The Early Math Experience Matters

posted by Dr. Bilge Cerezci

Traditionally, mathematics education has not been considered developmentally appropriate for young children (Battista, 1999). Math is abstract while young children are deemed to be concrete thinkers, and some cognitive developmental work done in the mid-twentieth century has been used to suggest that young children’s mathematical ideas develop on their own timetable, independent of environmental factors like teaching (Piaget, 1969). Over the past two decades, however, a growing body of literature has indicated that many mathematical competencies, such as sensitivity to set, size, pattern, and quantity are present very early in life (National Research Council [NRC], 2009). magnetic numbers plus symbolsYoung children have more mathematical knowledge, such as an understanding of number and spatial sense, than was previously believed. For example, research suggests that young children have a basic understanding of one-to-one correspondence even before they can count verbally (e.g., pointing to items in a collection and labeling each with a number) (Mix, 2001). Further, young children also enjoy exploring spatial positions and attributes of geometric shapes by building towers with blocks and cubes and by manipulating various materials, such as puzzles and two- and three-dimensional shapes (Clements, 1999; Clements & Sarama, 2008). They also demonstrate emerging awareness of measurement, when they begin to notice and verbalize similarities and differences in the size, height, weight and length of various objects and materials (Clements & Sarama, 2008). In addition, research also suggests that 3 and 4 years-old children engage in analytical thinking as they collect and sort materials by various attributes (e.g., color, size, and shape) and in algebraic thinking as they copy the patterns they observe in their surroundings and create their own patterns by using pattern blocks and other materials (Epstein, 2003; 2006). In fact, as research points out, most children enter school with a wealth of knowledge in early mathematics and cognitive skills that provide a strong foundation for mathematical learning (Clements & Sarama, 2009; Ginsburg, Lee, & Boyd, 2008; Mix, 2001).

There is also new evidence that achievement in early mathematics has a profound impact on later success. For example, Duncan and Magnuson (2009) examined the mathematics achievement of children who consistently exhibited persistent problems in understanding mathematics in elementary school and analyzed it in comparison to children who had stronger early math abilities. The results of the study revealed that 13% of the children with persistent problems are less likely to graduate from high school and 29% of them are less likely to attend college than those who had stronger early mathematics abilities. In other words, the initial differences in mathematics skills in early years may lead children to remain behind their more knowledgeable peers not only in primary grades but throughout their formal schooling (Geary, Hoard, & Hamson, 1999).

Studies also showed the predictive power of early math skills compared to other academic skills, such as reading. Lerkkanen, Rasku-Puttonen, Aunola and Nurmi (2005) investigated the relationship between mathematical performance and reading comprehension among 114 seven-year-old Finnish-speaking children during the first and second years of primary school. The results suggested that the level of mathematical knowledge children have before schooling is very important because these skills are predictive of their subsequent reading comprehension. In other words, early mathematics skills predict not only later achievement in mathematics but also later reading achievement. Similarly, Duncan and colleagues (2007) conducted a meta-analysis of 6 large-scale longitudinal data sets to examine the relationship between early learning and later school achievement. Of them, two were nationally representative of U.S. children, two were gathered from multi-site studies of U.S. children, and last two focused on children either from Great Britain or Canada. The researchers focused on the relationship between school-entry skills (i.e., reading achievement, math achievement, attention, internalizing behavior problems, social skills, and anti-social behavior) and later math and reading achievement while controlling for children’s preschool cognitive ability, behavior, and other important background characteristics such as, socioeconomic status, mother’s education, family structure and child health. Their meta-analysis revealed that only three of the six sets of school entry skills and behavior are predictive of school achievement: math, reading, and attention. Further, early math skills were consistently a stronger predictor of later achievement compared to reading and attention (Duncan, et. al., 2007). Consistent with the educational attainment analyses (Duncan & Magnuson, 2009), early math achievement was found as the most powerful predictor of later school achievement (Duncan, et. al., 2007).

Even though young children are natural mathematicians (NRC 2009) and capable of developing some complex mathematical ideas (e.g., addition) and strategies (e.g., sorting by multiple attributes to analyze data), it is also true that they do not become skilled in mathematics without adult guided rich and intentional interactions with those foundational math concepts. This month, we are going to focus on three of these foundational math concepts (e.g., number sense, sorting and geometry) and how you can provide your youngsters with rich and engaging math experiences that offer for opportunities and structures for the development of deeper math understandings.

Silly Putty Recipe Card

Measurement

posted by Stephanie Forsman

clockMeasurement is an area of my math curriculum that I often feel gets neglected, rushed through, and sometimes, at crunch time, overlooked all together.  As a result, I have worked on infusing small aspects of measurement into the routines of the day.  From linear measurement to volume, weight and mass to telling time, temperature and money, measurement is an everyday skill, “real life math.” It is important that children know how to identify appropriate units and choose the correct tools and technology for measuring those units.

One of my favorite topics that I consistently revisit throughout the year is Time. Even in 3rd and 4th grade, some children cannot tell time and rely on the adults in their life to tell them where they need to be and when. At the beginning of the school year, regardless of what grade I am teaching, I do a quick lesson on Time – 24 hours in a day, AM & PM, the short hand is the hour hand and the long hand is the minute hand. One of my favorite tools to teach Time is a Judy Clock. I have a class set and each student has one in which they practice telling time and learning the concept of elapsed time. A Judy Clock features easy-to-read numerals that show elapsed time in 5 min intervals. The clock makes learning to tell the time simple and fun for children and comes with visible functioning gears that maintain correct hour hand and minute hand relationships.

 

3-clocks

I will routinely ask the children to show me the time on their clocks or I will pose questions, “if it is 10:45 AM now and we have lunch at 12:00 PM, how much time does that leave us for snack and math?” Another handy time telling tool I have is a rubber clock stamp.
clock-stamp
I will routinely ask the children to show me the time on their clocks or I will pose questions, “if it is 10:45 AM now and we have lunch at 12:00 PM, how much time does that leave us for snack and math?” Another handy time telling tool I have is a rubber clock stamp.

When I put up the day’s schedule on the board, I will put the event and the time and then have a blank picture of a clock where the children will draw in the correct time using the hour and minute hands.  I will write times such as “Math – 10:45 AM” with a blank clock next to it and make sure that the child responsible for noting the time will make sure that the hour hand is closer to the number 11 than to the number 10.

Just like my parents did with my brother and I when we were growing up, I like to have a height chart located on the inside of my doorway. One of our beginning of the year activities is to partner up and mark your height on the door. I use a cloth tape measure for this activity and it does require a pre-lesson on how to use the measurement instrument. The first year that I did this activity, I just gave the children the tape measure and had them go at it. I quickly realized that the majority of the children did not know what to do when they had run out of tape measure but still had not completely measured their friend. I have a class set of 60 inch, cloth tape measures that the children use throughout the year. I find that the cloth tape measures are easier to manipulate, cheaper, and easier to store.  After a lessons in which we discuss “How many inches in a foot?” and “If a child measures 52 inches, how would we record that in feet and inches?”, we place our names, the date and our heights against the door. We do this activity 3 times a year and at the end of the year, each child figures out how much they’ve grown through the school year. In our end-of-the-year reflection, we include our physical growth as part of the child’s reflection, “This year, I have grown 3 ½ inches and have become a much more of a risk taker when approaching difficult math problems.”

It is also extremely important to allow them exploration of various types of measurement tools and educate them to which tool is best for which situation.  Measuring how long things are, how tall they are, or how far apart they might be are all examples of length measurements. I expose the children to all sorts of measurement units in which they can use to measure various objects. Centimeters, inches, feet, yards, miles, and kilometers are all the units we use to measure distance, height, and length.

We brainstorm items we’d like to measure and then categorize them according to the units of measurement we’d use.

units-of-measurement

I like to put this conversion chart up in the classroom for constant reference –

1 foot = 12 inches

1 yard = 3 feet = 36 inches

1 mile = 1,760 yards = 5,280 feet = 63,360 inches

Liquid measurement is another aspect of measurement that when I run across it, often need to look up a conversion chart to make sure that I am measuring correctly. I am not always certain that 2 pints equal a quart since I very rarely use these units of measurement.  Again, this is when a conversion chart comes in handy but we make our own “Gallon Man” with empty, recycled containers that the children bring in from home. We bring in one plastic gallon (milk), 4 quarts (milk or juice), 8 pints (ice cream, yogurt), and 16 cups (yogurt, sour cream). Preferably all plastic and clean. Before I put up a conversion chart, I essentially create a water table and see if the children can come up with the equivalents on their own. “How many quarts equal a gallon?”, “If there are 2 cups in a pint, how many cups in a quart?” After figuring out the conversions ourselves, we create “Gallon Man.” We actually create this by attaching the quarts to the gallon with holes and wires for the arms and legs and then 2 pints to each quart and finally, 2 cups to each pint. We should rename our creature “Gallon Robot” or “Conversion Robot.”

gallon-man

We hang up “Gallon Man” in our classroom for easy reference.

Teaching measurement or any concept for that manner, using hands-on activities, manipulatives, and real-life applications makes concepts more interesting, engaging, and fun for my students. I get a lot of my ideas from Pinterest and often, these “real life math” lessons take little time and don’t take away time from keeping pace with my mandatory math curriculum.

 

Cooking with Preschoolers

posted by Alison Balis Hirsch

Cooking with kids offers a wonderful array of learning opportunities for young children. It provides practice in language arts (vocabulary and “reading” a recipe), science (chemistry and exploring the senses), and developing social skills (cooperation and turn-taking). The kitchen also provides a range of math practice such as counting, measuring, and understanding order.

IMG_5879In my son’s pre-k class, the teachers and children cooked together almost weekly; the recipes coincided with their Letter Of The Week. So for B week they made banana bread and for O week they made omelets. The recipes were simple enough for the teachers and children (ages 4 to 5-years-old) to manage, each having a minimum number of ingredients. My son LOVED the rice pudding so much that I asked his teacher for the recipe. When I saw how simple it was, I suggested we collect ALL of her recipes and create a cookbook to share with other parents, whom I imagined were equally excited to cook with their kids at home. After all, these were recipes already vetted by our experienced and talented teacher.

IMG_5878

As a teacher in the classroom, we sought out parent/caregiver volunteers to assist with cooking projects. Having the child’s special grown-up allowed those participants to engage in the school life of their child, providing them with an opportunity to better know the other children and teachers and also observe their own child in the context of the classroom. It also allowed us to maintain good adult/child ratios while working with small groups of children (typically 4 – 5), in the kitchen. The children who cooked or prepared snack (sometimes it was simply designing bagel faces with cut fruit, vegetables and sprouts) usually delighted in their food and seemed proud to share their creations with their classmates. The learning continued through the service portion of the meal, since the child chefs, with the help of the grown-ups, needed to figure out how to divide what they made into equal portions for their friends. Something like bagel faces required counting and one-to-one correspondence; baking a quiche required cutting it into enough equal-sized pieces to serve everyone.

For recipes that were made frequently, we made recipe booklets that were much more readable for children. Play dough was something we made with children on a weekly basis and for that we created cards, bound by binder rings that had visual instructions and described quantities with pictures.january photos 070

Cooking presents children with plenty of opportunities to learn and is also a great way to teach principles of good nutrition and encourage an adventurous palate: in my experience kids are much more likely to try foods they’ve grown or prepared themselves.

 

April Showers Mean….

I’m sitting at my dining room table watching the rain pour down so hard that I can’t see out of my windows and although it is 4 o’clock in the afternoon, the sky is as black as the night.  My kids used to love to play in the warm rain and they especially loved to play in a sun shower.  We ran outside and splashed in the puddles and if it rained enough, we brought out the shampoo and washed our hair.  Inevitably, the rain would subside just enough that we could never get the soap out of our hair, but it was totally worth it.

These memories remind me that rain, just like the snow, is one of nature’s ways of providing us with an interesting topic to explore.  How can we create curriculum using the spring rains and support mathematical understandings in meaningful ways with young children?  I usually see preschoolers talk about the weather during circle time.  One of the children is a weather person. S/he walks over to the window and reports the day’s weather and sometimes s/he gets to mark the weather on a graph or the calendar.  Most often, it is an unremarkable part of the morning circle.

If we get a good amount of spring rains this month, how about shaking it up and creating an opportunity for a real exploration of rain?  If it is warm (and there isn’t any thunder and lightning) take the children outside to play in the rain. This will require some planning in terms of rain coats, rubber boots, and extra sets of clothes.  I guarantee the children will find new ways of exploring the same old space.

It might also be interesting to place different sized receptacles outside and near a window so the children can watch the rain accumulate.  You can create a graph so they can mark “how much” rain is in each container throughout the day.  This will also allow them to see that even though the rain is falling into the containers in the same way, the different sizes and shapes of the containers will make a difference about how high the water rises.

Next week, I will write about another exploration of rain that you can try this month.  Let us know what you think or if you have another idea about rain and math.

 

Inchworms

Inchworms

On Friday, Math at Home presented at the Opening Minds conference in Chicago.  We spoke about the Math at Home site and our upcoming Professional Development series, soon to be available through the Gateways to Opportunity ilearning system.(More to come about that exciting project over the next couple of weeks). At the end of our talk, we presented the attendees with buckets of counting worms and walked through some possible learning activities that could be done with them.  Although the worms we had were a bit different, they reminded me of this post from 2013.

Have you seen these?  These are called Inchworms and they are actually one inch long.  That means they are standard units of measure (because an inch is an inch is an inch) while looking like a non standard unit of measure.  When children use these to measure, they might say, “It is 3 inchworms long,” which also means that it is actually 3 inches long. This is an important step in children’s understandings of measurement, which can be reinforced by laying these inchworms out next to a ruler to show that they really are one inch each.

It might be fun to introduce the “Inchworm Song” as well.  If you don’t know/remember it, it goes like this.

“Inchworm, inchworm, measuring the marigolds
You and your arithmetic, you’ll probably go far.
Inchworm, inchworm, measuring the marigolds
Seems to me you’d stop and see how beautiful they are.

2 and 2 are 4, 4 and 4 are 8, 8 and 8 are 16, 16 and 16 are 32.”

How Tall is Your Pumpkin

A couple of years ago, when we explored apples and one of our Thursday Themes, we took a stab at figuring out the circumferences using string.  You can also use string to figure out how tall your pumpkins are.

Have children hold string next to their pumpkins from the top of their stems to the bottom of the body.  You can help cut the strings so they are about the right length of the height of the pumpkin.  Now, you have several strings of different lengths that can be laid next to one another to create a graph all on their own.

If you use yarn, each child could have a different color so they know which one is theirs.  Otherwise, you need to label the strings so the children can remember which one is theirs.  These varying lengths of string can be used for comparisons, for sequencing and for documenting the different heights of the pumpkins.

Another way to measure the height of the pumpkins is by using Unifix cubes.  The children can put the cubes together until they are the same height as their pumpkin.  You can then count the cubes to see which pumpkin is the tallest and which pumpkin is the shortest.

Hands as a Measuring Manipulative

Sometimes our very best tools are actually attached to our bodies.  I use my feet for all sorts of “opening and closing” needs, and my nose is the perfect choice when I need to turn the page of my ereader and my hands are inside of mittens.

Children’s hands are also a great tool for measuring.  Since each child’s hands vary in size, it is important to use the mathematical language “nonstandard unit of measure” so they all know that their answers will be different depending on the size of their hands.

What can their hands measure?  They can use their hands to measure length, by placing one hand down and then the other right next it, and continuing until they have spanned the length (or width) of whatever they are measuring.  This also works well if the children trace their hands and cut them out, so the cut-outs can be used as the measurement tool.

Hands are also a great tool to measure quantity.  Using the questions, “How many” or “How much” children can explore quantity in meaningful ways.  “How many Unifix cubes can you hold with one hand?” or “How much sand can you carry with one hand?” are realistic activities that can be explored with several children.  They will discover that bigger hands hold more and smaller hands hold less.  However, they will also find out that it takes fewer larger hands to measure a length and more littler hands to measure the same length.  Be prepared for how confusing this might be for them.

Try using hands as manipulatives and let us know  how it goes.

Cuisenaire® Rods Compare Length

Over the winter break from my teaching job, I spent quite a bit of time cleaning and organizing our Child Development laboratory.  We really dug deep, opening boxes that had never been opened and discovering materials that had never been used.  In our excavations, I found a brand new set of Cuisenaire® Rods, complete with a beautiful wooden storage box.  Glorious!

There are few manipulatives out there that are as interesting and beautiful as a wooden set of Cuisenaire® Rods.  Developed 75 years ago by Belgian teacher Georges Cuisenaire these “rods” come in beautiful colors in varying lengths.

Using Cuisenaire® Rods to compare length is as simple as putting shorter rods next to longer rods and seeing how your children observe those differences.  Although these manipulatives were designed for a very specific purpose (units of 1, 2, 3, etc.) I think  it is far more likely that children will explore the rods by laying them out, standing them up, and comparing them.

Most young children will be able to identify which rods are shorter and which are longer, especially when they are laid out next to each other.  It is far more difficult for children to compare several rods of differing lengths simultaneously.  Putting many of them in order from shortest to longest is really challenging because it asks children to think about 2 things at the same time; which rod is shorter than these – but longer than the others?

If you look carefully at the above photo, you can see that the units of 1 are white and the units of 2 are red, 3 are green and so on.  They provide a visual representation of number units, up to 10, or for today’s purposes shortest to longest.

 

Giant Chain Links

Since I had my babies long ago, people have developed really interesting things to hang on strollers. We were lucky to have working harnesses so our babies didn’t fall out onto the street.  Nowadays, folks hang all sorts of stimulating and educational things from the tops of their strollers to keep their children’s attention and to keep them entertained.

Links are one of those really versatile manipulatives that children will play with throughout their young lives.  As infants, they will use them as chew toys as well as to connect their other toys to something.  Later, children will connect them to make chains that are “long” or “longest” or to go across the room.

They can also be used to show numerical differences over distance. For example, stretching a 10-link chain next to a 5-link chain shows that it is twice as long.  You can also explain that the 5 link chain is half as short.

Links provide children with a nonstandard unit of measure. Here is a great lesson plan that uses links as way to measure common household areas.

Click here to see a video of a child using large connecting links in a whole other way.

 

The Common Core – Measurement & Data

The fourth area addressed in the Mathematics Core for kindergarten is “Measurement & Data”. I will unpack this one over this week and next since it is broader than the 2 previous areas.

Describe and compare measurable attributes.

  • CCSS.Math.Content.K.MD.A.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
  • CCSS.Math.Content.K.MD.A.2 Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

___________________________________________________

Children explore “attributes” throughout their young lives.  Attributes help children distinguish observable qualities of objects, people, ideas, etc. into smaller and more comprehensible groups.

Although this standard focuses solely on “measurable” attributes ( (i.e., length, height, and weight) I would broaden the exploration of attributes with younger children to include any observable qualities (i.e., color, shape). I believe that the value in providing opportunities for children to distinguish and define the attributes all around them will prepare them for the kindergarten skill of comparing measurable attributes.  Understanding that a stuffed duck is “yellow” and “soft” relies not only on appropriate vocabulary but the ability to apply attribute qualities to the item.  This is a beginning skill that will eventually lead to comparing attributes – the yellow, soft duck is bigger than the green, hard frog.