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# Category Archives: Math

## Montessori Monday–Working With Cut-Out Labeled Fraction Circles

I have to be honest and say that Dora has seriously started to rebel against using Montessori materials. Plus, she seems to be learning things in other ways much better, such as through imaginative play and listening to literature and music (for instance, I was quite surprised last week to learn that she could do basic addition, which we haven’t remotely begun to address in our “homeschooling”). Ironically, the child who really seems to thrive with the Montessori method is Gohan. Unfortunately, Gohan is in 8th grade and Montessori did not develop a curriculum for high school. At times, however, Montessori seems to the only way for him to learn a concept. Previously, I posted that I used a Montessori-inspired approach to teach Gohan how to subtract negative integers. That was in May and he hasn’t had any trouble with working with negative integers since! I only taught him the concept using the manipulatives once and it just clicked.

Last month, he began struggling with fraction word problems. He was doing review work of concepts that he mastered two years ago, or so I thought. Clearly he didn’t really understand the concepts, however, as he was unable to do the word problems. So, I finally broke down and bought some cut-out labeled fraction circles. I wanted the cut-out fraction circles, as opposed to the metal fraction circles, because a) they take up less space, and b) the metal fraction circles only go up to tenths, while the cut-out fraction circles go up to twentieths, and c) this method doesn’t work as well with knobbed pieces. Were I to do this again, I’d buy .pdf files and just print out the pieces. I taught Gohan how to do just one problem using the fraction circles and he hasn’t had any trouble with fractions since!

The problem was a pretty straight forward one. It was “What is 2/3 of 3/14?” So we laid out three 1/14 pieces. Just looking at the pieces made the answer pretty obvious, but we worked out the whole problem just to be sure. We separated 2 of the 3 pieces (or 2/3), which was 2/14. Of course, Gohan needed to reduce the problem and that he can do easily in his head, but we completed the final step using the fraction pieces, because I wanted him to really see how the paper worked answers and manipulative worked answers are the 100% the same. The 1/7 piece fit over the two 1/14 pieces perfectly. So we had our answer of 1/7! I really, really, wish I had understood how Montessori worked when Gohan was younger. He would have benefitted so much from using manipulatives to learn everything.

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Labels: Math, Montessori

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Montessori Monday–Geometric Cabinet, Teen Boards, and More Montessori Math

For the first time ever, Dora wanted to work with the geometric cabinet. The geometric cabinet is actually considered a sensorial material in Montessori education. It teaches visual discrimination, while also working on the pincer grasp, and introducing children to geometric terms. Dora has not wanted anything to do with the geometric cabinet, until this week. She breezed through the first drawer, but had some problems with the second drawer, as illustrated in the photo above (she doesn’t understand why I don’t want her just to force pieces into their spaces, my method seems so fussy and time consuming!). Even after struggling with the second drawer, she wanted to do more, so I brought out the third drawer, which was pretty challenging for her. We only worked with putting the pieces back into their spaces and using general terms (for instance, I used the term “triangle” for all of second drawer, instead of terms such as “isosceles triangle” or “obtuse triangle”).

It took a bit of encouragement on my part,

but Dora finally agreed to work with the teen boards. I used the beads in conjunction with the boards, as this was really an introduction lesson (I had presented the teen boards to her before, but she totally forgot everything, since she has not wanted to work with them in so long). She grasped the pattern of the teen boards, but I’m still not sure she understands the “one and ten makes eleven” concept (I don’t want to give the impression that I am worried about it or pushing her to learn it quicker).

She also worked with the wooden counters and cards. She knows most of her number symbols, but is confusing the pairs “2” & “5” and “6” & “9”, which seems pretty age appropriate. She is easily able to count, with one-to-one correspondence, up to ten.

She also worked with the spindle box some. The spindle box is easier than the wooden counters and cards, however, as two being next to one and so forth gives her some guidance, so she doesn’t confuse the symbols.

I also gave her the very first introductory presentation to the blue and red counting bars. She was much more interested in these than the long red rods. I could see that the gears were spinning in her head when I showed her these rods. Clearly, they presented a new concept to her, though I am not sure what it was (I didn’t show her that the one rod with the nine rod equals the ten rod or anything of that sort).

Finally, we completed grading all of color box 4. Honestly, this gave me a headache! I have always had good vision, but maybe my color discrepancy skills aren’t what they should be, because I had a really hard time sorting the browns and purples. I cannot imagine doing color box 3, which has seven grades of each color! Of course, I didn’t buy the highest quality color tablets, so maybe that was part of the problem… The color variances look more obvious in this picture than they did when we were working with them.

It was a good week overall in regards to Dora’s using the Montessori materials. It is all beginning to click with her and she is asking to use the materials on the weekends too! I really enjoy seeing her make the connections between one activity and another. I doubt I will ever give any thought to switching to a different method of teaching math until she is in middle school. I really wish I had used the Montessori method with Gohan. I greatly suspect that if he had been taught with Montessori materials from the beginning, it would have saved him a lot of frustration. As is, I am giving great consideration to buying the fraction circles for him. He has memorized how to work with fractions, but the fraction word problems in pre-algebra are tripping him up. He is not understanding when he is supposed to multiply or divide with fractions. For instance, if he got a problem that said, “Johnny took 2/3 of the apples and Mary took 3/4 __ of__ Johnny’s apples, what fraction of the apples does Mary have now?” he would not know if he should multiply or divide the fractions. When I was in pre-algebra, I was just taught that the word “of” means to multiply when working with fractions. So, I’d know that in order to solve the above problem, I’d multiply 2/3 x 3/4, just because of the word “of”. That is not exactly a true understanding of what is happening and I’d like for Gohan to have a better understanding of how fractions work.

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Labels: Math, Montessori, Summer, This and That

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Montessori Monday–Color Box 4 and Teen Bead Hanger

When I first started looking into using the Montessori method with Dora, she already knew most of her colors, so the first color box, which consist of just red, yellow, and blue, was definitely too remedial for her. I also suspected that the second color box, which consists of 11 pairs of basic color tiles, would be too easy for her and we wouldn’t get much use out of it. Yet, I felt that color box 3, which has seven shades of nine colors, which the child grades in order of darkest to lightest, would prove to be too difficult for her. Color box 4, on the other hand, only has four shades of eight colors, but also has a matching tile for each tile for a total of 64 tiles. I felt that given our limited budget, color box 4 was the best bang for our buck. A couple of months ago, I tried having Dora grade just one set of colors and that was way too hard for her. So this week, I presented her a tray with two shades of several colors. I went with the most extreme shades, darkest vs. lightest. She was able to do this tray fairly easily. In a few weeks, I will add a middle shade and see if we can’t slowly, but surely build up to grading the whole box. I have found that proper lighting is very important for this work. In some rooms of our house, it is impossible to tell some of the shades apart from each other.

Dora has also started choosing to work on more math work! She completed the teen bead hanger many times and has it down pat now. I’m not sure if she has memorized the order or if she truly understands the idea of “10 and 1 make 11”. I’ve been encouraging her to try the teen boards, which I think would help to cement the knowledge, but thus far she is resisting my efforts. She also continues to love to do the hundreds board, up until 20.

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Labels: Math, Montessori, Summer, This and That

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Montessori Monday: Math Work

I have a confession to make. During the last two weeks, I almost gave up on using the Montessori method with Dora. She seemed to have come to hate working in my office and refused to do anything there. So I tried to move some materials out into the playroom, but then she threw a fit and screamed that she didn’t want to do any of this “stuff” anymore. Finally, I started trying to take pictures of the materials in order to sell them, as they require a lot of storage space. Dora had an absolute fit and insisted that she never said she didn’t want to use the materials and that she promised she would work with them. It was then that completely surprised me! She saw some of the more advanced materials that I had set out to sell. She asked what they were and I decided that since I was getting rid of everything anyway, it wouldn’t hurt to let her do some of the materials out of order. First she really wanted to try the hundreds board. Though she finds the teen bead bars and teens boards to be very frustrating, she had no problem filling in the hundreds board all the way up to twenty!

She then asked to use a balance that I had purchased to use with Gohan to help with illustrating equations to him. It was somewhat of an expensive balance, but I decided that she is old enough now that she probably wouldn’t break it. Not only did she do tons of experiments with the balance, she even did some practical life work with it! It has a built in weights set that comes with tweezers to lift out the weights. She loved using the tweezers to take the weights in and out and did that over and over again. She also experimented with putting the weights away in different orders and I realized that the weight set was essentially a mini version of the knobbed cylinders. She had no problem sorting the weights, even though she still struggles with sorting the knobbed cylinders. I’m not sure why she was able to do these so much more easily, perhaps because they were shiny and new? Like the knobless cylinders, the color and texture remain the same for each cylinder, so there were no other clues for her to use to figure out how to sort them…

Finally, she asked to do the most difficult bolt board that we own. Previously, I had told her that she couldn’t do it until she could master the other bolt boards, because her frustration threshold is so low when she can’t do something. Then this week she said, “Mama, let me show you that I can do it.” So I opened the board and let her go for it. Not only could she easily sort the bolts by size into the correct slot, she could easily manipulate the flathead screwdriver, all while talking on the phone with my parents!!!!

So now I am left wondering if the problem was that I had not been leaving out challenging enough materials. She didn’t seem to have mastered the earlier materials and as I mentioned, gets easily frustrated when she can’t do something, so I had made a point of not setting out too difficult of materials. Now I don’t know what to think. Either she needed more of a challenge or she hit some developmental milestone, because suddenly she was able to do all sorts of activities that she wasn’t able to do last week.

__Please note my left sidebar for all the awesome link-ups that I am participating in. __

**Disclosure:** Some item links in this post are affiliate links. I will make a small amount of money if you click on them and make a purchase. All opinions expressed, however, are 100% my own.

Labels: Math, Montessori, Summer, This and That

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Montessori Monday–Assembling a Flashlight, Paper Punching, and Spindles, Oh My!

This week, Dora and I started her preschool work officially. She really loves having a room just for her homeschooling work. She was very enthusiastic about the trays I had set out and even did some other work that had been out all summer, but she was never interested in. Our first project of the week was a flashlight tray. To mix things up I put out two flashlights, one that uses “D” batteries and one that uses “AA” batteries. I am really glad that I set out both flashlights! I had thought that the smaller flashlight would appeal to Dora more, which was true, but I had not realized how much harder that flashlight was to assemble. The spring on the top was very springy and she just couldn’t master pushing down on it, while turning it at the same time. You can see in the above photos that she was very determined to master this and tried a variety of approaches. What you don’t get to see, is her falling apart, because she just couldn’t put the top on, no matter how hard she tried (I did not take pictures of this, as I was trying to help and comfort her). She was eventually able to assemble the “DD” flashlight, but it was just a consolation prize, as far as she was concerned. In the photo below, you can see how I laid out he tray.

We also tackled paper punching. Dora was not able to use the regular hole puncher, but fortunately I had set out some scrapbooking paper punches also. She spent hours on this activity. She would punch out a whole bunch of shapes and then glue them onto another piece of cardstock. I even promised to buy some more punches soon.

For math, she did some spindle box work. She pretty much has this work down pat and I will probably be retiring this from our shelves soon, which makes me feel a bit melancholy.

She also worked with the red rods. She got them lined up, but not quite in order. I asked her if something didn’t look quite right and she said, “Oh yeah”. She then proceeded to take one of the smaller rods and add it to a longer rod to make the longer rod the right length for the slot she had put it in. It was a creative solution and a great introduction to addition, if not quite the intention of the activity!

She worked with the graded nuts and bolts board and I was surprised by how much her fine motor skills have improved since spring!

Finally, she brought out the sound cubes, which really surprised me as she usually is very resistant to all of the auditory, tactile, and olfactory sensorial work.

This is another item that I will probably retire soon as it really is rather easy for her. Sigh… I know all to well how quickly kids grow up, given that Primo is turning 21 this week, and I am actually looking forward to Dora becoming more independent, but still it makes me a bit sad to be finished with any of her preschool supplies.

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Labels: Math, Montessori, Summer, This and That

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Beginning Counting Practice With Montessori Bead Bars

Dora’s counting skills have been getting stronger and stronger, but her ability to recognize numerals is not at the same level. I have had the sandpaper numbers, counting chips/number card set, and spindle box out for quite some time and she has expressed no interest in working with those items. So I finally brought down the color bead stair hanger. She immediately took to it and insisted I also get down the teen bead hanger. She was able to do the teen bead hanger with some help from me, but since that first day, has only worked with the color bead stair hanger. To do this work, she hangs a bead bar on the hook below the corresponding number. Ideally, she would also put all of the numbers in the slot on top of the hanger, but right now she won’t even try to do this (there is a control strip below in the form of a long wooden strip that shows the numbers in order). I’m just happy to see her making the connection between the numerals and quantity at this point in time and have no problem with just leaving the number symbol wooden chips up on top, in the correct order, for her. In addition (no pun intended!), when the bead bars are put in the tray below, they make a triangle.

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Labels: Math, Montessori

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Using Manipulatives to Teach Addition and Subtraction of Negative Integers and Variables

I believe that I have previously mentioned that Gohan has a language-based learning disability. When Gohan was originally diagnosed, he was quite advanced for math, but we had been warned that as he got into more advanced math, he might start exhibiting signs of dyscalculia also. Sure enough, in 6th grade, math started to become more of a challenge for Gohan. This year I switched to a Montessori curriculum for Gohan’s math, mid-year, and it has been working well. One thing he has struggled with, however, is subtracting negative integers. Recently, the program also started covering subtraction of negative variables and Gohan just kind of fell apart. So I did some research online and came up with a method of teaching this concept using manipulatives. We tried it today and it went fairly well. To do this, you need two colors of beads, beans, buttons, gems, etc. for integers and two colors of the item of your choice for the variables (if you want to teach variables also). We used glass gems on the light box. The yellow gems stand for negative units and the red gems stand for positive units. We started out with five negative units and five positive units, which offset each other and thereby equal zero (you can start with any number, as long as you have an equal number of positive and negative and your child must understand that –5 + 5 = 0).

The next step shows the very first integer in our equation, 2. We have the original 5 yellow and 5 red, plus 2 more red, resulting in a total of +2.

The next step shows 2 – (-2) = 4. We literally took away two of the negative units. So now we have three negative units and 7 positive units, for a total of 4.

We then take away 4 positive units: 2 – (-2) – 4 = 0.

Finally, we add (-3). The end equation being: 2 – (-2) – 4 + (-3) = (-3).

I then moved on to using turquoise gems to represent negative “x’s” and purple gems to represent positive “x’s” (sorry the photos don’t show the purple better ). Once again, I started with 5 positive and 5 negative gems. So, the above photo shows 0 + 0x.

The next photo shows 0 + x = x, since I added one positive purple “x” gem, but no unit gems.

We then subtracted (-2), taking away two negative unit gems, resulting in: 0 + x – (-2) = x + 2.

The next step was to subtract four negative “x’s”. So the equation now reads 0 + x – (-2) – (-4x) = 5x + 2.

The next step was to add 3 positive unit gems, resulting in: 0 + x – (-2) – (-4x) + 3 = 5x + 5

We’re almost done! We now add back some of those negative x’s, getting: 0 + x – (-2) – (-4x) + 3 + (-2x) = 3x + 5

Our final step was to subtract some of the positive “x’s” (yes, even my head was spinning at this point!). So we end up with: 0 + x – (-2) – (-4x) + 3 + (-2x) – 3x = 0x + 5, or simply, 5.

Hopefully, this makes some sense written out. Working with the gems in person, Gohan was able to understand the concept much better. By the end of the day, he was getting about 90% of these types of equations correct. Now, I just need to cross my fingers that it sticks (one of the joys of learning disabilities is that what a child knows today and what a child knows tomorrow are not always one and the same).

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Labels: Math

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Weekly Wrap-Up – Ladybugs and Middle School Montessori Methods

I’ve decided that I’m going to try doing more of a general wrap-up on Fridays that includes both Dora and Gohan, rather than trying to do two separate posts like I have been doing for some time. As Dora is getting older and I am incorporating Montessori into out lives, their “schooling” isn’t as separate as it once was.

Dora did focus on ladybugs this week. Our ladybugs arrived when our worms did, but were already hatched. They are still in the larva stage, but I expect them to go into the pupa stage sometime this week. We’ve had many more of the larvae survive this time. Last year, when we tried this, the larvae were very cannibalistic and ate each other, such that only two lived. I also brought out the insect puzzles that we have, which are only a butterfly and ladybug. I am hesitant to buy more, since Dora is so fearful of insects. In fact, we had planned to go to the Seattle Bug Safari, which is an insect zoo that I have never been to, but Dora was too scared to go. She now is comfortable with the three sets of insects that we are currently raising, so I’m very glad that we did this unit, which has allowed her to somewhat get over her complete terror of bugs.

In fact, we read Ladybug Girl this week and Dora is totally enthralled with Ladybug Girl. We went to the bookstore and bought two more books in the series, plus the last remaining Ladybug Girl doll. Jacky Davis and David Soman, the authors and illustrator, really have captured the spirit and personality of girls in this age group. Ladybug Girl is an excellent and realistic hero role model for young girls.

For our craft, we made finger print lady bugs on a “branch” background. Dora really enjoyed this activity, especially adding the details to the ladybugs.

Meanwhile, I am still trying to transition to more of a Montessori method with Gohan. Missing all of the Montessori foundation, is providing and added challenge, but I am going to try to use as many of the Montessori principles as I can. I find it interesting that in many ways, Maria Montessori, a developer of school curriculum and philosophy, was also one of the first unschoolers. While she believed in presenting a prepared environment and in having teachers present concepts to children, she did not believe in forcing children to do anything. Montessori students were free to learn what they wanted, when they wanted. Obviously, she was not an unschooler as homeschooled unschoolers are, in that living and learning were still took place in two separate environments, but certainly she was more of an unschooler than I am. So I am currently struggling with how much control over his own education that I am willing to give Gohan. Were I to completely absolve him of any school responsibilities and trust him to learn as he saw fit, he assures me that he would spend the day on Skype, chatting and gaming with his friends. I am not comfortable with such a schedule, so I am still assigning him some work.

He is currently working through the ETC pre-algebra cards. The cards are very basic and only require about 10 minutes of work from him. Yet, I find that they almost always result in serving as conversation starters for Gohan and I. As a result, he is actually learning more and having greater understanding, than when he was working through the textbook.

I have come up with what I hope is a good solution for the language arts conundrum that I found myself facing. I honestly believe that the best way to learn composition, vocabulary, and grammar is to read good literature. Yet, though Gohan is now reading, he is still not up to reading super challenging books. I finally decided that I will continue to have him read one book of his choice, but I am also going to have him start listening to audio books of higher quality literature. I will allow him to select this literature also, but he will only be able to chose from a list of about 25 books. We will then come back to grammar, composition, and vocabulary studies when he is in 10th and/or 11th grade, in order to finish preparing for the SAT and college.

That is a brief glimpse of our week? What about your week? Anyone else going through an educational approach identity crises right now?

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Disclosure: Several item links in this post are affiliate links. I will make a small amount if you click on them and make a purchase. All opinions expressed, however, are 100% my own.

Labels: Language Arts, Math, Montessori, Science, Wrapping Up Our Week

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Preschool at Home–Winter Birds

We kind of, sort of, studied winter birds this week. We made some bird treats using suet, peanut butter, and bird seed. Don’t they just look yummy! (That is sarcasm, in case it wasn’t obvious. If you’ve never worked with suet before, it is rendered beef fat and for some reason it is the “thing” to put into bird treats, though I doubt you will ever find these birds eating cows in the wild. Anyway, the stuff is greasier than anything that I ever worked with before. I had to wash my hands about 25 times to get it off of them. Then I had to clip all of my nails super short as the stuff had worked it’s way under my nails. Then I realized that my hands still had some suet on them, so I washed them about 10 more times. So I highly recommend using rubber gloves if you are going to work with it.) Unfortunately, there do not appear to be any winter birds in the area, so the treats remain untouched. I have been informed by a friend that if you want to feed birds that winter over, you need to start feeding them before winter sets in, so that they know where food is. I guess we may have to take our treats down to the duck pond that is right by our house (not even squirrels seem to be investigating our trees).

In addition to this messy craft project, we did a Montessori bird puzzle and Dora then insisted on not only doing all of the puzzles in the animal puzzle cabinet, but doing all of the puzzle activity cards that are labeled with the body parts’ names, some of which I didn’t even know how to pronounce! Finally, we finished the forest section of Maurice Pledger’s Animal World and are almost completely done with the book. We were supposed to go bird watching, but had to get some medical tests done instead. So no field trip this week, though we did go to our homeschool support group’s young kids’ park day.

Dora took one look at The Burgess Bird Book for Children and declared it “boring”. Nothing I said could induce her to give it a chance, so for our new literature selection for the week, we read poetry from The Classic Treasury of Children’s Poetry instead. She was not able to sit for some of the longer poems, but enjoyed many of the shorter ones, especially the ones that she had heard previously while watching Maurice Sendak’s Little Bear. It was like she thought the book had some magical ability to channel Little Bear or something. As soon as I read one of the poems that had been in Little Bear, she would sit up in a “I know that poem” way, and then her face would light up with joy and excitement when she realized it was from Little Bear.

For math, we read Anno’s Counting Book and played with the long red rods. Dora understood the concept of the long red rods, but since she did not line them up at the bottom, she was not really able to accurately compare the length of them all. In the above photo, she took them out of the stand, in order, and I lined up the first few, trying to demonstrate the process to her, but she just was not ready for that step yet. She did really enjoy working with the red rods, however, much more than the brown stairs or pink tower. In regards to Anno’s counting book, one thing that I really liked about the book was the way he managed to work the flow of the seasons into a wordless counting “story”.

Disclosure: Several item links in this post are affiliate links. I will make a small amount if you click on them and make a purchase. All opinions expressed, however, are 100% my own.

I’m linking this post to The Play Academy at NurtureStore,

Labels: Arts and Crafts, High School, Math, Montessori, Preschool, Science, This and That, Wrapping Up Our Week

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff

## Preschool at Home–Gingerbread Theme

I’m really late in this weekly summary, but our weekend was insanely busy and this is the first chance I’ve had to get anything done on the computer. This week, we looked at gingerbread. We read a version of the Little Gingerbread Boy, plus watched the Little Bear episode where they make gingerbread cookies (Little Bear being one of the few shoes that Dora will watch). We went to a gingerbread house making workshop. Afterwards, I learned that you can now buy the gingerbread houses pre-assembled. Part of the reason I do this annual workshop is so that I don’t have to assemble a gingerbread house, so next year we just may buy a kit and do it at home. The gingerbread house above is one of the ones they had on display that are made by local companies.

In addition, we went to Volunteer Park in Seattle to see their seasonal conservatory display.

We also did some seasonal button art, using popsicle sticks to make ornaments and making Christmas trees on paper. If I was doing these again, I’d paint the popsicle sticks white and/or glittery and let them dry before doing the craft with Dora.

I haven’t talked much about Dora’s math recently. In the last few weeks, we’ve started using “Living Books” for math as she seems to hate ALL workbooks and traditional math manipulatives. This week we read How Many Snails?, which was a bit too advanced for her. Previously, we have really enjoyed What Comes in 2’s, 3’s, and 4’s? and Teddy Bear Counting.

I’ve also been doing some Montessori math work with her, such as these wooden cards and counters that I bought from Kid Advance Montessori. I have decided to stick with these two approaches, Living Math books and Montessori math materials, for teaching Dora math for the time being. This is all very new for me as I have not used either method to teach math, or any subject for that matter, to any of my other kids. Dora just seems to naturally respond to a Montessori/Charlotte Mason approach, however, while my other kids have preferred more traditional learning tools or unit studies. Of course, Dora is only three, so who knows what we’ll be doing three years from now.

Disclosure: Several item links in this post are affiliate links. I will make a small amount if you click on them and make a purchase. All opinions expressed, however, are 100% my own.

I’m linking this post to The Play Academy at NurtureStore, Footprints in the Butter’s Reading Aloud Challenge, and…

Labels: Arts and Crafts, High School, Math, Preschool, This and That, Wrapping Up Our Week

Posted by Maureen Sklaroff

Posted by Maureen Sklaroff