Joanna Breaks Down Math from K to College

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  • joannarammell

    I don’t know if I will be able to write all that I have been ruminating on in one post or not. This might have to be done in parts, as I am rev-ing up to begin school here on the 29th.

    I also am not sure that this is wanted or needed. But I have felt that I needed to post this for a while now…so here it is.

    I am going to divide math education into three parts.

    Arithmetic (Basics)

    Algebra and Geometry (High School)

    Calculus and beyond

    In Arithmetic we have the four operations addition, subtraction, multiplication, and division …henceforth referred to as asmd(also-Add, Subtract, Multiply, Divide).

    And as I mentioned in a previous post, we learn this in deepening levels. We start learning about just regular whole numbers and we first add, then subtract, etc.

    I have checked in on the other posts. I agree that we want to understand concepts…but these are tools. You do not want to build a screwdriver or a hammer before starting a project. You do not want to have to go to an encyclopedia each time you need to use a wrench to build an understanding of it in order to use it. Before the building project, you need to know the how and why of the use of the tools. Same with the asmd. They are tools. So, YOU MUST KNOW THEM or they can’t be tools. They will be stumbling blocks without memorization. Balance here is key. Memorization without understanding is as meaningless as knowing a screwdriver and a screw go together but having no clue that they can screw two pieces of wood together to make a birdhouse. However, without memorization (w/the understanding in place) you will never be fluent in the language of mathematics. Can you imagine if you had to look up the definition to just about every single word you read? I know I wouldn’t read any more than I absolutely had to. Yikes. Can you imagine a child who in Algebra has to calculate every single basic addition or multiplication fact and who stumbles over every single division…4 divided by 2. Or how many 9’s are in 36?

    It is painful to think about. Success (in that and related subjects) would be denied that child. I am not saying that one can not live a valuable and productive life and not know their math facts. I AM saying that one can not have success and feel successful in math and the related subjects w/o the tools or the components of the language. 2+4=6 is a definition of 6. Just like each word in English has a definition. It would be crippling to not know our basic definitions. Same in Math. 9×4=36. No thought required. Understanding complete. Put in toolbox! Next.

    Almost all of Arithmetic is ASMD. Of whole numbers, fractions, decimals, percents–and even neg and pos numbers.. We also have the vital ingredients of place value, units–like inch, kg, oz., etc. measuring, and parts and whole, problem solving, etc. But the main math dish here is comprised of those four fundamentals ASMD applied across the different expressions of numbers.

    This is pretty much all of arithmetic. Yes, children learn to recognize shapes. A young child recognizes a triangle. And there are many things that have been added in to this stage, even introduction of variables, equations, geometry, etc.

    I have nothing against adding all that in. My 7 y/o does know the difference between parallel and perpendicular lines. But I assert, that IF you and your child are struggling in math. The BASICS, as defined here, w/o the extras, are ALL that are required and needed!

    Let’s talk math timeline.

    Basics (Arithmetic) starts early in normal living…so we will say K or even earlier.

    So from Kindergarten (or even Pre-School) to ??? are we learning the Basics?

    SCM takes us all through the 12 years to come back to the beginning to plan. Let’s do that here too.

    If a child is bent to math and wants to be an engineer, architect, mathematician, scientist, computer scientist, businessman, doctor, etc. We are talking college with calculus.

    If a child is wanting to be a nurse or non-technical field that requires a college degree…we are probably talking college algebra–and that will possibly be it.

    There are also children who never plan to attend a university or do anything that requires a college degree. And then the undecided ones need to leave some options open.

    Goal: College Calculus and beyond


    1. Through Pre-Cal (Also called Trig/Pre-Cal) in HS. Can go into college algebra and then into college calculus or sometimes straight into college calculus.

    2. Through Calculus in HS or Through AP Calculus w/concurrent college credit. If got college credit, can skip cal 1 and go straight into cal 2 if needed. If didn’t, can try to clep cal1 or just take cal 1 and get a feel for college and its requirements—the exact same things are not necessarily covered—depends on semesters vs. quarters, and other factors.

    Grade Range to Begin Algebra1:

    Algebra 1 needs to begin in 8th or 9th grade. 8th: Algebra 1, 9th: Geometry, 10th Algebra 2, 11th Trig/Pre-Cal, 12th Calculus or AP Calculus w/college credit. (Start in 9th if only going through Pre-Cal.)

    Goal: College Algebra


    1. Through Pre-Cal (Also called Trig/Pre-Cal) in HS. Can go right into college algebra.

    2. Through Algebra 2. Can go right into college algebra. Might have to take a placement test, maybe.

    Grade Range to Begin Algebra1:

    By 9th Grade ideally. 9th: Algebra1, 10th Geometry, 11th Algebra 2, 12th Pre-Cal or whatever math credit required for your state or if three is enough…none or problem solving or business math, etc. If three major maths are enough for your state, you could potentially start Algebra 1 in 10th and still make it. But I like that extra year for room. What if it takes two years to conquer one of those subjects?

    Goal: Maybe College

    Options: Through Algebra 2. Leaves options open. Can go right into college algebra if desired.

    Grade Range to Begin Algebra1:

    By 9th Grade ideally. 9th: Algebra1, 10th Geometry, 11th Algebra 2, 12th Pre-Cal or whatever math credit required for your state or if three is enough…none or problem solving or business math, etc. If three major maths are enough for your state, you could potentially start Algebra 1 in 10th and still make it. But I like that extra year for room. What if it takes two years to conquer one of those subjects?

    Goal: No College

    Options: Through Geometry. So Algebra 1 and Geometry…Geometry is the language of builders, contractors, etc. It is necessary to many many jobs.

    Grade Range to Begin Algebra1: This is going to depend on your state’s math requirements. But obviously, you could do other H.S. math and start Algebra 1 as late as 11th and then Geometry in 12th. I would personally give more leeway than that! What if it takes two years to conquer one of those subjects?

    My own personal recommendation: Start Algebra 1 in 9th grade unless child is definitely math bent and needs all the credits he/she can get…then 8th.

    If can’t start in 9th start as soon as you can—meaning mastery of basics has been accomplished and can now tackle Algebra 1. If this is 10th, don’t panic. If this is 11th, depending on above goals, don’t panic.

    So the basics. From pre-k to ?? 8th? 9th? 10th? 11th?

    You decide based on the above outline of goals and your own family’s make-up.

    But the BASICS (w/o the extras—for you who are mathematically challenged)…go from pre-k to at least 8th or 9th or even 10th grade. Wow. We have more than 10years to teach ASMD in all its forms.

    I wish we were in a room together. I am expecting to hear some sighs of relief. Take the pressure OFF! Master the basics. Spend the time. The curriculum’s schedule should not be your master or even your leader. Mastery should be the indication to move forward. AND put the tools in the toolbox…memorize the required definitions (math facts) so you don’t cripple your children. It doesn’t have to be stressful. Don’t get a whip out and beat it into them…but do get it done. There have been a lot of good suggestions offered in the other posts. I was proud to see the post from the mom who said it took 1 ½ years (I think) to master the multiplication tables…and then they moved on! YEAH! When she gets to grade 8, 9, or 10…her child will have a toolbox full of items to build with in Algebra. He/she will learn the language of algebra b/c they won’t be getting out the dictionary to define each three letter word they run across.

    I pray that this has put some of the math race/pressure/issues/upsets/complications/etc. to rest. I pray this clarifies things so that you can focus on the important stuff.

    I really do pray this is a blessing to you and leaves you peaceful instead of in turmoil.

    God Bless,



    I do appreciate this post. I have just been struggling with how many days a week to do math, how fast I need to move through levels, etc. Really what I needed was the end goal which you so nicely laid out! Thanks also leaving a range of end options open; recognizing that some of our kids will be engineers and others will pursue callings that don’t even include college. Very balanced and realistic.

    I am not a math loving momma but want my kids to have a good foundation so I appreciate moms like you who are willing to help with my weak area :o)


    I agree,  Thank you so much for breaking this down.  My children are young and don’t have plans for the future yet, but at least I have the information now for when they figure out what they want to do.  Smile  Thanks again.



    Thanks for taking the time to write this post.  I am going to put it away in my homeschooling references!  

    BTW, I pray your mom is better and will continue to pray for her:)


    Sara B.

    Thanks, Joanna!  I also appreciate this post.  I love math, and I have one child who for sure is going to do something with the sciences.  This was so helpful as I plan for her future!  It also gives me permission to SLOW DOWN!  Laughing


    Joanna – why is geometry always placed in between Alg 1 and 2?  Is it to combat “algebra fatigue”?

    This is very helpful, thank you for putting so much of your time into helping with this subject.


    Wow! Thank you so much.  The sad thing is that even some of the simple ex. you gave I could not do in my head as I never memorized them as a child myself! It has been a handicap and am looking forward to re-learning with my boys. 

    I was wondering, for those of you who plan to allow for early graduation at 16, approx. “10th grade”, how should the h.s. maths be worked in?  (after graduation the plan is to help them find meaningful work/apprenticeships in fields of interest, choosing further course work, etc. while still living at home a couple more years).

    I’m pretty sure at least one of my boys is very technically bent and science drives him.  So I would like for him to have the college option.  Since we school year round, with light summers, perhaps it wouldn’t be too much of a stretch to do Alg 1 in 7th (assuming asmd has been mastered in all forms, measurements, etc) and continue on from there, Geo in 8th, Alg 2 in 9th, Cal in 10th, but to take our time and stretch a little out into the summers to get thru all that material and gain understanding. What do you think? Of course right now it’s all theory since we are just starting 1st grade! ha!


    I also have to admit that I have a hard time reconciling the U.S.  math courses compared to what I remember (from quite a while ago) as math in the province where I lived at that time….

    I remember doing some geometry and pre-algebra (I think it would be called) in Grade 7…

    some algebra type stuff in Grade 8 and 9… but maybe it would also be a pre-algebra type thing?)   – there was salving simulatnious equations and stuff like that….

    Then Grade 10 we did some Trigonometry (although maybe that was mostly in my Physics course…) and Algebra for sure.  Probably Exponents, Logrhythms, etc.  

    Grade 11 we did Algebra, graphing of functions and our class did calculus as well but didn’t know that was what it was called… (we were an advanced class – and the teacher taught us how to do some calculus to figure out where the functions being graphed would “turn” – but it was on a “trust me and do this”.  Oh, and some Euclidean Geometry including calculating angles and stuff like that.  (Grade 11 was 2 math “courses”)  I think there was some statistics and probability in here too (including permutations and combinations.)

    Grade 12 we did Calculus, including the proofs of what we had done the year before,  Because we got done the course early and the teacher asked us what we wanted to do – we also learned how to do Tax Forms (done in the non-college-track math courses but not in ours) and also Financial Math like Loans and Interest.  Or maybe that was in one of our other courses (the Financial math….)

    The names of these courses….   Math 7, Math 8, Math 9, Math 10IB, Math 20IB, Math 30IB, Math 31IB.

    So I have to say when I hear “Algebra 1” “Algerbra 2”, “Geometry”  – well, I have no idea how to compare it to what I did – and no way to compare to what is probably done now…  I don’t know how math courses compare in Canada versus the U.S. or what is “required.”    Although my kids are young, I think my son might have the inclenation to go into science or engineering – and will need high math.  I loved math (until I got some crummy math professers at University… ugh)  

    Anyway…. don’t know why I went on and on on this….


    Everyone: Mom’s doing ok. So far since returning home, she’s managed to stay out of the hospital! She’s in good spirits, too. Thank you for your prayers.

    Gem: It could also be Algebra 1, Algebra 2, and then Geometry. This would help in some ways in that there would be less forgetting between the Algebras. Algebra 1 is the pre-requisite for both the other two. I took Algebra 2 and Geometry at the same time. They are so different, there was no conflict. And I taught Algebra 1 and Geometry at the same time…by the time they needed Alg in the Geometry course, we had it covered. So there is flexibility.

    For suzukimom: Don’t know. Here are the three books I taught from. Click on the book, then on front matters to see the table of contents for that book. But remember, these are considered a bit advanced.

    Heatherma: It would have to be a very special child for me to introduce Algebra in 7th grade. There is a shift from concrete to abstract thinking required in Algebra. I am sure there are plenty of exceptions to this, but, that shift generally happens in the brain’s capacity/methods of thinking around 13.5 to 14.5-give or take. That is my own rough estimate. But, many times, I would see the switch flip mid year (8th or 9th)…and suddenly, the ability to represent items and kind of mentally manipulate that representation was there…where even the week before..the student kept a huh? on his face and it was very obvious in his questions that the abstract part of the thinking wasn’t yet present. This ability shift happens at different times for boys and girls. And it isn’t something that is readily forced.

    Everyone: I’m glad some of you are relaxing a bit or even rev-ing up if you’ve been too lax to reach YOUR goal…I am glad that seeing the overview that helps you determine where, how much, and to what extent is being helpful to you.



    Both my sons have taken Algebra I earlier than 8th grade–one in sixth and one in seventh.  But they are pretty good at math in general.  They are now in calculus (college) in 12th grade and my 10th grader has about one-third of Algebra II left and then he will start Pre-Calculus.  We didn’t hurry anyone, just kept moving from wherever they were.  So kids with mathy heads don’t usually have much trouble with an earlier age. 


    I also know that our forefathers started higher math earlier…I think it is going to be very individual. Those forefathers seem so much smarter than us?! or is it just me? I like how you put it Bookworm…those w/ mathy heads 🙂


    Yeah, there has to be a term to explain how a poetic mom ended up with kids who MEASURE petals on wildflowers instead of sketching them in their nature books.  Laughing




    This is AMAZING! I am printing it out as well if that is OK.

    I have an English degree and math … well, I use it when necessary. Laughing

    DH has an engineering degree, minor in math, but no desire whatsoever to teach. He helps explain when necessary, but anyway, I still feel the pressure.

    So, I’m taking your advice!

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