Author Archives: mathmindsblog

Explicit Planning vs Explicit Teaching

Planning is like…..

How would you finish that sentence?

As a facilitator, I use this sentence starter to open Illustrative Mathematics’ 5 Practices Professional Learning. To be completely honest, when I designed the PD I was a little hesitant of using it because I was nervous it was opening a can of worms within the first 5 minutes of the day. I am, however, always surprised with all of the beautiful analogies participants share and feel challenged each time to come up with something new and better than the one I used in previous sessions. When I first delivered this PD, I started with analogies like a marathon or really hard workout – something that is exhausting, a lot of work, but ends with something I take pride in. While these analogies were accurate representations of how hard I think lesson planning truly is, I was continually unhappy with where students’ ideas fit into my analogy.

My most recent sentence was this…

Planning is like putting together a puzzle.

When sharing my reasoning with participants for the first time, I included a lot of beautiful words around mathematical connections but in the middle somewhere I used the phrase “making connections explicit” in relation to the puzzle pieces and saw an immediate reaction from a few people in the room. Of course, I had to pause and ask, “Was it the word explicit?” – answered by many nods in the room.

For a long time, the word explicit in relation to teaching held a negative, cringe-worthy connotation for me as well. If ever asked to paint a picture of what explicit teaching looks like in the math classroom, I would describe scenarios in which a teacher is either at the board telling students how to solve a problem or showing a struggling student how to solve a problem because they are stuck or “taking the long way there.” To me, being explicit meant telling students a way to do something in math class – typically in the form of a procedure.

Through teaching a problem-based curriculum [Investigations], designing and implementing math routines such as number talks, and reading Principles to Actions, 5 Practices and Intentional Talk , I realized that I was guilty of making mathematical ideas explicit every day in my classroom, but not in the way that made me cringe.

I was explicitly planning, not explicitly teaching.

To me, those two phrases indicate a big difference in how I think about structuring a lesson. I have found when teaching a problem-based curriculum, it is easy for ideas to be left hanging and important connections missed, forcing me to explicitly teach an idea to ensure students “get it” before they leave the class period without any understanding of the mathematical goal for the day. Many days, I would find myself frustrated because students would completely miss the point of the lesson, however now I realize this was because I was expecting them to read my mind of what I wanted them to take away from the problem. On the flip side of that coin, however, not teaching a problem-based curriculum and explicitly teaching students how to do the math in each lesson is not an option (and is a topic that could be its own blogpost).

This is exactly why I find the 5 Practices framework invaluable in planning. The framework forces me to continuously think about the mathematical goal, choose an activity that supports that goal, plan questions for students toward the goal, and sequence student work in a way that creates a productive, purposeful discussion toward an explicit mathematical idea. I have learned so much using this framework over and over again in planning for my 5th grade class, collaborating with other teachers and coaching teachers across different grade levels.

Explicit planning is how I would describe the new, open education resource (OER) by Illustrative Mathematics. As a part of the writing team, I explicitly planned warm-ups such as number talks and notice and wonder activities to elicit specific mathematical ideas that play a purposeful role in the coherent plan of the lesson and unit. But not only are the warm-ups explicitly planned, but each lesson and unit tells a mathematical story in which students arrive at a specific mathematical landing point. While they may not all arrive at that landing in the same way, the problems and discussions are structured to ensure students do not leave the work of the day without any idea of what they were working toward.

While I would love to think my blog posts paint a clear picture of explicit planning, I am not that naive. So, what does explicit planning look like in a 5 Practices Framework?

This lesson from Grade 7, Unit 2, Lesson 2 from Illustrative Mathematics’ Middle School Curriculum is one of many in the curriculum. (All images are screenshots from the online curriculum that is linked at the bottom of the post)

Practice 0: Choosing a Mathematical Goal and Appropriate Task

Screen Shot 2017-08-25 at 10.13.27 AM.png

Lesson Learning Goals

With the goals in mind, the lesson begins with a notice and wonder warm-up that engages students in thinking about tables, followed by two activities that build on those ideas and support the mathematical goals. While both activities demonstrate explicit planning, I am focusing on one for the sake of space.

Screen Shot 2017-08-24 at 10.29.24 PM.png

Task Statement

Practice 1 and 2: Anticipating & Monitoring

Activity Narrative

Practice 3 & 4: Selecting & Sequencing

Activity Synthesis

Practice 5: Connecting

Activity Synthesis

So..Planning is like putting together a puzzle. It is hard, takes time, and is sometimes difficult to figure out where to start. We know all of the pieces connect in the end, but making a plan for all of those pieces to connect takes an understanding of the final picture – the goal. There will be missteps along the way and some parts will take longer than others, but we know it is important to carefully connect each piece to another as one missing piece will leave unconnected ideas and the final picture unfinished. As you work alone, the way the pieces connect to form the final picture may not always be obvious, but as others help us see the pieces in different ways during the process, connections become explicitly clear and the final picture is something in which you can take a lot of pride.

The ‘others’ in my teaching journey have helped me see a difference between explicit teaching and explicit planning. Through explicit planning I have seen the importance in understanding the mathematical goal in a way that enables me to structure activities and lessons that enable students to make important mathematical connections through their own work and discussions. It is so exciting to see IM’s curriculum be a model for how I think about explicit planning in such a coherent, purposeful progression.

Link to Illustrative Mathematics 6-8 Math Curriculum.

Link to the 7th Grade lesson featured in this post.

Choral Count: Trusting Patterns

2 Replies

In the next round of 1st grade videos for my Teaching Channel Math Routines series, I am so excited to release a video of a Choral Count. When I taught 5th grade, I used Choral Counts a lot with decimals and fractions, so using them in the earlier grades was such an interesting parallel to that work.

I don’t want give away what you will see in the video, but Heidi’s recent post about trusting patterns when multiplying by 10s made me think about two particular student journal entries after the count.

In this count, students counted by 10s starting at 4 as I recorded their count and I stopped them at 154. They discussed patterns they saw in the count and afterwards, I asked them to journal about any new or extended patterns they think may happen if we continued counting. One thing I stress in their journal writing is the fact that it doesn’t have to always be written in words, we can explain mathematics with numbers as well. These two students didn’t include any description of what they noticed, but from their recording, I can assume some things about what they saw.

Through Heidi’s lens of trusting patterns, I watched this student record her extended count. Instead of writing every number in its entirety, this student wrote out all of the tens and then went back to add all of the 4s on. I asked her how she could show the pattern she noticed and she quickly went back to underline all of the tens. She trusted the pattern that the tens would continue going up by 1 and the ones place would stay the same throughout the rest of the count since we were counting by 10s. The pattern she noticed and trusted worked, would continue to work if she kept counting, and is something teachers could build on by counting by different multiple of tens.

Now, this student below invented another pattern that doesn’t keep the count in tact but brings up an interesting connection to the work above. Instead of looking at the tens going up by 1 and ones place staying the same, this student added 100s going down each column. In his work, the tens and ones stayed the same while the 100’s increased by 1 with each jump down.

The more I looked at these, I thought about how to support students in later grades who are unsure whether to trust a pattern or even why patterns work in the first place. For example, and not that counting by 7’s would be the way to go, but for Heidi’s problem of 2,000 x 7, I wonder how a choral count could support students who were struggling to explain multiplying by 10s. The student in Heidi’s example obviously has a grasp of the idea, but what about those who don’t?

Let’s see what we notice here:

7 14 21 28 35

42 49 56 63 70

77 84 91 98 105

112 119 126 133 140

Could students pull out where the multiples of 10 would show up if we kept counting?

Could they translate that pattern to equations?

Could they connect this additive thinking to multiplicative thinking?

Could they apply that same understanding to counting by another number but keeping the same structure of 5 in each row that is shown here?

All of these questions are so interesting to me and leave me wondering if we did more of this work in the earlier grades the impact it would have in later grades. Thank you Heidi for sparking such a great convo on Twitter and giving me a different lens for student work in the earlier grades!

True or False Equations: Kindergarten

1 Reply

Yesterday, the Kindergarten math routine videos I recorded were posted. This set of routines were so incredibly fascinating to me and completely out of my comfort zone – such an incredible learning experience. In addition to the 6 videos that are posted, there are about 12 more in my Google Drive that I had to chose between, each with its own unique student responses that would be so interesting to discuss.

With all of the videos and associated student work, I am just finding some of the work I thought I forgot to collect after the routine. This particular set of work is from the True or False Equation routine. This is probably my favorite routine in the set because it really pushes me to think about the language, recording, and understandings students have around the meaning of the equal sign.

The final equation, as anticipated, caused a bit of a controversy. Since the class was split on whether 2+3 equaled 1+4, I asked the students to explain their reasoning in their journal.

This response is reflective of the student’s experience with equations. How much do we record, or ask students to record, equations that only have three numbers? I would guess many students only see equations with two addends the majority of the time.

I liked the “same thing” and “I used my fingers” here. This is the language piece I find so interesting. What does it mean to be the same? In this case it could mean the same amount or looks the same. She could find the amount on her fingers or the two ways of showing the expressions on each hand would look the same in the end. A small, but important distinction I think.

This student appears to have related this to the first problem in the string, seeing both as the same since 5=5.

I really like this one because of the arrangement of dots. This student seems to think of exactly the same as the same amount since the dots look different in the way they are drawn. The dots are great because they look the way a student would easily subitize an arrangement.

This student broke the equation apart and set each side equal to 5 and showed with circles that each side was 5.

I was inspired by this number talk to dive even further into what it means to the be the same. I brainstormed some thoughts here and then tried an activity about what it means to be the same that ended in this work: (I haven’t gotten around to a blog on this one, but will soon!)

Today’s Number: Making Connections

1 Reply

The Investigations curriculum and Jessica Shumway’s book, Number Sense Routines contain so many wonderful math routines. Routines designed to give students access to the mathematics and elicit many ways of thinking about the same problem. One of the more open routines, is Today’s Number. In Today’s Number, a number is posed to the class and the teacher can ask students for questions about that number, expressions that equal that number, or anything they know about that number. I love this routine, and while it is more commonly used in the primary grades, I used it often in my 5th grade classroom. While I would capture so much amazing student thinking, I always felt like all of that great thinking was left hanging out there. I could see some students were using what they knew about operations and properties to generate new expressions for the given number, however I wondered how many saw each expression as individual, unconnected ideas.

After I read Connecting Arithmetic to Algebra, I had a different ending to Today’s Number, an ending that pushed students to look explicitly at relationships between expressions. I tried it out the other day in a 3rd grade classroom.

I asked students for expressions that equaled Today’s Number, 48. I was getting a lot of addition, subtraction and multiplication expressions with two numbers, so I asked students if they could think of some that involved division or more than two numbers. I ran out of room so I moved to a new page and recorded their ideas.

After their thinking was recorded, I asked the students which expressions they saw a connection between. This is where my recording could improve tremendously, but I drew arrows between the two expressions as students explained the connection.

In case the mess is hard to see, these are some connecting ideas that arose:

Commutative Property: 3 x 16 and 16 x 3, 6 x 8 and 8 x 6, 12 x 4 and 4 x 12

Fraction and Fraction addition: 48/1 and 24/1 + 24/1 and 24 + 24

Subtracting from 100 and 1000: 100-52 and 1000-952

Multiplication and Repeated Addition: 4 x 12 and 12+12+12+12

Adjusting Addends in similar ways: 38+10 and 18 + 30, 40 + 8 and 48+0

Other ideas that I don’t particularly know how to categorize:

10 x 4 + 8 = 10 x 4 + 4 x 2

58 x 1 – 10 = 58 – 10

The second page got even more interesting:

“Groups of” and Decomposition: 7 x 4 + 2 + 18 = 14 + 14 + 2 + 10 + 8 . This student saw the two 14s as two groups of 7 and then the 18 decomposed into 10 + 8.

Halving and Halving the Dividend and Divisor: 192÷4 = 96÷2. This student actually used the 192 to get the expression with 96.

Another variation of the one above was 200 ÷ 4 – 2 = 100 ÷ 2 – 2.

Other cool connection:

96 ÷2 = (48 + 48) ÷ 2; This student saw the 96 in both expressions since they were both dividing by 2.

I think asking students to look for these connections pushes them to think about mathematical relationships so expressions don’t feel like such individual ideas. I can imagine the more this is done routinely with students, the more creative they get with their expressions and connections. I saw a difference in the ways students were using one expression to get another after just pushing them try to think of some with more than 2 numbers and some division.

Number Talk: Which Numbers Are Helpful?

The Equal Sign

15 Replies

True or False?

5 = 5

5 = 4 + 2

2 + 3 = 1 + 4

After reading so much about the meaning of the equal sign and equality in books such as Thinking Mathematically and About Teaching Mathematics , I anticipated students may think each was false for different reasons…..

5 = 5: There is no operation on the left side.

5 = 4 + 2: The sum comes first or 4+2 is not 5.

2 + 3 = 1 + 4: there is an operation on both sides or because 2+3 =5 (and ignore the 4) or because 2+3+1 ≠ 4.

While I anticipated how students may respond, I was so intrigued by the number of students (probably about 75%) that said false for 5=5. They were about split on the second one, but for many reasons – not many of them being that 5 ≠ 6. The final one left many confused, in fact one student said, “Well now you are just trying to confuse people by putting two plus signs.” So cute.

As they explained their reasoning, my mind was reeling….

What questions do I ask to get them to:
- Think about what the symbols mean?
- Talk about what is the same?
- Realize the equal sign in the first one is not a plus sign, so there is no answer of 10?
- See the equal sign to not mean “the answer is next”?
- What wording do I use for the equal sign?
  - “The same as” felt wrong because the sides do not look the same in both cases….so, is “Is the same amount” a helpful way for them to think about it?

I got back to my room and starting thinking about what learning experiences would be helpful for students in building their understanding of the equal sign? I talked through it with some colleagues at school and reached out to those outside of school, I needed some serious help!

I started playing around with some cubes and realized how interestingly my thinking changed with each one. I didn’t take a pic of those cubes so I recreated them virtually to talk thru my thinking here.

The first set represents 5 = 5. I can see here where “the same as” works for the equal sign because there are 5 and they are all yellow. But what if I put 5 yellows on the left and 5 red on the right? Then they are the same quantity, but do not look the same.

The second set represents 2+3=5 and is definitely the one students are most comfortable seeing and representing as an equation. It looks and feels like composition to me so I can definitely see why student think the equal sign means “makes” or “the total is.” It looks like 2 and 3 more combine to make 5.

Something interesting happened with the green set. I made two sets of 5 and then broke one set to make the right side – felt like decomposition. I can see why it would feel differently to students. I also realized that when I look at them, I look left to right and much of that lends itself to the way I was thinking about what was happening.

The last set I made by taking my 2 sets of 5 connected cubes and breaking each set differently. Again, “the same as” doesn’t work for me here really well either because they don’t look the same.

Screen Shot 2017-04-26 at 6.27.05 PM.png

Still thinking of next steps because I always like to put context into play with these types of things, but I am finding that very difficult without forcing the way students represent their thinking which I don’t want to do.

Right now, things I am left thinking about before planning forward:

What do students attend to when we ask if things are the same?
Our language and recording is SO incredibly important.
How can these ideas build in K-1 to be helpful in later grades?
If I am thinking of moving students from a concrete to more abstract understanding, how does that happen? Is it already a bit abstract in the way the numbers are represented?
Do we take enough time with teachers digging into these ideas? [rhetorical]

I look forward to any thoughts! So much learning to do!

1st Gr Number String: Missing Number

1 Reply

Yesterday, I wrote a quick post as I was trying to decide which of two number talks I should do with a 1st grade class. I got some great feedback and went with the first one in the post! It was amazing and completely evident that the teacher, Ms. Williams, does a great job asking students to share their thinking regularly. The students were so clear in explaining their reasoning and asking questions of one another.

The first problem drew out exactly what I was hoping and more. One student shared counting on and a few students shared how they decomposed the 4 and added 2 and then 2 more. I was not expecting the use of a double, but two students used 8+8 in their reasoning. The use of their “double fact” reminded me of the solving equations conversations I have with Michael Pershan but in a much more sense-making way than I personally think about it. The students said they “knew 4 and 4 made 8 so they took 4 away and that changed the answer.” I tried to get out of them that they subtracted the 4 from the 16 as well, but it just made sense to them the 16 changed to 12 because he subtracted 4 from the 8. I am so glad I videoed this talk because I want to talk more about it after I re-watch it!

The second problem was as tricky, as I anticipated, and split the class between the answers 1 and 9. The students seemed very used to having the difference on the lefthand side of the equal sign which is great, but some still wanted to add 1 to the 4 instead of subtract the 4 from the missing number. I moved on to the final question because we were at a bit of a standstill at this point. Hindsight, I wish I did that problem last, but I had them journal about it after the talk.

The final problem, which I wish was my first problem – what was I thinking in this order? – was great! They decomposed the 5, made 10 and talked their way through the two incorrect responses.

I asked them to journal about the second problem when we finished. The prompt was to explain which answer, 9 or 1, they thought it was and why. Here are few examples:

I think I would love to post the following string (all at once) on the board to start tomorrow’s lesson:

? – 4 = 5

5 = ? – 4

? + 4 = 5

5 = ? + 4

Ask what the question mark is in each one and which equations seem most similar.

Such a great day in 1st grade!

1st Grade Number Talk

8 Replies

I am planning for a number talk tomorrow with a 1st grade class. I have been playing around with two different problem strings that I would love feedback on, because I can’t make a decision!

I would particularly like feedback on:

What could we learn about student thinking?
What would you be curious to find our about their thinking at the end?
Do you think one would be better before the other or doesn’t it matter?

Here are the two I am playing around with (sorry, I had them written on Post-its):

My thoughts:

1st problem – Do they add to 10 and then add on? For example, 8+2=10 and since 12 is 2 more the answer is 4 or do they subtract 8 from 12?
2nd problem – How do they do with the missing number on the right side of the equation? Do they visualize a 10 frame, taking 4 off of the bottom row to leave 5? Do they add 5 and 4?
3rd problem – Do they decompose the 5 into 4+1 to use the 1 with the 9? Do they count on from 9?

Prompt at the end – How are these problems different? Which was your favorite to do? Why?

Second option:

My thoughts:

1st problem: Set the stage for expressions on both sides of equal sign. Notice you can’t add more to the same number and stay equal. Did we need to solve both sides to know that?
2nd problem: Both equal 10 but did we need to solve both sides to know it is equal? Take one from one addend and add it to the other and still remain equal.
3rd problem: Commutative property.
4th problem: Now that I just wrote the commutative for problem 3, I want to switch the 8+5 on this one to 5+8 so that they might also think about taking 2 from one addend and adding it to the other.

Prompt at the end: Write two of your own equations that would fit something you noticed in our problems today. (wording is rough on that one).

Chances are I will have the opportunity to do both of them and I think they both hit on different, interesting things. I would love feedback on both and know if you think one is better before the other or if it doesn’t matter?

True or False Multiplication Equations

2 Replies

Today, I was able to pop into a 3rd grade classroom and have some fun with a true or false equation routine! This routine has become one of my favorites, not only for the discussion during the activity, but more for the journals after the talk. I haven’t figured out quite how to use them with the students, but it gives me such great insight into their understandings that I would love to think about a way to have students reflect on them in a meaningful way. I keep asking myself, what conjectures or generalizations could stem from this work?

I started with 4 x 3 = 3 + 3 + 3 + 3 to get students thinking about the meaning of multiplication and how we can solve for a product using repeated addition. I followed 6 x 4 = 8 + 8 + 4 to see how students talked about the 8’s on the right side. They could explain why it was false by either solving both sides or reasoning about the 8’s as two 4’s in some way.

My final problem was the one below, 8 x5 = 2 x 5 + 2 x 5 + 20. I chose this one because I wanted students to see an equation with multiplication on both sides. Up to this point, I structured them to be multiplication on one side and addition on the other. There was a lot of solving both sides – I think because of the ease of using 5’s – but, as the discussion continued the students made some really interesting connections about why the numbers were changing in a particular way. I really focused on asking them, “Where do you see the 8 and 5 in your response?” to encourage them to think relationally about the two sides.

I ended the talk with 8 x 6 = and asked the students to go back to their journals and finish that equation to make it true.

Some students knew it was equal to 48 right away and started writing equations that were equal to 48. For this student I probably would ask about the relationship between each of the new equations and 8 x 6.

There are so many interesting things in the rest of them, that I am not sure what exactly to ask student to look at more deeply.

In all of them, I see…

Commutative property
Multiplication as groups of a certain number
Distributive property
Doubling and halving & Tripling and thirding

The student below shared this one with the class during the whole class discussion:

8 x 6 = 7 x 10 – 3 x 10 + 2 x 4

From her explanation, she could explain how both sides were 48, but when I asked her how it related to 8 x 6, her wheels started spinning. You can see she played all around her paper trying to make connections between the two. That is the type of thinking I want to engage all of the students in, but based on their own personal journal writing – but what is the right prompt? “Where is one side in the other?” or “How are they related?” <—that one feels like it will lead to a lot of “They are both 48” so I need a follow up.

I actually left the room thinking about how I would explain how they two sides were related – in particular looking for either 8 groups of 6 or 6 groups of 8 on the right side. I found it was easier for me to find six 8’s, but now want to go back and find eight 6’s for fun. I can see how this could be so fun for students as well, but there is a lot of things going on here so I wonder how to structure that activity for them? Would love thoughts/feedback in the comments!