Math Puzzle Monday!

Time for a slightly belated Math Puzzle Monday! This is another “shape” puzzle, in which every copy of the same shape needs to contain the same number. These puzzles are an excellent way. to work on a mental model of variables!

This one only uses addition but is in fact rather tricky:

Click on the image to get a copy of the PDF file for easier printing! And here’s the solution:

Place Value

Ready for a paradox? I present you with the following two true statements about place value.

Many kids find place value challenging and don’t learn it well if at all.

Place value is a concept with an extremely simple mental model. It is an idea that gets absorbed very well with continued exposure.

How can those two statements be true at the same time? The issue is that most adults are so familiar with place value that they cannot separate the mental model from its many consequences. Therefore, they teach kids the consequences, expecting that as a result kids will grasp the concept. Unfortunately, that is not an effective method of teaching: it will neither work for all kids nor will be efficient with the subset of kids who are able to learn like this.

So what is the straightforward mental model I’m referring to? Restricting our attention to integers for simplicity’s sake, here it is:

To figure out a quantity represented by a given integer, all we need to know is that each digit represents how many copies of a particular quantity our number contains, where each such quantity is ten times greater than the quantity represented by the digit to its right. The rightmost digit represents how many ones our number contains.

To put it in more familiar language, the rightmost digit of a number represents how many ones the number contains, the next digit to the left represents how many tens it contains, the next digit to the left represents how many hundreds (which is equal to ten copies of ten) it contains, and so on, so forth.

“That’s it?” you say. “Everyone knows that. Kids know that starting from Grade 1 or 2. They learn about ones, tens, and hundreds, and that’s all this is saying.”

Actually, as it turns out, kids are largely uncomfortable with place value, and so are many adults. They are comfortable with many of the consequences of place value, such that as the fact that when we start counting, the sequence begins with

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …

and the fact that adding a 10 to a number between 10 and 89 simply increases the first digit by 1, and the fact that multiplying an integer by 10 tacks a 0 onto the end of the integer, but they are uncomfortable with the mental model itself. This becomes clear when you try to teach them a new application of this model. Suddenly, kids who seemed to be totally fluent with place value when using it to add can’t figure out how to use it to subtract. Or if they can subtract, they can’t figure out how to use the model to multiply. And don’t even get me started on dividing.

What’s going on? Unsurprisingly, it’s the usual issue. Instead of being presented with a single, clear, coherent model, kids are bombarded with a wide variety of tools, tips and techniques. If a child has a naturally good memory, they can use all these techniques to get right answers. If they aren’t as good at this, then they can’t. Either way, they are getting no practice with the model itself.

So what can we do? We can present them with tools that let them explore the immediate consequences of the mental model itself. Once the model is robust, we can build on top of this model. When approached in this way, all the shortcuts and algorithms are easy to remember because they make intuitive sense.

Of course, as usual, absorbing a mental model takes a surprising amount of time! In my next post, I’ll explain how I introduce place value, and in later posts, I will explain how to use it in a way that builds facility with the model.

This is a post in a series about our basic approach. Here’s the introduction to the series. Here’s my second post about place value. Here’s a post about arithmetic operations. I will soon also add posts about equality and variables, as well as a couple of detailed examples of our daily lessons.

Math Puzzle Monday!

It’s time for our weekly puzzle! My apologies for not posting last week﹣we were on the road.

This week’s puzzle, like the hexagon puzzle I posted a while back, can easily be done via counting on, but this time, there’s a twist: consecutive squares on the path need to be connected by the move of a chess knight instead of being straightforwardly adjacent! This makes for an appealingly tricky puzzle apart from the arithmetic.

Click on the puzzle to open a copy of the puzzle as a pdf.

And here’s the solution to the puzzle as a pdf, if you want to check your student’s work: