Introduce the problem or the conceptual understanding first?

I was working with a student during period one today and she was having a hard time understanding how to subtract a negative number. For example, 18 – (-3). Why does the answer get bigger? She was absent yesterday and I saw that the teacher was going over range on the board. The teacher wrote on the board:

Lowest number is 3 and the highest number is 18. Range is the highest – lowest. He then drew the number line below to show the answer is 15. He was representing subtraction as the distance between two numbers.

As I was working with the student an idea popped in my head and I drew the same number line the teacher had on the board and right below it, I drew a second one that represented -3 as my lowest number. See below. This clearly showed how the distance increased, therefore, I must add to get my answer which shows why we can “add the opposite”.

While it did help the student make sense of the problem, I could see she needed more help and further explanation.

I had the fortunate opportunity to work with a student the next period on the same thing. This time, I chose to give her the problem 18 – (-3) first and let her make sense of the problem herself. After a minute, she came up with the answer 15. I then showed her the first number line of 18 – 3. I then showed her the second number line below it of 18 – (-3). The students eyes lit up and realized why subtracting negative three is like adding three.

What was the difference? The first time I did not allow the student to make sense of the problem themselves, I went right into teaching the concept. The second student was able to make sense of the problem themselves first and then see the concept which enabled her to correct her own thinking and understand of why the answer is bigger than 18. Very interesting. Boaler talked about this in her book Mathematical Mindsets. Allow students to make sense of the problem first and then they have a reason to want to understand it. It was also a visual way to show them and anytime I can make something visual, I believe it helps the students.


Estimating Square Roots

I had the fortunate opportunity to work with a small group of students to teach them how to estimate square roots.  As I was teaching the students, I quickly realized through questioning that a student did not understand the place value of decimals.  I asked the student to see me at the end of the day so I could help them with decimals.  During that time I showed him a number line such as the one below:


I only wanted them to estimate to the nearest tenth.  Once they gained an understanding of decimals, I then proceeded to teach about estimating square roots.  The student was able to understand how to find the two perfect squares that the number fell between, but could not understand how to decide which tenths place value to estimate to.  He was estimating √51.  As he was struggling, I saw the number line I drew and made a copy of it directly below except I replaced the 7 and 8 with their square roots (see picture below)

I then asked the students to place the √51 on the number line I drew and he placed it beside the √49 and when he looked at the original one above it, it clicked that 7.1 made sense as a reasonable estimate.  It was amazing to see his eyes light up when he could see the visual.


I reflect on what Jo Boaler said, that drawing out our mathematics is a powerful tool to help kids make sense of mathematics.  This is something I know I can continue to improve upon.

Problem Solving: Complicated or Complex?

I recently read a post by Robert Kaplinsky on whether problem solving is complex or complicated?  You can read it here!  He does an excellent job of explaining the difference between the two by using the idea of programming a remote verses driving a car.  Here is his explanation:

“To make this clearer, think about the differences between programming a TV remote control and learning how to drive a car.  Programming a remote control can certainly be a pain, but as long as you follow the instructions it can be completed.  Now think about what happens when someone learns how to drive a car.  While instructions on how to drive can teach you the basics, there are so many variables you can’t control, from icy roads to road construction to defensive driving.  This results in no instructions covering it all.”

This is very profound to me as we used to use a reading strategy called ACE to help student answer open ended problems on the PSSA test.




The math and reading coach helped develop this and the math coach realized this did not work for math so we made the letters be CAE which placed the word Cite first and meant that the students needed to find all of the key information through circling and underlining.  This was 10 years ago.  That strategy was to help students get the right answer on the test and really only worked for the original PSSA test because they were not allowed to ask more than one concept per question.

Today, with the state core standards and the core PSSA tests, problem solving and open ended questions have become much longer and contain multiple concepts.  The questions are much more complex and I think involve a different approach.  I never believed in following the ACE approach and told my students this past year that you need to read and make sense of the problem yourself because no recipe like ACE will help you solve these.  Some students still tried to use the strategies they learned in the past, but it did not help them very much.

As I reflect for this year, maybe providing a structure of what I really wanted the students to do last year is the way to go.  A structure is not a strategy, but a pathway for kids to lay down their thinking.  I believe the structure below would help them and also give me some great feedback on their thinking.

Hopefully this sheet will give the students and edge on how to think through complex situations.  I really like the guess box as I did not have the students do that last year.  This can be applied to most problems and it fits into the goals of having students make sense of the problems, but now they have a pathway to do it and it should help.

Integer subtraction: Removal or Difference? How do I teach it?

I have struggled with teaching integer operations to students for as long as I can remember, in particular, subtraction. I have taught using rules and having the kids write the rules in a graphic organizer and apply them. I have taught using integer counters and number lines to no avail. The compression and long term understanding just hasn't been there the way I would like for it to be with my students. Why is that?

What is the difference between teaching subtraction as removal verses difference?

Mark Chubb describes it as follows:

Subtraction can be thought of as removal…

We had 43 apples in a basket. The group ate 7.  How many are left?  (43-7 =___)

Or subtraction can be thought of as difference…  

I had 43 apples in a basket this morning.  Now I only have 38.  How may were eaten?  (43-___=38  or 38+____=43)

The hard part for me is to teach subtracting a negative such as 3 – (-7). I show the students the following method with integer counters (picture taken from Mark Chubb)

This is teaching subtraction as removal, however, the students don't really seem to grasp this unless the chips that need to be removed are already there.

Next I tried using a number line and teaching removal like this:

This past year, I tried using a number line and teaching difference to the students like this using 5 – (-2):

This really seemed to confuse the kids because they had to know that you start at the second number (-2) and move right to 5 which is a distance of positive 7. If the problem was -2 – 5, the answer would be -7 because they would reverse their direction.

I think in the end as I reflect on this year, I realize that in all three of my examples above, I keep saying "I show them"! Did I ever give them a chance to make senses of these problems on their own? How do they see it? For the sake of time, I try to SHOW them how to do it and get the right answer verses letting them make sense and sharing with each other. Having them make sense of the problem and listening to them will give me a better understanding of where each of them are at and I might be able to help them with their understanding better which will help them in the long run. Knowing that there are two ways to think of subtraction and having a deeper understanding myself will be invaluable to helping the students as they make sense of this. We know rules do not produce the long term results that we need. It might be worth it in the end if the kids will remember it since we use integers and rationals throughout the school year.

I have been challenging myself to gain a deeper understanding of the mathematics I teach and I saw a post of another way an elementary teacher sees and teaches students about subtracting integers. His post is below:

My concern with representing difference on the number line is that there is no way other than a rule for a student to make sense of the difference as positive or negative. For instance, why is (-4)-(-7) = (+3) and not (-3). To me, the idea of difference distinguishes magnitude, but not direction. After all, (-7)-(-4)=(-3) both have a magnitude of 3, but in opposite directions. I use number lines and integer chips in modelling integer operations with my students. I use addition (more) and subtraction (less) along with the quantity being subtracted to indicate direction. For instance, +(+)=more positive, -(-)=less negative with both pointing toward the positive end of the number line and -(+)=less positive and +(-)=more negative with both pointing toward the negative end of the number line. I think this is in line with the idea of difference from a slightly altered approach. (-4)-(-7) can then be interpreted as (+3) is 7 units less negative than (-4).

Do our students see it this way? Would they make sense of integers this way. It is another way to think about it and a way our students might try to make sense of the problem. I would not show every student all 4 ways as that confuses them in my experience. I would make sure I understanding all 4 ways deeply myself so I can help students as they make sense of problems. Who knows, I might learn a different way of thinking about integers from the students as they share if I give them a chance to.


Should I stat this lesson with sense making and see where they are at?

How much time do I give to this? A day. Half a period?

How will we start our school year?

I remember teaching 6th grade 4 years ago and the first thing we did after the first day was give a pretest.  The test was long and the hope was to gain information about each student and what the students mathematical needs were.  It was about understanding their gaps and their weaknesses based about the Pennsylvania Core Standards.  I now feel bad that I started the year that way.  I was so focused on weaknesses and data and my goal was to hammer down what the students needed.  My goal was fill in those gaps throughout the year.

As I type this, that doesn’t sound like a bad goal, it sounds quite noble and what a teacher should strive for, however, after studying and reading math blogs and books, I realize just how skewed my goal was.  What is it that I really want for my students to be able to do?  What is it that I want to convey to my students that I value from them in my classroom?  What is the mindset I want my students to have throughout the school year?  I know I can see any of those gaps through rich mathematical tasks.  Formative assessments are taking place all of the time which is like a streaming video I capture everyday of where the students thinking and learning are.  In giving a pretest, the only goal and atmosphere I conveyed to the students was:

  1. Sit quietly
  2. Don’t talk to each other
  3. Correct answers are the most important thing

Ouch!  That is not what I wanted, but that is what I believe my decision and action conveyed to the students.

What I really want to focus on as my goals for students as I start the year is to let them know that I value:

  1. Curiosity
  2. Mistakes
  3. Risk Taking
  4. Sense Making
  5. Growth
  6. Collaboration

This set of goals sets the tone for what I want to set for the students.  The start of the year is invaluable to me.  My grade level partner and I started the year off with a paper that relies on students using their number sense to figure out what number each letter stands for.  It is neat to see the kids working together and reasoning as to why they believe each letter equals the number they think is correct.  The students also get to share at the end of class and I gain a lot of wisdom from listening to their answers.  The paper is below.  I picked this up from a PCTM workshop a few years ago.


Reflective Questions:

  1. What are you going to value as you start the year?
  2. What are your goals for the students?
  3. Are the 8 mathematical practices a more worthwhile goal to start the year knowing they should be intertwined with the standards and the rich mathematical tasks you choose?
  4. Could I add a component that the students create their own mathematical patterns using letters for other students to solve if they finish he paper early?  Would this make it more of the low floor/high ceiling task?

Math CC Statistics Curriculum 2nd to 8th! Adjustments to make?

I have recently posted about the geometry curriculum at our school in this post.  I have now decided to post about my observations within the statistics curriculum.  The common core changes and the idea of mastery has had a dramatic impact on scores across states in the higher grades and I am starting to see why.  I posted the curriculum map below for you to look at:

Statistics Curr Cocalico Gr 2 – 8

My big take aways are:

  • Students never hear or explore mean, median, mode, or range until 6th
  • There is a two year period of no statistical information between 4th and 5th
  • Circle graphs are not mentioned at all across the whole curriculum, however, they are expected to be able to decide which graph displays their data better in 8th grade and a circle graph is part of that.
  • The progression flows nicely into grade 3 and then stops, I wonder why?

Triangles: The sum of the interior angles is 180 degrees can be relational to transversals?

Teaching that the interior angles of a triangle equal 180 degrees is a 7th grade CC standard.  This is a heavy chapter for 7th grade because it is the first time they will hear or see this property for triangles.  Knowing that the students do not have a background in this topic, I was wondering how I could teach this unit for better understanding and more importantly, how can I get them to compress this concept so that they will remember it into their school careers to relate to other concepts?

In the past I have taught this as a quick lesson and I used to have the students draw a triangle on paper and then rip the corners off and put the vertexes together to show them that the bottom of was a straight line and the students knew that a straight line is a 180 degree angle.  That was pretty much my attempt to give the kids the understanding of why.  Not that there is anything wrong with this approach, I just wonder if I could show this a different way and have it relate to another concept.

Taking into account the 8 mathematical practices, I have been transitioning to teach this lesson much differently over the last 3 years.  I really like the flexibility of asking an open question to students and allowing them to make sense of the problem.  I decided to try this with my 4th and 6th grader at home to see how I might question my students this year.  I asked the students to draw a scalene triangle.  This is what they drew.


From here I asked, what can you tell me about the interior angles of this triangle?  I liked how my 4th grader noticed that each angle was acute making it an acute triangle.  My 6th grader said that they all add up to 180 degrees.  I asked if he could prove that to which he responded with two answers.

  1. A square has 4 right angles which makes 360 degrees.  Since two triangles make up a square that means 1 triangle would b half of 360 which is 180.
  2. An equilateral triangle has angles that are each 60 degrees and 3 times 60 is 180.

At this point, my 4th graders was starting to make some connections listening to his brother, but he didn’t quite reason enough for him to understand.  Now, here is what I think I could change for this year to make it more relational to the students.  I believe I can tie parallel lines and transversals into understanding that the interior angles of triangles equal 180 degrees.  I can also tie into a straight angle being 180 degrees.

I started out drawing a line parallel to a base of the triangle and asked him to color in the angles each with a different color as you can see below:


From here, I asked what they noticed about the orange and purple angles below in relation to the top.  They struggled with this problem for about 5 minutes until my 4th grader saw a relationship between the alternating interior angles.  the conversations in those 5 minutes and watching them trying to make sense of the problem was fantastic.  I asked him to draw in color the angles he saw that matched.  He immediately got to work and drew this:


He then realized that the three angles together made a 180 degree angle and he knew that because he related what he knew about a straight line being a straight angle which is exactly 180 degrees.

I like this approach better because I am indirectly teaching some properties about parallel lines and transversals and straight angles.  To me, this is a minor change in my teaching that can make a big difference.  The other piece that I think that makes this powerful is color coding.  I posted about this previously and still think that this could be a powerful tool to help students see relationships.