemonic SOHCAHOA.

2nd ETA: Another thought: does she understand that rational and irrational numbers are fundamentally different? Without knowing the words, does she see that the property of being able to be expressed as a fraction is something that is non trivial? I am wondering to what degree you are dealing with retention of specific words and to what degree it is about understanding concepts.

Edited by regentrude
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I think the biggest problem is definitely the retention of specific words. I am completely puzzled as well. LOL! Roots do help her. Sometimes the concept takes a while, too, but she gets the concepts sometimes years before the terms stick with her. It took a long time (several years) to know what a variable was. She could work with them, substitute values for them, solve for them, but if I said what is the variable she couldn't remember what that word meant. I always remind her that a variable can vary. Same with coefficient.

She can work right triangle trig problems and even doesn't have trouble when finding the angle rather than one of the sides and seems to understand the idea of inverse, but every time she would have to look up the ratio's. If much time has passed she might not remember that the ratios exist and apply to a specific problem, but once reminded she then remembers what is involved. I used SOHCAHTOA  with her for two years whenever it came up. She could never get that to help her remember the ratios.  After attending a homeschool conference and hearing about "right brain methods" I VERY skeptically tried them. I drew a picture and told a story and she added things to the story and picture and we wrote the ratio in the picture and taped them to the wall. In a few weeks time she knew them without looking. I just have to accept that her brain doesn't work like mine and we need to do these kind of things to help her with terms. It is just time consuming and there is only so much wall space LOL!

Because it is time consuming to get words down I need to pick the most important ones to focus on so that we can move forward in concepts. You have confirmed my gut feeling that I should continue naming the properties such as commutative but not spend our time drilling them. The names such as irrational and rational etc I will work harder with her on. It may be picture time for those.

She can tell me the degree of a polynomial easily if I remind her what the word degree refers to. She can identify exponents (again sometimes I have to remind her which part is called the exponent) and knows that you add the exponents of the term with the highest total of exponents. She seems pretty solid on identifying which expressions are polynomials and that they can only involve the three operations on a variable. That was new this year, or at least hasn't been seen since Algebra one  2.5-3 years ago. Whether she will remember that the term polynomial applies to this concept I can't yet say. But that term comes up enough that I will work on getting that one.

Thanks for the input!

Edited by matermultorum
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• 4 weeks later...

So a little update. She took the ACT and based on practice tests I was hoping for a 19 on the math. She got a 24!  I'm more excited about this than the 34 and up my older kids got. Ruth reminded me to look at the long haul and it is nice to see some progress after months and months of plugging away steadily.

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