Teaching mathematics part IV - laying the foundation

In parts II and III, I discussed why mathematics needs a good teacher, and what that good teaching entails.

Now, let's look at the how.  And let me point out that, at this point, we are mostly concerned about how to teach the subject from the beginning, in the early years.  This is when good teaching is so critical and can make or break a child's attitude toward and understanding of math.  A young math student needs to have a solid foundation upon which to build if he or she is going to be successful in the later years.

Children should not be rushed through mathematics, but should be allowed to take it slowly - slowly enough for each individual child to fully grasp the topics.

"...In arithmetic, above all other studies of the common school course, it is of the utmost importance that one step shall be thoroughly understood before the next is attempted.  The first two years' training is of more importance than all the rest the child receives." (The Eclectic Manual of Methods, p.107) - emphasis mine

Charlotte Mason also understood the importance of laying a solid foundation in the early years.

"Carefully graduated teaching and daily mental effort on the child's part at this early stage may be the means of developing real mathematical power, and will certainly promote the habits of concentration and effort of mind." (Home Education, p.257)

"If the child does not get the ground under his feet at this stage, he works arithmetic ever after by rule of thumb." (Home Education, p.259) 

So where do we start?  How do we lay this so-important solid foundation?  What steps do we take to begin our children in the study of mathematics?

It all begins with numbers - learning numbers as whole quantities and learning all the different combinations that make up that quantity.  This is how the child's mental math ability is built up.  He is taught good number sense, mentally, which at the same time teaches him to think more deeply about mathematics.  A child who does not come away in the first year (or two, if needed) with good number sense has not really been trained to think on a deeper level and understand how math is unfolding.  And that will be detrimental to the rest of his math instruction.  If he doesn't learn to think early on, he'll struggle.  Period.

So, let's go through the method of teaching the first year and we'll see how Ray's Arithmetic mirrors Charlotte Mason's writings on teaching math.

Here we go!

1.  Teach through oral instruction, focusing on the concrete.

"Do not teach the figures in the first lessons, and do not allow the children to do any written work; but teach orally, illustrating every operation, at first, by means of various objects. -- The instruction should be entirely oral, and should deal altogether at first with concrete numbers." (Eclectic Manual of Methods, p.107-108) 

CM also advocates oral instruction and, as we've already discussed, beginning with the concrete.

"Give him short sums, in words rather than in figures..." (Home Education, p.261)

If we begin with written instruction, then we're really beginning with abstract ideas.  Having a child complete a math problem like 2 + 4, on a worksheet or on the board, is an abstract idea, which, as I discussed in an earlier post, is much more difficult for a child to grasp than asking the child how many are 2 beans and 4 beans.  The latter is a concrete idea and is much easier to understand.

2.  Teach through the use of objects to manipulate.

"Begin the teaching of arithmetic, then, with objects, -- blocks, balls, marbles,...etc." (Eclectic Manual of Methods, p.108)

 CM also recommended using objects in the beginning.

"A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to 'do sums' on his slate." (Home Education, p.256)

This not only allows the child to understand the concrete ideas first, which will eventually pave the way to understanding the abstract, but will get the student involved and, in turn, get him to grasp the rationale of numbers and how they work himself, (rather than just being spoon-fed by the teacher).

3.  Teach numbering, first with objects, then without.

"The first step is to teach numbering; that is, so to train the child that he can instantly give the number of any group of objects not exceeding ten, at sight, and without counting...Do not allow a child to count by ones to find how many objects there are in a group, but teach him to recognize the group as a whole. -- Teach what three means by repeatedly combining two and one, and one and two, into groups of three apples, three blocks,...etc,...teach four in the same manner...As soon as the class is sufficiently advanced, have the children do the combining and separating of objects for themselves." (Eclectic Manual of Methods, p.108-109)

This teaching of numbering is so important - it is the first step toward training a child to think rapidly and accurately.  It is strengthening a child's mental math ability from the get-go.  The child sees all the different combinations that make up a particular number, which prepares him for addition and subtraction.  He also learns to recognize each quantity as a whole rather than teaching him to rely on counting.  A child who learns to rely on counting is not learning to think mentally.  There is also a good chance that the child will carry this habit of counting over to adding numbers - he'll rely on counting to add instead of his mental ability.

This is very similar to CM's method.

"He may arrange an addition table with his beans, thus --

OO   O          = 3 beans

OO   OO       = 4 "

OO   OOO    = 5 "

and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2+7=9, etc." (Home Education, p.256) - emphasis mine

The place where the two methods differ is when to have the students learn to recognize the combinations of imaginary objects and eventually abstract numbers.

"From combining and separating objects they can see, lead them to combine and separate groups of objects that they cannot see, but can readily imagine, such as animals, houses, trees, tools, toys, or any objects with which they are familiar." (Eclectic Manual of Methods, p.110-111)

So the Manual of Methods recommends introducing imaginary objects after mastering the combinations of objects before their eyes; then abstract numbers after mastering imaginary objects.

"When you are satisfied with the results of the work thus far, take the next step by simply dropping the names of the objects, and teach the abstract digital numbers orally in the same order that you taught the concrete." (Eclectic Manual of Methods, p.111)

It seems that CM, however, recommends teaching with imaginary and abstract along with using objects that the children can see.  Continuing with the above CM quote from p.256 in Home Education:

"Thus with 3, 4, 5, - each of the digits:  as he learns each line of his addition table he is exercised upon imaginary objects, '4 apples and 9 apples,' '4 nuts and 6 nuts' etc.; and lastly, with abstract numbers - 6+5, 6+8." 

4.  Teach addition and subtraction together.

"The one is the reverse of the other, and when taught together they help the child to understand each process more readily than if they were taught separately." (Eclectic Manual of Methods, p.110)

"A subtraction table is worked out simultaneously with the addition table.  As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, instead of adding, until he is able to answer quite readily, 2 from 7? 2 from 5?  After working out each line of addition or subtraction, he may put it on his slate with the proper signs, that is, if he has learned to make figures." (Charlotte Mason, Home Education, p.256-257) 

5.  Teach the written characters last.

"...when the children are thoroughly proficient in the preceding steps, teach them the written characters that stand for the numbers which they have learned to use orally." (Eclectic Manual of Methods, p.111)

It seems that CM would agree with this as well - mastering the material mentally through oral teaching before any written work.

 "...the child should be able to work with these freely {objects}, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to 'do sums' on his slate." (Home Education, p.256)

Note, however, that this does not mean to teach the children to write the numbers themselves.  It's just teaching them to recognize the written numbers.

"It is granted that much greater apparent advance can be made at this time, and that children can be taught the names of numbers as high as a hundred or more, and to write the figures representing them; but the learning of names and the making of figures do not of themselves imply the gaining or developing of ideas, and classes forced too rapidly over the preliminary ground without thoroughly understanding each step as they advance, will sooner or later show the bad effect of this method of teaching." (Eclectic Manual of Methods, p.113-114)

All that is required in the first year of mathematics is mastering groups of numbers, and addition and subtraction of numbers, through 10.

What this does is present the concrete ideas of mathematics which the child can understand, as opposed to presenting mathematics abstractly, which he cannot.  Instead of beginning with the abstract idea of 2 + 1 = 3, for example, we begin with the child combining and separating 2 beans and 1 bean.  And in allowing the child time to manipulate and work freely with these concrete ideas, he develops an understanding of how mathematics works - its logical nature and the truth and beauty and exactness of it.  Using this method to teach subsequent topics - addition and subtraction with larger numbers, place value, multiplication, division, fractions, weights and measures, etc. - will hopefully develop in the child an appreciation and respect for the subject as an immutable law of the universe, worthy to be studied deeper.

~~~

So, we can see how Ray's Arithmetic so far seems to be very conducive to the way CM taught mathematics.  I think that as long as these general steps are taken early on, allowing the child enough time to fully understand each step, we'll see our children begin to flourish in the subject.

One last question!  How will we know that our children are progressing the way they should be?

"Accuracy and rapidity are the important aims, and the children should be drilled until they can give the answers to all the possible combinations and separations instantly, and apparently without stopping to think." (Eclectic Manual of Methods, p.111)

 "...excite him in the enthusiasm which produces concentrated attention and rapid work.  Let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring." (Charlotte Mason, Home Education, p.261)

Accuracy and clear, rapid thinking from the child is how we'll know that they're on the right track, and that they're building that solid foundation so important to future success in mathematics.

Other posts in this series:
Part I - why study math?
Part II - good teaching
Part III - good teaching continued
Part IV - laying the foundation <---you are here
Materials, manipulatives, and activities

Bullfrogs, bats, and snakes! {Nature Study}

Here are some pictures from the last couple of weeks.  There are some wonderful places near where we live to explore nature.  There's our neighborhood, of course.  Or our backyard.  The other day we were in the backyard and a red-headed woodpecker came flying down, pecked around in the grass, found a worm, and few off.  Ten feet in front of us!  It was so cool!

There's a park about 15 minutes away that has a good sized pond and a little creek.  We go there every so often (I've posted pictures from that park on this blog before) and the following pictures were taken there.

Eating lunch.

We saw a pretty blue jay flying around - such beautiful detail and design on him.  We never see them at our house.  Please excuse the blurry photo!

We watched the turtles swim around and fed them some bread.

I had checked out the take-along guide, Frogs, Toads & Turtles from the library and we tried to identify the turtles.  I think this one below is a slider, according to this website.  It wasn't in the book, but it was still fun to look and try to identify it.

Wild strawberries.

Trying to feed the ducks.

Izzy spotted a big ole' bullfrog.  The bullfrog was in the above-mentioned take-along guide and the girls were fascinated by the fact that they sometimes eat snakes.

See him in the water?

I tried to get a picture of his webbed feet.

These next few pictures were at one of Izzy's tball practices down by the river.  When we got there the other night, there was cotton stuff flying through the air and all over the ground. 

It was from a cottonwood tree right next to the field.

See the cotton buds?

Here are some of the cotton buds.  I don't think I've ever seen/noticed a cottonwood tree before.

Addy thought it was pretty cool.  She looks so much older in this picture!

The next few pictures were taken in our yard.  We watched a bunch of ants carry around a dead bug.  It was quite fascinating.

Jared was weed eating and spotted a tiny frog on our air conditioning unit.  He was probably half an inch long.  We looked him up in that trail guide and determined that he's a spring peeper.

Last Friday we went to a conservation area about 20 minutes away.  This place is so neat and I really hope we visit it often.  They have lots of trails, a pond, and a large creek, plus they do some cool things for kids in the summer - they teach them archery, to fish, and every Wednesday they do little nature "classes."  The day we visited we went on one of the trails.  It was about a mile and took us like an hour and, even though we went before lunch, it was SO HOT.  Thankfully we were in the shade most of the time, but the stretch back to the pavilion where we ate our lunch was in direct sunlight.  And man, the sun was brutal that day.  And Addy had been cranky for quite awhile (note to self:  bring a snack and don't leave it in the car!)  So I gave her a piggy-back ride.  Nothing like giving your 4 year old a piggy back ride in the summer heat, with no shade, when you're 27 weeks pregnant.

Fun times!  Definitely memorable :)

Before we went on this trail, we got to see a couple of really neat things at the visitor's center - of which I did not take any pictures, ugh!  First, one of the workers showed the girls some bat scat scattered around, then proceeded to take down a clock that was hanging outside and, lo and behold, there was the bat!  He had made his home behind that clock, ha.  We watched him just hanging there, and he eventually flew off.  Pretty crazy.  Bats are really hairy, by the way.

Second, there were barn swallow nests all over the place and we saw two with baby birds in them.  We also got to watch a mama swallow feed her babies.  So cute!

On to the trail!

Lots of mushrooms next to a big log.

Snail shells are so cool with their spiral.

Excuse the blurriness, but here's what we guessed was some kind of cocoon.

We saw lots of lizards.

And a snake in the water.  The girls freaked out.  But then when we got back to the main building, they were fascinated with the snakes in the little aquariums.  One of the workers got a black rat snake out and let the girls touch it.  He also let us take home a little snake identification guide and the girls fought over that thing for like 3 days.

We also saw some very large schools of fish.

At one point, there was a butterfly that kept landing on Izzy's hat over and over again.

Nature is so full and the complete opposite of boring.  There is always something to see, if we're willing to look.  I love this quote by Charlotte Mason:  

"We were all meant to be naturalists, each in his degree, and it is inexcusable to live in a world so full of the marvels of plant and animal life and to care for none of these things."  (Home Education, pg. 61)

I understand and value the importance of nature - it is full of beauty, of diversity.  It brings such a sense of awe at God's creation, which I think is most important for children.  And it's up to us, as parents, to introduce and encourage our children to become aware of what's around us and to excite that sense of awe.  We have so much influence on our little ones.

"Some children are born naturalists, with a bent inherited, perhaps, from an unknown ancestor; but every child has a natural interest in the living things about him which it is the business of his parents to encourage; for, but few children are equal to holding their own in the face of public opinion; and if they see that the things which interest them are indifferent or disgusting to you, their pleasure in them vanishes, and that chapter in the book of Nature is closed to them...Audubon, the American ornithologist, is another instance of the effect of this kind of early training.  'When I had hardly learned to walk,' he says, 'and to articulate those first words always so endearing to parents, the productions of Nature that lay spread all around were constantly pointed out to me...My fauther generally accompanied my steps, procured birds and flowers for me, and pointed out the elegant movements of the former, the beauty and softness of their plumage, the manifestations of their pleasure, or their sense of danger, and the always perfect forms and splendid attire of the latter.  He would speak of the departure and return of the birds with the season, describe their haunts, and, more wonderful than all, their change of livery, thus exciting me to study them, and to raise my mind towards their great Creator.'"  (Home Education, pg. 58-59)

I let myself worry some because Izzy (6), with whom I'm about to begin first grade, has always preferred to be playing and running around outside than slowing down and paying attention to her surroundings.  But I've decided that I can't worry; I just need to do.  It's just like teaching a child to be polite or to read.  Of course there will be some resistance, but we know what's best and what's important for our children so we keep at it.  We must be consistent - develop the habits.  And we trust - trust that God will see our efforts and produce in our children the fruits of our labors.

And I can already see some fruits.  I can tell a difference between now and, say, six months ago.  By being intentional about putting our kids in nature's path, Izzy and Addy (4) are both more inclined to spot and show us random things they found, and be excited about it.

We are laying a foundation upon which to build.  And in the coming weeks, when we officially begin first grade, we'll go on to the next level and utilize a nature journal - which is a whole other form of intimidation for me!  But that's another conversation :)

Have you been enjoying the outdoors lately?  Share with me!

Teaching Mathematics part III - good teaching continued

Last time I discussed two steps that math teachers should take in helping students pursue math as a delightful subject to study.  Let's talk about two more important things teachers should do.

3.  Demonstrate the hows and whys.

This is very much related to what I mentioned in my previous post about introducing the concrete before the abstract.  In doing so, these concrete ideas must be demonstrated to the children.

“The next point is to demonstrate everything demonstrable. The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4x7=28 may be proved."  (Charlotte Mason, Home Education, p.255-256) - emphasis mine

I remember as a teacher, teaching that "mile wide and inch deep" curriculum in public schools, really only having enough time to just tell my students what to do in order to solve a problem because, like I said in my previous post, we had to cover a certain amount of material before the end of the year (really, before the big test).  But this method only allows for a "monkey see, monkey do" kind of learning, with the students just memorizing steps, not gaining deep understanding.  And this results in children quickly forgetting the material.

I needed time to give my students the opportunity to go behind the scenes, so to say, and discover the how and why of the material.  And sometimes - very rarely - we would have a little time to do that.  But, honestly, it would not go over well.  My students (remember they were 8th and 9th graders) either 1) did not care about the how and why and just preferred me to tell them what to do, or 2) got lost - the demonstrations and proofs of the mathematical processes would completely go over their heads.

So that told me that they were not used to this - to getting underneath the surface and grasping the rationale of the material.  And this not only results in a lack of understanding the material, but a lack of understanding math for what it is.

For example, if children could see how multiplication works it will provide for a much greater understanding than just memorizing the multiplication table will ever do.  And a firm understanding of the early concepts and how they work will have a domino effect - it will make it easier for the child to comprehend more and more mathematical processes because he is being trained to think, not memorize.

"In introducing these new processes of multiplication and division, the principle of developing ideas before words should control, and the knowledge already gained should be used as a stepping-stone to the acquisition of the new ideas.  For example, the child already knows that 2+2+2=6; also, that 3+3=6.  With this knowledge as a basis, it is very easy to show him that three times 2 are 6, and that two times 3 are 6.  Thus, the child's knowledge of addition is used in teaching him multiplication.  This seems much more rational that at once plunging into the multiplication table, and, by dint of incessant repetition, memorizing the combinations of a host of factors and products.  By this first method, the child learns how the product is formed, and why 3x2 or 2x3=6.  By the second method, he simply remembers - if he can remember - the formula."  (Eclectic Manual of Methods, pg.124) - emphasis mine

Even though these two quotes are about multiplication, the idea still applies to any mathematical topic.  Taking the time to demonstrate and allow the students to discover the hows and whys of these mathematical processes is so important in deepening their understanding and developing an appreciation for the logical nature of mathematics.

4.  Focus on real, contextual problems.

Mathematical drill is important.  Students need to practice being accurate and quick in their calculations.  However, too much focus on drill will not give us the results we want to see in our students.  Instead of encouraging deep understanding and a sharpening of a child's reasoning abilities, it only encourages rote memorization.

"Multiplication does not produce the 'right answer,' so the boy tries division; that again fails, but subtraction may get him out of the bog.  There is no must be to him; he does not see that one process, and one process only, can give the required result.  Now, a child who does not know what rule to apply to a simple problem within his grasp has been ill taught from the first, although he may produce slatefuls of quite right sums in multiplication or long division."  (Charlotte Mason, Home Education, p.254)

So how do we really train a child's ability to reason and problem solve, and encourage deeper understanding?

“How is this insight, this exercise of the reasoning powers, to be secured? Engage the child upon little problems within his comprehension from the first, rather than upon set sums. The young governess delights to set a noble 'long division sum,'––, 953,783,465/873––which shall fill the child's slate, and keep him occupied for a good half-hour; and when it is finished, and the child is finished too, done up with the unprofitable labour, the sum is not right after all: the last two figures in the quotient are wrong, and the remainder is false. But he cannot do it again––he must not be discouraged by being told it is wrong; so, 'nearly right' is the verdict, a judgment inadmissible in arithmetic. Instead of this laborious task, which gives no scope for mental effort, and in which he goes to sea at last from sheer want of attention, say to him–– 

'Mr. Jones sent six hundred and seven, and Mr. Stevens eight hundred and nineteen, apples to be divided amongst the twenty-seven boys at school on Monday. How many apples apiece did they get?'

Here he must ask himself certain questions. 'How many apples altogether? How shall I find out? Then I must divide the apples into twenty-seven heaps to find out each boy's share.' That is to say, the child perceives what rules he must apply to get the required information. He is interested; the work goes on briskly; the sum is done in no time, and is probably right, because the attention of the child is concentrated on his work. Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.” (Charlotte Mason, Home Education, p.254-255) - emphases mine

In addition to being able to add and subtract, for example, children need to be able to apply these processes in order to solve problems.  This not only strengthens a child's reasoning skills and builds up his ability to do mental math (which is so important in training the child to think), but also keeps him interested in the work.

“That he should do sums is of comparatively small importance; but the use of those functions which 'summing' calls into play is a great part of education.” (Charlotte Mason, Home Education, p.254)

As usual, let me reminisce!  When I would assign problems to my students, which ones do you think 90% of them would skip?  That's right...the real world, contextual, word problems - the problems in which they had to think and figure out how to apply what they knew.  It was disheartening.  And it wasn't just my students - this was the norm, at every school and for every math teacher I ever worked with.  The kids claimed they had no idea how to go about solving problems like these.  They wouldn't try.  Many of them wouldn't even read the problems (I'm sure there was some laziness at play here as well).  It was apparent that this was not something they were used to.  They were used to being drilled, not to think.

"Consider how mathematics was introduced to children a few years ago.  To pupils of Junior School age only arithmetical ideas were presented.  Algebraic and geometrical ideas were thought to be beyond the understanding of young children.  Great emphases was laid on mechanical work and learning by rote." *

Let me stop here and say that this is what I'm talking about.  Our children are not given the time to think and discover the ideas of mathematics, and so we're left with teaching rote memorization.

Continuing on...

"...What was the result of this preoccupation with mechanical 'sums'?  It was true that a few pupils attained an almost machine-like accuracy and came to enjoy the subject in which they could achieve such perfection.  However, the majority managed only partially to understand the elementary mathematical concepts.  There were some, indeed, who gained no real knowledge of the subject; who remained completely mystified by the language of number and size.  The unfortunate pupils in this latter category usually developed an early antipathy toward mathematics.  Everyone who has been involved in the teaching of 'Secondary Stage' maths will have encountered the child, otherwise able, who has no understanding of Number." *

Wow.  This whole paragraph - all of it - describes my experience teaching math in the public school system to the T.  The kids had no knowledge or respect for mathematics.  They saw math as confusing.  They disliked the subject.  They didn't know how to think.

"...The basic ideas of mathematics must be understood if we are to be able to follow modern science, but undue emphasis on 'mechanical drill' will not achieve this end." * - emphasis mine

There must be a balance of drill and solving real, contextual problems - aka word problems; problems in which the child must apply what he has learned.

(*Quotes taken from the PNEU article Knowledge of the Universe by G.L. Davies)

~ ~ ~ ~ ~ ~

So to summarize, the successful teaching of mathematics requires, above all else, a teacher who understands how to teach mathematics.  And this good teacher will:

• Give the individual student plenty of time to develop the ideas of mathematics.
• Focus on the concrete before the abstract.
• Get under the surface and allow the student to understand the hows and whys of mathematical processes.
• Include plenty of word problems for the students to solve to encourage deeper thinking.  

These are the things to not only think about and remember as we teach our children math, but also as we search for an appropriate math curriculum.  

And hopefully, as a result, the child will be trained to think, will gain a solid understanding of mathematics, and will develop an appreciation for it as a delightful subject worthy to be studied.

Next time, I'll outline the first year of mathematics instruction.  And you may be surprised to see how much Ray's Arithmetic mirrors CM's thoughts on math.

Other posts in this series:
Part I - why study math?
Part II - good teaching
Part III - good teaching continued <---- you are here
Part IV - laying the foundation

Healthy chocolate no bake cookies

I do not remember where I found this recipe, but these little cookies are awesome, and of course the best part is that they're healthy!  They have no sugar, butter, oil, or flour (even though I have nothing against flour, especially whole wheat flour.  I do lots and lots of baking with flour.).  Honey or maple syrup is used instead of the sugar; we prefer raw local honey or pure maple syrup.  I usually use honey because pure maple syrup is so expensive.

These cookies are quick and easy to throw together, but be sure to make them ahead of time because they need to set up for about 2 hours in the freezer (of course you could just eat the batter with a spoon if you wanted!).  They're rich and chocolatey so if that's your thing (it's definitely mine!) I bet you'll like these.

printable recipe

Healthy Chocolate No Bake Cookies

makes about 24 cookies

Ingredients:

¼ cup cocoa powder

¼ cup honey (or maple syrup)

2 tsp vanilla                          

2 bananas

½ cup peanut butter

2 cups oats

Combine the cocoa powder, honey or maple syrup, and vanilla in a bowl.  Whisk until combined (it’ll take a minute or so and should look like chocolate syrup – see pic below).  Add the bananas and mash into the mixture.  Then add the peanut butter and mix well.  Add the oats and mix.  (You may have to add more oats – you don’t want the mixture to be too dry because the cookies won’t stay together, but you don’t want it to be too wet either because it’ll just be a gooey mess – see pic below.)

Spoon cookies onto wax paper, as big as you like.  Place in the freezer until set, about 2 hours.  Transfer the cookies to a container with a lid and keep them stored in the freezer.

Now for some pics!

This is what the cocoa powder/honey/vanilla mixture should look like when it's thoroughly combined - this would probably be a good topping for ice cream!  *Note - I halved this recipe this time around, for what reason, I have no idea.  They did not last long.

Izzy was my helper.  She likes to help me cook and bake, especially when chocolate is involved.

Adding the oats after mixing up the banana and peanut butter.

Here's the consistency we ended up with.  Like I said above, I halved the recipe, only needing 1 cup of oats, but we ended up adding just a tad more - maybe 2 tbsp or so.  It's still gooey, but not dry.

And here's Izzy licking up as much chocolate as possible.

Goob.

We ended up with 13 cookies.  Izzy spooned them out, so some ended up on the small side.

More chocolate leftovers.

And more.

Pretty yummy!

We love love these cookies.

I hope you like them!

Water fun and how I take better pictures with a cheap(er) camera

It's been warming up around here, getting into the mid-80s, so we broke out the ole' wading pool and sprinkler (on separate days).  The girls, of course, had a big time, and I got to take lots and lots of pictures.

I do not have a fancy DSLR camera or whatever they are, so I've been really focusing on ways to create the best photos with what I've got.  My pictures are definitely not perfect or of professional quality, but by keeping some things in mind, I can manage some pretty decent ones.

Here are some things I've learned (in no particular order):

1.  Natural light is best.

My best pictures are taken outside.  I have a difficult time taking good pictures indoors, mainly because our house does not have a lot of natural light filtering in.  I have to rely mostly on artificial lights (lamps and what not) and it's just not the same.  With sunlight, however, I can snap action photos that are usually not blurry.  Indoors, not so much.

Love this pic - they're both jumping.

Her expressions crack me up!

2.  Avoid direct sunlight.

Even though getting outside and using natural light will produce higher quality photos, it's best to stay out of direct sunlight.  Instead, if at all possible, I try to take pictures when the kids are in a shady spot or when it's cloudy.

Notice the difference between the next two pictures.  In this one, the sun is shining directly on the girls' upper halves.  It's still a decent picture, but a little hard on the eyes where it's so bright and the shadows on the girls are a little unappealing. 

In this picture, however, the sun wasn't shining directly on Addy.  There are no unappealing shadows and no super bright spots.  It's pretty well-balanced.

This was a cloudy day.  The picture is a tad blurry and I cut off part of Izzy's head and arm, but as far as lighting goes, it's much better than the first picture.

3.  Get down.

Most of the time when I'm taking pictures of the girls, I squat down.  Getting down on their level just makes for a better picture.

For example, in this picture I squatted way down to take this shot.  I could have just stood there, with the camera pointing down, but the picture wouldn't have been as neat as this one - it makes it feel like you're right there in the action.

4.  Edit.

Of course, the most ideal picture-taking experience would be to center my camera on whatever object I'm taking a picture of, to make sure the lighting is perfect, and to avoid an unnecessary and/or unappealing background.  But most of the time, that amount of planning doesn't happen.  So, after I download photos to my computer, I usually upload them to Picasa, a free photo editing program that is downloaded to my computer.  With this program, I can crop, touch-up, play with lighting, etc.  Many so-so photos can turn into much better ones after doing a little editing.

For example, here are two different copies of the exact same picture, the second one being cropped.  By cropping, I was able to center Izzy in the picture and also cut out the blue part of the trampoline and the yellow water hose.  Although the legs of the trampoline are still in the picture, it's not as apparent as the blue padding was.  Also, it's very subtle, but I lightened up the second picture as well.

The difference in the next two is the tuning.  The first picture's lighting and coloring is a little dull.  In the second, I sharpened the shadowing and color temperature.

For this one, I zoomed in, cutting out the unattractive water hose in the background, and sharpened the color temp.

I usually just play around, but I try to be careful not to make the photos look too touched up.  I want them as natural looking as possible.

5.  Take lots of pictures!

One of the best ways to get a good picture is to take lots of them!  

My photos are definitely not perfect, but who cares?  They're mine and I enjoy them.  Happy picture-taking!!