# 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:

# Math Puzzle Monday!

Time for another magic square! This one requires a good understanding of place value:

Click on the picture to pull up a pdf of the puzzle! And as usual, here are the solutions:

# Math Puzzle Monday!

This week’s puzzle is a math crossword! To go with my most recent post, it’s easiest to do with a good understanding of place value:

Click on the picture to access a pdf copy of the file.

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:

# Math Puzzle Monday!

This week’s puzzle is a “shape puzzle.” To fill in the shape puzzle, you need to fill in every blank shape. Moreover, the same shape in the puzzle must always contain the same number: so, for example, if one of the squares contains a 3, so must every other square.

Shape puzzles are a staple in my teaching, because they both gently introduce children to algebraic concepts and allow kids to use backwards instead of forwards reasoning. Backwards reasoning always requires more fluency and understanding, and as a result, it’s very worth practicing once an initial mental model is in place.

You can click on the picture above to open a pdf of the puzzle.

Here’s the solution to the puzzle:

Please let me know if you have requests for specific puzzles (types of puzzles, operations used in puzzles, numbers used in puzzles, etc.) in the comments!

# How to Use Clear Mental Models

In my last post, I discussed the pitfalls of fuzzy mental models﹣mental models that do not provide a clear, unambiguous picture of what operations and symbols mean. But getting a clear mental model is only the beginning of the story.

Let’s come back to our example from last time, multiplication. Say you decided to give a simple, unambiguous definition: 3 ⨉ 5 is equal to 3 5s, 6 ⨉ 7 is equal to 6 7s, and so on, so forth. What happens next? How do you move from a clear mental model to actually being able to multiply with fluidity?

The first thing you’ll notice when you start being mindful of mental models is that they take surprisingly long to absorb. It may take a child a few weeks to become comfortable with a new definition. Even after the definition can be fluidly recited, you’ll see that some “obvious” things may not actually seem obvious to your child. For example, anyone who’s used to multiplying knows that 1 ⨉ 5 is 5, and that in general, multiplying a number by 1 results in that same number. However, this may very well not be immediately obvious to your child! That’s because it can take a while to get comfortable with the idea of “putting together copies of a number.” At this stage, any kind of gentle practice will help the mental model become fully integrated and absorbed.

After the model is at your child’s fingertips, it’s time to start making observations using the model. Some observations turn out to be a lot closer to the surface than others, and they should be worked on first. However, it’s imperative to keep the mental model in mind when deciding what’s obvious and what’s not. For instance, the fact that

3 ⨉ 5 + 6 ⨉ 5 = 9 ⨉ 5

turns out to be much more obvious than the fact that 3 ⨉ 5 = 5 ⨉ 3! That’s because 3 copies of 5 and another 6 copies of 5 clearly make 9 copies of 5, whereas the fact that 3 copies of 5 is the same as 5 copies of 3 requires something else: the realization that an array shows a multiplication. The benefit of having an explicit mental model is that it becomes clearer what things should and what things should not be apparent to your child.

As you and your child make observations, it’s important to constantly refer back to the mental model. People already fluent with a model have a tendency to think that once a student has seen a definition a few times, the student understands it and is comfortable with it. Nothing could be further from the truth. Mental models take a long time to be internalized, and it takes even longer even for all the “obvious” observations to be made. And only after the “obvious” observations are made are you ready for the more sophisticated observations that make calculations easier.

You may well wonder whether this means that progress using mental models is very slow. Won’t a student take too long to learn to learn elementary mathematics if every single observation needs to be built on top of other observations, and if you constantly need to refer back to the original mental models?

The short answer is “no.” The longer answer is that the sequence of attained skills using these methods is very different. At the beginning, you sacrifice facility with calculation for conceptual fluency. However, in the long term, the student gains robust knowledge that is retained well and easily applied. While this starts out slowly, your student will eventually speed up and surpass learners who can calculate quickly immediately but whose mental models are fuzzy.

As a small case study, I have applied these methods with my 8-year-old since she was about 4.5. In the first few years, she was neither ahead nor behind: she had started on all four operations young and she had early exposure to algebraic concepts, but she had not spent time on the standard algorithms and she was slow on many of the standard 1st grade skills. We spent our math lessons making sure her mental models were well-integrated and making observations about how the various concepts we were working with related to each other.

The further we moved into mathematics, the more rewards we’ve reaped from her clear mental models. When we finally covered the standard algorithms, they each took about a day to learn, due to the clear models of the operations and place value. When we covered fractions, the time we had spent developing models of division made things much simpler. Despite our “slow” start, she’s currently working on high school topics such as algebra and proof-based geometry. Furthermore, she is very confident in her mathematics and she is able to write down clear explanations for everything she uses.

Of course, not every child will wind up accelerated through these approaches ﹣ acceleration in mathematics is a mix of both nature and nurture. However, every child will reap the benefits of the confidence and logical thinking that clear mental models bring. Clear mental models make math make sense.

The next question, of course, is a practical one. How does one use this idea to actually teach one’s child? In the coming weeks, I’ll provide lots of examples of the (deceptively simple) lessons I give to my children and other children I teach, and I will explain how they fit into this framework. This should make the ideas clearer. Stay tuned!

This is the third and last post in a series of posts about mental models. Here are the first and second entries in the series.

# Math Puzzle Monday!

Time for a new math puzzle! This week, we have a true or false equation grid: each square contains an equation, and the job of the puzzle-solver is to spot the wrong equations and then to color them in. If done correctly, the wrong equations form a pattern! For my classes, this pattern is often a letter or a number, although that’s not the case in the puzzle below.

These grids are an excellent way to make sure that a student has a robust, relational view of the equals sign. They are also a good spot check for whether they understand the difference between expressions like 2﹣3 and 3﹣2.

Staying on theme with my recent posts, this week’s grid only uses addition and subtraction:

Click on the image to open a pdf of the same puzzle.

And as usual, here’s the solution if you need. In this solution, the green squares are correct and the red squares are incorrect:

# Math Puzzle Monday!

Today’s puzzle is a magic square! These are really fun: the sum of every column, row, and diagonal must be the same.

In the square below, it’s possible to figure out this common sum without too much trouble. However, these can get much trickier. The trickiest ones require some algebra!

And here’s a solution to the puzzle:

Click on the picture to open a pdf of the puzzle for printing.

Here’s the solution to the magic square with the missing numbers filled in in red:

If you have requests for a kind of puzzle (or for a given puzzle difficulty), please let me know in the comments! These are easy to generate.

# What’s Wrong With Fuzzy Mental Models?

In my last post, I started talking about what it means to have a good mental model of something. As you may recall, a mental model is an internal representation of an abstract concept, and a good mental model must be clear and unambiguous. It must be easy to work with. It must be a satisfying answer to the question “But what is it?”

I call models not satisfying the description above “fuzzy.” So what happens when someone has a fuzzy mental model? Let’s take a peek:

Student: Professor, can I use this rule?

Professor: Well, what do you think? Does that make sense?

Student: Hmmm. I’m not sure. How would I figure that out?

As you can see, the student has no faith that they can figure out what’s true or false. They have no confidence in their ability to reason. They think that their professor holds the keys to the mathematical rules.

A mental model is what gives you the ability to interact with a concept. If your initial mental model of the concept is fuzzy, then you never gain the ability to reason about it, since you don’t fully understand it. And if you don’t fully understand it, then all the rules you learn feel “handed down from above.” They can’t possibly make sense, because you can never check them against a robust internal representation. In the long run, this leads to a loss of mathematical confidence and an inability to tell the difference between correct and fallacious arguments.

A mental model is the foundation that you build your mathematical reasoning on. Without a clear model, any structure we put on top is shaky.

Sadly, we often neglect to mindfully form these models. And the reason why is both very understandable and yet very wrongheaded.

The example of multiplication will make this reason clear. In my last post, I criticized the definition of 3 ⨉ 5 as “either 3 5s or 5 3s” because it didn’t produce a clear mental representation. It is not at all obvious to a beginning student that those two quantities are identical. An ambiguous mental model is by definition fuzzy.

So how would I define multiplication to make the mental model clear and not fuzzy? Simple. I would define 3 ⨉ 5 as 3 5s. Period. End of story. Nothing more to see here.

I can immediately hear the objections to this definition, because I’ve heard them many times before. “But how are they supposed to actually multiply then? What, are they expected to calculate 100 ⨉ 2 by adding up a hundred 2s? No, thank you. I’ll stick to my ambiguous definition! Look how much easier it is to use!”

And to that I say: I see your point. It is absolutely true that choosing a clear initial mental model conflicts with the goal of having kids achieve quick computational proficiency. However, it produces much better results in the long run. Remember, a clear mental model is merely the foundation of a student’s engagement with the concept. Once the foundation is laid, computational proficiency can be built on top of it. And unlike with fuzzy models, this proficiency will endure.

In my next post, I’ll explain how to build on top of a clear mental model and how rigorously obeying these principles has turned out with my 8-year-old.

This is the second in a series of posts about mental models. Here are the first and third entries in the series.