Algebra Textbook for Younger Students

[mathmarm]

Do you have an algebra textbook that you like for when you have younger than usual student?

I'll be teaching from and assigning problems from the book, it's not for the student to read and study independently.

[EKS]

Jacobs

[8filltheheart]

For a first run through, I have used MUS alg and geo.  I follow it with another yr of each.  The MUS book has a lot of white space so it isn't overwhelming with text for a younger student.

[Guest]

I used Key to Algebra as a first run through, after Hands on Equations, and then moved into AoPS. Key was nice because it moved to doing work on paper, but still was very gentle.

 

[Crimson Wife]

I haven't used them myself but I've always been intrigued by the Arbor Center series.

[EKS]

25 minutes ago, Crimson Wife said:

I haven't used them myself but I've always been intrigued by the Arbor Center series.

I agree--I think they would have worked well for my younger son.

[daijobu]

AoPS

[mathnerd]

We started AOPS prealgebra with my young son. I taught it to him and he was able to handle it (though, it was real slow going for a couple of years).

[Cake and Pi]

My younger-than-usual student's algebra-related path started with the DragonBox Algebra apps, moved to Algebra Lab Gear (books and blocks), followed by AoPS Prelagebra, then AoPS Intro to Algebra.

[wendyroo]

My oldest did/is doing:
1. Hands on Equations (especially the Verbal Problems book), 
2. AOPS Prealgebra (and Alcumus) interspersed with Zaccaro's Becoming a Problem Solving Genius when he needed a break,
3. AOPS Algebra (and Alcumus) interspersed with Zaccaro's Real World Algebra when he needed a break.

We made it through chapter 9 of AOPS Algebra and then DS hit a wall when we got to factoring polynomials, so he is working through the third Arbor Center book, Chuckles the Rocket Dog.  After that we might go back and finish AOPS Algebra chapters 10-13, or he might just reach mastery of those chapters on Alcumus, or we might call it good and move on to AOPS Intro to Counting & Probability.

[Clemsondana]

We started with AoPS pre-A at a young age.  Kiddo understood the math but paid too little attention to detail so had tons of mistakes (losing exponents and negative signs, etc) because their problems are complicated.  We took a break and did some of the Arbor Press books until kiddo grew up a bit.  We would have probably been OK if we'd moved more slowly - I should have set a time limit early on.  We've continued with the routine of doing AoPS a couple of days each week and then doing Life of Fred on any busy days and not worrying too much about how quickly we finish the book - we spent 1.5 years each on pre-A and Alg.  For geometry, we did that a couple of days each week and continued on with LOF algebra as a weekly review of algebra concepts.  We also took some time to do number theory and introductory probability, which kiddo found to be fun.  Now in the second AoPS algebra book, we still do LOF on busy days, usually around once/week.  Kiddo has found it to be useful to see the same concept presented 2 different ways and see different types of problems for the same material.  

[mathmarm]

On 3/9/2020 at 11:32 PM, square_25 said:

I think that's a tricky one, because kids at this age, kids can manage SOME of the things but not others. And most textbooks aren't really designed for that, since they are meant for older kids. 

Personally, I'm currently working on algebraic manipulations with my daughter -- so far, we've done both linear equations and quadratics. The idea of doing the same thing to both sides of an equation seems to make total sense to her, so that's gone well. She's had no trouble with expanding and factoring quadratics, either, but then I've basically spent all of our arithmetic years working on the properties of the operations, so it's all well-integrated for her. 

We've also been working on functions which, again, has been pretty intuitive and easy. But we aren't doing anything like domain and range yet -- just defining functions and calculating them. 

I'm also planning to quite a lot of graphing with her -- kids not understanding graphs is a pretty big problem, so I think it's a good idea to graph lots of fun functions. I personally think it's a good idea to start by graphing all sorts of functions, instead of focusing on lines to the exclusion of everything else. I'll let this Math with Bad Drawings post speak for me: 

https://mathwithbaddrawings.com/2017/03/08/lines-beyond-y-mx-b/

Specifically, the following bit: 

"If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”

Anyway, I'm not really helping with the question, and I apologize for that! But I do think it's a hard problem, because if you go through a textbook, you'll run into things that will be too hard for no reason other than age, and then you'll have to slow down on going through concepts in order to beat your head against a wall that may very well turn out to be a door in a few years, anyway. And I personally prefer grabbing the concepts first while keeping the material age-appropriate :-). 

Thanks.

We've also done arithmetic from a properties first point of view and he's easily branching into algebraic concepts as well. I'm fairly confident in directing his math education, but I don't want to write out all the problem sets myself all the time.

He is already doing a great deal of algebraic skills. He graphs (and transforms) curves in 2d and is learning to graph 3d curves, he plays with functions, solve equations, manipulates and simplify expressions etc. He can work with quadratic equations and expressions as well.

I want a book that will help me consolidate and solidify his ability to use algebra in a real world sense of the word.

[mathmarm]

So I picked up a copy of 2nd edition Foerster Algebra 1 and Algebra 2 for a good price locally.
The Algebra 1 guides the students in how to write expressions for specific situations and it's exercises for word problems.
Exercises for proofs are written out and required to be fully explained by the student, which is an excellent exercises for building those math skills.

Foerster uses bar-models in the early chapters then weans the kids off of them so that they can write expressions intelligently and includes diagrams throughout for word problems.
Chapter 7 is solving and evaluating equations with BASIC programming which is a little obsolete-but over all it seems like a solid text. I like that by Chapter 8 it's expected that students are able to read and work the text independently, because Ch1-6 are building them up to that skill. There is oral drill in each section and even in Algebra 2 there are Do These Quickly drills for each (most) section?

It seems like a well put together book. It's not flashy or cute, but I like it. Unfortunately the texts are thick and heavy, but I think that we can work with that fairly well.

I will keep my eyes out for something better, but I really like these Foerster books. I may pick up a copy of his Precalculus with Trigonometry text to see how the whole series reads.

[mathmarm]

40 minutes ago, square_25 said:

Hmmmm, OK. I'm not sure what algebra in a real world sense of the world means. You mean doing word problems? I think what he's doing sounds like algebra already :-). How do you graph 3D curves, by the way? Do you use a computer program?

No, I'm am old fuddy-duddy. We do all our graphing by hand. We graph on a XYZ coordinate plane that we draw on the board.

We are holding off on introducing technology beyond pencil and paper for the majority of elementary. I want him to go through calculus and differential equations by hand first.

[8filltheheart]

4 hours ago, mathmarm said:

So I picked up a copy of 2nd edition Foerster Algebra 1 and Algebra 2 for a good price locally.
The Algebra 1 guides the students in how to write expressions for specific situations and it's exercises for word problems.
Exercises for proofs are written out and required to be fully explained by the student, which is an excellent exercises for building those math skills.

Foerster uses bar-models in the early chapters then weans the kids off of them so that they can write expressions intelligently and includes diagrams throughout for word problems.
Chapter 7 is solving and evaluating equations with BASIC programming which is a little obsolete-but over all it seems like a solid text. I like that by Chapter 8 it's expected that students are able to read and work the text independently, because Ch1-6 are building them up to that skill. There is oral drill in each section and even in Algebra 2 there are Do These Quickly drills for each (most) section?

It seems like a well put together book. It's not flashy or cute, but I like it. Unfortunately the texts are thick and heavy, but I think that we can work with that fairly well.

I will keep my eyes out for something better, but I really like these Foerster books. I may pick up a copy of his Precalculus with Trigonometry text to see how the whole series reads.

What yr is the copyright of your alg 1 book? Mine are 1994.  Chpt 7 in my book covers expressions and equations containing 2 variable. Chpt 8 is linear functions, scattered data and probability. (a few word problems scattered throughout the book are still BASIC problems, but only a handful.

Edited March 18, 2020 by 8FillTheHeart

[mathmarm]

5 hours ago, square_25 said:

I think what he's doing sounds like algebra already :-).
Let us know how Foerster's goes! I'm always interested in gathering reviews :-). 

Yes, he's definitely doing algebra already.

1 hour ago, 8FillTheHeart said:

What yr is the copyright of your alg 1 book? Mine are 1994.  Chpt 7 in my book covers expressions and equations containing 2 variable. Chpt 8 is linear functions, scattered data and probability. (a few word problems scattered throughout the book are still BASIC problems, but only a handful.

 

Mine is second edition, copyrighted 1990. It says second edition on the cover and on the title page There are 14 chapters and a Final Exam, on 679 pages not counting the indices, tables, etc.
01-Expressions and Equations
02-Operations with Negative Numbers
03-Distributing: Axioms and Other Properties
04-Harder Equations
05-Some Operations with Polynomials and Radical
06-Quadratic Equations
07-Evaluating Expressions by Computer
08-Expressions and Equations Containing Two Variables
09-Properties of Exponents
10-More Operations With Polynomials
11-Rational Algebraic Expressions
12-Radical Algebraic Expressions
13-Inequalities
14-Functions and Advanced Topics
---Final Examination

[8filltheheart]

48 minutes ago, mathmarm said:

Yes, he's definitely doing algebra already.

 

Mine is second edition, copyrighted 1990. It says second edition on the cover and on the title page There are 14 chapters and a Final Exam, on 679 pages not counting the indices, tables, etc.
01-Expressions and Equations
02-Operations with Negative Numbers
03-Distributing: Axioms and Other Properties
04-Harder Equations
05-Some Operations with Polynomials and Radical
06-Quadratic Equations
07-Evaluating Expressions by Computer
08-Expressions and Equations Containing Two Variables
09-Properties of Exponents
10-More Operations With Polynomials
11-Rational Algebraic Expressions
12-Radical Algebraic Expressions
13-Inequalities
14-Functions and Advanced Topics
---Final Examination

Looks like the 94 ed removes chpt 7, makes chpt 8 chpt 7, and adds a new chpt 8 on probability.  The 94 addition includes all of the other chpts/exams.

7 hours ago, mathmarm said:

No, I'm am old fuddy-duddy. We do all our graphing by hand. We graph on a XYZ coordinate plane that we draw on the board.

We are holding off on introducing technology beyond pencil and paper for the majority of elementary. I want him to go through calculus and differential equations by hand first.

By the middle of the book, Foerster does assume calculator usage on most word problems.  It sort of comes and goes, but there are going to be many problems that would be a royal pain without a calculator.  It sort of bothered me with an older kid who is solid on arithmetic, but I would find it even more troublesome for a younger child.  I liked the text well enough to get over it, but thought I would share.

[8filltheheart]

22 minutes ago, Mom2mthj said:

By the middle of the book, Foerster does assume calculator usage on most word problems.  It sort of comes and goes, but there are going to be many problems that would be a royal pain without a calculator.  It sort of bothered me with an older kid who is solid on arithmetic, but I would find it even more troublesome for a younger child.  I liked the text well enough to get over it, but thought I would share.

I look through the answers to determine which ones I allow the calculator for.  Mostly the cumbersome radical problems.  

[mathmarm]

On 3/18/2020 at 7:02 PM, Mom2mthj said:

By the middle of the book, Foerster does assume calculator usage on most word problems.  It sort of comes and goes, but there are going to be many problems that would be a royal pain without a calculator.  It sort of bothered me with an older kid who is solid on arithmetic, but I would find it even more troublesome for a younger child.  I liked the text well enough to get over it, but thought I would share.

Yep, that's right.
The author states in the Foreword to the Teacher:
"...The first part of the book is carefully sequenced to lead to the Quadratic Formula by mid-year. This departure from the more traditional sequence of topics is made possible by technology, specifically the use of a calculators to evaluate radicals. As a result, students are able to work more realistic word problems in which answers are decimals. Students must check their answers based on whether or not they are reasonable, not because they came out as small, whole numbers."

 For now, I plan for him to do everything by hand. It will keep his arithmetic and mental calculation skills razor sharp. 🙂  The book includes Table of Square Roots and a Table of Trigonometric Functions in the back, and some of the examples say "by calculator or by tables" when they explain a step, so I'll be photocopying and laminating those tables for easy and continual reference throughout.

I haven't gotten a chance to look through the whole book yet, but I do anticipate skipping 12-8 and 12-9, and adjusting the 12-Test accordingly by removing that material from the exam because the higher powers chapter would require another chart, or a calculator and I'm okay not getting that deep into the weeds on it with this first pass since it'll come back around in the Algebra and Trigonometry book.

[mathmarm]

On 3/18/2020 at 10:40 AM, square_25 said:

Let us know how Foerster's goes! I'm always interested in gathering reviews :-). 

It's not a review YET, but here's where my thinking is. I'm planning to read through chapters 1-6 more closely this week. Honestly he can do most of the calculation or manipulation type stuff in ch1 - ch4 now, and is learning the ch5 and 6 material already so I expect we'll get through the first few chapters easily. The learning objective for him during the early chapters will be:

• reading a textbook correctly
• identifying the important stuff
• taking notes from a textbook
• copying problems from a book to a paper and
• showing his work

We tend to do school sessions 2x a day, and my thinking is that for the first part of the book, we'll work by the timer for about 20 minutes, then have him do 5-15 minutes of table time. I am thinking it'll look something like:

Session 1: I'll sit with him on the couch and we'll buddy-read through the lesson and work the examples on a mini-whiteboard (if needed.) I'll  have him do the oral practice immediately. He'll use a sticky-note to mark something he feels is important information. Then his table work will be to write down 1-3 important things from the lesson. (this can be an example, a definition, whatever he determines is important from that lesson)

Session 2:, I'll sit with him on the couch and he'll work the straight-forward exercise problems on the mini-whiteboard (if needed) or orally for about 20 minutes. I'll assign 3-5 problems for him to do during his table time. So he'll do 3-5 problems on a sheet of notebook paper.

I think we'll be able to get through most lessons in a day with this approach, but
I plan to spend 2 days on
1-8, 1-9, 2-8, 4-3,
and I think we'll spend 3 days on
3-5, 3-6, 4-6
This way he's able to work through all or nearly all of those exercises. Especially 3-5 and 3-6 which gets him started with simple proofs. (I intend to come back around to his list the steps and justification proofs later and show him how to write a paragraph-proof at the end of the year, so I want to be able to use his earlier work for this)

 Chapter 5 has 528 polynomial drills grouped in 10 sections and organized according to pattern and arranged so that you learn to factor them in your head by the time that you're done.  Chapter 6 has 299 quadratic drills grouped across 8 sections before you get to word problems in 6-9 and 6-10. 
I want him to do all the drill in Ch 5 and 6, but not in a way that it creates a bottle neck or becomes a drudgery. Since he's familiar with but not fluent with polynomials and quadratics, I'm thinking that I may start him on chapter 5 and 6 at the same time that we start Ch 1. Then when we finish Ch4. He should be able to smoothly run through Ch 5 and Ch6, and the math-sessions should stay short and fun, because there should be minimal struggle with the math itself, but in being careful with his reading, copying, etc.
 

There is a test for each chapter, and a cumulative test for Ch 1-6. I haven't worked the tests myself, so I don't know if I'll assign them or not. I don't plan to skip much in the first half of the book, but plan to skip Ch 7 (BASIC) entirely. So I will be looking at Ch8-14 within the first weeks of April, and I will re-visit them probably at the beginning of May as well to come up with a plan for those chapters.

 

I'm thinking we'll be able to do almost a lesson a day for CH1-4, so we'll get through most of Ch4 before May
 I think that he'll finish Ch 5 and 6 at a steady pace once we get to them in early-or-mid April, but if he's needing to slow down and spread the drill out then I may start Ch 8 while he works on those chapters to keep him engaged. But I'm not sure. For now, I'd like to focus on getting through 1-6.

Edited March 23, 2020 by mathmarm
correcting the April -- > May

[mathmarm]

24 minutes ago, square_25 said:

What degree polynomials are they factoring and what are they using to factor? Since I'm not using a program, I could probably stand a reminder of what we're supposed to be doing 😉.

Given that it's at the beginning of an Algebra 1 text, I'm certain that they're just second-degree trinomials. That's pretty standard for first year algebra in the US. Typically you don't get into the factor theorem until the 2nd year Algebra (or at least the end of Algebra 1)

  

1 hour ago, square_25 said:

I'm curious how he'll do using a textbook! We haven't tried that experiment :-). 

DD7 has become way more able to show her work this year, although personally, I would rather have work on explaining her work in words (that is, writing proofs) rather than showing mechanical calculations. And that one is definitely dependent on her ability to write, which is really coming along, but still needs work. 

As I said, I'll be helping him through it step by step, so I'm expecting that he'll do great using a textbook. No way am I going to set my 1st grade Big Boy in front of an Algebra text all by himself. We're going to sit on the couch, buddy-read it, discuss it together, etc. I'll be supporting and scaffolding him all the way. He'll only have table tasks when I feel certain that he can do the work successfully.

As for formal proofs written paragraph style, yes, that's the end-goal.  As I mentioned, we'll loop back around to paragraph proofs later on this year. But he's not ready to successfully write paragraph proofs on his own so I'm not going to ask him too.

Math is a secondary subject for Jr at this time, Hubby and I prefer that his main academic focus remain on writing, drawing, geography and music.

We'd start the book, not so much for the math, but for all the other skills that I already listed.

[mathmarm]

10 minutes ago, square_25 said:

Ah, OK, I see. Has he worked with other textbooks? No. Even as a math person, I find math textbooks about the least pleasant textbooks out there -- math tends to be much denser to read than lots of other subjects. We've started taking notes on textbooks this year, and it's been pretty fun. But we haven't tried a math text yet. Would you work towards taking notes after you work through buddy-style? Yes, as I outlined this in my earlier post. Go back and read where I describe how I would like each session to go and you should notice that my thinking is that after we've read the lesson, done the examples, and he's identified something important, he'll take a few notes on it during his table time. 

Oh, got it! For what it's worth, DD7 seems to have just intuited Vieta's formulas and is factoring random monic quadratics without any drill, so he may not need that much, either. (I didn't really expect her to, but she surprised me.)  I'm looking for him to build speed and fluency so even though he's able to factor polynomials, it takes him a little bit of time. My hope is that some concentrated drill will get him to go build speed and fluency.

I'm not really familiar with the US algebra sequence... for one thing, I took math in Canada, and for another, I pretty much knew most of it by the time I got to Canada in grade 6 (it was taught way earlier in Ukraine, and my grandmother is a math teacher, anyway). So I apologize for the clueless questions :-). 

What do you guys do for geography? I'm pretty bad at geography, so we haven't started doing anything for it, really. We've been doing reading, writing, music (piano), math, and recently Russian. Everything else is pretty unschooled.

Quote

Geography: learning about the worlds physical geography using a Read-Draw-Write learning pattern. This study will culminate in creating a hand-drawn world atlas.
    Read - he'll read daily about a specific continent for a week.
    Draw - each day he'll draw a map of the world, and he'll draw the continent of the week 3x. Which brings me to the
    Write - I'll make a photocopy of his best continent map and he'll write geography-themed copywork on it.
    Write - Beginning in the 3rd cycle through the continents, he'll take 2-column notes on his reading and use them to write a paragraph in later cycles.
The focus is mostly on the 6 permanently inhabited continents but Antartica will be included too. For the 3rd cycle through the continents we're shortening the time we focus on a continent to 2 to 3 days at a time. After he's got each map down solid, we're going to make an atlas by compiling his best maps, copywork and paragraphs into a book.

 

[bensonduck]

I was going to suggest Foerster! I presented it a little differently - I don’t think she ever even saw the book. I presented the lesson on a big whiteboard and then wrote some of the exercises in a notebook for her to solve. While I agree that learning to take notes from a textbook is important, I didn’t want to ruin DD’s zeal for math by trying to impose note taking or other executive function skills on her in that subject. We’ll do that in some other subject. (I should add that she did write down geometry formulas in a small formula notebook during geometry - she was fine with that and found it fun.)

we are now in Precalculus with Trigonometry and it is by far my favorite of the 3 Foerster books we have used so far. It is so clearly written! There are little explorations in most sections that act kind of like the discovery method or at least a hands on way to experience the topic.
 

i love Foerster for a young student because it’s deep but not weedsy, if you know what I mean. (DD has done AOPS summer camps and I have personally worked through all of AOPS PreA - that approach is not for her.  At least not yet.)

[mathmarm]

8 hours ago, bensonduck said:

I was going to suggest Foerster! I presented it a little differently - I don’t think she ever even saw the book. I presented the lesson on a big whiteboard and then wrote some of the exercises in a notebook for her to solve. While I agree that learning to take notes from a textbook is important, I didn’t want to ruin DD’s zeal for math by trying to impose note taking or other executive function skills on her in that subject. We’ll do that in some other subject. (I should add that she did write down geometry formulas in a small formula notebook during geometry - she was fine with that and found it fun.)

we are now in Precalculus with Trigonometry and it is by far my favorite of the 3 Foerster books we have used so far. It is so clearly written! There are little explorations in most sections that act kind of like the discovery method or at least a hands on way to experience the topic.
 

i love Foerster for a young student because it’s deep but not weedsy, if you know what I mean. (DD has done AOPS summer camps and I have personally worked through all of AOPS PreA - that approach is not for her.  At least not yet.)

Yes! Deep, but not weedsy! I love how it scaffolds and guides so neatly. The problem sets are well designed and the explanations and the text is readable--I honestly feel like Jr. can read it and follow what he's reading. The more I read these 2 books, the more I want to buy the precalculus and calculus books.

There are a couple of (probably nit-picky) things that I don't like, but I'm more than willing to supplement and scaffold around them. Over all, this seems a really well built series. I have to read Algebra 1 and Algebra 2 more carefully, but so far I like what I'm seeing pretty well.

I wasn't going to start him with the book until 2nd grade started, but he's been dying to have a math book and when he saw it he wanted to start right away and to have a math book. . I'm a mean-mom because I told him he has to wait until April 1 to start.

As for taking notes, that's probably the part he's most looking forward too. We lucked out in that we have a little boy who loves to write and is always asking for "writing homework" so we've bought a special notebook for him to use when he writes math notes and he's so excited.

This isn't exactly what I'd planned, or exactly as I'd prefer it, but it is what it is.

[8filltheheart]

If you decide you want the precal book, let me know. I have a copy and solutions manual that are like brand new that I would be happy to sell after the  current crazy. I cant teach precal (beyond my abilities). Only 1 of my kids used it with Kathy in Richmond as her teacher.

[mathmarm]

5 hours ago, 8FillTheHeart said:

If you decide you want the precal book, let me know. I have a copy and solutions manual that are like brand new that I would be happy to sell after the  current crazy. I cant teach precal (beyond my abilities). Only 1 of my kids used it with Kathy in Richmond as her teacher.

Excellent! What is the copyright or publication date on your edition?

[8filltheheart]

1 hour ago, mathmarm said:

Excellent! What is the copyright or publication date on your edition?

I'm not sure and it is in a moving box in my closet (we moved this summer and I didn't have room to unpack all of my boxes.  I bought it new when my 25 yr old dd was in 11th grade, so I bought somewhere around 2010 or 2011.

[mathmarm]

2 minutes ago, square_25 said:

I'm super curious how your DS likes it -- keep us updated! I may very well get a textbook someday, so reviews with little kids are helpful :-). 

If you wind up getting a textbook, I recommend that you import one from Eastern Europe or a country with a similar approach to mathematics. You won't like most of the texts designed on this continent.

[mathmarm]

2 hours ago, square_25 said:

Oh yeah, why?

Just a suggestion.

 

[mathmarm]

6 hours ago, square_25 said:

Could you please explain where you're coming from? I think you must be misinterpreting my background. I haven't worked with Eastern European textbooks since I was in elementary school and am not biased towards them. 

Oh, maybe I have you confused with another poster.

What do you want out of a math textbook for your young child?

What would be the benefit of using one in your homeschool given that you reject the Standard Scope and Sequence and prefer to create their worksheets and problem sets on demand, by hand for what interests them/you?

[mathmarm]

On 3/23/2020 at 10:32 PM, square_25 said:

Really, mostly to have example problems and make sure I’m covering everything? I don’t reject the standard topics, I just tend to like covering them in a different order!! But I’d absolutely want to make sure I covered the standard stuff in an algebra text. I teach online at AoPS, but even though DD7 is ready for the material their textbooks cover, I doubt she’s ready for the format... so I like hearing reviews of other books!

We use BA occasionally and I like using old Math Kangaroos for word problems, so it’s not like we never use outside sources :-). 

I hope I haven’t offended you with my suggestions!! 

Yes, I said:  you reject the standard scope and sequence. I did not say that you reject standard topics. What I mean is that you go through material to a different depth-level (scope) and you go through material in a wildly different order (sequence) than is typical.

That's super common among mathematically inclined parents, as far as I know. It isn't a weird or bad thing in my opinion--it's natural. Bookworm Parents that love to read share more books and share books differently than parents who are literate but indifferent to reading. Athletic parents share their love of sport differently. They interact with their kids and sports in a way that is inherently more focused and intent than most other parents are when they're playing catch with their kids. It's not weird or unusual that a mathematician engages their kids mathematically in a different way than is standard. At least I don't think so.

If you just need a scope and sequence to make sure that you're covering everything then a Table of Contents will do just fine (and take up a lot less space). You can view many of them online for free or download a counties scope and sequence. If you want a few example problems, then yeah maybe invest in a text but not always. I have found many textbooks for the elementary crowd to be...really generic.

Also, you haven't offended me--Too me personally, you're posts have --at times-- come across as braggy or boastful, which is a bit...much?, The boastful tone of your posts sometimes makes it awkward to read and know how to respond to but I have realized that that's more to do with the way you talk, and less to do with your actual intent. You're very passionate about math education and very outspoken, and that's just your personality.

That's not something to be offended by. Honestly,  you seem to be doing a pretty good job teaching your young daughter mathematics. You're documenting and publishing details about your approach. Such data is always something too celebrate and nothing to be offended by. 

I have a colleague who is a linguist. She's raising her kids in 5 different languages just because she can. She teaches her kids to read and write 3 other languages outside of their spoken languages. She documents her kids language acquisition quite closely and takes extensive notes on their development.

I have another acquaintance who is a high performance athlete and her daughter is being raised exceedingly physical. She's got all sorts of charts, alarms, timers and stats on her girl.

Several of the mathematicians that I know of have taught mathematics in wildly different ways to their kids and many of their kids are advanced well and above what is typically expected of their age/grade. Some of them share their experiences openly, others guard their "trade secrets" fiercely, lol.

It's just natural to many parents (especially those in Academia, I think) to include their kids in their world view and their specialties.

Edited March 25, 2020 by mathmarm
ordering paragraphs for clarity

[mathmarm]

7 minutes ago, square_25 said:

 

Hmmmm, I'm sorry if it comes out that way. ..

I apologize again for input you found unhelpful. ...

. I'll moderate my tone :-). 

I don't think you need to apologize or be sorry.

I realize that it's not your intent to boast. You aren't bragging--but if you sensed weirdness in my replies in this thread, it's because I was thrown at first and couldn't determine if you were, but once I determined that it's just your personality, your enthusiasm, your communication style, I realized that you're not bragging. You're just communicating.

[mathmarm]

20 hours ago, square_25 said:

We did talk a bunch on your other thread, so I think I was also assuming we would both kind continue that conversation (in which I made an unreasonable number of suggestions, lol.) 

Ooooohhhhh. That was over a month ago. I haven't re-read that thread and I didn't realize that you were continuing on from then.

I'm just excited to find someone else doing an algebra experiment with a little kid :-). We weren't even going to start algebra this year, but DD7 pestered me because of her Murderous Math books, so here we are :D. It's been really fun. I don't even know how long we'll spend here, since I feel like we need to circle back to some stuff (like long division, lol. We still haven't done long division!) I hope you enjoy having a structured approach to algebra and that your DS enjoys the book!! I really do want to hear reviews. 

I know families in real life who explore algebra or geometry simultaneously (or first) and have met a few others online. It's not unheard of for me. I myself had a very non-standard math education, so doing math according to an alternative scope and sequence with my kids feels like less of an experiment and more of just an experience.

It doesn't feel like this big mysterious unknown to explore geometry and algebra with my young kids (no pun intended). To me, even though I feel uncertain about the exact tools to use on this journey, I feel certain about what I'm doing. It's like solving an equation from a class of equations that you know how to systematically approach. You know some theorems and properties that inform the approach you take to the problem, even if you can't recall a particular algorithm at that instant. You know if you play with the different parts of the equation, something will occur to you.

I will keep my eyes out for interesting math texts that might be fun for or doable by a 5-8yo. From reading through it, there are things that I like about Foerster and things that I don't like about Foerster, but I haven't used it enough to have anything concrete to report. I've already shared the details of how I plan to use it. When we're a month or so in, I'll try and remember to come back and give an update.

[mathmarm]

On 3/26/2020 at 12:27 AM, square_25 said:

What was your math education like? 🙂I don't know if mine was standard or nonstandard, but it was practically all self-motivated past the age of 10... I was an obsessive math contest kid and I did a TON of math in high school. And then I got a math bachelor's and a math Ph.D, lol. I've always loved it, although I wound up not loving research math enough to do it as a job. 
 

My mother was a high school math teacher. She ran a math tutorial and enrichment program as well. She taught her kids math far beyond the depth and pace of the public school schedule. I was exposed to a lot of quality, enrichment math from the beginning. By Jr. High/middle school, I was being exposed to a lot of undergraduate and graduate level math, as well as helping in her Math Program with other kids.

Quote

Oh, totally random question -- I was looking over the thread, and you said you were doing graph transformations. How are you guys approaching that? How did you introduce them? We haven't done anything like that yet... just barely started graphing. 

We played with functions, mathematical curves and reading the coordinate plane as separate and intertwined skills for a couple of months. I don't have time to write up the approach right now, but I'll share more when I can.

 

[mathmarm]

Does you DD like manipulatives in math?

[mathmarm]

22 hours ago, square_25 said:

No, she never has. Visuals, yes, but not manipulatives. (And with fractions, she spurned visuals, too, and decided to talk every calculation out out loud. It was a little excruciating to listen to, to be honest, but it worked for her.) 

How about your son? 

Yes, Jr. is very handsy. He typically likes to use manipulatives, but I have to be very conscientious about the manipulatives when we use them during a lesson or they just become distractions.

In general, I will teach with manipulatives and lots of discussion. We will work with the manipulatives a lot, talking through what's happening and why. Making observations. Writing down what we noticed. Posing questions and using the manipulatives to investigate them.

Even though we use manipulatives, I think it's the discussion that's key, because as he becomes "fluent" in the discussion of a topic, he's mastering the concept and internally dominating the ways it can be processed. Once Jrs reached the point that he's able to tell me why something happened in a previous step, or tell me what's going to happen after some step, then the visual will be mostly redundant. For some topics, I let him create the visuals when we make doodle-notes for the math. But we often go from manipulatives + disucssion, to written work.

Some times, as we go deeper into a concept or if he begins to stumble at some point doing the written work, I will use visuals in place of manipulatives. We draw it and discuss it and he's back on track quickly then. Some times if I've moved him to written work too quickly, then while he's still processing a concept, I will toggle between manipulatives and visuals to help talk him through what he's doing and eventually just go to only/mostly visuals.

When he is satisfied that he can get it right without the manipulatives--like, really, really sure that he knows it--he doesn't want to use the manipulatives anymore. 🤷‍♀️

We used manipulatives a lot in the early stages of graphing, but he's resisting the manipulatives for the graphs more now, even if he's stuck, he'll screw his eyes shot and imagine it.

Jr. didn't use visuals for fractions either. I just extended the concept of units, and the operations. We talked through them.
JR, understood the prerequisite concepts well enough that fractions took like...4 days once we begin formally covering them.

[mathmarm]

On 3/29/2020 at 11:01 PM, square_25 said:

What kinds of manipulatives do you use for graphing? 

 

😞 I had a massive reply with color coding, indented lists and I lost it. I'm completely demoralized...

😔

The short answer is I used a wooden board and toys, then transitioned to a home made template on a whiteboard which was too slippery, so I finally landed on using a big XY-plane on cardboard. I got a lot of tips and suggestions from @Gil when I was starting out. If you want more detailed advice, you could reach out to him directly.

We played first with placing the toy and reading the coordinates.
Then placing the toy, reading the coordinates and then I would move the toy to a different place on the plane and JR would read the new coordinates and work out how the coordinates had changed and tell me what path(s) I took or could've taken to get there.
Another exercise was to place the toy, then I'd give him the new coordinates of where the toy should be and he'd work out how far left/right or up/down he needed to move the toy.

We talked about the "somersault shadow" also, and where the "somersault shadow" would be (reflections of the graph across an axis) and when we'd do the the What Was My Path version of the game,  I mixed in "somersault shadows" from time to time.

Later I transitioned to using cutouts of functions and we did the same thing. We just worked our way up to it.

In our pen-based math, we were working on functions a lot at that and I had him doing 1 or 2 graphs per day by transforming in stages. We use the pages with 4 seperate XYplanes on a single side. He'd pick a graph card and then we'd read the graph and say what kind of graph it was, what it looked like and how we knew. Even though we'd talked through the entire graph, I made him graph it in stages.

1st the parent function. On the 2nd XY plane, we'd say if it had wiggled away from "home base" left/right and graph that. Then we'd say if it had shimmied up or down from the main-line, and finally we'd say if it was fatter or thinner than usual based on the 4th XY-plane, we'd have the completed graph.

Flip the sheet of paper and do a second one.

From the math play and the pen-math exercises, when it was time to merge the steps, formally, the transition was very smooth. He deduced logically that the x +- h, notation would need to be different from the vertical shift notation.

It gave him a fantastic sense of spatial awareness for 2D graphs.

When he was comfortable with the basic functions, I introduced the math curves and as he was getting consistent with circles, I introduced the Trig functions. He loved graphing trig functions. Something about unwrapping a circle, and then "phase shifts" just absolutely tickles him. I think it sounds "cool". In hindsight, I should've introduced the word Transformations to him sooner.

This isn't nearly as good as my Lost Reply...

😪

[mathmarm]

@square_25. We had our first day with Foersters.

We read through and discussed the first 3 lessons, because he didn't want to stop. When he picked what was important to remember, I had him do notes for 2 lessons, and apparently he did notes for the  3rd lesson all on his own. He's so proud of his math notes. Algebra 1 CH1 - JR.pdf

He understood but missed the very last problem that he did in 1-3. The instructions were to write the amount as a power and he didn't. I won't mention it directly, but I will highlight the habit responding to the question asked or the directions given if this pattern continues.

[mathmarm]

1 minute ago, square_25 said:

Is that his handwriting? It's lovely! 

Thank you! Yes it is.

I wouldn't have termed it lovely, 😊 !

He was a real wiggle worm today so it is wobbly-er than I like it to be.

[mathmarm]

#45 for 1-3 got out of control. He was writing all over the page--he knows that's a no-no!  😏

He told me "It's sloppy but I checked it!" How can you be mad at a kid who checks their math? Answer: You can't be.

The answer was supposed to be written as a power so he should've written just 2^18, but he missed that and worked it out long hand.

[mathmarm]

32 minutes ago, square_25 said:

If you have the energy, I'd LOVE to see you post the notes from the lessons, at least occasionally. Was this stuff he knew already or new stuff? (I'm guessing he knew most of it.) 

Well, I'll probably be home a lot more for the next few months so I will try and update the notes every so often.

Yes, this was material that he has seen similar to before (even though its been a while for some things) and we read the book together and discussed the lessons we did many problems orally just snuggled on the couch together.

 

[mathmarm]

17 minutes ago, square_25 said:

Weirdly enough, we were doing exponents today, too... 

Oh? What was the lesson?

How'd it go?

[mathmarm]

I think that the exponent properties make for good first proofs. They are easy for young kids to experience by using the definitions themselves.

Exponentiation is often less familiar than addition/multiplication, so the properties aren't so obvious you can't tell the point of what you're doing by manipulating them.

Jr. had a lot of fun with exponents back when we did them.

Today, when he saw the lesson on exponents he grinned and said "yay! Exponents!"

 

[mathmarm]

18 minutes ago, square_25 said:

(she lost serious writing ground when I sent her to public kindergarten, grr.)

Fortunately we weren't able to find a suitable academic setting for Jr.

Hubby was telling me recently that he believes that us not finding a school for him was one of the main reasons that JRs core skills are so strong--that their development wasn't disrupted by school. He wasn't taught or allowed psuedo-alternatives or destructive practices like sight words.

He learnes--and practices--writing, reading and drawing without the disruption of disjointed/contrary instruction.

[mathmarm]

26 minutes ago, square_25 said:

So I'm excited to work on them with her! They still take a while for her to phrase correctly... I think that page took like an hour for her to put together, with input! But I was happy with her overall.

That's impressive that she was able to focus for an hour. Do you trained and developed her focus intentionally, or is that a personality thing?

[mathmarm]

So, I have a quandary: What to do about the quadratic equation?

It's in Ch. 6 of my text, but I don't like the way it's introduced.
The book does guide the students to explore it and explain it more through out the chapter. But I just don't like the texts coverage of it and their introduction is just so...uninspired.

I will be reading Ch6 of the text more closely over the next few days. I don't know that I want to use Ch6 as it's written. I may write some alternative lessons for that chapter.

[8filltheheart]

8 minutes ago, mathmarm said:

So, I have a quandary: What to do about the quadratic equation?

It's in Ch. 6 of my text, but I don't like the way it's introduced.
The book does guide the students to explore it and explain it more through out the chapter. But I just don't like the texts coverage of it and their introduction is just so...uninspired.

I will be reading Ch6 of the text more closely over the next few days. I don't know that I want to use Ch6 as it's written. I may write some alternative lessons for that chapter.

I know nothing about math other than what I have taught my kids over the yrs, but I do like the way the quadratic equation is introduced.  IIRC, the equation is introduced as a formula at the beginning of the chpt so that they start memorizing it as a formula, but they do not use it at all.  He spends the rest of the chpt introducing the concepts that lead to being able to derive the equation.  My kids have all been able to write a proof deriving the equation by the end of the chpt, so for my kids it has worked well in solidifying their understanding of it.

[mathmarm]

3 hours ago, 8FillTheHeart said:

I know nothing about math other than what I have taught my kids over the yrs, but I do like the way the quadratic equation is introduced.  IIRC, the equation is introduced as a formula at the beginning of the chpt so that they start memorizing it as a formula, but they do not use it at all.  He spends the rest of the chpt introducing the concepts that lead to being able to derive the equation.  My kids have all been able to write a proof deriving the equation by the end of the chpt, so for my kids it has worked well in solidifying their understanding of it.

Yep. That's about it. I don't like that it's just plopped down for them to memorize without wider context, but I do like that the book spends the next several lessons developing the ideas needed to derive it/understand the derivation of it and that the book highlights each connection as it crops up. I am not sure how I feel that they just keep asking the kids to regurgitate the QF from memory (it's in the exercises for each lesson in Ch6  But that could just me being picky or hard to please.

6-1 is the shortest lesson so far. About 1/2 a page.
It's pretty much just "This is the Quadratic Formula. Learn to read it and say it. Here is a strategy to help you memorize it."

6-2 is all about evaluating radicals and exploring the root properties for + and -, * and /.
They lean really heavily on the calculator for this lesson, but the exercises are well thought out. We can get around this by reasoning with properties and factoring the radicands then using a reference chart to calculate the square roots to two decimal places as requested. They call on the student to produce the QF from memory.

6-3 they formalize the whole concept of absolute value. They introduce the +- notation and note it's appearance in the QF.
They call on the student to produce the QF from memory. I do like the problem sets. I already know which ones are "can't skip" for us.

6-4 They connect absolute value and squares. They make the connection between absolute value and principle or positive square roots of expressions.
They now expect solution sets, instead of just "a solution" so that is good. (Note to self: Teach some introductory set theory before this lesson). They call on the student to produce the QF from memory. The problem sets are solid, and the final question is a discussion question.

6-5 This lesson is all about making the connection between trinomials (extensively covered in Ch5)  and quadratics. This is absolutely a stepping stone lesson because it bridges very smoothly to the next lesson. They call on the student to produce the QF from memory.

6-6 They work with their base in trinomials and begin completing the square. The exercises for this section are very sound and I like them. They call on the student to produce the QF from memory.

6-7  They solve quadratic equations by completing the square. I like how they've built up to this through 3 lessons and this section see's  the return of interesting problems to discuss. They call on the student to produce the QF from memory.

6-8 By this point they've explored all of the fundamental ideas used in proving the QF, so they are told that they can now learn what it means and why it works so that they can use the formula to solve quadratics, the final exercise in this section is deriving the quadratic formula from ax^2 + bx + c = 0. By now the student should have a good idea of what they're trying to prove and should have both some intuition and some insight to guide them through the whole process.

6-9 They get to use the QF to work out expressions, equations and solutions for 18 multipart scenarios/word problems using vertical motion. 

6-10 Guides the student to consider the discriminant of a QE and what the discriminant can tell you about the QE itself

6-11 is the Chapter Review and Test.

6-12 is a cumulative review for Ch1-6.

This is NOT a weak chapter. The development of the  QF taken as a whole is good--or even very good.
But there is *something* about it, that I can't quite put my finger on--that I just don't love.

This is a really good text. But there is something about this chapter that just doesn't...pop for me.

Edited April 1, 2020 by mathmarm

[8filltheheart]

1 hour ago, square_25 said:

 

I have never, ever gotten good results out of teaching a formula first and then explaining it to kids. I've experimented with this at AoPS -- if you start with the formula, you run a much higher chance of someone treating something like a black box they not only will not understand, but will not really feel motivated to understand. 

I would probably work up to this in the opposite order -- I'd start with solving equations like (x+2)^2 = 9 (weirdly enough, that's what I did today with DD7, lol), and then I'd move to completing the square and finally to the quadratic formula. Setting things up so that things are motivated will lead to, yes, slower memorization of the quadratic formula, but a very minimal chance that someone will be unable to rederive it. Starting with the formula first will lead to a higher chance that you will need to seriously remediate something that has become a black box. 

(I found these results fascinating when I saw them in my classes, by the way. I always thought it'd be fine to rigorously introduce the formula, use the formula, and remind kids where it came from along the way to keep them functioning in a logical way. However, it turned out that once most kids thought of something as a formula, their brains shut off to other understandings of it for a good long while. Somehow, the formula became primary. Probably once the formula becomes fully internalized and part of long-term memory, one should be able to go back and really, honest-to-goodness explain it. But I'm skeptical that you'd be able to switch from a procedural to a logical orientation by the end of the chapter, even with an explanation.) 

I have had excellent outcomes with mathematical understanding in terms using  nothing but Foerster's methodology with my 6 kids that have moved on from alg to cal+. (7th is in alg 1 right now.) So, I know it teaches well bc my kids have had nothing but Foerster's since I dont have the ability to teach it any other way. So, it is hardly a completely flawed approach.

Kids do nothing with the formula other than write it down until they know how to derive it. It isnt memorized and followed by simply explaining how it works. The formula is simply presented as a formula to memorize and then students spend 6 sections working through step by step processes until they encounter a lesson that explains now they are ready to understand what it is and why it works.

It would be very easy to skip the 1 question in the lesson that tells them to write it from memory until after they have learned to derive it. But while they dont use the equation until the later lesson bc he does explain how the step by step processes look like parts of the equation.