I know you see it all the time, when you ask a student what a number is made of, and they instantly throw place value out the window. That is why so many of us are doing this “Squashing Misconceptions” blog hop about place value.
I was working with a second grader that was struggling to add some two digit numbers (12 and 24). I was thinking that if we could split the 12 she could add a ten and 2 more to 24. So I asked her: “What two numbers go together to make 12?”
“1 and 2!”
Now you might be thinking that I tricked her with the question or that it was a matter of not understanding what I was asking, so I posed the question in a few different ways (“What is 12 made of? What two numbers can you add to get 12?”). Each time, she maintained that 12 is 1 and 2 put together.
So, what do you do when there is a struggle with place value? Get out the MATERIALS. ALWAYS. Don’t waste time or think that this will make it more difficult or confuse them. Just get out manipulatives, counters or cubes. I love cubes because you can group and ungroup them at any time.
I gave her some unifix cubes and told her to prove to me that 1 and 2 make twelve. The second she pulled out those cubes her face lit up. “That’s only 3!”
So I asked her again what makes 12. Do you know what that little cutie did? She pulled out a ten and 2 cubes without even having to count the ten. She was so used to seeing those concrete tools that it was a no brainer. We had to have those materials out to make it concrete, she was not ready for bare numbers yet.
We constantly make those leaps too soon, and then time and time again (myself included!) it doesn’t occur to us to pull out tools, manipulatives or materials. Our young learners need this, as they are very concrete and it can help them make the connection to number sentences.
So let them have those materials! Have them out for ALL students, so that no student ever feels “babyish” having to use them. This is essential for students to understand place value conceptually.
Want to read some more about misconceptions in place value? Check out the next stop in the blog hop:
Love the post! I am curious…did she pull out 10 individual cubes or was it already stuck together in 1 ten stick? I am wondering if it would make a difference? A lot of kids see the link of 10 cubes as just a big ‘1’, so if it was already stuck together I would be curious to see if she could do it when all the cubes are pulled apart and are just a pile. I love this video by Marilyn Burns that shows sometimes we think kids ‘get it’ but then they show us in a different scenario they really don’t: https://www.youtube.com/watch?v=_ofQ_WnQiZ4.
This is a great discussion of place value. In reading your post, I am realizing that I need to promote a greater use of a variety of materials. My student’s favorite go-to materials are base ten blocks, counters, money and hide zero cards. The linking cubes are such a great tool because rather than just “trading in” students can see a new unit being built or taken apart in the group of ten. I am going to begin a “linking cube campaign” tomorrow to promote use of this powerful tool! The Math Spot
Great post as always! Your points about going back to manipulatives are so timely. If we take the time to make sure everyone has a deep understanding, we will actually save time later.
Love your insight! It is so important for kids to get their hands on materials and go back to the concrete. I think sometimes we ask for the abstract to early!
I would say that your last statement above is an understatement.
Numbers as things or objects ARE abstract. Numbers, or number words, first appear as adjectives as in three dogs, ten bricks and so on. Observations are made such as “I have three bricks, you have four bricks. If we put our bricks together how many bricks will there be?”. The answer is “seven” , which is short for “seven bricks”, still a description. Only when it becomes clear to the child that it doesn’t matter that the situation is about bricks, but works for (almost) anything then it is time to consider the idea of a number as a thing, but abstract.
Numbers as adjectives have names, and can also be represented by symbols, but the symbols are NOT the numbers. The question at the start about what two numbers go together to make 12 is not a clear question. As it stands the answer given is one of the many correct ones and the child is seeing 12 as a pair of symbols comprising a 1 and a 2. The questioner is expecting the child to see the place value interpretation of the symbols, and to come up with 10 and 2. However, as it stands the question has other answers, such as 9 and 3.
Great post title–says so much with so little! :0) Thanks so much for sharing your insight!
Sarah – thanks for your comment. I have read several of the place value blogs and comments. I was very surprised by one, which said that a child, when asked to read 27 said 2 and 7. Maybe the rush to symbols is too fast. Number names after 19 say what the individual digits mean, so it is essential to be able to go both ways, 234 is two hundred and thirty and four. This of course may have no practical meaning for some, which is a shame !
One more, that’s it !
In the mid 1960’s teacher training in the UK was extended from 2 to 3 years, and my first job was teaching maths as an academic subject in a training college. I soon got involved with infant (K-2) math, and the approach was exactly what these posts are describing. They had joinable blocks, cuisenaire rods, an abacus, and more, for place value. This was nearly 50 years ago !!!!! There was much emphasis on Piaget and his theories as well. Did he die ?