Tag Archives: TMM Weekly Series

The Math Map Lesson 3: Comparison

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Our Classical Conversations year has started, and as I take my Challenge class through The Math Map (TMM), as well as using it at home with my four youngest children, I’m sharing some tips each week from things I’ve learned over the last 3 years of using TMM.

Tips for Lesson 3: Comparison

  • On the i2 page, note the top left box about 1D: The First Dimension. Compare it to what what said about 0D in Lesson 1 on the i2 page. (Every time you move into a new dimension, it’s helpful to look back at these and compare.)
  • Note the special numbers listed on the number line (square root of 2, phi, Euler’s number, and pi). At some point (not necessarily your first time through), these are helpful to memorize.
  • The Graphing Axes won’t mean much until we get to 2D, but the Invention (i2) pages will always show both rectangular and polar coordinates, so it’s good to note.
  • Note that the “Rectangular Axis” (which is shown horizontally) shows Real numbers (though its labels may only be in a narrower domain), while the “imaginary axis” is shown vertically. Depending on which domain your students are working in, they will probably be asked to label these.
  • On the i3 page, the “International System (SI) of Units: Prefixes” has lots of patterns, so take some time to attend to attend to it and see which ones are familiar/unfamiliar.
  • For Challenge students, it’s especially helpful to discuss scientific notation and significant digits, as they’ll need to be familiar with these ideas for the Research strand. (For younger kids, or those brand new to TMM, I probably wouldn’t give these much time on the first time through this lesson.)
  • Depending on which domain you’re working in, there will probably be a few places in the lesson pages where you’d be lost without the Charts. When completing pages 3, 7, 11, and 15, C3:2 has divisions on a ruler that will be a great reference and C2:3 has a section on “Significant Temperatures” that will be needed as well.
  • Remember the Flashcards! If you’ve been doing them since Week 1, your student has probably mastered at least a few. Add 2-3 new ones in but keep practicing the ones that have become easy.

Other helps: Here are a couple extras that might be helpful as you explore Lesson 3:

  • If you want to go a little more in depth, I like this one (that was deep enough for me, but there are plenty that keep going!)
Posted in Classical Conversations, The Math Map

The Math Map Lesson 2: Number Notation

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Our Classical Conversations year is starting, and as I take my Challenge class through The Math Map (TMM), as well as using it at home with my four youngest children, I’m sharing some tips each week from things I’ve learned over the last 3 years of using TMM.

Tips for Lesson 2: Number Notation

  • Euler diagrams are introduced on the i2 page. You’ll see them pop up throughout TMM, as well as Euler’s number. Euler was German, so his name is pronounced like “Oiler,” not “You-ler.”
  • Fun fact: Speaking of Germans, the reason Z is used for the domain Integers is because it comes from the German word “Zahlen,” meaning “numbers.” (If you want to learn more about domains, you might find this article interesting.)
  • On the i2 and i3 pages there are lots of symbols or concepts that may be new to kids who haven’t done TMM before. Be sure to start by asking them to look for something familiar. It helps these pages feel less overwhelming!
  • If you’re overwhelmed, remember you don’t have to master everything on your first trip through TMM. Remember Andrew Pudewa’s EZ+1 concept from IEW? Pick one new thing to try to understand this time. I promise, next time through, more things will feel familiar. (My own children and the Challenge students I’ve tutored who’ve done two or more tours of TMM can all attest to that!)
  • If you’re not familiar with systems outside of base 10, take some time to attend to the top box on the i3 page before looking at the other base systems. Find as many patterns as you can. then compare what you’ve discovered to one of the other base systems on the page.
  • While trying to understand bases, I found this example helpful. If you think about it, we use things other than base 10 whenever we talk about time. With seconds or minutes, we count in base 60 (starting over again after 59); with hours, we count in base 12 (or base 24 if we use military time); etc.

  • Regarding the Charts. If you’re overwhelmed by the i2 and i3 pages, proceed to the Charts with caution. They are detailed and thorough, as their purpose is to explain all the concepts related to each lesson that one would hope a Challenge IV student would be able to understand and articulate after years of going through TMM. Because of that, they can be a bit intimidating. While sometimes they are necessary for filling in the lesson pages (look at the title bar across the top of the lesson page and then find the corresponding chart), I would estimate about 90% of what I’ve needed to complete lesson pages can be found on the i2 and i3 pages.
  • Remember the Flashcards! Even if you and your students just learn 1-2 a week, that’s more than you would know if you did none at all! There’s a box to check them off on the first page of each set of four lesson pages (1, 5, 9, and 13).

  • Don’t forget about The Math Map Companion on CC Connected. You can pull up the pages from each lesson in any available domain and click on the different sections of each page to get more explanation about the concepts. (I’ve blurred these screenshots to protect copyright but still give you an idea of what I’m talking about.)

Other helps: When I first started to dive into TMM, I jotted down notes of things that might be helpful as I taught each week’s concepts. Here’s what I found for Lesson 2:

  • On p. 1 the students are told to study and memorize the pattern for imaginary numbers. I find it really helpful to walk through each step and help them understand the reasoning behind each step. (A Challenge A tutor shared something similar to this on Facebook, and I’ve tweaked it a little as it made sense to my kids).
    • i0 = 1 (because the “zeroth” power of anything = 1)
    • i1 = i (because the first power of anything = itself)
    • i2 = -1 (because the whole basis for imaginary numbers is that we have to go beyond Real numbers and imagine the square root of -1)
    • i3 = -i (because i3 = i2  x i1 =, or -1 x i)
    • i4 = 1 (because i4 = i2 x i2 , or -1 x -1) 
    • i5 = i (because i5 = i4  x i1 =, or 1 x i)
    • And the pattern just continues…
  • Finally, I love this visual for Euler diagram of domains. I’m collecting boxes as we speak to try to create this for my kids at home. I want to write some numbers of scraps of paper and have them figure out which box to put them in.
Shared on TMM Facebook page, originally by a Challenge director named Carrie. If it’s yours, please let me know so I can give you credit!
Posted in Classical Conversations, The Math Map

The Math Map Lesson 1: Ex Nihilo (0D)

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Our Classical Conversations community starts next week, and as I prepare to take my Challenge class through The Math Map (TMM), as well as using it at home with my four youngest children, I thought I’d share some tips each week from things I’ve learned over the last 3 years of using TMM.

Tips for Lesson 1: Ex Nihilo

  • The Lesson 1 booklet is really helpful because it explains the different components of TMM and what to expect. However, it’s laid out differently than all the other booklets, so it tends to throw people off. I encourage parents to read through the entire Ex Nihilo booklet before attempting to do it with their students.
  • Be sure to spend some time attending to the top left corner of the Invention page (p. i2). Every lesson that introduces a new dimension (Lessons 1, 3, 9, 19, 23, and 27) will follow this format.
  • Note the way the domains are listed using a different font. Once students are aware of this, they’ll see them popping up throughout TMM.
  • If you have younger students, make this a light week. Pages 1-16 are normally the lesson pages that the students work on (usually 4 pages each day), but in Lesson 1, it’s a lot of reading. Just pick 1 or 2 things to discuss each day.
  • Page 6 discusses place value and introduces some of the ideas around base 10, which is helpful to understand before heading into Lesson 2, which will look at other base systems. I practiced this with my Foundations students by pulling out some math manipulatives. I told them we can only use 0-9 in each place, and then we have to move over to the next spot. So we played around with a thousand, hundreds, tens, and ones.
  • Be sure to introduce Lesson 29 Seeing the Unseen, which is a blank booklet intended to be worked on all year. Lesson 1 Ex Nihilo explains how to use it on p. 1, and then the bottom portion of each of the remaining pages gives examples of the kinds of things students might add to their Seeing the Unseen booklets. In my experience, this takes a lot of practice and encouragement (few of my students did much the first year until the last minute), but it’s also a great way for the kids to engage with math in the world around them. My first year, I tended to see things online and I would copy and paste them into a file, which you can see here. (We all got better at noticing things and thinking about adding them to our Seeing the Unseen booklet. I’ve already added 3 things to my book for this year.)
  • Don’t skip the Flashcards! See p. 4 of Ex Nihilo. Even just 5 minutes a day makes a huge difference. We set a timer and everyone goes through a set of 5-15 cards they’re working on. It’s important not to do too many at one time! On Thursdays, the kids give me the ones they think they’ve mastered, and then on Friday, I meet with each of them and check them on ALL the cards they’ve mastered so far. (Well, once they’d finished memorizing their addition facts, we dropped those. But we review multiplication facts if they’ve just learned them in the last year, and then any Notations cards they’ve learned. (I love the sets sold by CC. My ChB student has memorized all the Fractions and Notations cards, so now he’s working on the Trigonometry set.)

Other helps: When I first started to dive into TMM, I jotted down notes of things that might be helpful as I taught each week’s concepts.

  • I wanted to review some of the concepts for Lesson 1 by watching the corresponding Book Club videos. I decided to make a playlist of all the episodes that are about Lesson 1 or 0D, and I figured I’d share it. So if you’re looking for videos discussing Lesson 1: Ex Nihilo, you can find them here: YouTube Playlist of Book Club videos about Lesson 1 or 0D. (The Book Club has gone through The Math Map several times, with a different focus each time. I’ve made a list of all those tours with a brief description in this post: “The Math Map Book Club Videos: Where Do I Start?“)
  • In one of the Book Club episodes, someone recommended eventually memorizing the table on p. i3 of Ex Nihilo, starting with the first column (a list of dimension characteristics). I recorded a simple song to help us get started with that. (Looking back, that was way too ambitious for our first tour through TMM. But hey, someday I might find it helpful!)

Just for fun:

  • In the very first Book Club episode, in discussing the idea of zero, someone brought up Zeno’s Dichotomy Paradox, which could make for an interesting discussion about how an infinite series can be contained in a finite sum. (I think this paradox could be fun to tie in to the idea of an asymptote, which is introduced on p. 8)
Posted in Classical Conversations, The Math Map

New Blog Series Coming: Weekly Notes on The Math Map

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Ever since diving into The Math Map when it was rolled out for all Classical Conversations Challenge A classes a few years ago, I have spent countless hours trying to figure out how to best utilize it both at home with my own kids and as a tutor in Community. Consequently, I’ve had lots of people in the various CC communities around us come to me with questions because they know I’ve already wrestled through some of the things that come up when looking at The Math Map for the first time. As we prepare to start this school year, I realized the easiest way to share what I’ve learned would be to blog week by week as I plan out what I’m going to discuss with my class and/or with my students at home.

If you’re tutoring in a community that’s already started, I’m sorry! I probably won’t post more than a week ahead of our own schedule. But hopefully this series will be helpful to at least a few CC parents out there trying to figure out The Math Map as they tutor or use it at home with their own students. (Be sure to check out CC’s document “The Math Map: Where to Begin.”)

One of the resources I used to help me prepare to tutor with TMM was the Book Club. For the last few years, the people who helped create the curriculum, as well as people using it in the pilot program and families wanting to jump in, have met weekly to discuss the lessons. That makes for a LOT of videos to weed through, so I wrote a post explaining what the focus was each time the Book Club cycled through, which might be helpful if you’re overwhelmed. “The Math Map Book Club Videos: Where Do I Start?“

Just for reference, here’s my experience so far with various domains of TMM, so if it looks like I’ve done something you have a question about, ask away!

  • Spring 2023 Before introducing it to my children or tutoring TMM, I did all the pages of the Naturals domain myself. (I highly recommend this for all parents using the curriculum)
  • Summer of 2024 We did one lesson a day for 6 weeks before CC started. My youngest 2 did the first 4 pages of Digits, and 3 of my older kids did the first 4 pages of Naturals in preparation for the school year. (Our family takes May and June off, starting up again after 4th of July and easing into the habit of schoolwork before our CC Community starts meeting in August.)
  • 2024-25 School year, I used TMM as our sole math curriculum for my younger 4 and tutored it in Challenge A.
    • Complex: I did all the lesson pages myself so I could better help the families in my class. I just had my own ChA student copy 4 pages a day from the Solutions. (There are just so many new things coming at them in ChA, and I didn’t want her to feel overwhelmed and want to give up. I did about 90% of my own pages this way as well and learned SO much. Don’t be afraid to just use it as copywork!)
    • Fractions: My 11-year old was prepping for ChA the following year. He went through the lesson pages pretty independently.
    • Naturals: My 7 and 9 year olds both went through Naturals. My 7 year old flew through all 30 lessons by Christmas and went back to her former curriculum second semester because I didn’t feel she was ready for Fractions. (Note: this was when the lower domains were available digitally on CCConnected, so we didn’t have the booklets with the front and back pages, which I’ve since learned are CRUCIAL to using TMM successfully. This meant we weren’t having a lot of “math discussions,” so my daughter just kept working through all the practice pages in the book I’d printed and bound for her.)
  • Summer of 2025 Again, we did one lesson a day (4 pages from each) for 6 weeks before CC started. My youngest 2 worked in Decimals, my 11-year old (about to turn 12 and start ChA) worked in Rationals, and my rising ChB student worked in Reals. (Those domains were pretty easy for the older two, but I wasn’t looking for a challenge, just a review.)
  • 2025-26 School year
    • My ChB and ChA students went through Monomials and Complex. I also used these domains to lead discussions on TMM with my ChII class in community each week since none of them were using Saxon Algebra 2 at home. We went over concepts from the i2 and i3 pages, then looked at examples from the lesson pages. SO many amazing Aha! moments came out of these conversations!
    • I purchased the Integers booklets for my 8- and 10-year olds. I wanted to keep the family together on the same week, so if either of them wanted more, I had them do the Decimals pages for that week as well. (I had downloaded them from CCConnected when they were available digitally.)
  • Summer 2026 As we had done the past two summers, we did a quick run through the entire Math Map, touching one lesson each day in the 6 weeks before CC. After several years, I finally felt like we were using TMM as intended. Each day we explored the i2 and i3 pages as a family, asking each of them to find something familiar on each page (starting with the youngest and working up). I was amazed at some of the things they noticed, and thrilled at how my Challenge students remember things they’d discussed in class. We took about 15-20 minutes to go through those each day, and then each student went back through their booklets from last year and either filled in incomplete pages or did the Careful Calculations page in the back. (Neither of my Challenge students had completed 100% of their booklets.) If my 9- and 10-year olds had already finished a booklet, they did 4 pages in the Decimals since I had made the year before.
  • 2026-27 School year I’m hopeful that we’ll be able to continue having TMM discussions as a family each day. I’m thinking we’ll focus on a different component each day (e.g. the art, the i2 and i3 pages, something from the inside back cover, etc).
    • My ChI and ChB students are going through Binomials and Monomials. I’m tutoring Challenge II, and since again, none of my students are following the Saxon schedule from the Guide, I’ll be planning class conversations around TMM, going through my own set of Binomials books since at least one of my students is doing that at home.
    • My 9- and 11-year olds will be going through the booklets from Fractions and Reals. (I don’t know that their Decimals books will ever be finished, LOL.)

All that to say, I’m pretty familiar with The Math Map and have quite a bit of experience using it both at home and in Community. If you’re ready to dive in, here’s my post about Lesson 1 Ex Nihilo.

Posted in Classical Conversations, The Math Map