I'm Bon Crowder and the photos above are both of me - in 1989 and today. I'm a Generation X mom of Generation Z kids.

I began peer tutoring in high school in 1984. MathFour.com is the 2015 version of me helping peers be comfortable in math.

If you're a Gen-X parent, you're in the right place!

# Tag Archives: cognition

### Singapore Math is Thinking Math

I’ve been hearing about Singapore Math for years now. Every time I investigate, it just looks like it’s based on pictures.

In some senses, this is true.

It goes deeper than that, though. This “picture math” gives students freedom. It allows them to use paper as working memory!

In my classes, I encourage students to use tons of paper for pictures and math sketching. A professor of mine used to say, “You have to get your hands dirty.” Thinking inside your head is not enough.

Yes, some people do it. And some people are very good at it. But for the vast majority of us, to keep all the thinking inside our head just doesn’t work.

And that’s why Singapore Math (the thinking math) is so cool.

### The Standards Require Thinking

I attended a presentation last night hosted by my local school district. They’re moving to Singapore Math to better align with the new Texas state standards (TEKS). Like the Common Core State Standards (CCSS), the new Texas standards ask that students be able to think.

It’s not enough to calculate based on algorithms anymore.

But thinking is hard. And it’s even harder to teach. Especially if the teachers and parents were never taught to think about math when they were young.

Singapore math, or picture math, gives teachers and students a tool to articulate and demonstrate thinking.

### More than Just Pictures

Also included in the Singapore Math method is basic numeracy practice. Instead of teaching children rote memorization and algorithms, they encourage playing with numbers.

They call them “number bonds.” It’s decomposing and recomposing numbers in all sorts of beautiful ways.

Like many teachers already do with “number talks,” number bonds encourage flexible thinking.

For example, you can think of adding 8 and 5 in these two ways:

8 + 5
8 + (2 + 3)
10 + 3
13

8 + 5
(3 + 5) + 5
3 + 10
13

Of course, there’re many more ways to do this. Singapore math encourages students, and teachers, to do this more – and often!

### The Basis of Algebra

The Singapore Math number bonds is a brilliant way to prepare students for Algebra 1, Algebra 2 and College Algebra. And the picture math gets them prepared to tackle any variety of word problem.

The numeracy play allows students to see the algebraic properties at work with regular numbers. Which leads to more comfort and better understanding of algebraic manipulations later.

And the picture math gives them the confidence and perseverance to keep trying new and novel ways to solve problems. Even when they don’t have an example in the book that matches it.

### Possible Issues

You might’ve seen the Facebook meme about a crazy subtraction method. This was in an effort to bash the Common Core State Standards, not the new Texas TEKS. But I fear something like this might happen for us with Singapore Math.

In the presentation last night, the teacher showed the decomposition of the problem 52 – 4 as 40 + 12 – 4.

One of the parents asked, “How do you know to change 52 into 40 + 12?” The teacher didn’t explain that 40 + 12 was only one way of decomposing 52 to make this work. That many others were possible.

Hopefully the teacher’s explanation would be clearer to a child. But I worry that teachers won’t get that 30 + 22 would also have worked. Or 50 + 2, then 48 + 2 + 2.

And there’s the problems with the pictures. Again, graphical representation of thinking comes in many forms. My pictures weren’t like the teacher’s. Or the parent next to me.

For many teachers, seeing thinking represented in pictures is new. And some might not see the various ways it could be done.

So we still might end up with a set of algorithms for drawing the pictures. And children getting “points counted off” because they drew pictures wrong.

I certainly hope this doesn’t happen. Especially because I think that this is one of the best ways to teach math.

But we do have to look for it. And encourage teachers to expand their own thinking when we see it.

Have you seen Singapore Math? Do you use pictures and number bonds in your class? Share your thoughts in the comments.

### 2 Responses to Singapore Math is Thinking Math

1. Great post Bon. I can help with this one:
“52 – 4 as 40 + 12 – 4. One of the parents asked, “How do you know to change 52 into 40 + 12?””

The teacher/presenter should have shown that when using a place value chart, a student:
-Starts with 5 tens and 2 ones.
-Moves a ten from the tens column and regroups it as ten ones, leaving 4 tens and 12 ones on the place value mat.
-Now they can subtract 4 ones from 12 ones.
-(Or borrow, as I bet the parent thinks in their head as heir doing subtraction.)

The manipulatives guide students to the standard algorithm most parents already know! It’s actually quite intuitive for first graders once they work with concrete objects.

Other methods of decomposing come later.

2. Bon says:

Thanks so much! That totally makes sense. And parents would understand it (especially when the “borrow” word is used).

### Rush Hour Traffic Jam Game – Low & High Tech

Rush Hour Traffic Jam Game by Think Fun has tons of math in it. AND it can be played low tech, high tech or both together! Continue Reading

### 2 Responses to Rush Hour Traffic Jam Game – Low & High Tech

1. I am pretty crazy about the low-tech version. I like TOUCHING and MOVING the pieces. too bad manipulatives weren’t popular when I was learning Math–I might have done better!

• Bon says:

It IS nice to feel these things. I use my iPad, a fancy app and a stylus to take notes on some stuff. But on other stuff I still need to TOUCH the paper and pen.

I wonder if the next generations will feel the same way we do.

Thanks for stopping by, Heather!

### AT&amp;T 'In My Day' Commercial is Killing Math Students

AT&T should consider where they’ll need these students before they help to destroy their engagement, problem solving and creativity in math. Continue Reading

### 4 Responses to AT&amp;T 'In My Day' Commercial is Killing Math Students

1. This is terrible! File under: “what were they thinking?” How many kids even get a chance to see a mathemagician, anyway?

• Bon says:

Absolutely, Evelyn!

2. Terrible. I never even paid any attention. Our kids are already learning just to take a test. Then this?!? Great message!

• Bon says:

We have so much math negativity that often we let it go without thinking about it. My job is to point it and and to call them on it.

Thanks for stopping by, Denise!

### Teaching Intuition in Math

Do you teach intuition in math? Do your students understand what you’re doing or are they confused with your teaching style? Continue Reading

### One Response to Teaching Intuition in Math

1. I think this has always been my problem with math. I was never taught to use my gut with it, and that’s what I do with just about everything. As an adult I’ve learned, what I like to call, work arounds that make math easier for me, and when I think about them they are all gut things that I do.

### A Different Way to Teach College Algebra

If your sense of number doesn’t get established, your brain can’t compress math learning. I’m thinking about using this principle to teach college algebra. What do you think? Continue Reading

### 4 Responses to A Different Way to Teach College Algebra

1. I’ve often thought the key to math is doing a lot of it until it becomes second nature. I tell my students to compare it to learning to walk. Watch a toddler and see how much effort it takes to learn, but you haven’t given it a thought in years.

How do you get that number sense? With our games we try to have lots of different activities of different types, so that students see the same concept over and over but in different situations.

However you do it, I agree, number sense is a foundation too many people do not have, and just like in construction, you can’t build very high without a foundation.

• Bon says:

Repetition is important, but only if the foundation gives you the ability to make it compress. Otherwise you’re just trying to be a computer running a gazillion subroutines!

2. suehellman says:

Great illustration, Bon. The only thing I’d suggest is to replace is the “younger” vs “older” labels in your comparison with “novice” vs “expert”.

Have you read any of David Tall’s stuff about mathematical thinking (http://homepages.warwick.ac.uk/staff/David.Tall/themes/three-worlds.html) ? He says that as expertise in math develops, processes become encapuslated (aka: compressed) into what he calls “procepts” — “thinkable concepts” or hybrids of process + object which can be manipulated in their own right.

Working memory (WM) is known to have a very limited capacity (newer research has lowered this from 7 plus/minus 2 items to 3 plus/minus 1). Without encapsulization/compression, WM fills to capacity with the calculations and single steps which all must be juggled individually and sort of held in suspension there until they can be assembled into a problem solution.

When students have to deliberately recall and then ‘drive’ each little step in a more complex process, WM capacity is exceeded and cognitive overload ensues. Some of the little pieces may be displaced by other little pieces which need attention. Unable to recall how the individual steps contribute to the solution, the student may become lost inside the bigger process. Unable to ‘see’ a solution taking shape, the student may shut down.

Developing fluidity is essential, and facility in fractions especially is now seen a sort of bellwether of future success in math.

• Bon says:

Very nicely articulated Sue!

I had no idea that working memory was so small. I was still back on the 7.

My brain just realized that it can’t take much more and is wanting to shut down. Time for coffee!

When you struggle to remember things, do you reach for google? Or do you strain your brain? Could the strain be helping you build math skills? Continue Reading

### One Response to The 'Just Say It' Challenge

1. I love it!
A big part of counseling is helping others develop a sense of empathy. The difficult part of that is this: It’s easy for us to empathize with others about something we struggle with, ourselves, but not so easy if that thing comes easy for us. As a counselor, I ask folks to empathize with the feeling, instead of the experience.

It looks like your doing a great job in creating this for yourself and showing your appreciation for those who are fighting the “good fight” in using positive math-talk. I think your readers will appreciate it!

### Ellipse vs. Ellipsis – And Other Similar Math & English Words

Ever wonder why math has an ellipse (oval) and English has ellipsis (…)? Me too! Continue Reading

### 3 Responses to Ellipse vs. Ellipsis – And Other Similar Math & English Words

1. I love this! Before I read it, I just “knew” the words by rote, but now I really know them because I see how the meaning is connected to the angles. Thanks for sharing.

• Bon says:

Thanks for stopping by Denise! It is kinda, cool, huh?

### 6 Responses to Math Basics – The Real Ones!

1. Math4kids says:

Awesome article about math and math facts. Liked to read how parents and educators need to care more about Logic, Joy of failure and Familiarity rather than forcing kids to memorize math facts.

Math is everywhere around us and math 4 kids can found in the kitchen in the form of pizza slices, cookies, recipe. Math can be found in every part of our body. Look at fingers and toes, kids can learn to count to 20 using them.

• Bon says:

Thanks for stopping by, Math4kids!

Math can be found everywhere – but sometimes people don’t see it. But we’re getting there.

2. This is a good article. I involve my children in balancing our check book. Everyday life is a great way to teach.

• Bon says:

You can’t get much better than balancing your checkbook for everyday life!

Thanks for stopping by!

3. The “joy of failure” point is particularly poignant. I remember about 3 years into my undergrad when I finally (mostly) got over the fear of saying something wrong to my professors. Fear of failure is incredibly crippling–beyond making it harder for you to come up with ideas, it also makes it very difficult for someone to give assistance in a meaningful way.

Saying the “wrong” thing gives tremendous insight into where the gaps in your understanding are. It’s the most helpful thing in the world.

• Bon says:

Wow, Cory! I somehow never have thought about the fear of failure inhibiting communication. You’re totally right.

And your comment makes me want to periodically write a post on featured comments.

Thanks so much for sharing!

### The Prime Directive in Education

There’s one thing that trumps all other advice – The Prime Directive in Education. Continue Reading

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