# Helping Children Master the Basic Facts

During the Power Point, I worked on many different strategies to help me learn to teach basic math skills.

Addition Strategies:  Using COMMUTATIVE PROPERTY such as 5 + 3 = 3 + 5, called Turnaround Facts, can be helpful to help children learn.  Also, counting on is a beginning step for learning addition facts.  Do not discourage children from counting on their fingers as this is a good beginning strategy and they will stop once they are ready.  Counting on and counting on fingers is useful but it is time-consuming.  So, we want to give children other strategies to help them.  Such as:

• Adding Zero (especially using manipulatives, so they don’t get confused)
• One More Than
• Two More Than
• Near Doubles
• Combinations to Ten (using tens frame is helpful for this)
• Adding to Ten and Beyond

Subtraction Strategies:  Again, there several ways to help children master subtraction besides just counting back, which again is very time-consuming.

• Down over 10
• Take from the 10 (ex: 17 – 9 …. 10 – 9 = 1 + 7 = 8) This was a great addition to my subtraction repertoire that I will definitely remember.

Multiplication Strategies:  Using books is especially helpful for laying the foundation for learning multiplication.  I loved this idea and will definitely use the books suggested:  Each Orange Had 8 Slices, Anno’s Mysterious Multiplying Jar, What Comes in 2’s, 3’s, & 4’s, and Amanda Bean’s Amazing Dream.  Then once you have read about things that come in groups, have the class make a list of things that come in groups.  Remind them to use usual groups and not crazy things 🙂  Then, make a class book.  I have done this in kindergarten and it didn’t seem that powerful.  But, I can definitely see how having student create a class book in an older grade (2nd) would be wonderful!  Love this!  Other strategies to use:

• Circles and Stars Game
• How Long?  How Many? (with the rods)  I have never seen this before and LOVED the game – how much fun to be learning multiplication without even realizing it!
• Doubles

Division Strategies:  I used multiplication/division skills when I divided the list of fractions.  I didn’t even think about it.

Finally, what about Time Tests????  They will benefit fewer than they will impede from what I heard because of things like:

• they will improve skills children already know
• they will not promote reasoned approaches because students don’t have enough time to think through strategies
• they will produce few long-lasting results
• they will reward few but punish many
• they can give some students a strong dislike for math and a faulty idea of learning math

# Storyline Online

Library Lion by Michelle Knudsen

Nina read some books in the morning.  Then she read 4 more books.  Now Nina has read 7 books today.  How many books did Nina read in the morning?

The Library Lion had 3 books.  Then he bought some more books.  Now he has 8 books in all.  How many books did the Library Lion buy?

Olivia checked out 10 books from the library.  She returned 4 of the library books.  How many books does Olivia have left to return?

The librarian had 20 cookies.  She gave some cookies to the good listeners at story time.  Now she has 6 cookies left.  How many cookies did the librarian give away?

# Solving Word Problems & Math Matters

In this first article by R. Charles, we are presented with the two traditional teaching strategies for math word problems:  key words and steps.  Using key words teaches students to always use a particular operation whenever a word problem contains a certain word or phrase.  This is problematic because key words can be misleading leading students to chose a wrong operation or nonexistant leading students to confusion.  When assessment use word problems that don’t have any key words, it is not because they are trying to trick students but rather because everyday real-world problems are not usually asked in such a neat and tidy formula.  Charles’ ultimate goal is for teachers to help students understand that word problems are a process and that it takes logic and reasoning to figure them out.

A new “Visual” approach to teaching word problems helps students identify the known and unknown quantities using a bar diagram and arranges them in such a way to show relationship.  Using “meaningful representation” to understand word problems instead of just writing down an equation can be much more useful and help students understand a wider range of ways to solve problems.  These bar diagrams can be used to help solve joining, separating, part-part-whole, comparison, joining equal parts, separating equal parts and comparison problems.

### Charles’ most convincing statement was:  “One of the powerful attributes of this set of bar diagrams is that they are connected to parts and wholes.  This consistency in visual relationships helps students see not only the connections between the diagrams but also connections between and among operations.  An important part of understanding operations is to know all relationships between and among the four operations.”

Suggestions for teaching this method are:

• model bar diagram representations on a regular basis
• discuss and connect them to quantities in the  word problem and to operations meanings
• use them to focus on the structure of a word problem, not surface features like KEY WORDS
• encourage students to use them to help understand and solve problems

In our second reading, Math Matters, the focus was on the importance of students attaching meaning to mathematical operations through the manipulations of concrete objects and then connecting their actions to symbols.  As teachers, we need to give students a variety of word problems to work and gain experience.  Though this article did focus on the same types of problems (join, separate, part-part-whole, and compare) they broke these specific types up further into subsets of each type of problem.  This was done to further identify what the question is asking for and which quantity the student is asked to solve.  I had never thought through addition and subtraction problems with this depth before.  And, some of the problems that were put into the “join” category I would have put in the “separate” category because I would have changed information around.

For example:  Laina had 4 dolls.  She bought some more dolls.  Now she has 6 dolls.  How many dolls a did Laina buy?  I absolutely see NOW how this is a “join” or addition problem, but I would have originally said it was a “separate” or subtraction problem because instead of writing 4 + __ = 6  I would have written 6 – 4 = __ to find the solution.  I see the difference now and will need to be VERY CAREFUL when teaching younger students so that I do not confuse them.  What do you think about my problem?

Classifying Addition and Subtraction Word Problems

1.  Carlton had three model cars.  His father gave him four more.  How many model cars does Carlton have now? Join Result Unknown:   3 + 4 = __
2.   Juan has nine marbles.  Mary has six marbles.  How many more marbles does Juan have than Mary?  Compare Difference Unknown:   9 – 6 = __
3.   Janice has three stickers on her lunch box and four stickers on her book bag.  How many stickers does she have in all?  Join Result Unknown:   3 + 4 = __
4. Catherine had a bag of four gummy bears.  Mike gave her some more.  Now Catherine has seven gummy bears.  How many gummy bears did Mike give her?  Join Change Unknown:   4 + __ = 7
5.   A third grader has seven textbooks.  Four textbooks are in his desk.  The rest of his books are in his locker.  How many books are in his locker?  Separate Result Unknown:   7 – 4 = __
6.   Vladimir had some baseball cards.  Chris gave him 12 more.  Now Vladimir has 49 baseball cards.  How many baseball cards did Vladimir have before he received some from Chris?  Join Initial Quantity Unknown:   ___ + 12 = 49

# Spatial Ability/Reasoning

That was a lot of fun!  I really liked I Took a Trip on a Train and found it fun and fairly easy to see the order in which the pictures could be taken when beginning at start and circling around.  Plots Plans and Silhouettes was not too difficult and it was fun to “build” the structures that we were given front and side views of.  I did find Shadows challenging at first.  But, once I saw the shadows that were produced, I “got it.”  I initially had trouble seeing what the shape of the shadow would look like if the light was coming from above or in front of the cube.

Spatial competence is important for every day life.  I know that I can’t pack my bookbag or a lunchbox well if I don’t take the time to look at it but instead just start shoving items in.  This is the same as spacial intelligence.  We all need to be able to “see” different situations in our every day life.

I really think that students from K and 1st would benefit from being introduced to these concepts.  Probably not as difficult as the tasks that we experienced, but other more simple work.  I really believe in schema building.  That being said, I really think that the more experiences we give students, be them “educational” experiences or not, the better their understanding of everything will be.  So, let’s give students more opportunities to “play” with shapes and work with them, providing them with building blocks, geometric shapes, everyday items, etc. and ask how can you fit as many as possible into this box.

# Case Studies 18-22 – Shapes

As a point of interest, as I googled to find a picture of “geometric shapes” to place here, 99% of all the shapes were PERFECT TEXTBOOK  shapes.  So, it is easy to understand why children have a misunderstanding about “what is a triangle?” when many of the examples are all equilateral triangles arranged with one side “sitting” horizontal and its point at the top.  Wow!

What are my definitions for these geometric terms?  Prior to reading the articles, I believe that a triangle is a closed figure with three straight lines and three angles that all add up to 180-degrees; a rectangle is a closed figure with 2 pairs of sides that are congruent and parallel and 4 90-degree angles; and, a square is a special rectangle that has four straight sides that are congruent and parallel.

What is the difference between a definition and a list of properties/attributes?  Before reading the articles, I would think that a difference between a definition and a list of properties or attributes is not much.  Maybe one might be more technical that the other, but both should contain the same type of information.  When the students were asked to write the name of a shape and the properties, they rephrased it as “the rules of each different shape.”  Interesting that children would refer to these as rules to be followed, just like rules of the classroom and rules of the game.  Interesting …..

What is the purpose of a definition?  Definitions provide us consistent agreed upon descriptions so that everyone understands what is being discussed.

What specific issues do they (students/children) need to consider in order to make sense of definitions for triangle, square, rectangle, and parallelogram?

When students are considering how to categorize a shape and define its properties, they must include their “scheme” meaning that they can only apply these RULES as much as their experiences provide.  So, if some students have more experiences with more and different triangles, they will probably understand the overall concept of a triangle better than others.

What is the process they (children) go through as they learn to apply their definitions?

Looking beyond the specific geometric content of this set of definitions, how do children develop a sense of the purpose of a definition?

Natalie’s Case 20 – In this case with 1st and 2nd graders, the students are realizing that as they discuss the definition for a square, it is the same definition that they had designated for a rectangle “four sides, four corners, four angles, and it’s a square.” But then “it’s a square” becomes part of the definition to help in understanding.  But, then, one of the children points out that “Actually, you don’t need to say four corners and four angles; they’re the same thing.”  This is a true statement, but can the students see it this way?  When the square is turned so that it now looks like a diamond, some of the students say it is different, not a square anymore.  And, then Charlie draws a random figure on the board that fits the “definition” but is not a square as they all agree.  So, the children understand that they must refine their definition.  Through their continued discussions, they decide that a square must have “angles that are the same.”

Delores’s Case 18 and Andrea’s Case 19 – In the 3rd grade class (Delores) Mei-ying did not like that the triangle was “upside down” and that it was “skiny” but did agree that these nontypical figures were triangles.  She did not let the sideways orientation of the rectangle bother her understanding or identification.  Francie would not identify shape 1 as a triangle because “I know my shapes” but she would concede that shape 2 was a rectangle.  Louis did identify shape 1 as a triangle but did make sure to note that is was positioned “upside down.  It kind of doesn’t look like a triangal.”  But he was not worried about the orientation of the rectangle either.  Finally, Shannon wouldn’t identify shape 1 as a triangle but rather insisted that it “looks like a carrot.  It has two long sides and 1 short side.  I think it’s a real shape but I don’t know the name of it.”  Again, the orientation of the rectangle did not bother this student in identification.  Interestingly enough, the 2nd grade class (Andrea) seemed to have a more “defined” discussion of the shape dilemma despite their younger ages.  “Well, I think tri means three, so triangle means three.”  When asked three what, the response is “angles.”  It was further discussed that an angle is when two “straight” lines meet.  So the discussion about STRAIGHT lines again becomes a focus.  “And triangles have to have points.  They’re pointy, not like curves on the ends.”  “Three points.”  And then the children begin to discuss the overall LOOK of the triangles.  One child disagrees that a shape is not a triangle because “It’s too stretched out.”  But, then another child (Evan) makes a profound statement that “If I’m all stretched out and turned upside down, I’m still Evan.”  WOW!  Now that seems like true understanding to me.  The students go on to discuss “slanty” sides and the ability to use these triangles as ramps, and that if you turn them around, they look different and more triangle like.  So they are grappling with the idea of right triangles and how it looks “different” when they are joined.  Finally, one child describes them as “It has 3 side, 3 corners, 2 slants, 1 strate side.  If you turn it all around it will still be a triangle.”

# NRICH Early Years Activities

Incey Wincey Spider Gamehttp://nrich.maths.org/content/id/8863/Incey%20Wincey.pdf

I drew a giant “drainpipe” with sidewalk chalk on the pavement.  The two girls chose to be either the sunny spider or the rainey spider.  They took turns throwing the dice and moving themselves (the spider) up or down the drainpipe according to their role.  They use giant dice with dots to determine how many jumps to make and remember to direct the “spider” to count with each jump after landing.

As we played, I asked questions:

• where are you now?
• how many jumps did you make?
• can you put something on paper to show what numbers you threw or to show someone what you learned from the game?
• why did the sun win, do you think?
• why did the rain win, do you think?
• you’ve thrown two:  what do you have to do now?
• how many steps more have you got to go?
• you’ve thrown three: will that get you to the end?
• how many turns do you think you need to get to the end?
• what do you think will happen?  why?
• could we make another game like this?
• what could we use?

These are the mathematical concepts that we practiced while playing:

• counting
• remembering the order of the number words
• saying the number as they landed on each square
• remembering the “stopping” number
• knowing how many more steps remain (part, part, whole)
• recognizing the number of dots on the dice
• associating the number of dots with the number of jumps along the track
• knowing that a bigger number means going further

The girls were highly engaged in the activity.  They enjoyed “playing” math with me and didn’t want to stop.  Their understanding of the number concepts was closely matched despite the differences in their ages (4 and 6 yrs old).  The older girl did have a better understanding of how to represent the information if we wanted to record it (I have a picture of the paper that the 4 year old used to record our information).  Also, the older girl could instantly tell me what number was rolled on the dice while the younger one had to count the dots each time.  Otherwise, I was very surprised by both their number sense of how many more, why did you jump that many, why did she win, how many turns, etc.

# van Hiele Levels of Geometric Understanding

Level 0 – Pre-recognition – inability to distinguish between figures

Level 1 – Visualization – recognize by appearance alone

Level 2 – Analysis – see shapes as collections of properties but don’t see relationships

Level 3 – Abstraction – see relationships between properties and between figures

Level 4 – Deduction – able to construct proofs, understand definitions, and know the meaning of conditions

Level 5 – Rigor – comparing mathematical systems

Geometric Understanding – By using van Hiele’s levels of geometric understanding, students progress through a level of understanding from 0 to 5.  This understanding is dependent upon educational experiences and not age, grade level, or maturity.  So, I understand this to mean that students must truly understand the material before they can progress to the next level.  Moving on from memorizing information rather than understanding it will keep students from progressing.

Teaching Above a Student’s Level – If a teacher tries to teach content that is above a student’s level, they will not be able to master the information, will try to memorize but may easily forget the material, or be unable to apply the information to a given situation.  So, the importance of knowing your students’ understanding level before proceeding with new material is crucial.

Instructional Practices – Students will be able to learn when they actively experience the objects and when they engage in discussion and reflection.  So, this means that just lecture and reading will not help students learn.  They must be afforded opportunities to investigate and experiment with the geometrical figures.  Once investigations and exploration have taken place, then students can make use of information from lectures or books to aid in their learning.  The final piece of this must be reflection and their understanding.

The Role of Language – Each level of geometric understanding has its own language associated with it and an interpretation of that language.  It is important to discuss and have students talk about geometry using this language.  As student understanding increases, so does the use of more specific language.  I understand this to mean that initially students will understand that a triangle has 3 sides and talk about it in those terms.  Later students will learn that triangles have 3 angles and they all add up to a certain measure.  As understanding is gained, language will increase to improve and specify the learning.