Tag: teaching math

  • Why Learning to Subitize Is Important

    Why Learning to Subitize Is Important

    Previously, I gave a definition and some resources for subitizing – assessing quantity without counting. The questions I posed in the previous article were

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    Having thought a great deal about it, I wondered if it was even relevant to teach it. And Husband asked the question, “Don’t they need to count and know their numbers before learning to subitize?”

    You don’t have to learn to count to know “how many.”

    Although evidence indicates that subitizing and counting happen in the same part of the brain, you don’t have to know how to count to subitize.

    Subitizing up to 4 or 5 using recognition bypasses the “counting” of numbers.

    Daughter is currently subitizing 2 – without counting. I haven’t taught her how to count objects at home and they don’t do it at school yet. So she has no concept of one. But “two bows” (one for my hair and one for hers) is very important to her. Likewise “two bowls” (one with goldfish and one with Rice Chex) is an amazing thing.

    This article by Ernst von Glasersfeld explains how number words can be associated with the quantity without ever counting. Children give the appearance of counting because they’ve memorized the number words and they use the cadence to tap their finger on objects. But they’re usually doing this to be rewarded with praises and cheers. They don’t really understand the concept of counting. This is what I saw the little girl doing at the playground.

    It IS important for kids to subitize.

    What’s the point, anyhow? If I can quickly see 8 things  – does it matter? And is it important to tell the difference between 8 things and 9 things? If there are 8 hungry tigers heading for me and I subitize that there are 9, will it make a difference in how fast I run from them?

    I researched more and found out that  subitizing does matter to the understanding of numerical concepts.

    Subitizing 2 items and putting it with a subitzed 3 items can yeild the perceived 5 without ever knowing what addition is. And having the group of 4 items and seeing within it a group of 3 or 2, leads quickly to subtraction and division.

    Thus subitizing “teaches” various mathematical concepts without ever getting into the language or construct that grownups have put on it.

    So how did we learn it?

    My initial guess was that we’ve learned to do it the same way we’ve learned to read without sounding out the words. Over the years we’ve seen and counted many hundreds of bundles of 3 things that when we see another bundle of 3 things, we know there’s 3 of them.

    Upon further research I’m now of the opinion that I learned it from Ma saying to me “there’s three of them” while I looked at a collection of three objects. I bypassed the counting and went straight to learning the cardinality – with the help of grownups.

    The cliffhanger…

    So I’ve yet to answer the question, “How do we teach subitizing?” Alas, it’s coming. (UPDATE: it’s here, complete with downloadables!)

    In the meantime, where do you observe subitizing and does it seem important?

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  • Can You Teach All K-12 Math in 8 Weeks?

    Can You Teach All K-12 Math in 8 Weeks?

    Elizabeth, @Ser3nd1pity, tweeted an excerpt from a book by David H. Albert called Have Fun. Learn Stuff. Grow.

    Reading through it, the thought that came to mind over and over was, “Finally.”

    Here was my favorite part:

    …the subject matter itself isn’t all that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step.

    Indeed there is a gracious sufficiency of beating – and resistance – when teaching math.

    But everyone already knows math!

    In the book, Albert wrote:

    If you never teach a stitch of math, in a mathematical culture your kids will learn heaps of it anyway. …learning math along the journey is a difficult thing to avoid.

    You can’t not do math and exist. It’s like not breathing. Impossible.

    We survived for a very long time without the written word. But we’ve never existed without math.

    Early humans knew that to divide a chunk of meat between two people would yield too little:

    meat2<what I need\frac{\text{meat}}{2} < \text{what I need}

    He might not have have had the fancy way to write it – but he could learn how to write it in 8 weeks.

    Daughter playing with washers: counting them and learning math in her world.

    Can you teach all of K-12 math in 8 weeks?

    The short answer – it depends on to whom. In Albert’s article, he’s teaching children. You can totally do this with kids. And cavemen, I suspect.

    Grownups are a different story. Innumerate adults already have anxiety, anger or fear associated with math. There’s a whole lifetime of un-doing that would have to take place.

    But once you undo this, then the 8 weeks would work.

    The way I teach college classes is this: we’re going to spend 48 contact hours together – I’m going to wow you in ways you’ve never expected. When we’re done, you’ll be much calmer. You’ll be able to learn math much better, both on your own or in a classroom.

    I don’t force learning. I don’t even require learning. I let it happen. It’s the closest I can get, in the classroom, to what Albert is saying.

    Can you get closer to the 8-week method?

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  • What is Subitizing?

    What is Subitizing?

    Suzanne at Guided Math Study Group emailed me about subitizing the other day.

    I used perceptual subitizing to “know” that there were three green hair curlers here.

    Alas, I had never heard this term. So I did some research and thought I’d share what I’d learned.

    Subitizing is a way of instantly counting. In fancy math terms it would be getting to the cardinal number of a set (how many) without without going through the ordinals (counting each one’s position).

    There are two types, perceptual and conceptual. Perceptual is perceiving the number of objects immediately. Conceptual is putting a little effort into it.

    For instance, I glanced at the hot rollers Daughter had scattered on my side of the bed. I perceived there were three on the nightstand. But I had to do a little conceptual subitizing to arrive at the number of curlers on the floor (see the pictures).

    I had to mentally “chunk” these hair curlers into smaller perceivable bundles to conceptually subitize the number here.

    Here is a list of resources I found about it:

    In the next article, I’ll attempt to answer the two questions I see associated with subitizing:

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    What do you think of these questions? Do you do it? Do you teach it?

    (By the way, it is pronounced with ooo – like “Ooo! Subitizing is neat!”)

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  • How Calculators Inhibit Learning the Distributive Property in Algebra

    How Calculators Inhibit Learning the Distributive Property in Algebra

    Do you wonder if your children should be using a calculator “at their age”? Are you a fan of calculators, but have friends who aren’t? Are your friends “into” calculators while you oppose them?

    I often hear people say that children 50 years ago understood math concepts more quickly. Although our parents weren’t taking classes called algebra in the 7th grade, they were doing algebra in the 7th grade.

    Algebra is arithmetic.

    There are two fundamental and rarely understood facts about algebra:

    1. Algebra is arithmetic with one or more numbers in disguise.
    2. Algebra has exactly the same  rules as arithmetic.

    Which means if you can do arithmetic you already know how to do algebra!

    Our parents or grandparents, 50 or even 30 years ago, weren’t using calculators. They had to apply all the rules of arithmetic to get the job done. Which means that they had to apply all the rules of algebra.

    Teaching them a class called “Algebra” was much easier because of this.

    What are the rules?

    The basic rules that non-calculator users must apply are the distributive property and the order of operations. The distributive property is the thing that calculator use eliminates.

    Children could get practice mentally multiplying things like 3 x 86 and do 3(80+6) = 240+18=268. With this practice, they are ready for 4x(3y+2z) = 12xy+8xz.

    If they never have to multiply 3 x 86 in their head, they never get the experience of the distributive property. Which means teaching them 4x(3y+2z) = 12xy+8xz will cause anxiety and frustration. They see it as “magic” or “something you made up just to confuse me.”

    Give them the tools they need.

    Refuse to let students have the calculator. Let them have the tool of the distributive property for algebra before you teach them “Algebra”. Give them the benefit our parents and grandparents had!

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  • How to Tell if Your Child Is a Top-Down Learner

    How to Tell if Your Child Is a Top-Down Learner

    Does your child struggle when you put him down in front of his math book? Are you frustrated in your attempts to get him to do math classwork or homework?

    Maybe he’s a top-down learner. If so, you might not know based on his current “regular” work. It will help in his education if you know he needs the big picture before the details – or the big theories before the steps.

    I remember learning to do derivatives when watching the foster kids that lived with us. I was eight. I’m a top-down learner. Here’s how to find out if your child is one too:

    How the “green beans” con works.

    My mom used to leave green beans open in a can on the table. We would walk by and eat them. If she put them on our plate, we would refuse them. So she got us to eat vegetables without asking us to, by just making them available.

    You can use the “green beans” con as a test.

    For math, put out the harder stuff. Find some books at Half Price Books or someplace cheap in your area. Open up the book. Sit and do some of the math yourself while mumbling aloud. Then walk away.

    If he sees his folks (or older sibs) working through those problems, he might be interested. Watch to see if he goes up to the book to check it out (steal a green bean). Be available to answer questions if he asks.

    If he can grasp some of  that “higher level” stuff, he’s probably be a top-down learner. He won’t want the building blocks until he sees the plans for the whole house. This could be the cause of some of the struggle and frustration – he’s been given the building blocks instead.

    Let him have the big stuff – start “allowing” him to do more of the advanced books. He’ll back up on his own to learn the “lesser” stuff so he can understand the big stuff better. You won’t have to force the work on him anymore.

    Share your experience with your top-down or bottom-up learning in the comments!

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  • 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?

  • Be the Village – How to Help Every Kid with Math

    Have you seen the kid that isn’t getting the attention he needs at home? Wonder what’s going to become of him?

    Well, the “it takes a village” concept isn’t far off. You might not be able to change his world, but sharing a little about math might help with one corner of it.

    Share math in small bites.

    Every time you see the child, pose a fun puzzle. It doesn’t have to be blatantly mathematical. Anything to get his logic working helps. I love the Childcraft Mathemagic book for puzzle ideas.

    Ask her random math facts. Tell her you’ll give her something for each one she gets right – or each 10 she gets right in a row. Find out what’s important to her that you can’t get in trouble sharing. Like money, chocolate, baubles, etc.

    Tell him that when you were growing up you wanted to be a mathematician. Ask him to find his favorite mathematician online. Find one yourself that you can discuss – I’m partial to Abel and Galois.

    Be positive about math.

    There might be many things wrong in a child’s life. Let math be the one that isn’t. Be positive. Help make math the thing a kid can hold on to.

    Be the village because it takes a village. Share your stories in the comments!

  • 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.

  • The Only Reason to Do a Word Problem

    The Only Reason to Do a Word Problem

    I’ve avoided teaching word problems for years. Mostly because it’s hard to teach word problems.

    I’ve figured out a great way to do it, but I still don’t like it.

    I’ve learned of a thing called “What can you do with this?” from dy/dan. This teacher sets up a situation so that students can ask questions.

    I’ve been pondering the effectiveness of this for a while.

    The thought is that if you allow students to observe something interesting and ask them “What can you do with this?” then they’ll create their own word problems.

    This is in response to the fake or “made up” word problems from a textbook which mostly don’t work for teaching thinking skills.

    But the issue remains the same. If someone presents a student with a video of Coke vs. Sprite and the student lacks curiosity about that subject, then it’s still a contrived problem. Or a contrived situation.

    The only reason to do a word problem is if you’re emotionally attached to it.

    Husband and I were talking about word problems the other night. After my demonstration about using to teach math, he said he wished he learned math that way. He needed something to hold on to. A reason for doing it.

    He’s a set dresser in Hollywood for part of each year. Which means that he has to hang pictures on movie sets. And they have to be 55″ above the ground – at the center of the picture.

    Not hard to measure, but there’s also the wire on the back to consider. Is the wire dead center? No. It’s probably above the center of the picture.

    It becomes one giant word problem. But it isn’t written in a book. And it isn’t videoed by a teacher. It isn’t fake. There’s a real reason for him to do it.

    Which made me realize that there’s only one reason to do word problems: if you’re emotionally attached to it.

    If you need an answer to a question, you attach to it emotionally.

    Parent: You’ve got 45 minutes to clean the kitchen before we leave for softball practice.

    Kid: If I finish the kitchen before we leave, can I watch TV?

    Parent: Sure, but the kitchen better be spotless.

    Most likely the kid has a plan for TV – like watching his favorite cartoon on DVR that takes about 30 minutes. So he works out how fast he needs to clean the kitchen so he can get in his cartoon before leaving.

    This is a real problem. His problem.

    Watch your kids intently. See where they are doing word problems in their heads. Ask them to explain them. Give credit for work done – especially when self-created.

    If someone else needs an answer, you attach to it.

    Being helpful is a powerful motivator. Try this: with a pencil and paper sit in a public place. Act like you’re writing something important. Then ask out loud, “What’s 87 minus 13?” $5 says that at least four people will chime in to be helpful.

    Let your kid help with balancing the checkbook or creating the budget. If you’re a classroom teacher, let the kids help figure out what teacher supplies to buy. Give them a limit on what to spend and the catalog and some guidelines.

    If someone you like wants an answer, you attach to it.

    I couldn’t have given a feathery duck’s tail about biology, but the teacher was crazy cute. So I wanted to please him. So I worked. Hard. And had a 100 average.

    I suspect this is why the teacher at dy/dan is so successful. He’s cute, compelling and cool. Who wouldn’t want to engage with him?

    If you have carisma and charm, use it. This might not work as a parent but will definitely work as a classroom teacher – at least for some students.

    Give it a try. Tap into the emotion. And share your success below!

  • 5 Philosophies of Teaching Math

    5 Philosophies of Teaching Math

    I discovered this video about the philosophies of teaching math from the blog Learning to Teach, The Empty Set of Education. Having degrees in math and not teaching I found it very interesting from a theoretical point of view.

    And I totally love his conclusion – the 5th philosophy!

    What is your philosophy for teaching math? Share your thoughts in the comments.