Tag: cognition

  • An Observation of Cardinals and Ordinals at the Playground

    An Observation of Cardinals and Ordinals at the Playground

    We were hanging at the playground on Mother’s day. Daughter had taken up with a cute little girl – a good playmate for the teeter-totter. I watched Daughter’s new friend do something interesting. She was “counting” some gumballs that had fallen off the sweetgum tree.

    Here’s what she said: ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT

    I looked and there were ten gumballs. I watched her more and saw that while she was reciting the terms and pointing at the gumball pile, she wasn’t associating each number word with the next gumball in the pile.

    Developmentally, I’m not sure if this is okay or not. I wonder.

    Mathematically, I know that there’s still a disconnect with this girl and what numbers mean to her.

    There are two types of numbers.

    In the realm of counting numbers, there are two types. Cardinals and ordinals. A cardinal number tells us how many we have. Like this:

    And ordinal number tells us the order:

    When we count, we use both types of numbers.

    The group of gumballs has the cardinal number of 10. But to arrive at that cardinal number, to determine how many there are, the friend had to “order” them. She was trying to point to each one and assign it a position. “YOU, I label 1st, you are 2nd … and since you are 10th, I know I have 10 gumballs.”

    We grownups take advantage of this procedure. Indeed most people don’t know what cardinals and ordinals are.

    But knowing this when you teach counting is quite helpful. At home you can do something similar to the potatoes in the pictures above. At school, you can buy some great posters (I found one at Teacher’s Heaven last night) that demonstrate this.

    Will you change the way you think and talk about numbers?

  • How to Train the Brain to Understand the Transitive Property

    How to Train the Brain to Understand the Transitive Property

    Remember the ol’ “if A equals B and B equals C, then A equals C” deal? At parties it’s a great line to drop. In math, it’s officially called … cue music…

    The Transitive Property

    Saying it is fun, teaching it is curious, learning it can be weird.

    Grownups think it’s intuitive. But to a kid, it isn’t. It takes experience and experimentation to learn all the bits that we think are “common sense.”

    The transitive property is really thinking things through. Starting from one place and moving along through another and then arriving at a third place.

    There are many ways to help kids with this learning. Word problems simulate thinking stepping stones. But they can be rather stressful. If you do it through play, you reduce the stress that they face and give them skills they need to tackle advanced thinking, forever.

    This video shows a nifty “toy” from Discovery Toys that can get kiddos using those brain stepping stones.

    Notice the flow is

    1. Choose the number tile with the question number.
    2. Read and answer the question.
    3. Correspond the answer to the letter in the answer box.
    4. Put the number tile with the question number in the corresponding letter box.

    Thinking through from question number to answer letter while avoiding the pitfalls is the challenge.

    Have you played with these? How do you train your kids’ brains for the transitive property?

  • Two Reasons to Memorize Math Facts

    Two Reasons to Memorize Math Facts

    I learned my math facts by “singing” them while looking at flashcards.

    Having these facts ingrained with chanting or singing isn’t a bad idea. It might not “feel right” because we’re so into experiential learning these days. But if a kid can’t immediately access and use things like 8 x 7 = 56, he’s going to be slower than if he can.

    And if he’s slower, he might get frustrated and start to think that he’s not good at math.

    Also, knowing these cheap and dirty math facts helps with confidence. Even if a kid’s struggling with other things in math, knowing that he has this one thing (the “facts”) will help out.

    I fight this battle often. Some people feel that math facts shouldn’t be memorized. But there’s so much value in it.

    How about you? Which side of the fence are you on?

  • 5 Phases of Learning Math

    5 Phases of Learning Math

    This post is an answer to a question Michelle, a high school math teacher, sent. She writes:

    I explain a new concept then give an example and finally do an example with the class directing me on how to solve. What I have noticed is that the bulk of the questions occur during or after the chapter assessment. Most of the students’ questions are great questions. It’s just that the questions they are asking are ones they should have asked much earlier.

    Think of knowledge as a bag of shapes – all of which have different colors and textures. Once you get the bag, you have to sort through them. Some people want to count them, some people want to sort them by color, some by shape. Some people need to sort by texture. Some people need to sort by all three or just lay all of them out in rows to get a good look at them.

    Everyone sorts the bag of shapes differently. Likewise, everyone sorts information differently.

    Turns out, the bag of shapes isn’t the knowledge. It’s merely the information. And it takes each person “sorting” it in their own way to turn the information into knowledge.

    In teaching and learning math, I’ve noticed a cycle. I’ll use the bag of shapes to illustrate it:

    1. Exposure

    Exposure is usually in the form of lecture and examples done by the instructor. It can be the first time the student has seen it or the first time in a long while.

    This is where the student receives the closed bag of shapes.

    2. Activity

    Any activity following the exposure. This typically is in the form of homework or classwork practicing the concept and problems.

    This is the “peeking into” the bag of shapes. The student gets to remove a few of them and start looking at them.

    3. Settling

    Allowing the subconscious to work. The brain does this all on its own.

    Often mathematicians will go for long walks, go to the movies, hang out with their kids, talk to non-math people or do any number of non-math things to force the settling phase.

    Children don’t know how to force the settling phase, nor do they need to. It just happens between when they do the homework and when they start to study for the exam.

    In the bag-o-shapes analogy, this is where the students dumps all the shapes onto the floor and sorts them in various ways. It takes a while to get through all the shapes and see what kinds of sorting can be done.

    4. Re-engagement

    This is typically in the form of studying for the test and taking the test. It can be a heightened emotional situation where the learner is under stress.

    This is a revisit of the concepts. It becomes easier because the settling has occurred and the information (the bag of shapes) is already organized.

    The learner at this point will attempt to modify some of the conscious thinking to best fit with what the subconscious has done. The added stress will allow them to connect with what they’ve done better – as the “feeling” state induces a different type of learning.

    5. Application

    This is using the concepts for something else. This will often be the next class or next term of the math curriculum. If you learned graphing functions, you will likely use graphing functions in the future.

    To wrap up the analogy, this might be a student realizing that the bag of shapes is needed for something – not just a random bag of shapes. Therefore he can re-organize them to be of use in the new situation.

    Let the learning flow.

    This flow of learning is natural. It will happen and has to happen. The only thing you can do to artificially speed it up is cycle it more often.

    The students ask the good questions, as Michelle said, “during or after the chapter assessment.” This is in phase 4 – Re-engagement.

    Instead of going through a single cycle, do it two or three times. Like this:

    1. Monday: Lecture, chapter 1 (exposure)
    2. Monday: Classwork and homework, chapter 1 (activity)
    3. Friday: Test, chapter 1 (re-engagement)
    4. Monday: Lecture, chapter 2 (initial exposure to chapter 2 and application of chapter 1)
    5. Monday: Classwork and homework, chapter 2 (activity)
    6. Wednesday: Test, chapter 1 (yes – chapter ONE; re-engagement again)
    7. Friday: Test, chapter 2 (re-engagement)
    8. Monday: Lecture, chapter 3 (initial exposure to chapter 3 and application of chapters 1 and 2)
    9. Monday: Classwork and homework, chapter 3 (activity)
    10. Wednesday: Test, chapters 1 and 2 (re-engagement again)
    11. Friday: Test, chapter 3 (re-engagement)
    12. Monday: Lecture, chapter 4 (initial exposure to chapter 3 and application of chapters 1, 2 and 3)
    13. Monday: Classwork and homework, chapter 2 (activity)
    14. Wednesday: Test, chapters 1, 2 and 3 (re-engagement again)
    15. Friday: Test, chapter 4 (re-engagement)

    What do you think? Share your experiences with the cycle below in the comments.

    Thanks to Michelle for requesting this tip. Michelle is one of only two math teachers in a rural private school. She teaches Algebra I, II, Geometry and Calculus.

    Do you have a question? Ask it in the comments section.

  • 6 Techniques to Brain Training from a Pro Brain Trainer

    6 Techniques to Brain Training from a Pro Brain Trainer

    This is a guest post by Dr. Vicki Parker of The Brain Trainer.

    If your child has always done well in math but has recently had difficulty in one area of math, such as geometry, then tutoring on specific information may be helpful.

    However, if your child has struggled with math year after year, it may be time to look at underlying cognitive skills, the building blocks of thinking. The specific skills that drive math include

    • Attention
    • Memory
    • Visual processing
    • Logic and reasoning
    • Processing speed
    • Number fluency

    If there are weaknesses in any of these areas, there will be learning struggles.

    Attention is the ability to stay focused over time.

    Attention is important for math because you have to be able to focus and attend over time to information, especially as problems get more complex. You can tell if your child has trouble paying attention if he understands the concept of the problem but adds instead of multiplies, or subtracts instead of adds.

    A simple deck of playing cards can be magic for reinforcing cognitive skills. To build attentional skills, have your child raise his/her hand or hit a bell whenever s/he sees the targeted number or suite of card as you flip through a deck of playing or Uno ™ cards.

    To further challenge your child, s/he must say the targeted card or quickly add, subtract or multiply a number to the card. To build sustained attention, add another deck of cards.

    Memory is the ability to store and retrieve information.

    Memory is important to recall number facts and sequence. What’s your child’s ability to hold on to the first steps of a problem or the initial calculation?

    If she cannot hold this information long enough to move on to the next step of the problem, progression will be difficult. She may need to retrieve previously learned information from long-term memory to execute the problems at hand.

    Try showing your child a numbered card, then turning it over, hiding the number, then have your child say the card number. Present another card in the same way.

    Next, have your child remember the two numbers and then add the numbers. Repeat this process with two new cards at a time.

    As s/he gets better, have him/her work on serially adding in this sequence:

    1. See 1st number & hide
    2. See 2nd number & hide
    3. Add the two numbers

    The child will recall last number shown (not the sum), you will show & hide another card and the child will add this new number to the previous number recalled.

    Continue, but remember: don’t add the sum number, only the numbers presented visually.

    Visual processing is the ability to see and manipulate visual stimuli.

    Visual processing is helpful to see shape, size, and relationships. We use it to see groups, understand angles, and other activities in math.

    Quick matching of similar shapes or numbers is helpful here. You can make small tweaks to this activity by sorting by size with various sizes presented and the same for the orientation of the shape – a triangle upside down or at an angle matching a triangle presented in the vertical position.

    Logic and reasoning allows us to see patterns and trends.

    It allows us to order events. You need logic and reasoning to understand bigger concepts. When we decide what’s needed and how to set up a story problem we’re using logic and reasoning.

    Practice copying patterns with young children using such items as beads or blocks. You can even have fun and have them create a pattern for a crown, flower pot border or placemat for dinner.

    For older children, start a pattern and see if they can finish the pattern. This can be easily done with building blocks and Leggo’s ™.

    Processing speed is how efficiently and quickly we can process information.

    Processing speed is very important to be able to do the basics quickly and move to second or third steps.

    To work on processing speed, try timing your child working his/her way through various paper and pencil mazes. Your child will love the competition when you make it a race between multiple participants!

    Number fluency is recognition of written numbers.

    Number fluency is a coding process normally developed by age three or four. If we are delayed with recognition of numbers, it slows us down with calculation.

    You need two decks of cards for this fun task. Deal out one deck of cards, an equal amount of cards for each player. Use the second deck to flip the target cards over.

    The players must match the number on the card, being pulled from the second deck. The first person to get rid of all their cards by matching the numbers is the winner.

    To push number fluency that is more than visual recognition, have the participants say the number before they place their card on the target card and then the game moves on.

    Conclusion

    Knowing your child’s unique cognitive profile will help you understand their performance and take you one step closer to solving their math challenges.

    The good news is weak cognitive skills can improve if targeted and trained. Brain training is a type of mental exercise, carefully designed to stimulate the brain and make lasting changes in cognitive abilities.

    The idea is to improve one’s ability to learn, rather than focusing on one concept of math. It is analogous to learning how to play an instrument (which is a process) and not just a specific song (which is knowledge or data – one concept).

    Vicki Parker, Ph.D. is the founder and director of The Brain Trainer and writes for their blog.