Forming First Impressions: Kindergarten Math

The Explorer, 5, recently told me that his favorite subject is Math.  At this age he’s forming early impressions on the blank canvas that will become his Math Education.  Developing a strong base is important, but so is lighting a spark of interest.  Led partly by his fascination levels, I focus on three areas with my Kindergartener: math facts, new concepts, and problem solving.

1. Math facts

Mastery of the math facts will greatly reduce frustration for him down the road with things like muti-digit multiplication, long division, and many other areas.  It’s important that he learn the concept behind the idea.

I start him on skip-counting by introducing a hundred number chart.  After he’s learned to count to 100 (by ones), we learn the two’s, five’s and ten’s.  We then cover both odd and even numbers and what digits each type will end in.

We recently learned the three’s. For example:  I have him start at zero and count up three, and then cover the number with a rectangular counter stopping at 30 – for starters.  The idea is that he’ll begin to notice a pattern.  [He observed that the three’s have both odd and even numbers.]

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Hundred number chart, counting by 3’s starting at 0.

Then he removes the counter and writes down the numbers – beginning at zero then up to 30.  (If he gets frustrated writing the numbers down, I just have him read them while I write them for him.) What I like about this is that not only does he learn to count by three’s, but he also realizes that it starts at zero.  He reads them forward and backwards a few times.  Then we focus on memorizing.

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Recording the 3’s as he removes them

After he’s mastered the 3’s starting at zero, I then have him pick another starting number – say one.  We then count up by 3’s but notice that our numbers are now different depending on our starting point.

So far this year, we’ve covered the one’s (0 to 100); two’s (0 to 100); three’s (0 to 30); five’s (0 to 100); ten’s (0 to 100), the hundred’s (0 to 1000) and the odd numbers (1 to 21).  Next week we plan to start the four’s (0 to 40).  How many we’ll cover depends; I use his interest level to help guide me about how far to go.  It is not a race, though, and it’s far more important to me that he understands the idea.

2.  New Concepts

By this I refer mainly to the learning of the arithmetic operations:  addition, subtraction, multiplication, and division.  I often use a curriculum as my guide, and some of my favorites are Singapore Math, MEP Math, and Miquon Math.  However, I believe that any curriculum is a tool that should help and not hinder.  I often find myself filling in concepts that I think a curriculum should have spent more time on.  I use a Place Value Activity Kit to demonstrate all four operations.

We start by learning the plus and minus facts up to 10 (all the combinations), then we expand it up to 20.

Interestingly, we recently learned three different methods for subtraction. For example, consider:  15-8 = 7.

 Method 1: Start at 15 on our number line and count back 8 steps.  We stop at 7.

 Method 2: Think of 15 as a number bond of 10 and 5.  I ask my son if we can take 8 away from 5. (We’ve not learned negative numbers yet.)  He replies that we cannot, but we can take 8 away from 10.  So we calculate (10-8) and we still have our 5 from before so our answer is 2 + 5 = 7.

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Method 2 demonstrated with place value kit

 Method 3: Start at 15.  Think about where our nearest ten is when subtracting. Answer:10.  Think about how many units we would take away from 15 to get to our nearest 10.  Answer: 5.  Then subtract 5 from 15 and next subtract 3 more. (15-5-3=7)

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Method 3 explained.

Having learned three different methods hopefully causes him to analyze, compare, and develop a greater number sense regarding subtraction.

So far we’ve worked though the concepts of addition and subtraction mainly, and also a good start in multiplication.

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Multiplication demonstrated with place value kit, and noting that ‘x’ means ‘groups of’.
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More multiplication showing that multiplying can be thought of as repeated addition.

Having already introduced multiplication (with skip counting),  I then use examples like the one above with a place value kit to go into more depth by showing that multiplication is repeated addition and that ‘x’ can be thought of as ‘groups of’.  For division, I only introduce it with the place value kit and show how division is related to multiplication.

[As an aside: If a student is confused by a concept, I recommend trying another curriculum to explain it a different way.  Use the curricula that best helps your child understand OR simply explain it in your own words how you understand it.]

3.  Problem Solving

How I interpret it, problems solving means that a student has enough mastery of a concept that he can apply it to solve a problem.

Thus far for this type of activity, my favorite curricula are Singapore Math Intensive Practice and Challenging Word Problems workbooks.  What I like best about these are the way they encourage a child to visualize the problems in order to better understand and solve them.  Also, questions are asked from many different angles so that the child has to really think about what is being asked.

Recently we worked on a problem where he had to think about how a larger number can be made out of two smaller numbers to solve puzzle-like problems related to subtraction.  We pulled out our Cuisenaire rods to help visualize the combinations that make 10: (1,9); (9,1); (2,8); (8,2); (3,7); (7,3); (6,4); (4,6); and (5,5) – to remind us of our math facts. (Hopefully he sees the pattern that addition is commutative). Manipulatives can be a great addition when solving a complicated problem.

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Looking for a pattern and using C-rods to visualize.

Of course, problem solving can also bring feelings of frustration. During these trials of deep thought, he grows in knowledge and develops math endurance.  I’ve found that giving praise when he’s solved a problem works wonders, and it’ll likely light a fire and make him want to learn more.

Overall, our plan is that we cover math fact practice and new concepts daily.  Problem solving (puzzle-type questions) may only happen two or three days per week.  We touch on all three areas every week.

It’s my intention that Math time be fun for him at this age.  Demonstrating that problem solving can be both rewarding and enjoyable [hopefully] paints a positive first impression.

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MEP Reception – A Thoughtful Introduction to Math

Are you looking for an introductory math curriculum with a strong focus on thinking for your 4 to 6 year old?  Consider MEP Math – Reception, a FREE printable curriculum based on a Hungarian Kindergarten Math program.  MEP has been developed by the Center for Innovation in Mathematics Instruction “to enhance the mathematical progress of students in primary schools.” See this page for more background information.

Last month my four year old completed the 60th and final lesson, and here are some things I like about MEP Reception:

Counting:  Students are introduced to numbers up to ten, but the emphasis is not on the reading or writing of numerical digits.  Instead, for example, when the number six is discussed a student may write six dots or tally marks, clap six times,  knock on the table six times, or count out six objects (counters).  The writing of digits is postponed until Year 1.

Observation:  In an early lesson, a student is given colored sticks or toothpicks.  He is asked to look at a picture and study shapes made out of colored sticks and think about how many of what color toothpicks are needed to make the table, chair, tree, etc.  He then recreates the pictures on his own with care to make the same shape and use the same color sticks.

Mental Operations:  Students are asked to listen to a story, think of an answer but wait until called upon to say it.  For example, an earlier lesson says, “Two rabbits were playing in the clearing, then a rabbit joined them.  How many rabbits are there now altogether?” (Pause and wait and then ask for the answer.)  This is my one of my favorite parts of Reception, because it trains him to think of the answer and hold it in place before saying the answer.  Over time the difficulty level of the these questions increases, and towards the end of  Reception comes a question like this:  “A monkey had ten bananas.  He ate two bananas first, then three bananas, and then one.  How many bananas remained?”  This all takes place mentally without writing it down, which I see as a plus.  (It is okay if the child wants to draw a picture of what is happening after thinking about it for awhile to help him solve it, but this program really gets him to think about things rather than simply being told what to do and how to solve it immediately.)  Another question asks the following: “Ann, Ben, and Celia collected shells.  Each of them collected three shells.  How many shells had they collected all together?”  In the last example, multiplication is being introduced.

Games: Reception uses games throughout to make learning more engaging.  When played well, the student forgets its math time and focuses on winning the game (and meanwhile is learning addition or subtraction depending on the game.)  For example, in one game the student is to put a dog (or some counter) on a colored square to begin.  She throws the dice and moves as many steps as the number of dots she rolls on the dice.  Later the game is played again but modified so that now the player moves one more space than the number on the dots rolled on the dice.  To win, you must throw the exact number needed to get to the bone when you get close enough or you will lose your turn.  The student will be so focused on winning that she will forget that she is learning the +1 addition facts!

In another game, a student throws the dice and if a one, three, or five (odd number) is rolled, he may move that many place to the right or left.  If a two, four, or six is thrown, he may move that many places up or down.  He must also say aloud the number of spaces and the direction.  The first player to reach his colored square on the opposite side wins.  This game mentally separates the odd and even numbers on a dice without formally discussing ‘odd’ and ‘even’ numbers.  It is guiding the student, I think, to make a distinction between those two groups of numbers for a reason to be revealed at a later time.

Ordinal Numbers: Students are asked when looking at a line of people about who is the first, second, third, fourth, and then it jumps to the question of who is the fourth from the end of the row, first from the end of the row?, etc.  It goes from simple to more complex questions gradually.

Sets: Set theory is gently introduced.  For example, one lesson asks the student to notice a certain set of mugs in a group.  It asks how many are colored, and then how many are not colored.  Also the mathematical symbols often used with set theory, intersection and union, are creatively placed on sheets to be traced – building a student up to these ideas by using both thoughts and symbols.

Geometry: Geometric shapes introduced are mainly circles, quadrilaterals (including squares and rectangles), triangles, and few others types to show the distinction from the former.  Students are expected to analyze the shapes that make up a steam engine in one lesson.  Then they are asked to color them the same color they are on the train.  Next they’re asked how many circles, quadrilaterals, triangles, and total number of shapes to build it.

* Decomposing:  A student is asked to think of how many different ways can we divide ten counters between two people.  All the combinations are demonstrated: 10,0; 9,1; 8,2; 7,3; 6,4; 5,5; 4,6; 3,7; 2,8; 1,9; and 0,10.  Students will possibly see the pattern of what is happening after arranging the first few combinations.  Also, if the student notices that combinations can be reversed (3,7 and 7,3) , this would be an opportunity to demonstrate the Commutative Property of Addition – that is – the order of addends doesn’t change the sum.

Sequences: Sequences appear towards the end of Reception.  Any counters/shapes/etc. can be used to make a pattern that repeats, and the student is asked which shape or color of counter would come next.  (I used Teddy bear counters of different colors to make patterns for my son.)  Later, numbers are said in patterns, and the student is asked to repeat (say) the number pattern. Singing it in a chorus may help him remember it.

* Making Figures: This was my son’s favorite part.  Paper is folded and shapes are drawn on the fold to be cut out by the student.  The last lesson contains many of these, and it promotes spatial skills and the idea of symmetry.

What I view as Reception’s greatest strengths are the mental development of operations and the focus on thinking throughout the program before the writing of words and numbers is ever introduced.  Ideas are developed in small, purposeful steps with the depth expanded leading to more complicated concepts over time.  Things are happening for a reason even if the child may not see where he is being led.  I was impressed with the thinking skills it encouraged in my son, especially with the development of mental operations.  We had fun playing the games, and he was so determined to win that he forgot he was learning math.

It is teacher intensive, and I do recommend reading the lesson ahead of time for preparation to understand where the authors are going with the concepts and to locate any needed manipulatives, counters, etc.

MEP Reception is a gentle and thoughtful introduction to Mathematics for little ones.  If you would like to learn more about it, I recommend starting here and spending a little time reading through the introductory material.  You may also read an earlier review I wrote, MEP Math – For a Change (An Early Review).

Week 9 – First Quarter Reflection (2012-2013)

As the first quarter of our school year comes to an end, it’s time to reflect on what we’ve covered and focus on where we are headed.  Looking back over the previous nine weeks with both its trials and joys, I am overall pleased with what we have learned, and it has been a blessing to be a part of it.

DSC_0042The Artist, 12, is off to a promising start this year.  In CLAA Latin IIB, she is continuing to progress through both Latin to English and English to Latin translations of Ego Minus Saepe.  In the next week she will be intensifying math contest preparation – dedicating both Thursday and Friday to prepare for her first exam which will take place in under two weeks.  Also, beginning next week I’ve decided to replace AoPS Number Theory with Counting and Probability, since the content of the former appears to be mostly review for her.  Having completed both The Hobbit and Tom Sawyer in Reading earlier this quarter, this week she is continuing through Huckleberry Finn.  She has completed the first nine weeks of Writing With Skill, and what she most likes are the helpful tips it provides for specific topics – like ‘description of a place’ – for example, when it suggests that “space and distance words and phrases can help you create a clear picture of a place”.  As the one who grades her WWS, I find the Instructor Text’s weekly rubrics helpful for guidance on specific grading criteria.  Rod and Staff English 7 is a dry, but solid coverage of Grammar for her.  I am impressed with her tenacity; when it comes to approaching difficult subjects, she plows through them.

“Unless you try to do something beyond what you have already mastered, you will never grow.” ~ Ralph Waldo Emerson

Ever the Investigator, my 10 year old is usually the one in the room to notice something everyone else has missed; and when it comes to school he is excited to point out any nonsense, but will also spy seemingly hidden patterns and relationships. No changes are planned for his MEP5/Singapore Math 5 combination as I think both of these curricula are effective at guiding him to see patterns and relationships with numbers.  Books he has read this quarter include Half Magic, Gregor the Overlander, A Wrinkle In Time, Aesop’s Fables, and he is currently reading Mrs Frisby and the Rats of Nimh.  For Writing he is working through Writing With Ease 3 and he likes the summaries, however we bend the rules a bit on the dictation and slow down on these ‘fun’ sentences.  He tolerates Rod & Staff English 5, and I know he is learning consistent Grammar.  In Spell to Write and Read he has learned all the phonograms and spelling rules up front – an example of my own adaptation of the program, and now we will focus on the WISE Guide T-lists words and above.

The Investigator working on Mapping the World with Art.

One thing I can say about The Architect, 7, in addition to learning many things these past nine weeks, is that break time is building time.  He quickly flies through his daily lessons so that he can get to the fun of building fascinating Lego structures.  His favorite subject is Reading, and he has enjoyed reading various Magic Tree House books.  In Math he is currently working on about Lesson 70 in MEP Year 2 along with Singapore Math Intensive Practice 2A, one of my favorites.  In Spell to Write and Read, he has learned most of the rules (all except the Dismiss L and IE/EI rules) and knows all the phonograms including the advanced ones.  He will resume the WISE Guide spelling words next week starting at list J.  He is fond of Rod and Staff English 2 and thinks it’s great to have his own Grammar notebook.  No changes are planned for his schedule – although as he finishes up some of his books early for the year, he will add in Building Thinking Skills Level 1 in January.

The Architect studying shapes in Math.

No longer a toddler, The Explorer, 4, has really developed into quite the learner this quarter.  How he has grown in knowledge from a year ago!  Last week he completed MEP Reception (all 60 Lessons) which was a thoughtful and engaging math experience for him, and he continues to work through Miquon Orange and the end of Singapore Math Essentials B.  He is also reading Bob Books and currently working through ETC 2.  For penmanship, he is learning Cursive First, and he will likely have learned all the cursive letters by December.  In Spell to Write and Read he knows a few rules and many of the basic phonograms including some of the multi-letter phonograms.  For a puzzling extra, he has also worked through most of the the first Lollipop Logic book.  In addition, he also loves break time – especially building Legos with his older siblings, playing with cars, or simply relaxing on the couch with a book.

The Explorer coloring in Math.

For all we have been working through K12 Human Odyssey Ancients Volulme 1, Mapping the World with Art, and a Nature Center Science class which meets once a month.  Our group subjects are an important time for us to come together and share ideas, and these topics have all been a hit!  We have covered through the end of Part1 of K12 HO, the first 7 Reading and Map Lessons of MtWwA  (up through the Iberian Peninsula), and our first Nature Center class on the topic of seeds.

What an opportunity I have had this quarter to observe these leaps in understanding my children have taken.  Being with them during the ups and the downs has helped me notice and appreciate what I have and has given me a chance to [hopefully] help them develop their gifts.

“Gratitude is not only the greatest of virtues, but the parent of all others.”  -Cicero

MEP Math – For a Change (An Early Review)

For elementary math I want to shake things up a bit this year.  So I have been experimenting with an online free British math program called  Mathematics Enhancement Programme. From the website, MEP is a resource from the Centre for Innovation in Mathematics Teaching. “The Centre for Innovation in Mathematics Teaching (CIMT) was established in 1986. The centre is a focus for research and curriculum development in Mathematics teaching and learning, with the aim of unifying and enhancing mathematical progress in schools and colleges.”  

For the past two weeks, I have been trying out Years 5, 2, and Reception with my soon to be fifth, second and preK grade children.   To get started, I first went out and bought some new printer cartridges and printed out the course pages for the first thirty lessons for each of the three levels.  For Reception color ink is necessary for the Copymasters, but black ink cartridges are all that is needed for the Lesson Plans, Practice Books and Copymasters for the other levels.  (Note: color ink is necessary if you print off the OHP Transparancy Collection.) I then printed off the Number, Shape, and Sign Cards, Shape Cards with Dots, and Number Lines and had them laminated.   Once all of the printing was done, I then read the lesson plans the night before and get out materials that will be needed.  I store the ‘not in use’ Lesson Plans in binders for each level so that they can possibly be reused at some point, and I keep the daily ‘weeks worth’ of Lesson Plans, Copymasters, and Practice Book worksheets in separate folders so that I can quickly get to them for each child.

My four-year-old has made it through the first twenty lessons of Reception.  I so far have liked the focus on thinking and the fact that the lessons have only a little writing.  The colorful Copymasters include conversations about a family we meet over and over throughout the advancing lessons and games that make the math more engaging for him.  I do substitute some of my own manipulatives from other math curriculum I own to adapt the games at times.   After two weeks of experimentation,  I have decided to continue on with MEP for him and finish the next forty lessons.

My 7 year old has also made it through the first twenty lessons of Year 2.  As a positive, MEP uses proper mathematical vocabulary and introduces other concepts (like Roman Numerals, less than or equal to, greater than or equal to, and combinatorics) much earlier than I have seen in other programs.  It is spiral, so the concepts get revisited often, but more thinking is needed from the student when concepts are introduced compared to other spiral programs I have used.  There are some problems where the student has to really think about and look for a pattern versus being told how to do it, and then being expected to just memorize what to do.

My 9 year old has completed the first ten lessons of Year 5. Each lesson at this level has a many concepts going on throughout, so I’ve found it best to only do one/at most two lessons per day with him.  I have found things come up in MEP Year 5 that he has not yet seen – for example, graphing intervals on a number line including endpoints.  For this reason, we are progressing at a slower pace when new concepts come up.

As with the other levels of MEP, I see the necessity for the student to think or figure out the pattern when an idea is initially introduced to be a positive.  My favorite parts of this program are the consistent use of correct mathematical vocabulary and set notation from the start, the way the program encourages thinking with the presentation of concepts, and the continual application of the concept over time.

With what I have seen so far of Year 5 (the first ten lessons 😉 ), I think the CWP and IP books in Singapore Primary Maths contain more challenging word problems.  For this reason, I plan to continue using them alongside MEP Year 5 as we continue moving forward with this program.

Overall I have been happy with what I have observed in MEP.  It is very teacher intensive, and I have found it helpful to be ready both mentally and with having any manipulatives set up for a smooth lesson.  There have only been a few small errors (possibly typos) so far in the Year 5 Lesson Plans and some of the Lesson Plan problems do not line up exactly to particular problems in the Copymasters and Practice Books, but I did not consider these to be major issues.  We also replace the money signs in MEP with $ signs, and I cover money in more depth with Singapore Math Standards as well.

For these reasons, I have decided that we will continue on with MEP in the fall combined with the more challenging parts of the Singapore Math Programs and the other adjustments mentioned.  I am sure I will have more to say about it as we get further along.

There is also an MEP Homeschoolers Yahoo group which I have found to be helpful in getting started with tips and files which answer many questions.