Sorting and grouping

The thing with children is they change so quickly and if you blink, you feel like you missed the moment when a major milestone was achieved. For example, Rory used to say he wanted a hangaber for dinner. Alan and I thought it was adorable, and did nothing to encourage the proper pronunciation of the word! But then one day, we noticed he was asking for hamburgers instead of hangabers, and the moment was gone.

This week, I was playing with Oliver and noticed that he now knows some of his colours! This is a very recent development and one we’ve been anxiously waiting for. Rory knew all his colours by 2 years old, and Oliver is almost 3 and was showing no signs of progress; but then, just like that, he got them all right! This is so exciting for me as a mathematician because it now opens up so many more informal sorting activities!  

“As the children’s vocabularies increase, they will be able to label and describe how and why they are sorting and grouping things.” (Charlesworth, 2012)

Here is an example of Oliver engaged in naturalistic play. Notice how I commented any time he knowingly (although usually unknowingly!) put things into groups. Also notice he learned a new word (rectangle!) and now has additional sorting power for next time!

Because Oliver is now ready for more informal instruction on sorting, I started looking for articles about this important stage of development and was surprised when I couldn’t find many. I couldn’t even find agreement on what strand of math sorting falls into! In some books, classification was stuck under geometry, but the content was directed at a higher age level. For example classifying polygons versus nonpolygons; or triangles with the same area versus different areas.  Another resource I looked at, clumped sorting under data analysis because organizing data into groups is important for graphing. I myself, would have linked classification with logic and pre-algebra, because sorting involves reasoning and logical thought. It is also the precursor to addition (putting groups together) and subtraction (taking groups away).

In addition to the controversy over what strand this falls into; sorting and classification only really appears in the pre-k to k curriculum, and as a result it is minimized in the teaching resources or believed to develop naturally. This surprised me because classification is such an important skill not only at school, but also in our daily life. This skill, although it may appear basic, is the basis for further logic and reasoning. It provides an introduction to graphic organizers such as Venn diagrams and to me, it is a life-skill that may even precede executive functioning ability! (New research project?!) Think of the importance of learning how to sort and classify in this day and age, with all the information we have access to.

Now that I have convinced you of the importance of this seemingly natural ability, I want to share with you how to nourish this skill in your child. In the early years, classification activities fall into three categories:

Stage: Your responsibility:       Example:
Naturalistic: Provide free time, material and space
  • Blocks, cars, farm animals, nature things
Informal instruction Provide comments or tasks that identify or encourage sorting
  • Your picture has lots of red.
  • Can you separate the forks from the knives?
  • Could you put your cars in the car bin and your balls in the ball bin?
  • I see you’ve arranged your dolls from smallest to largest.
Guided instruction Give specific objects and guide classification strategies
  • Find some things that are___.
  • Tell me why these belong together.
  • Sort these into groups, how did you decide?
  • Is there another way to sort these?

 Rory has a larger vocabulary than Oliver and a larger understanding of the universe. For his sorting activity, I used guided instruction. You’ll notice he came up with interesting ways to sort things: by function (button, sticker), by colour (red, blue, yellow, green) and by category (animal, vehicle, shape).  I guided him by encouraging him to think of different ways to sort his materials; however, it was ultimately his decision.

Next time, I might choose different objects that force him to make different decisions. For example, choosing all cars but different sizes, or choosing all art mediums (canvas, paper, felt etc.) and let him sort by texture, or all natural objects and have him sort by common features. I would also provide objects that relate to different content areas. For example, objects that float or sink (science), pictures of workers and different materials (social studies), or sorting plants into edible and non-edible. The possibilities are endless! The only thing to keep in mind is that classification activities should follow the same progression as manipulatives (see my post on this here), so start with 3-D objects and then move to cut-outs and then to pictures.

Although I couldn’t find much on how to teach classification, I found bucket-loads of activities that involve sorting.

Click here  and sort through these for starters!

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Estimation is easy!

Anyone who has had me as a teacher in Grade 6 or 7 has heard me echo, “Estimation is Easy!”…but is it?! Estimation can be a difficult skill to learn because it involves number sense, spatial sense, measurement sense and lots of mental computation. This important skill is often left out of the curriculum, or is inserted as an insignificant add-on because approximate answers are not valued as much as correct answers; but they should be!

that was easy

That was easy!

Why is it important?

Estimation may not be easy, but it is essential! We use estimation every day, whether estimating how much sand to buy to fill the new sandbox, or guessing whether our suitcase is overweight, or figuring out if we have enough money to buy something. Estimation helps develop number sense and fluency and is a great way to get children to visualize amounts mentally.

“The emphasis on learning in math must always be on thinking, reasoning and making sense.” Marilyn Burns

And what better way to emphasize these skills than to begin problems with estimation!

What does it accomplish?

Mathematicians agree on these four things:

  1. It helps children focus on the attribute being measured (length, time, volume etc.)
  2. It provides intrinsic motivation for measurement because children want to see how close their guesses are.
  3. It helps develop familiarity with standard units, if that is what is being used to estimate.
  4. It develops referents or benchmarks for important units and as a result, lays the groundwork for multiplication.

When should you start?

My belief is that once your child can count with meaning and he or she can comprehend the language of comparison (more, less, the same), you can start estimating with small amounts. Why not start when the kiddos are small and not yet pre-programmed to believe that right answers are more valued than close answers?

That being said, Rosalind Charlesworth suggests that children can’t make rational estimates until they have entered the concrete operations stage (ages 7-11 yr). This is because she believes children should have already developed number, spatial and measurement sense before they can make educated guesses.

Well, let’s see what Rory thinks; will he make a wild guess or a rational estimation? He’s 4 1/2 years old and in the pre-operational stage of development, but I believe he can make good estimates through motivating activities, coupled with appropriate phrasing of questions.  Let’s see how he does.

The Problem

We are going to our new house on the weekend to measure the rooms so we can plan where to put our furniture, but oh-oh…Daddy forgot the measuring tape! What could we use to measure instead? Oliver! And he is a very willing helper…at least in the beginning!

As you can see from Rory’s first attempt, he made a rational estimate even though his estimate wasn’t that close. He thought it would take 10 Olivers to line the wall, but it only took 5.  I knew his guess was rational because he explained it to me and it made sense. His mistake was that he counted 10 steps initially, instead of 10 Olivers.

Notice how he has already improved his understanding and his next estimate was much closer; he guessed it would take 1 Daddy and it actually took 2. I give him a thumbs-up for his first estimate activity and look forward to doing more with him to see how he improves.

Teaching Tips

Here are some tips so your pre-school children meet with success also:

To start, provide numbers for the children to choose from so they don’t have to pull numbers out of thin air.
Stick with numbers they can count up to.
Begin using benchmarks (5 and 10).
Use and teach proper words: about, around, estimate.
Ask good questions that encourage comparisons:

·         Will it be longer, shorter or the same as _____________?

·         Will it be more or less than ____________?

·         Will it be closer to 5 or 10?

Ask good questions to ensure understanding:

·         How did you come up with your estimate?

·         How can we find out whether your estimate is reasonable?

Start with length, weight, and time.
Let the estimate stand on its own; do not always follow with the measurement.
Develop the idea that all measurements are approximations (thus estimates!), the smaller the units – the more precise but still approximate.
Incorporate estimation activities into your every-day life so it becomes second nature.

It is easy to incorporate estimation activities into your daily life and the more your child practices, the better they will become! Need help getting started?

Click here for activities!

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Making Math Meaningful with Manipulatives!

If you’re not a teacher, you may not be familiar with the term manipulatives, but you can probably infer what they are. Manipulatives are models that help children think and reflect on new ideas in math. They include resources that allow children to explore, question, guess and check, but more importantly, to play with the problem. Counters, toys, linking cubes, abacuses are just a few examples.

Why use manipulatives?

We all know the old adage: we learn better by doing and math is no different! Manipulatives give students, of all ages, opportunities to have a hands-on approach and develop deeper understanding of concepts. Research has shown benefits to using manipulatives all through life! That means, don’t be in a rush to move your child into more abstract ways of solving problems. There is a natural progression to manipulatives and you need to assess your child’s readiness before pushing them to a more abstract level. When choosing materials, they should be sequenced from concrete to abstract and from 3-D to 2-D. See the chart below for more information.

Transitioning from concrete to abstract manipulatives (Charlesworth, 2000)

1) Start with real objects. Sensorimotor stage.
2) Move to real objects supplemented by pictures. Pre-operational stage
3) Once the first two are mastered, you can use cutouts of real objects. This is the transition from 3-D to 2-D, but the objects can still be manipulated. Pre-operational stage.
4) Now move to pictures. Transitional stage
5) Finally (and much later!) use paper and pencil. Concrete operations stage

So where do virtual manipulatives fit on this spectrum? Good question! I’m not sure! My guess is that they act like real objects because you can move them, but because they are 2-D, they might be more on par with the cut-outs level, in terms of concreteness (see step 3 above). Let’s see what Rory thinks. I’m going to get Rory to do a task with real objects and then do the same task with on-line manipulatives. Then we’ll see what he has to say! This task is an introduction to addition but it would also be great to use for lessons on one:one correspondence, decomposing numbers, counting on and  cardinality.

Well it looks like Rory prefers virtual manipulatives. It may have been the novelty of it or the fact that the computer images acted more life-like than the real objects! He claims that the boat was more real compared to my egg carton version and he liked that the bears kept looking at him (in case you couldn’t tell)! The important thing is that children are given the freedom to choose their own manipulative so that they aren’t restricted to one method. That way, they can discover their own way to reach a solution that makes sense for them. If he likes the on-line tool, on-line tool it is! But I’ll make sure he has the real objects on stand-by in case he’d like to use them as well.

Looking for manipulatives? Look no farther!

Click here for a list of manipulatives that teachers often use with this age group!

Are you a parent? The great thing is that anything can be a manipulative! You don’t need to run to a teacher supply store in order to help your child.

Click here for a list of great things to use at home!

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C is not just for cookie!

Although some would argue that I am a cookie monster and thus not able to think outside of the “C is for Cookie” box, I say: have you met my friend Mahsa?! Hee hee.

Today I will prove that I can distance myself from cookies long enough to tell you about the 5 C’s of mathematical engagement. (That being said, I should probably admit that as I write this, I am waiting for the oven to pre-heat so I can make cookies! Coincidence? Aha! Another C word! But I digress…)

How do you get children inspired about learning math? You get them excited about solving problems! And how do you get them excited? Get them engaged! And how do you get them engaged? With the 5 C’s of mathematical engagement (Jo Boaler, 2015)!

Curiousity!

Find a problem that they want to find the answer to.

Math Milestones-005

CURIOUSITY!

Connection making!

Find a problem that has connections with other subjects as well as connections with other strands of mathematics.

Math Milestones-006

CONNECTION MAKING!

Challenge!

A problem is not a problem unless it poses a challenge to the learner. Find a problem that will lead to a productive struggle.

Math Milestones-003

CHALLENGE!

Creativity!

Find a problem that is open-ended in its methods or in its solutions, or even better, in both! A problem that encourages them to think critically AND creatively is ideal.

Math Milestones-004

CREATIVITY!

Collaboration!

Find a problem that requires group work to solve. Encourage conversations. Make math social!

Math Milestones-002

COLLABORATION!

and if all else fails….EAT COOKIES!

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