Building Vocabulary in Math Class

A draft I didn’t realize I had made me #pushsend today! This is from pre-school planning in August!

Math has a lot of vocabulary terms and for the past three years I’ve pretty consistently complained about how my students can’t solve problems because they don’t know what the words mean without really doing anything about it. During planning time today, I asked one of my colleagues (English teacher currently pursuing a master’s in literacy) to help out. Here are some suggestions she gave with what I hope are correct examples… I made them math examples when I could think of one easily!

Quad(Tri) Text

  1. Introduce math term using pictures or video first.
  2. Read a narrative passage that uses this math term out loud to the students (they can read along with you if you’d like, but she emphasized it’s important for the teacher to read it out loud in context).
  3. Read the math term in informational text passage (textbook, article, etc.).
  4. The fourth step of this would be to read a longer text (like a short story, novel, study,  full article, etc.) but this would most likely be skipped for most secondary math terms.

Word Lines

  1. Determine what you want to be ranking/sorting. Type up and print out.
  2. Create top and bottom spots (usually greatest to least) in a physical location in your classroom.
  3. Pass out cards to students. Have them sort themselves on the word line with guidance from the class.
  4. EXAMPLES: sorting probability words (generally, usually, impossible, guaranteed, etc.) from a probability of 0 to 1; sorting types of polynomials (linear, quartic, etc.) from highest degree to lowest.

Shades of Meaning

  1. Have paint chips of one color varying in shade.
  2. Choose a basic vocabulary word/term. Make this the lightest shade paint chip.
  3. Determine synonyms for the word and rank them on intensity. The darker the shade, the higher the intensity of the word.
  4. EXAMPLE: This is easier to see than read about.

Four Square

  1. This is basically a Frayer model but you can change the titles of each box to suit your needs.
  2. EXAMPLE: If you were teaching about Federalism, you could use “definition, people, events, and beliefs” as your boxes instead of “definition, characteristics, example, and non-example”

Concept Mapping

  1. Choose a broad vocab term. Place this at the top of or in the middle of a piece of paper.
  2. Identify the key concepts that directly relate to that term and branch them out from the original concept.
  3. Repeat step two for each branch as needed.
  4. EXAMPLES: A nice blog post from Tina on concept maps with a few pictures.

Semantic Feature Analysis

  1. Choose a broad vocab term/category.
  2. Determine subgroups. Identify characteristics of each subgroup.
  3. Create a table with subgroups in the far left column and characteristics in the top row. Check off each box that holds an always true statement (could use for A/S/N also!)
  4. EXAMPLES: A partially filled out example on quadrilaterals. I’m imagining you could do this for features of graphs as well.


  1. Select a broad vocab term. Create a list of terms that relate to the vocab term.
  2. Cut out all of the terms, including the broad vocab term, and have enough copies for each group.
  3. Have students group the terms based on similarities, looking for the broad vocab term (label) and creating titles for each group (related to their reasoning for creating the groups).
  4. EXAMPLES: An example of how to do it independently/without cutting out cards.


  1. Choose a vocab term. Identify two examples and one non-example of the term. Write these in a random order in a list.
  2. Students circle the vocab term, cross out the non-example, and leave the two examples as-is.
    1. f(x) = ax² + bx + c
    2. g(s) = |14s – 6|
    3. h(t) = 5
    4. {Polynomial}

General Tips

  1. Have discussions involving the math term.
  2. Take virtual field trips (less possible in math).
  3. Provide context clues.
  4. Teach root words.
  5. Provide examples and non-examples.
  6. Use kid-friendly definitions.
  7. Students need 8 exposures (them actually using the word) to a word before it’s a part of their vocabulary.
  8. DO NOT ask if students know what a term means – an incorrect response sticks as the correct response.
  9. Require students to use correct spelling as often as possible.
  10. Teaching Tier 2 words (academic but content-neutral: characteristics, accurately, illustrates, etc.) has the greatest effect on students’ vocabulary and reading comprehension skills.

This was from a 15ish minute conversation around lunch time on the topic. There are examples of each out there – some ideas are definitely easier to adapt to secondary math than others. I will change this list based on what I find useful in my classroom!

Moral of the story though: USE YOUR RESOURCES – English teachers and reading specialists (and elementary teachers) are a wealth of knowledge when it comes to vocabulary and are almost always willing to help as it’ll make their lives easier too!


#UnsexyMath Days 3/4: Bias and Sampling

A recent instagram post lead to four people (both IRL colleagues and #mtbos members) complimenting and asking for more info. I had a conversation or two with Kate Nowak and others about the need to share more “unsexy math” – the regular-but-still-pretty-effective things we do more often in our classroom than the super awesome projects or really pretty foldables. So here goes my first one.

To start of our new geography/statistics course, our students are creating a survey inspired by some aspect of human-environment interactions (if you didn’t know this (like me), this is apparently a HUGE part of what is learned in HS World Geography). After administering the survey, they’ll create propaganda and administer it, then re-administer their survey to compare the results.

The entire idea behind project-based learning though is that most of what we do in the classroom should be tied to the projects our students are currently working on. So to hook our students, we decided to try to survey them on the cuteness and strength of hippos, show them a video of Fiona the hippo, and re-administer the survey. It worked! For some, at least. There were a handful in each class that changed their rankings after the video. So then I told them about the project and the majority got really excited. “You mean we’re going to try to brainwash our peers? Sweet!” I think they might get disappointed when we have to turn down some of their propaganda ideas for legal and ethical reasons later this month but it’ll still be good. Another lesson to learn.

Anyway, after the project intro, we talked about bias. I asked students what they thought bias was, then I projected four questions with varying levels of bias on the screen:

  • Do you think bike helmets should be mandatory for all bike riders?
  • Do you prefer the natural beauty of hardwood floors in your home?
  • Do you eat at least the recommended number of servings of fruits and vegetables to ensure a healthy and long life?
  • Do you want your kids to receive a faulty education by having their school day shortened?

After reading them to themselves, they offered up thoughts on which ones were more biased than others, what made them biased, what the questions were trying to imply or what responses they were leading you to, and how to eliminate bias as much as possible from them.

Then we did some vocab matching. You can see the list of terms they are learning about here. These students mostly know each other from their PBL courses last year, but in the first few days we’ve spent time doing a lot of collaborating on Geography (and Biology and English) course work, low-floor/high-ceiling math warm-ups, talking points, etc. I was still shocked by how much conversation was happening within most groups. The goal was to see what they could match based on the context clues – could they figure out “nonresponse bias” and “size bias”, for example, just based on knowing what “nonresponse” and “size” were? (For the record: 18/20 groups could. The fewest amount of correct matches was 3, greatest was 8.)

After they checked and copied down the correct vocab terms and definitions, I projected four examples and explained them a little more. Students identified at least one correct type of bias for each example. I was happy.

To end class we started talking about sampling methods. I had each class come up with a question we could ask students (“How often are you late to school?”, “Do you like the school’s food?”, “What’s your favorite food?”, and “What’s your favorite color?” were the suggestions) and we created Frayer models for each sampling method with our “example” going along with the question the class wanted to ask and how we could use that method at school. I thought this would be important as they are actually going to be surveying students (or other things at the school… I’m really hoping one group does something on amount of trash/recycling or use of our refillable water bottle stations!), so hopefully when they’re doing their survey design they can choose an appropriate method.

We didn’t get through all of the survey methods in 3/4 classes, so we’ll finish that slideshow up Monday/Tuesday along with some/all of the questions on the student work document linked above.

This wasn’t anything monumentally new. They wrote or typed a lot of notes while I talked up front and projected info on the screen. But there was a lot of peer-to-peer and whole-class conversation and for that, this unsexy lesson was a win.

First Day Plans

I still have more than three weeks before school begins but I spent hours each day this week working on plans for the upcoming school year. Nothing is set in stone yet, and all of these plans are totally dependent upon the assumption that I’m teaching certain subjects in certain rooms with certain people, so things will probably change… BUT here is the plan:

Man vs. Earth – 85 students for 3 hours every other day

  • Students will be randomly assigned to tables in groups of 3-4. We’re planning on doing visibly random grouping but changing every 2 weeks because we’ll be seeing students every other day. They’ve got fun names and logos which I’ll share once they’re nicely printed.
  • We’ll be starting the year’s warm-ups with the “Week of Inspirational Math”. We probably won’t do all of the activities associated with each day but will definitely do at least one every day for the first three weeks. Anyway, Day 1 is Four Fours. We have a decent amount of whiteboard up front (trying to figure out more VNPS in this new space) so I plan on having each group of students share a few of their solutions and especially having conversation about solutions that were found using different methods.
  • At this point we’ll be about ready to split the class into two groups, so we’ll briefly have a conversation about our program rules. The students will be a part of creating more norms, suggesting ways to celebrate success, and have a conversation on punishment vs. consequences in the next few days.
  • SPLIT:
    • I’ll be completing two team-building activities with the students. The first will be a small-group one about Cooperative Logic. Haven’t decided which of those activities we’re doing yet though so if you have suggestions please share. After that we’ll do a team-building activity as a larger group called Meaningful Adjacencies. I am super excited for the possibilities that can come from this activity.
    • My co-teacher will be working through map basics at this time with the other have of the class.
  • Return to the whole class setting and complete a conflict management style assessment. The plan is to have students take the assessment and determine their preferred conflict management style based on the assessment results, then somehow discuss pros/cons of their style with other students that got the same one before having a whole class discussion about it. As I’m typing this I’m thinking talking points would be a good idea but that means I’d have to either replace one of the team building activities with an intro to talking points or move conflict management to a later date so we’ll see.
  • Finish up with name tents.


  • Same warm-up as Man vs. Earth
  • Cooperative Logic activity
  • Meaningful Adjacencies activity
  • I’d like to do “Angles Around a Point” from Henri Picciotto’s Geometry Labs next but I need to know if I have access to pattern blocks. If I don’t have pattern blocks we’ll have lots of arguments surrounding things like “Is a hotdog a sandwich?” I know there are other Geometry teachers in MTBoS planning on doing this and I hope there’s a way we can share student reasoning!
  • Finish up with name tents again.

Hopefully all of this will work as planned but we all know nothing ever does!

2017 – 2018 #Goals

I know that you’re not supposed to take on too many things, so I just have three goals for this year. Just three.

  1. Be better at using effective, evaluative feedback. It’s the chapter I’m reading from Never Work Harder Than Your Students by Robyn Jackson. It was the chapter I needed to focus on based on my self-assessment and I know that it’s important and that I’m not great at it. I haven’t made an action plan yet (haven’t finished the chapter yet) but once I do I’ll blog it out. If you have suggestions for how to give effective feedback please let me know!
  2. Actually have discussions about the daily warm-ups. I think this will be easier to do if I have them all created ahead of time, so here is the PowerPoint with the year’s warm-ups. They follow a pattern but it’s not the same one each day of the week – I have two classes that will be every other day all year and two other classes that will be every day for a semester, so I felt like this would be the best way for me to be prepared. I can also have slips of paper printed for each type of warm-up and leave them in the classroom for students to use whenever they pop back up (or they’ll respond on Canvas).
  3. I’ve seen this blogged before but I don’t remember by who (I think it’s been multiple people). I want to be more conscious about how I help students, so when students are seated in groups and I see a hand raised for a question, I will go to the table to help but ask someone else at that table what the question is. Ideally this will create some forced collaboration (and/or some forced research either in their notes or online) that will eventually get them used to asking others for help. It seems to work well for other teachers so I hope I can get myself to stick to it!

There are a lot of other things I’m probably going to try but won’t beat myself up over if they don’t pan out or if I stop because it’s too much work. I’m teaching ProbStats and building another cross-curricular PBL course as a teach it this year (hopefully more will be done up front but we’ll see…) so I know there’s still a lot of content I have to re-learn and figure out how to best teach!

Online stats resources

Compiling this so I can get rid of the piece of paper where these are written down but also to hopefully help teachers and students looking for places to find/analyze/visualize data!

Fall 2016 Teacher Report Card

Matt Vaudrey (@MrVaudrey and has been using his version of a Teacher Report Card for a few years now, and I have been using a different one for the last couple of years but decided to give his version a go in January as a part of our end-of-course reflection.

My co-teacher and I only used this for Physics by Design, our project-based and blended learning pilot class. The responses given were for both of us, which may create some biased data; it would be interesting to receive this data individually because I’m not sure what (if anything) would be different! We also only received 25 responses out of 48 students and I’m not sure how that affected the end results.

Each response prompt begins with “My teachers…” and a rating of 1 means “strongly disagree” while a rating of 5 means “strongly agree”. Below you’ll see question-by-question results and my thoughts on each. Overall I’m much happier about these results than I was at the end of last year’s and I’ve got thoughts about that at the end.

  • respect each student. respect-each-student

This is good to see. I wonder what the six students who responded with either “disagree” or “unsure” believe that respecting each student looks like and if we did anything in particular to them that may have made them feel as though we didn’t respect them.

  • try to see the students’ point of view. see-the-students-point-of-view

What I found interesting here is that there are 3 more students who responded negatively to this than to the previous prompt, which again makes me wonder what our students feel a teacher should say/do to respect students. We did not do many of the typical first week activities because they had a week-long “bootcamp” the week before school but maybe we should have included more so that we could try to be on the same page a little more.

  • explain the math/physics concepts clearly. explain-physics-concepts-clearly

This is one where I wish we would’ve split it; I did the majority of math instruction while my co-teacher did most of the physics instruction. Did they believe one of us did a better job than the other? Also, we started off the year with much more direct instruction while our last unit was by far the most online-content heavy. Did that shift in teaching and learning style effect the responses to this prompt or would students have responded similarly in October?

  • use language that we can understand. use-language-that-we-understand

Why are the responses to this prompt mostly agree/strongly agree compared to the last prompt? What is the difference, in our students’ eyes, between explaining concepts clearly and using language that they can understand?

  • do a good job of treating all students equally. do-a-good-job-of-treating-students-equally

I am happy that our results are anonymous but I really want to know who put they strongly disagree with this statement and why.

  • seem to enjoy teaching. seem-to-enjoy-teaching

Well these are much better results than the results I got at the end of last year, and I think you could definitely see that if you walked into my classrooms to compare them. There were definitely still bad days (and weeks, if we’re being honest) but overall the first semester of this year was so much more enjoyable than the second semester of last year. Exponentially so. Spending half of my day all year long with the same students, who are freshmen/sophomores/juniors in Algebra 1, does not make me enjoy teaching.

  • show interest in students’ lives. show-an-interest-in-students-lives

I know I didn’t do as good of a job of this compared to my first two years of teaching. This year I have gone to significantly fewer sporting events and concerts but I tried to at least know what school activities each student was involved in.

  • make me feel important. make-me-feel-important

This question intrigued me the most because, reflecting on my high school experience, this wasn’t something I feel as though I thought about. In fact, one of my best and favorite teachers occasionally called me out for being dumb and/or full of myself (she did this with everyone; I chose to take her classes for three straight years). So maybe I didn’t need that but some students do. Or maybe I didn’t think I needed it but my life would’ve been altered in some way if I felt my teachers made me feel important. I’m not sure.

  • keep the class under control without being too tough. keep-the-class-under-control-without-being-too-tough

No disagreement here. I’m only in my third year so classroom management isn’t my strong suit, but that was compounded by the nature of this course. I ended up writing multiple referrals on the first day of the same class this semester so I think I may have swung a little too far in the opposite direction. We’re working on it.

  • answer questions completely. answer-questions-completely

I don’t always want to answer questions completely, especially in this more exploratory course, so I’m glad to see that most students agreed but I’m not broken up that we had a couple of who disagreed to varying degrees.

  • praise good work. praise-good-work

This makes me happy. I think it’s very easy, especially when your classroom management is lacking, to only focus on negative behaviors and students who aren’t doing work well. I am glad most students seem to have thought that we praised good work enough and I hope we can keep that up.

  • encourage me to be responsible. encourage-me-to-be-responsible

Oy vey, this is one of the hardest things about teaching freshmen. Suddenly they’re not forced to carry around a planner and they don’t have to have a locker and they get a personal school laptop/charger and calculator and it’s like all of the good work the middle school teachers did goes out the window. These results are encouraging but I know somehow we can do better.

Matt had two other prompts that I found interesting. These were fill in the blank and I’ve included a sample of student responses for each.

  • Sometimes the teacher _____, but not always.
    • gets frustrated
    • is nice
    • is crazy
    • didn’t understand what we were saying
    • does a good job of integrating online and face-to-face learning
    • has too many students to help at the same time
    • says bad words
  • Sometimes the teacher lets the class _____, but not always.
    • socialize
    • work outside in the hallway
    • choose who to work with
    • get loud

Overall I think this was a much more concise survey than I was giving to students previously and I appreciated the specific fill-in-the-blank responses at the end. There were a few responses to both that didn’t make sense, so that makes me wonder if those students created some misrepresentation in the previous ratings as well. The world will never know.

Why are word problems so hard?

Before going any further, I have no answer for the question posed in the post title. If you have good research on this subject please share.


My students really have gone above and beyond my expectations for our unit on systems of linear equations and inequalities, but there is still so much struggle happening, and not all of it is productive.

We worked on systems of inequalities and linear programming for about four days this past week and based on their quizzes the students procedurally understand how to solve these problems. But when I gave them their assessments on Friday there was some serious struggle.

The assessments were word problem heavy. I allowed students to do this open note so they could view the word problems we’d been working on all unit and hopefully use them as a guide. The majority of my students did not finish this assessment in the time allotted, and as they were turning them in I saw a lot of mistakes. I’m going to provide feedback on them and allow them to fix/finish on Monday before I grade them and hopefully that will help, but I’m still not pleased with it.

<blockquote class=”twitter-tweet” data-lang=”en”><p lang=”en” dir=”ltr”>So many tiny misconceptions about expressions and equations that I wasn't tuned into until now.</p>&mdash; Kent Haines (@KentHaines) <a href=”″>December 1, 2016</a></blockquote>

I’ve been following more and more elementary and middle school math people on Twitter trying to understand where these ideas and issues come from. Kent Haines, above, is seeing similar struggles with expressions in his middle school math classes that I am still seeing with juniors in my AFDA class. These teachers have helped give me ideas for supports and different types of lessons but I haven’t found answers. I teach a lot of students with special needs; more than a third of my AFDA class has some sort of educational plan in place. But even my non-diagnosed students struggle with word problems. Even my readers, the students who love and excel in English, struggle with word problems. WHAT GIVES?!?

This might be what finally kicks me into a master’s degree, or at least into reading more educational research. I need to understand. I don’t understand what it is about word problems that seems to cause everyone, even students with at- or above-grade-level reading levels, to suddenly lose math focus and ability and I want to.

P.S. I don’t really want to be a special educator (shout-out to them; good special educators are the hardest working and best teachers I know) but I honestly hope that I never stop teaching collaborative classes. These kids teach me more than they’ll ever know.