Alternative Algorithms

I don’t know why I can’t get this link for Ann Arbor MI schools – parent handbook on alternative algorithms – to come up, but it won’t.

So, I did a little digging and found some other sites that were helpful:

http://everydaymath.uchicago.edu/teaching-topics/computation/

Click to access Fischer_AlternativeAlgorithms_2008.pdf

Click to access section12.pdf

http://www.nychold.com/em-arith.html

http://www.mathatube.com/everydaymath-motheds.html

I’ll keep checking back to the Ann Arbor site and try to catch it again 🙂

Katie – My mind was blown away last night after learning new ways to “do math.”  How incredible!  I just happened upon this site during my work today and thought it might be of interest to you also:  http://instruction.aaps.k12.mi.us/EM_parent_hdbk/algorithms.html

Have a great weekend!

Place Value – Questions for Katie

Katie, here are my questions for the Place Value Module:

1. Overall – I found it difficult to put it into words or completely ascertain the problems behind student misunderstanding and/or confusion.  What did you think?  Did you find it easy to determine why children did or did not understand the concept of place value, and, if so, did you find it easy to explain?

2.  PowerPoint – I loved the Base Ten Riddles and had a lot of fun seeing how I could use this in the classroom to help strengthen place value understanding.  What did you think of the problems?

3.  Textbook Reading, Developing Whole-Number Place-Value Concepts – What did you see as the biggest “take-away” for us as future teachers?  And, what did you think about the different stages of understanding place value and how to support children at each level?

4.  Digi-Blocks – What did you think of Digi-Blocks and how would you use them in the classroom?

5.  Videos – What did you think of Cena’s understanding?  I had a hard time determining how and why her understanding was lacking.  She seemed to understand the concept of place value fairly well when she was instructing the teacher in the whole class setting; but she seemed to completely miss the understanding when working one-on-one.  Do you think that had to do with feeling more pressure?

6.  Ten Frame Tiles – Katie, what did you think of using ten-frames instead of base-10 blocks?

 

The Power of Ten-Frame Tiles

Students need models that provide (1) countable, visually distinct model for each number, (2) can be organized in obvious and predictable ways, and (3) provide a clearly defined context of ten.

Ten-frame tiles give students the visual model to make the connection between each number name and the quantity it represents.  It also shows the “unique configuration” for each number quantity from 0 – 10 – giving the students a “shape” of the quantity to recognize.  The dots show the “completeness” of each ten easily without students needing to recount each group to confirm that there are ten.

Ten-frames do not require students to internalize the concept of tens and ones and that one unit can more than one countable unit such as a base-10 rod.  These models also do not provide students with manipulatives in which they can “build” things and “play.”
Ten-frames also provide students with a visually distinct, instantly recognizable picture of the number – which leads to subtizing (instantly recognizing a number of things without counting each individual piece).  This step is missing if teachers skip using ten-frames and jump right to using tens-rods.

When computing, these frameworks give students a model that “invites number composition and decomposition as part of visualizing a quantity.”  So, this is evident when viewing the example “figure 4” in the literature.  Two ten-frames show a 5 and an 8.  If we wanted to add these two numbers together, it is easy to SEE that two of the units can easily move to the 8 frame to complete it, becoming a 10-frame, leaving 3 on the 5 frame; 10 + 3 = 13.  It is easy to see this and visually make that understanding while working with the oral and written number as well.

Place Value Videos

In class, Cena seems to have a good understanding of place value and how to write a two digit number.  However, when working one-on-one with a teacher, she is unable to count by 10’s and must count each block individually to come up with a total of 24.  Further, she is unable to show that the 1 in 18 equals 10 even though she did show that the 8 in 18 equals 8 one’s.  This tells me that she has some beginning understanding of place value but has not yet reached base ten understanding.  Cena may have been able to understand the concept better and articulate it when she was not so close to the manipulation – meaning that perhaps as she watched the teacher grouping the stars, it was easier than actually grouping the blocks herself.

Jonathan had a greater understanding of place value and how you count by 10’s then count how many 1’s are left over.  He has a much more developed understanding of base ten.

Digi-Blocks

This product can be a great help in teaching children place value.  The use of this system mandates that students attend to the 10-ness of place value and cap the 1’s and change them to the 10’s place when they have reached the number 10.  For beginning math students, this could be a very powerful tool to set them up on the path of understanding.

I can especially see how these could be very useful in teaching subtraction with regrouping.  As I was in a 2nd grade class recently, there were a few students who had a very difficult time understanding regrouping and the process of “borrowing” from the column to the left.  I tried to draw out the concept of taking a ten and giving it to the one’s but that was difficult.  I can see how this could be very useful as a visual and concrete model to assist in understanding with this concept.

Written Numerals and The Structure of Tens and Ones

Case 11 – Number of Days in School

Andrew’s understanding of what day of school it is as fifty-ten instead of 60 makes perfect sense logically speaking.  He knows that as you count “up” you move to the next ten.  But, he doesn’t quite make the connection of going up in the “ten’s place.”  He is understanding what comes next but is missing the connection of 10, 20, 30, 40, etc.  Andrew is making the connection of 10’s but is missing the next step.

Case 12 – Groups and Leftovers

As I grab a handful of beans and start to count out different groups, I have difficulty keeping track of how many beans there are in total.  But, as I count out groups of 10, it is easy to “sum” the beans.  This is the idea of using manipulatives to make groupable models and then use the benchmark number of 10 to count to a total.  The children came to the same logical conclusion that it was easier to count making groups of 10’s than it was other size groups.

Case 14 – Who Invented Zero Anyway?

In this case, the children are trying to figure out the meaning of zero in numbers and trying to find a way to understand and show that it does have meaning and can’t be left out but just what that meaning is is eluding them.  They are trying to understand that at the end of the number is not quite the same as a zero all by itself.  They are all talking around the usefulness of the zero but not understanding it enough to say that it is holding a place value.

Case 15 – One Hundred Ninety-Five

When students were asked to write one hundred ninety-five, they gave many different answers with differing understandings and explanations:

1095 and 10095 – these students knew that 100 meant something and that it included zeros, they just didn’t remember exactly how it worked

195 – was given by one student who just couldn’t explain but knew it to be true.  Maybe this child just has more number experience and has a beginning understanding that just can’t be articulated as of yet.

1395 and 1295 are just completely missing the mark but understanding that these are big numbers that are being discussed.  There does seem to be some understanding with the “95” portion of the number.

It was very difficult to make a prediction about how these children were understanding and how they weren’t.  Also, sometimes I could “see” their thinking but then others I was just not understanding any sort of logic that they were following.  Maybe this is when they were just making up their own “new insights” to try very hard to grasp something out of their current understanding.

 

Place Value PowerPoint

Teachers are typically instructed to use the standard computational algorithms (similar to how adults solve problems) when teaching math.  But, this method doesn’t prepare children for more flexible approaches to computation such as student invented strategies.  Place value understanding should be based on the patterns and relationships in the number system.

As teachers, we can use the Digit Correspondence Task to assess students’ understanding of place value.

To help students understand place value, the following should be used:

• groupable models (the most powerful as students must build)
• pre-grouped models
• proportional and non-proportional (coins) models
• hundreds chart and 99’s chart

I found this 99 Chart to be incredible ….. in that children can see right away the connections that are made with 10 and 11-19, 20 and 21-29, etc.  Otherwise, using a 100 Chart, these connections are not as apparent.

Base Ten Riddles – These were fun and I can see how this would help develop place value understanding more than just using 100’s, 10’s, and 1’s for 534.  Great ideas!

Developing Whole-Number Place-Value Concepts

So, as children develop place value understanding, there are different phases that they must go through and successfully accomplish/understand before they can move on to more complex understanding.  These are the phases:

• Pre-Place-Value Understanding – This is when children can count to a certain number as they count by one’s but may not necessarily understand what the numbers stand for.  For example:  a child may be able to count 23 jelly beans but doesn’t really understand that there are twenty and three jelly beans, or two sets of ten and three jelly beans.  This is also known as Pre-Base-Ten Understanding as which required unitary counting.
• Base-Ten Understanding – At this level, students understand groups of 10 and can count these as a single object.  Students will be able to count by tens and then continue counting in what is “left over” as the ones.  With this understanding, it is important to give students lots of opportunity to count in different ways and express these numbers with numerals and words to make that connection.  This is also when place-value understanding is specified.  It is important to have students explore how to group objects for counting in this phase.  So that they “know” that there are groups of ten because they created them, such as banded straws or cups of counters, or rods of cubes.
• Equivalent Understanding – At this stage of understanding, children see units of ten and hundred and do not need to singularly count each object to know that they can count with these objects as groups, but they must have had adequate time learning with groupable models first.  For example, 1 hundreds-flat , 3 tens-rods, and 2 ones-cubes is equal to 132.  This is also the stage at which children learn to count and work with numbers that are represented with nonproportional models.

Learning and Modeling – It is important that teachers provide students with adequate work and experience with each stage before they move them forward.  The suggestions of counting cubes, links, crayons, shoes, students, toothpicks, buttons, beans, plastic chips, craft sticks, beans, washers, etc. was great.  I wouldn’t have thought of counting anything and everything really.  Also, linking number word forms with numerals is an important factor which needs to be included and continued as learning progresses.  Once students have mastered equivalent understanding, the activity 11.2 in which students count various objects and record them as a number word and as tens and ones is powerful.  Additionally, with that same worksheet, understanding of ten-ness is reinforced with counting “ungrouped” objects and recording the numbers, counting out a given number of objects and filling in tens-frames to represent, and “looping” objects in groups to create a certain number.

Benchmark Numbers – Benchmark numbers are key number of recognition such as multiples of 10, 100, and later on special numbers such as multiples of 25.  This will assist students as they learn to work computations with numbers.  Using a hundreds chart or a number line can assist in this understanding.  For example, when students are multiplying 45 x 4 they can also see that as 45 + 45 + 45 + 45.  It would be easy for students to figure out this by using benchmark numbers.  So, students could say I am multiplying 50 x 4 or adding 50 together 4 times and then I am taking away those 5’s that were missing so taking away 20.  50 x 2 = 100 x 2 = 200 – 20 = 180.

 

Questions for Katie (Meaning of Operations and Basic Facts)

Solving Word Problems and Math Matters

1.  I had trouble with the addition/subtraction word problems and sometimes categorized the problem incorrectly.  So, I’m thinking I’m working out an addition problem when it is really a subtraction problem.  I think it had more to do with the way I would choose to solve the problem.  Did you have this issue with any of the word problem categories?

2.  I really liked the use of Bar Diagrams.  I felt like it gave a visual (almost tangible if you will) meaning to Part-Part-Whole problems.  I really saw the relationship building that is explicitly shown with this method.  It gives students more meaning than just addition or subtraction and shows relationships between the operations and how they work together.  What do you think about using this method?

I think that using a variety of these teaching strategies for helping students learn to understand and solve word problems would be my preferred approach?  Katie, what do you think, would you prefer to use one method over all others or a combination?

Storyline Online

3.  What book did you use from the Storyline Online?  I used Library Lion.  It is a cute book that I had read before (I realized once I was half way through) but I didn’t think it particularly lent itself to “math.”  I think there are other books that are so much more of a tie to math?  What do you think Katie, have you used other good math books?

Overall

4.  After everything we have read, what do you think about Timed Facts Tests for math?

Case Studies

5.  After reading the Case Studies, what do you think about giving students the choice of which manipulatives they can use to help problem solve?  Would this cause more problems with keeping students focused on the problem solving versus walking around the room investigating available manipulatives?

 

 

Final Articles – How can we help our students learn/know/recall their basic facts without resorting to memorization?

We should use the Chinese philosophy and us 10 as a bridge because of its importance in the base-ten numeration system.  Also, we could introduce the basic facts in units, such as 6+ unit, the 7+ unit, and so on.  These units are categorized by the know entity (addends) instead of the unknown entity (sums).  Lastly, the article reveals that we should teach different strategies that are not in textbooks but that can help children see the patterns among addition and subtraction facts (still using 10 as a bridge).

In the second article (The Road to Fluency and the License to Think), the author/teacher started out the year using a whole new method to teach basic math facts.  She followed a schedule of teaching addition strategies:  doubles, doubles plus one, doubles plus two, doubles minus one, doubles minus two, combinations of ten, counting up, add one to nine, make ten, adding ten, and commutative property.  Following this schedule, she would follow a certain “routine” as she taught each new strategy:  introduce and explore using manipulatives, create illustrations, use math journals, and assign homework including a parent information page.  Finally, she would complete each day with a Mental Math session in which students would share with the class their “strategy” for solving the problem.  This helped give other students strategies that they had not before considered and gave a whole “thinking math” feeling to the classroom environment.  Students would even name their particular method for figuring out problems – which made things more interesting for them and created more of an investment on their part.

In the final article, many different “types” of multiplication problems were discussed:  repeated addition, scalar, rate, cartesian produce, and area.  And the importance of helping children understand that multiplication has a variety of meanings and is not just a sequence of isolated facts.  This article that the the most effective sequence of instruction for multiplication facts is:

• introduce multiplication with real problems involving repeated addition
• use hands-on materials that truly represent the problem (ie markers in the ex)
• then, substitute blocks for problem items as a next step
• then, substitute student drawings for tangible manipulatives
• then, advance to students using tally marks or other representations instead of drawing true items
• give students many thinking strategies to simplify problems such as 8 times a number can be 5 times a number added to 3 times that number or 4 times a number added to 4 times a number
• give many opportunities for practice with games, flash cards, dice, calculator, computer-based practice, concentration cards, etc.
• finally, reteach those who really struggle by beginning showing them how much thy DO KNOW …. using a known and unknown multiplication fact table so that they can feel proud and confident in their current abilities and not be discourage about what they need to improve upon