So this post is a day later than I had planned, but I finally have time to sit down and write it.
Mathematics is one of those things that I have found people think about in very different ways. It sometimes amazes me the way that people can solve a mathematics problem in a way that seems so bizarre to me, but it works. The reason is because our brains don't all solve problems the same way. Therefore, I find that mathematics must allow for each child to discover what way works best for his/her brain.
Also remember that children with APD (auditory processing disorder) need to have visuals and/or kinesthetics to help due to the difficulties with auditory processing.
So keeping all this in mind, I have been very eclectic with mathematics. All of my children think differently about it and solve problems in their own ways. My sweet girl Hailey (who has APD) is very visual and so conceptualizes as well as solves the problems really by picturing things in her mind. For this reason, we have done a lot of hands-on, real world learning with mathematics as well as used manipulatives and pictures.
The Early Years:
When Hailey was a little one, we played with math like we would play anything else. We counted objects and we shared by dividing up our spoils. We bought enough apples for each family member by counting. We found out we are the second house on our block. We subtracted how many cookies we ate from the total we had to find out how many we had left. We would cook together and get 1 cup of milk or 1/2 cup of milk. You get the point; we just incorporated mathematical thinking and language into everyday life and made it fun.
I also liked to read aloud mathematics stories like Measuring Penny and 12 Ways to Get to 11. There are a ton of great story books that introduce math concepts and really get kids interested in trying to "do math". Hailey's twin brother absolutely adored books about money!
Introducing Numerals and Paper Problems:
At some point, we started writing the numbers down on paper and making written math problems. We used the objects we were adding and wrote them down so there was a one-to-one correspondence with the manipulative (what we were adding) with the numeral associated with it.
Sometimes I would write them as a problem like 2+3=5 and sometimes I would write them on a number line. Othertimes we would draw them as pictures or stick figures. This way, the kids would have seen a variety of ways to write it.
Having the Child Solve a Written Problem:
After having done the above for many times, then I would ask my sweetie to solve a problem by giving her the written problem. She would have access to objects, which was her preferred method at first. Then later, I would not have the objects out and she would have to decide what way she could do it without the objects. This generally led to using her fingers as objects. Her fingers then didn't become an obstacle until she got to numbers higher than ten and she needed to implement other ideas. As she would get stuck, I would show her options like the number line, counting up, counting down, drawing pictures, drawing stick figures, etc. I would only show one at a time to see if that made sense to her. If it did, then great. If not, then I'd try another one. However, if I had to do more than two, then she was frustrated and it was time to stop and try again another day.
Introducing New Concepts:
As we were doing mathematics for everyday living all the time, new concepts were usually not completely new. So when I wanted to add them to her written abilities - mathematics on paper - I would reintroduce it as "remember when we.......(ex. baked the cake and had to measure 1/2 a cup)". Then we would discuss what 1/2 meant or whatever the new concept was.
I would always be sure to have visuals and manipulatives as needed. So for fractions, for example, we would have measuring cups and spoons; we would have something to cut into equal pieces; we used fraction blocks (a plastic manipulative you can buy); we drew pictures (half the people are wearing hats).
I always taught Hailey the concept and its practical use before how to solve any problems. This is the way her brain best stores information: she needs the "why is this important and how does it all make sense" first.
Then she could start to solve problems both in real life and on paper.
Don't Make it Hard or Complicated:
Mathematics can be easy and it is best served in small chunks. I never spent too much time working any mathematics problems or concepts. If it started to become frustrating and we couldn't fix the frustration quickly, then we stopped and did something else. We could always go back to it another day.
My biggest concern was that Hailey would get frustrated with math and then decide she wasn't good at it and just shut down. I did not want that to happen, so I made sure to make the steps small enough for her to always feel successful. I really felt there was no rush to learn anything at any particular time and her perception of her skills was more important than any race to do things quickly.
The Workbooks We Used:
Mostly I made up my own mathematics curriculum and used the state mathematics as a guideline to make sure I covered everything she needed (but not at the time frame set by the state - at her pace).
I did use workbooks as ready made problems to solve as she got a little older. We also did some Time 4 Learning, but she found she liked the workbooks better. Here are some I liked:
School Zone Publishing Company Math Basics: These are organized by grade level and you can see how the skills are repeated but with a little more added to each year. I tended to use them without regard for the grade level. So I might do a multiplying fractions unit and use the Grade 4 to introduce them and then do Grade 5 and Grade 6 the next few days.
Educators Publishing Service Attack Math: These are books all about addition, subtraction, multiplication, and division. They are the basic skills only, but worked well for repetitive practice.
Educators Publishing Service It's Elementary! 275 Math Word Problems: This is workbook filled with word problems.
Singapore Math Practice: These are something that I just found this year and she has been doing well in them. She is technically a 7th grader and she is currently working in the 7th grade level workbook. I decided to mix up the flow of this workbook as it makes more sense to me to do the geometry sections all together and since we haven't done as much geometry before, I am saving them for last.
Algebra:
Hailey has not taken algebra yet, but she did do an app on her iPad that really taught her a lot about solving algebraic equations. The app is called DragonBox and the basis is that whatever you do to one side of the equation, you have to do to the other side as well. The point is to solve for x.
After completing the app, I gave Hailey some paper algebra equations and she could do the problems easily. So I guess it worked.
In her Singapore Math workbook this year, there is a small algebra section but it is even simpler in that they give you a value for x. However, the writing algebraic expressions from word problems was tricky.
So Hailey and my plan is for her to take algebra next year. (Remember she wants to take community college courses as a high schooler, so she is wanting to get prepared for that.) We probably will use the homeschool classes that she is enjoying this year as they have an algebra class. However, if they go about it in a way that makes it seem complicated (as sometimes classes and people make math more complicated than it needs to be), then I'll take her out and do it with her at home.
Mathematics is one of those things that I have found people think about in very different ways. It sometimes amazes me the way that people can solve a mathematics problem in a way that seems so bizarre to me, but it works. The reason is because our brains don't all solve problems the same way. Therefore, I find that mathematics must allow for each child to discover what way works best for his/her brain.
Also remember that children with APD (auditory processing disorder) need to have visuals and/or kinesthetics to help due to the difficulties with auditory processing.
So keeping all this in mind, I have been very eclectic with mathematics. All of my children think differently about it and solve problems in their own ways. My sweet girl Hailey (who has APD) is very visual and so conceptualizes as well as solves the problems really by picturing things in her mind. For this reason, we have done a lot of hands-on, real world learning with mathematics as well as used manipulatives and pictures.
The Early Years:
When Hailey was a little one, we played with math like we would play anything else. We counted objects and we shared by dividing up our spoils. We bought enough apples for each family member by counting. We found out we are the second house on our block. We subtracted how many cookies we ate from the total we had to find out how many we had left. We would cook together and get 1 cup of milk or 1/2 cup of milk. You get the point; we just incorporated mathematical thinking and language into everyday life and made it fun.
I also liked to read aloud mathematics stories like Measuring Penny and 12 Ways to Get to 11. There are a ton of great story books that introduce math concepts and really get kids interested in trying to "do math". Hailey's twin brother absolutely adored books about money!
Introducing Numerals and Paper Problems:
At some point, we started writing the numbers down on paper and making written math problems. We used the objects we were adding and wrote them down so there was a one-to-one correspondence with the manipulative (what we were adding) with the numeral associated with it.
Sometimes I would write them as a problem like 2+3=5 and sometimes I would write them on a number line. Othertimes we would draw them as pictures or stick figures. This way, the kids would have seen a variety of ways to write it.
Having the Child Solve a Written Problem:
After having done the above for many times, then I would ask my sweetie to solve a problem by giving her the written problem. She would have access to objects, which was her preferred method at first. Then later, I would not have the objects out and she would have to decide what way she could do it without the objects. This generally led to using her fingers as objects. Her fingers then didn't become an obstacle until she got to numbers higher than ten and she needed to implement other ideas. As she would get stuck, I would show her options like the number line, counting up, counting down, drawing pictures, drawing stick figures, etc. I would only show one at a time to see if that made sense to her. If it did, then great. If not, then I'd try another one. However, if I had to do more than two, then she was frustrated and it was time to stop and try again another day.
Introducing New Concepts:
As we were doing mathematics for everyday living all the time, new concepts were usually not completely new. So when I wanted to add them to her written abilities - mathematics on paper - I would reintroduce it as "remember when we.......(ex. baked the cake and had to measure 1/2 a cup)". Then we would discuss what 1/2 meant or whatever the new concept was.
I would always be sure to have visuals and manipulatives as needed. So for fractions, for example, we would have measuring cups and spoons; we would have something to cut into equal pieces; we used fraction blocks (a plastic manipulative you can buy); we drew pictures (half the people are wearing hats).
I always taught Hailey the concept and its practical use before how to solve any problems. This is the way her brain best stores information: she needs the "why is this important and how does it all make sense" first.
Then she could start to solve problems both in real life and on paper.
Don't Make it Hard or Complicated:
Mathematics can be easy and it is best served in small chunks. I never spent too much time working any mathematics problems or concepts. If it started to become frustrating and we couldn't fix the frustration quickly, then we stopped and did something else. We could always go back to it another day.
My biggest concern was that Hailey would get frustrated with math and then decide she wasn't good at it and just shut down. I did not want that to happen, so I made sure to make the steps small enough for her to always feel successful. I really felt there was no rush to learn anything at any particular time and her perception of her skills was more important than any race to do things quickly.
The Workbooks We Used:
Mostly I made up my own mathematics curriculum and used the state mathematics as a guideline to make sure I covered everything she needed (but not at the time frame set by the state - at her pace).
I did use workbooks as ready made problems to solve as she got a little older. We also did some Time 4 Learning, but she found she liked the workbooks better. Here are some I liked:
School Zone Publishing Company Math Basics: These are organized by grade level and you can see how the skills are repeated but with a little more added to each year. I tended to use them without regard for the grade level. So I might do a multiplying fractions unit and use the Grade 4 to introduce them and then do Grade 5 and Grade 6 the next few days.
Educators Publishing Service Attack Math: These are books all about addition, subtraction, multiplication, and division. They are the basic skills only, but worked well for repetitive practice.
Educators Publishing Service It's Elementary! 275 Math Word Problems: This is workbook filled with word problems.
Singapore Math Practice: These are something that I just found this year and she has been doing well in them. She is technically a 7th grader and she is currently working in the 7th grade level workbook. I decided to mix up the flow of this workbook as it makes more sense to me to do the geometry sections all together and since we haven't done as much geometry before, I am saving them for last.
Algebra:
Hailey has not taken algebra yet, but she did do an app on her iPad that really taught her a lot about solving algebraic equations. The app is called DragonBox and the basis is that whatever you do to one side of the equation, you have to do to the other side as well. The point is to solve for x.
After completing the app, I gave Hailey some paper algebra equations and she could do the problems easily. So I guess it worked.
In her Singapore Math workbook this year, there is a small algebra section but it is even simpler in that they give you a value for x. However, the writing algebraic expressions from word problems was tricky.
So Hailey and my plan is for her to take algebra next year. (Remember she wants to take community college courses as a high schooler, so she is wanting to get prepared for that.) We probably will use the homeschool classes that she is enjoying this year as they have an algebra class. However, if they go about it in a way that makes it seem complicated (as sometimes classes and people make math more complicated than it needs to be), then I'll take her out and do it with her at home.
Hi there,
ReplyDeleteMy name is Christian. I'm a 25 year old student. I'm interested in academic research that has been done in the area of primary maths with APD children, could anyone assist me? I am doing a literature review for my Masters of Teaching and as someone who was diagnosed with APD I am motivated to develop strategies for effective teaching.