Monday, November 26, 2012

Strip Boards and some thoughts on "Testing"

Both boys have been hard at work, side-by-side.  Their work parallels one another so often it is humorous sometimes.  For example, today they BOTH received introductory strip board presentations...Me Too on the addition strip board and Kal-El on the subtraction strip board.

Sometimes you see the strip board presented with the strips laid out above the board, sometimes with the strips on either side of the board.  We tried the "above" presentation today (makes it easier to find a place on the mat to do the written work).  It was a "fail" because the placement of the strips enticed Me Too sit to the side of the board or worse above the board.  Oddly, Me Too reads just as well upside-down and sideways as he does right-side up.  However, his ability to write simultaneously upside-down and backwards so that it is legible to a person sitting across the table from him is starting to make other adults think he is dyslexic.  Based on the fact that he can successfully read right-side up, upside-down and sideways as opposed to a lack of success reading in any position I'm going go out on a limb and say he's not dyslexic.  However, I am going to try to make sure that he reads and writes more often from the correct side of his work.

Moving the strips to the sides of the board did the trick!  Me Too experiments for a long time building his own equations.  Then, he asked what all of the "other rows" on the board were for. So, I showed him how you can use the strips to build all possible ways to create a particular sum.

He went ahead and built all of the ways to reach a sum of 10.

Then, he moved on to 9, 8, 7, 6, 5, 4, 3, and 2!

Kal-El worked with the subtraction strip board for the first time today.  He first learned how to build equations of his own choosing as well as some loose prepared equations.  Afterward he asked if he could start the subtraction tables booklet.  He completed tables one through five today.

Out of curiosity I looked it up today and discovered that it took Kal-El six months to do all of the addition memorization works.  However, three of those months were summer months in which very little of this type of work is actually accomplished.  The subtraction work will go much more quickly by virtue of the following:

  • fewer boards
  • the advantage of already knowing his addition facts
  • it's not summer

Hypothetically he now knows all of his addition facts through 20.  This is "behind" by Montessori standards (as the child would theoretically do the memorization materials for all all four operations by age six) however it is apparently quite good by "first grade" standards.  Kal-El has asked for addition flashcards because he has been hearing about them from his traditionally-schooled friends.  He is particularly intrigued by a friend who reported having to do "30 facts in 30 seconds." When I tried to oblige by picking some up at the Learning Shop (I didn't like the style I was finding in Wal-mart, etc.,) I was surprised that I was unable to find any sets that went beyond 12.  When I asked for assistance the former elementary schoolteacher who helped me was downright disturbed that he knew his fact through 20.  I found that odd.  As for the "30 facts in 30 seconds" I don't think we will be timing ourselves.  To make Kal-El happy we tried that ONCE (with some unsatisfactory Wal-mart flashcards).  I wish I had it on video so I could show you, and would tape it next time except I am NEVER DOING THAT AGAIN.  He was sooo emotionally and physically upset during the timed trial that he ASKED FOR that I thought he was going to be sick. He developed a stutter, waved his arms around, flapped his hands, developed facial ticks and almost vomited.  I am going to make him some flashcards myself (because he asked for them and because whenever the sum is in the teens I suspect he is doing fast mental addition rather than having the fact memorized), but we won't be using the timer again.  I am super glad that because I am his teacher he doesn't have to face that type of thing as a "daily educational requirement" and that he has never been "tested."  I don't think it meshes well with Kal-El's personality.


  1. How funny on the flashcards! I wonder if he had a different impression of them from his friend.

    My son likes flashcards sometimes and other times the experience is a lot closer to what you describe ;) There are several ways to use them too, so perhaps there is a more enjoyable way for him ;)

    What is that disturbed the other teacher about him knowing his facts? Now I'm curious!

  2. The same thing happened with me when I first timed DJ who is into flashcards. He wanted so bad to use them though. Whenever we use them now we only do about five at a time. Them he comes to me when he feels he knows them and ask me to time him. I add more cards to his stash as he has learned them, he loved the moment he got his full pack of flashcards to himself.

    I like to make the flashcard myself too when I can. Otherwise they all look weird to me and I can't dish them out because there is a fact on both sides of the card.

    I love all the math works that you guys do. Thank you for sharing.

  3. Try these triangular cards. They are great because you can teach fact family with them as well.

    Our D(6) doesn't like to be timed either. She's not fond of memorizing but can decompose and compose to do mental math. I prefer that over memorization because it shows me she has good number sense and can manipulate numbers well.

  4. Wow! Sounds like your expirience with flash cards was close to the same as mine with sound test! I asked Bunny to write the sounds on a paper as I said them and the result was a DISASTER! She did alot of what you describe Kal-El doing except that it ended in a screaming fit the had her running up the stairs and slamming the door to her room! :) Also on my never do again list!
    We are having an intersting thing with math right now. I keep trying to get Bunny to memorize her facts, but she has no interest in it all. However, she has been doing her addition and subrtration metally with the golden bead cards! This means she is doing up to 4 numbers and regrouping and borrowing as well all in her head! I am letting it go since I figure with all the paractice she is getting plus having fun with it, the memorization will follow! Great work by both your boys! I love how they are working side by side!

  5. My boy likes to time himself and is highly motivated by racing and winning. Not so motivated when he doesn't win though :(
    I had given up on the idea of getting some speed into this work until a friend suggested we use the timer but simply record how many he got right in that time. So we set the timer for 60 seconds, he does as many as he can, he checks the answers and records his total on a bar graph we made specially for this work. He doesn't get upset when he doesn't get a specific number (as in the traditional timed test). Instead he looks forward to beating his previous records with gusto! This may work for you?

  6. Thanks for the feedback everyone! As always, super thought-provoking.

    At Home with Montessori: What you described is what we tried to seems so logical and non-threatening. Something about the timer just gave him horrid anxiety. I thought he would like improving the number of equations over time but...

    Jessica: I don't know why that woman was disturbed. She was just visibly disturbed. The only thing I could think of is that she mentally has the bar set in a particular place and didn't like the idea that someone had moved it.

    Steph: I'm so glad Bunny is having so much fun with math and reading now! When they she takes off she really flies!

    DM: I think what you described is what is going to wind up fitting him pretty well. He is so much like DJ so that convinces me even more. If I do the flash cards without a timer on he gives me the answer in 1-2 seconds. WITH the timer that changes to 5-6 seconds full of stammering and hand flapping. I think the ever-increasing deck size you suggested will be a better motivator than the timer.

    Joyful Learner: I just looked those up! Those look perfect for when he gets a little further down the road with the subtraction boards. Thanks!

  7. My first reaction to this was "I couldn't agree more". As a kid, flash cards with pressure to respond quickly basically created a mental block for me with arithmetic. I read the post aloud to my husband, and he reacted that no one should get that upset about anything academic. While I do not necessarily agree with him, I made a mental note to myself to talk with our kids about tempering their reactions and realizing that no academic activity is a judgment on them, but merely an activity to enjoy, or possibly stop. Clearly this is not the best activity for your little guy, and I can relate to that!

    Today, I was looking at a new site that may be of interest. You do have to register to read the content but it is free and I have found it interesting.

    This free PDF offered by the same author on an open access site was very interesting to me
    He talks about the goal of education and math education. In todays age with the computing power of....computers, is quick mental arithmetic really that valuable? The montessori hands on approach is so lovely because it is more focused on the process and the problem solving than rote memorization.


  8. My experience has shown that quick mental math IS still necessary in our modern lives. When two young boys are playing a game together such as Mille Bornes - and one needs a calculator to add up his milage and the other can quickly glance, see the patterns of 25, 50s, 100s, and 200s and know in an instant... is that not saying something about the state of modern living? That there is something wrong with an over-dependence on computers?

    Ok, not every kid plays Mille Borne - but let's take a look at the gas station. You know you have a small amount in your checking account or available on your credit card - and the price of gas fluctuates, yet you NEED a certain amount of gas to get to work -- you have GOT to be able to do quick mental math to assure you get enough gas, but don't go over your spending limit. Sadly, many people canNOT do this and have a lot of financial problems because of it - BECAUSE they rely on computers so much.

    It's not about memorizing every single math fact out there - it is about knowing the basics and being able to apply them in particular patterns. So one only needs to know up to 9x9, 9+9, 18-9 and 81/9 - after that it is a repetition of a pattern.

  9. @Jessica - I think the point is to focus on higher levels of thinking, such as problem solving and critical thinking, rather than just rote memorization and regurgitation.

    Memorizing is an activity not without merit, but can be fun (for example games, as you suggest, can provide motivation for learning math facts without pressure, cribbage and many other games are great for that) or check this out

    My kids, even my 2 and 3 year old, LOVE to dance to this and sing their math facts (well, not the singing part for the 2 year old). I posit that the knowledge of arithmetic facts can be a bi-product of a well rounded foundation in mathematics rather than a focus of it. My question (is quick mental arithmetic really that valuable?) was perhaps not fully capturing my thoughts on the subject. It should have said: Is quick mental arithmetic really a valuable goal of mathematics education? I say NO, it is a by-product of engaging in extensive problem solving in mathematics and other entertaining math related activities.


  10. That makes sense! No, memorization is not the GOAL, but is a step along the way as well as a "by-product". Just to memorize the facts doesn't mean you can then apply them to various situations - but to have proper critical thinking and analytical skills (usually developed in tandem with recognition of patterns), coupled with quick-recall of the most basic of facts leads to wonderful results :)

  11. One of the wonderful things about Montessori math is that the focus is 100% on higher levels of thinking and that memorization is more a side-effect of the work instead of the focus of the work.

    For those who are not Montessori-obsessed my posts lately may seem over-focused on the memorization. The memorization strand is only one of five strands in the primary experience and it's one I happen to be posting about A LOT lately because both boys are at new points in that strand. There are probably 10-15 FUN activities in the Montessori memorization of addition sequence alone that are made to let the child acquire the memorization without pressure. Also of note, in the album I use the memorization strand does not even begin until the child has COMPLETED all of the higher-level activities for that operation in all of the other strands.

    The boys' experiences with math have been nothing but a joy up until this point. It was only that one day, when I tried a non-Montessori activity (flashcards with a timer) that we had had a horrifying 30 seconds. My point was that I won't be doing it quite that way every again because as Laura's husband said "no one should be that upset about anything academic." As I said, I was totally shocked because I only made the cards and put on the timer because it was something Kal-El asked for. All we planned to do was note the number of facts he completed in the 30 seconds. The pressure of the timer ruined the experience.

    I am not throwing the flashcards in the trash because I do believe that basic math facts (within the limits Jessica detailed) are necessary even today. I am not a "math" person and don't use a ton of math in my daily life (I'm not an engineer for example). I'm not even particularly good at mental math. However, I have noticed lately how surprisingly frequently I use my basic math facts on a daily basis. Because Kal-El is doing his math facts a little past the absorbent mind stage, the flashcards were an easy way for me to check if he knew all of his facts. We have since changed the way we do it to the way Discovering Montessori suggested. I was pleasantly surprised to discover that of the forty-something facts that I put on the flashcards (I put that commutative property to work so I wouldn't have to make 80-some cards. The Montessori sequence has left Kal-El very well-grounded in that property.) he only was counting out five or so facts in his head instead of just "knowing them." If I don't use a timer and just use the cards he recites the fact quite quickly for all of the other cards almost as if a metronome is on. So, the upshot is that the fun, low-pressure Montessori materials did their job quite well. He has a tidy little stack of flashcards in his "secret drawer of cool stuff" in his room that he seems to be using on his own time that will clear up the facts he didn't learn.

    Me Too has nearly finished his fact families tables with the addition board. Interestingly, he never ever used the strips. He already knew most of the facts from the work he has already done. As for the rest, the pattern of the tables made the answers too obvious. This will be taken care of when we switch to loose equations. It is quite obvious that he still has an "absorbent mind" and is going to pick the facts up almost effortlessly. He does however, at his younger age, have quite a bit more trouble doing things like checking his work with the control chart. He forgets what he's looking for, records his work on the wrong lines, checks off the wrong equations, looks for the sum as an addend on the chart and starts checking the wrong equation, etc.,