How to explain addition vs. place values to a 4½-year-old?

Question

My 4½-year-old son has had some questions recently about numbers. On and off for months, he's been asking questions along the lines of "What does four and two make" or "What is a four and a two"? At first, it was unclear whether he was asking about 4+2 or 42. Eventually it seemed to be that he was asking about what the two numbers written side by side meant, so that's how I started answering him.

More specifically, he asked one recent morning about 5 and 5 and I told him that would be fifty five. He said in response, "But Grandma said it was ten." I did my best on the spur of the moment to explain place values but my explanation was lacking.

I mentioned all this to the guides at his Montessori school prior to the end of the spring semester, but I didn’t revisit it with them before the summer break. The questions are persisting over the break and his summer camp is mostly play-based.

At the very least, I’d like to clear up any confusion that might have been introduced by having him pose the same question to various people in slightly different or ambiguous ways. Plus, I’d like to get him on the right track about the difference between addition vs. place values. Any tips or ideas for at-home activities, resources, games, etc. that might help him differentiate between addition and place values and otherwise reinforce each concept?

Answer 1

The crux of your issue is an English problem, not a Math problem!

I feel the question asked in the title and the issue described in the body are quite different, so I'll address them both, one at a time.

The English Problem

As you've described, your son is asking questions in an ambiguous way, and there is no point in teaching him the concepts behind one of the answers until he can clearly identify when that concept applies.

I see 3 occasions in your description where this was the problem:

At first, it was unclear whether he was asking about 4+2 or 42.

Eventually it seemed to be that he was asking about what the two numbers written side by side meant, so that's how I started answering him.

I did my best on the spur of the moment to explain place values but my explanation was lacking

At all of these points I wouldn't recommend answering the question until you have established clearly what it is he is trying to achieve. Instead, put the question back to him:

Hmmm, I'm not sure what you mean by "What does four and two make?" Do you mean a four and a two added together or do you mean a four and a two side by side? Because the answer will be different depending upon which question you are asking.

Lot's of good things going on here: We indicate to him that his question is unclear even though we heard it fine, we give him options he can use to make it clearer (which we'll be hoping he sees the benefit of and starts using in the future), and we establish that if grandma is giving him a different answer it could be because she didn't understand the question.

You can also use this as a chance to point out that if he does get an answer that he didn't expect then maybe the person answering didn't understand the question and he might want to double check with them.

Then you can drive the difference home, tell him the answer to both regardless of which one he clarifies he wanted (although if he does indicate a preference then start with that one):

Four and a two added together is six, because I had four and I put another two in as well. Four and a two side by side however, is forty-two, which is a lot more than six.

Once we've established this clear difference we have set the ground work for what you are hoping to teach him: why is 42 a lot more than 6!?

The Math Problem

Of course you still want your son to know about place value of numbers, and I reckon once we've established a clear difference between addition and position this will be made easier. Try not to combine teaching about one with teaching about the other too much (for example, 10 + 10 is 20 because the 1s are in the tens position...). At this early stage this will just muddle things and make it harder to establish the foundation which will be built upon as his understanding develops.

At 4.5 I would also avoid the term 'place value', instead talk about position and keep it between 0 and 20 to start with.

Use physical cues, such as counting blocks suggested in Rory's answer, even better if they are the type you can stick together, so when counting them out you can show that even though it is 1 'block' (by sticking 10 blocks together) it is still worth 10 'blocks'. This will help build the idea that a '1' can equal '10'.

Also make use of visual cues: write the numbers down from 0 to 9, one above the other, then when you get to 10 make a point of emphasizing that you've ran out of numbers so need to put a 1 in the next space (to the left) and then start from 0 again. We're talking about position after all, let's show it! Get all the way to 19 and then explain that since there is already a 1 there you now need to add 1 it to get 20 and start again. As he progresses you can ask him himself what he thinks happens at 99 and then explain the next position to him. Once he is comfortable with this I'd start talking about position and how much each position is worth, but I'd say you are a wee bit away from that yet so will hold off elaborating in this already lengthy answer.

Failing that just plunk him in front of this YouTube video I made some years ago on the very subject! Sure, it's intended for adults and is a precursor to my follow up video on hexadecimal and binary numbers, but who wouldn't want their kids to understand those?

Answer 2

There are actually many ways to do this so I'm not sure if this is ideal for this site, but the method I like for that age group is to start with the wooden counting blocks - the ones that include tens and units, so you can easily show the difference between "5 tens and 5 ones" and "5 ones and 5 more ones"

Definitely bring in the duration between units and tens - try using fingers and hands.

Answer 3

Because you mention that your son is at a Montessori, and assuming it's a traditional Montessori (and not a school using the name without the method), I would be careful to teach place value and addition, and explain the difference, with a consideration for how the Montessori method teaches it. For the most part I would talk to the guides as you have and ask them how they approach it; this has worked very well for me with my son, who's in a Montessori (entering Lower Elementary this year). Teaching in a consistent manner with how he's learning will make it much easier for him, both to learn from you and to translate the learning to the next school year.

The below describes how our Montessori teaches the concept, which may be different from yours (there are multiple schools of Montessori teaching), though this is the most common and most similar to the traditional methods of Maria Montessori. Your child can likely tell you if he uses these methods at school, and at 4.5 he should have been exposed to this already most likely (depending on his ability level, but I believe all of the 4 year olds in my son's school had already learned this).

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Place value for Montessori is most commonly taught using the Golden Beads, which is part of the Montessori method's focus on tactile methods of learning mathematics. You have single beads for 1s, strips of 10 beads for 10s, squares of 100 beads for 100s, and cubes of 1000 beads for 1000. This helps the child understand how to count a large number - 1852 is one thousand, 8 hundreds, 5 tens, 2 ones.

The next step then is Symbols, which is how this is translated into the decimal number system. Here instead of beads they have number cards - 1s cards, 10s cards, 100s cards, 1000s cards - and they get out the right cards for the number. 1 1000 8 100s 5 10s 2 1s for 1852. (The cards have the actual number on them - so 8 100s means you have 100, 200, 300, 400, 500, 600, 700, 800.)

Then they combine them in Formation of numbers, which takes a set of golden beads AND a set of symbol cards, and takes a golden beads result and translates it into a number with the symbol cards.

This is fairly straightforward, and something your son will naturally learn at the Montessori. If he stays through Kindergarten at least, he'll likely continue to use these golden beads for addition, subtraction, multiplication, and perhaps even division - my son used all four, and it became a nice link to a new concept using an old method.

Even if you don't have these materials, you likely can mimic some of them for lessons at home if he's asking for them. Maybe not the golden beads, but the symbol cards are easy to make.

If he's then interested in addition, these allow for easy teaching of addition also - as you can use either of these methods to learn it. Just keep the addition to small numbers to start with (numbers that don't require carrying), you can have him add things like 1432 + 2136 very easily using the symbol cards - just get out the first set of symbol cards (1432) and then for each place count out however many more you need.

I would definitely ask him which of these he's done already. At 4-4.5 I would expect at least some of this to have already been covered for many children, so you can skip to the point he's at and let him practice with his current level or explore a bit of the next level. If your montessori guides are available over the summer for questions, it may be worth asking them also for tips.